LFM Pure and Mechanics

If \(x=\log_bc\,\), express \(c\) in terms of \(b\) and \(x\) and prove that $ \dfrac{\log_a c}{\log_a b} = \ds \log_b c \,$.

1. Given that \(\pi^2 < 10\,\), prove that \[ \frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} > 2\,. \]
2. Given that \(\ds \log_2 \frac{\pi}{\e} > \frac{1}{5}\) and that \(\e^2 < 8\), prove that \(\ln \pi > \frac{17}{15}\,\).
3. Given that \(\e^3 >20\), \, \(\pi^2 < 10\,\) and \(\log_{10}2 >\frac{3}{10}\,\), prove that \(\ln \pi < \frac{15}{13}\,\).

A function \(\f(x)\) is said to be concave on some interval if \(\f''(x)<0\) in that interval. Show that \(\sin x\) is concave for \(0< x < \pi\) and that \(\ln x\) is concave for \(x > 0\). Let \(\f(x)\) be concave on a given interval and let \(x_1\), \(x_2\), \(\ldots\), \(x_n\) lie in the interval. Jensen's inequality states that \[ \frac1 n \sum_{k=1}^n\f(x_k) \le \f \bigg (\frac1 n \sum_{k=1}^n x_k\bigg) \] and that equality holds if and only if \(x_1=x_2= \cdots =x_n\). You may use this result without proving it.

1. Given that \(A\), \(B\) and \(C\) are angles of a triangle, show that \[ \sin A + \sin B + \sin C \le \frac{3\sqrt3}2 \,. \]
2. By choosing a suitable function \(\f\), prove that \[ \sqrt[n]{t_1t_2\cdots t_n}\; \le \; \frac{t_1+t_2+\cdots+t_n}n \] for any positive integer \(n\) and for any positive numbers \(t_1\), \(t_2\), \(\ldots\), \(t_n\). Hence:
   1. show that \(x^4+y^4+z^4 +16 \ge 8xyz\), where \(x\), \(y\) and \(z\) are any positive numbers;
   2. find the minimum value of \(x^5+y^5+z^5 -5xyz\), where \(x\), \(y\) and \(z\) are any positive numbers.

To nine decimal places, \(\log_{10}2=0.301029996\) and \(\log_{10}3=0.477121255\).

1. Calculate \(\log_{10}5\) and \(\log_{10}6\) to three decimal places. By taking logs, or otherwise, show that \[ 5\times 10^{47} < 3^{100} < 6\times 10^{47}. \] Hence write down the first digit of \(3^{100}\).
2. Find the first digit of each of the following numbers: \(2^{1000}\); \ \(2^{10\,000}\); \ and \(2^{100\, 000}\).

Let \(x=10^{100}\), \(y=10^{x}\), \(z=10^{y}\), and let $$ a_1=x!, \quad a_2=x^y,\quad a_3=y^x,\quad a_4=z^x,\quad a_5=\e^{xyz},\quad a_6=z^{1/y},\quad a_7 = y^{z/x}. $$

1. Use Stirling's approximation \(n! \approx \sqrt{2 \pi}\, {n^{n+{1\over2}}\e^{-n}}\), which is valid for large \(n\), to show that \(\log_{10}\left(\log_{10} a_1 \right) \approx 102\).
2. Arrange the seven numbers \(a_1\), \(\ldots\) , \(a_7\) in ascending order of magnitude, justifying your result.

Let \[\mathrm{f}(t)=\frac{\ln t}t\quad\text{ for }t>0.\] Sketch the graph of \(\mathrm{f}(t)\) and find its maximum value. How many positive values of \(t\) correspond to a given value of \(\mathrm f(t)\)? Find how many positive values of \(y\) satisfy \(x^y=y^x\) for a given positive value of \(x\). Sketch the set of points \((x,y)\) which satisfy \(x^y=y^x\) with \(x,y>0\).

Let \(a_{1}=3\), \(a_{n+1}=a_{n}^{3}\) for \(n\geqslant 1\). (Thus \(a_{2}=3^{3}\), \(a_{3}=(3^{3})^{3}\) and so on.)

1. What digit appears in the unit place of \(a_{7}\)?
2. Show that \(a_{7}\geqslant 10^{100}\).
3. What is \(\dfrac{a_{7}+1}{2a_{7}}\) correct to two places of decimals? Justify your answer.

Sketch the graph of the function \(\mathrm{h}\), where \[ \mathrm{h}(x)=\frac{\ln x}{x},\qquad(x>0). \] Hence, or otherwise, find all pairs of distinct positive integers \(m\) and \(n\) which satisfy the equation \[ n^{m}=m^{n}. \]

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If \(n^m = m^n \Rightarrow \frac{\ln m}{m} = \frac{\ln n}{n}\) so if \(n < m\) we must have that \(n < e \Rightarrow n = 1,2\). \(n = 1\) has no solutions, so consider \(n = 2\) which obviously has the corresponding solution \(m=4\). Since \(h(x)\) is decreasing for \(x > e\) we know this is the only solution. Hence the only solution for distinct \(m,n\) are \((m,n) = (2,4), (4,2)\)

1. Let \(r\) be a real number with \(\vert r \vert<1\) and let \[ S = \sum_{n=0}^\infty r^n\,. \] You may assume without proof that \(S = \ds \frac{1}{1-r}\, \). Let \(p= 1 + r +r^2\). Sketch the graph of the function \(1+r+r^2\) and deduce that \(\frac{3}{4} \le p < 3\,\). Show that, if \(1 < p < 3\), then the value of \( p\) determines \(r\), and hence \(S\), uniquely. Show also that, if \(\frac{3}{4} < p < 1\), then there are two possible values of \(S\) and these values satisfy the equation \((3- p)S^2-3S+1=0\).
2. Let \(r\) be a real number with \(\vert r \vert<1\) and let \[ T =\sum_{n=1}^\infty nr^{n-1} \,. \] You may assume without proof that $ T = \ds \dfrac{1}{(1-r)^2}\,. $ Let \( q= 1+2r+3r^2\). Find the set of values of \( q\) that determine \(T\) uniquely. Find the set of values of \( q\) for which \(T\) has two possible values. Find also a quadratic equation, with coefficients depending on \( q\), that is satisfied by these two values.

The two sequences \(a_0\), \(a_1\), \(a_2\), \(\ldots\) and \(b_0\), \(b_1\), \(b_2\), \(\ldots\) have general terms \[ a_n = \lambda^n +\mu^n \text { \ \ \ and \ \ \ } b_n = \lambda^n - \mu^n\,, \] respectively, where \(\lambda = 1+\sqrt2\) and \(\mu= 1-\sqrt2\,\).

1. Show that $\displaystyle \sum_{r=0}^nb_r = -\sqrt2 + \frac 1 {\sqrt2} \,a_{\low n+1}\,$, and give a corresponding result for \(\displaystyle \sum_{r=0}^na_r\,\).
2. Show that, if \(n\) is odd, $$\sum_{m=0}^{2n}\left( \sum_{r=0}^m a_{\low r}\right) = \tfrac12 b_{n+1}^2\,,$$ and give a corresponding result when \(n\) is even.
3. Show that, if \(n\) is even, $$\left(\sum_{r=0}^na_r\right)^{\!2} -\sum_{r=0}^n a_{\low 2r+1} =2\,,$$ and give a corresponding result when \(n\) is odd.

Let \(x_{\low1}\), \(x_{\low2}\), \ldots, \(x_n\) and \(x_{\vphantom {\dot A} n+1}\) be any fixed real numbers. The numbers \(A\) and \(B\) are defined by \[ A = \frac 1 n \sum_{k=1}^n x_{ \low k} \,, \ \ \ B= \frac 1 n \sum_{k=1}^n (x_{\low k}-A)^2 \,, \ \ \ \] and the numbers \(C\) and \(D\) are defined by \[ C = \frac 1 {n+1} \sum\limits_{k=1}^{n+1} x_{\low k} \,, \ \ \ D = \frac1{n+1} \sum_{k=1}^{n+1} (x_{\low k}-C)^2 \,. \]

1. Express \( C\) in terms of \(A\), \(x_{\low n+1}\) and \(n\).
2. Show that $ \displaystyle B= \frac 1 n \sum_{k=1}^n x_{\low k}^2 - A^2\,\(.
3. Express \)D \( in terms of \)B\(, \)A\(, \)x_{\low n+1}\( and \)n$. Hence show that \((n + 1)D \ge nB\) for all values of \(x_{\low n+1}\), but that \(D < B\) if and only if \[ A-\sqrt{\frac{(n+1)B}{n}}

The first four terms of a sequence are given by \(F_0=0\), \(F_1=1\), \(F_2=1\) and \(F_3=2\). The general term is given by \[ F_n= a\lambda^n+b\mu^n\,, \tag{\(*\)} \] where \(a\), \(b\), \(\lambda\) and \(\mu\) are independent of \(n\), and \(a\) is positive.

1. Show that \(\lambda^2 +\lambda\mu+ \mu^2 = 2\), and find the values of \(\lambda\), \(\mu\), \(a\) and \(b\).
2. Use \((*)\) to evaluate \(F_6\).
3. Evaluate \(\displaystyle \sum_{n=0}^\infty \frac{F_n}{2^{n+1}}\,.\)

The Fibonacci sequence \(F_1\), \(F_2\), \(F_3\), \(\ldots\) is defined by \(F_1=1\), \(F_2= 1\) and \[ F_{n+1} = F_n+F_{n-1} \qquad\qquad (n\ge 2). \] Write down the values of \(F_3\), \(F_4\), \(\ldots\), \(F_{10}\). Let \(\displaystyle S=\sum_{i=1}^\infty \dfrac1 {F_i}\,\). \begin{questionparts} \item Show that \(\displaystyle \frac 1{F_i} > \frac1{2F_{i-1}}\,\) for \(i\ge4\) and deduce that \(S > 3\,\). Show also that \(S < 3\frac23\,\). \item Show further that \(3.2 < S < 3.5\,\). \end{questionpart}

I borrow \(C\) pounds at interest rate \(100\alpha \,\%\) per year. The interest is added at the end of each year. Immediately after the interest is added, I make a repayment. The amount I repay at the end of the \(k\)th year is \(R_k\) pounds and the amount I owe at the beginning of \(k\)th year is \(C_k\) pounds (with \(C_1=C\)). Express \(C_{n+1}\) in terms of \(R_k\) (\(k= 1\), \(2\), \(\ldots\), \(n\)), \(\alpha\) and \(C\) and show that, if I pay off the loan in \(N\) years with repayments given by \(R_k= (1+\alpha)^kr\,\), where \(r\) is constant, then \(r=C/N\,\). If instead I pay off the loan in \(N\) years with \(N\) equal repayments of \(R\) pounds, show that \[ \frac R C = \frac{\alpha (1+\alpha)^{N} }{(1+\alpha)^N-1} \;, \] and that \(R/C\approx 27/103\) in the case \(\alpha =1/50\), \(N=4\,\).

How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5? What is the average value of these integers? How many integers greater than or equal to zero and less than 4179 are not divisible by 3 or 7? What is the average value of these integers?

My bank pays \(\rho\)\% interest at the end of each year. I start with nothing in my account. Then for \(m\) years I deposit \(\pounds a\) in my account at the beginning of each year. After the end of the \(m\)th year, I neither deposit nor withdraw for \(l\) years. Show that the total amount in my account at the end of this period is \[\pounds a\frac{r^{l+1}(r^{m}-1)}{r-1}\] where \(r=1+{\displaystyle \frac{\rho}{100}}\). At the beginning of each of the \(n\) years following this period I withdraw \(\pounds b\) and this leaves my account empty after the \(n\)th withdrawal. Find an expression for \(a/b\) in terms of \(r\), \(l\), \(m\) and \(n\).

Find the sum of those numbers between 1000 and 6000 every one of whose digits is one of the numbers \(0,\,2,\,5\) or \(7\), giving your answer as a product of primes.

1. By using the formula for the sum of a geometric series, or otherwise, express the number \(0.38383838\ldots\) as a fraction in its lowest terms.
2. Let \(x\) be a real number which has a recurring decimal expansion \[ x=0\cdot a_{1}a_{2}a_{2}\cdots, \] so that there exists positive integers \(N\) and \(k\) such that \(a_{n+k}=a_{n}\) for all \(n>N.\) Show that \[ x=\frac{b}{10^{N}}+\frac{c}{10^{N}(10^{k}-1)}\,, \] where \(b\) and \(c\) are integers to be found. Deduce that \(x\) is rational.

1. If \(\mathrm{f}(r)\) is a function defined for \(r=0,1,2,3,\ldots,\) show that \[ \sum_{r=1}^{n}\left\{ \mathrm{f}(r)-\mathrm{f}(r-1)\right\} =\mathrm{f}(n)-\mathrm{f}(0). \]
2. If \(\mathrm{f}(r)=r^{2}(r+1)^{2},\) evaluate \(\mathrm{f}(r)-\mathrm{f}(r-1)\) and hence determine \({\displaystyle \sum_{r=1}^{n}r^{3}.}\)
3. Find the sum of the series \(1^{3}-2^{3}+3^{3}-4^{3}+\cdots+(2n+1)^{3}.\)

From the facts \begin{alignat*}{2} 1 & \quad=\quad & & 0\\ 2+3+4 & \quad=\quad & & 1+8\\ 5+6+7+8+9 & \quad=\quad & & 8+27\\ 10+11+12+13+14+15+16 & \quad=\quad & & 27+64 \end{alignat*} guess a general law. Prove it. Hence, or otherwise, prove that \[ 1^{3}+2^{3}+3^{3}+\cdots+N^{3}=\tfrac{1}{4}N^{2}(N+1)^{2} \] for every positive integer \(N\). [Hint. You may assume that \(1+2+3+\cdots+n=\frac{1}{2}n(n+1)\).]

If \(\left|r\right|\neq1,\) show that \[ 1+r^{2}+r^{4}+\cdots+r^{2n}=\frac{1-r^{2n+2}}{1-r^{2}}\,. \] If \(r\neq1,\) find an expression for \(\mathrm{S}_{n}(r),\) where \[ \mathrm{S}_{n}(r)=r+r^{2}+r^{4}+r^{5}+r^{7}+r^{8}+r^{10}+\cdots+r^{3n-1}. \] Show that, if \(\left|r\right|<1,\) then, as \(n\rightarrow\infty,\) \[ \mathrm{S}_{n}(r)\rightarrow\frac{1}{1-r}-\frac{1}{1-r^{3}}\,. \] If \(\left|r\right|\neq1,\) find an expression for \(\mathrm{T}_{n}(r),\) where \[ \mathrm{T}_{n}(r)=1+r^{2}+r^{3}+r^{4}+r^{6}+r^{8}+r^{9}+r^{10}+r^{12}+r^{14}+r^{15}+r^{16}+\cdots+r^{6n}. \] If \(\left|r\right|<1,\) find the limit of \(\mathrm{T}_{n}(r)\) as \(n\rightarrow\infty.\) What happens to \(\mathrm{T}_{n}(r)\) as \(n\rightarrow\infty\) in the three cases \(r>1,r=1\) and \(r=-1\)? In each case give reasons for your answer.

The sequence \(a_{1},a_{2},\ldots,a_{n},\ldots\) forms an arithmetic progression. Establish a formula, involving \(n,\) \(a_{1}\) and \(a_{2}\) for the sum \(a_{1}+a_{2}+\cdots+a_{n}\) of the first \(n\) terms. A sequence \(b_{1},b_{2},\ldots,b_{n},\ldots\) is called a double arithmetic progression if the sequence of differences \[ b_{2}-b_{1},b_{3}-b_{2},\ldots,b_{n+1}-b_{n},\ldots \] is an arithmetic progression. Establish a formula, involving \(n,b_{1},b_{2}\) and \(b_{3}\), for the sum \(b_{1}+b_{2}+b_{3}+\cdots+b_{n}\) of the first \(n\) terms of such a progression. A sequence \(c_{1},c_{2},\ldots,c_{n},\ldots\) is called a factorial progression if \(c_{n+1}-c_{n}=n!d\) for some non-zero \(d\) and every \(n\geqslant1\). Suppose \(1,b_{2},b_{3},\ldots\) is a double arithmetic progression, and also that \(b_{2},b_{4},b_{6}\) and \(220\) are the first four terms in a factorial progression. Find the sum \(1+b_{2}+b_{3}+\cdots+b_{n}.\)

A firm of engineers obtains the right to dig and exploit an undersea tunnel. Each day the firm borrows enough money to pay for the day's digging, which costs £\(c,\) and to pay the daily interest of \(100k\%\) on the sum already borrowed. The tunnel takes \(T\) days to build, and, once finished, earns £\(d\) a day, all of which goes to pay the daily interest and repay the debt until it is fully paid. The financial transactions take place at the end of each day's work. Show that \(S_{n},\) the total amount borrowed by the end of day \(n\), is given by \[ S_{n}=\frac{c[(1+k)^{n}-1]}{k} \] for \(n\leqslant T\). Given that \(S_{T+m}>0,\) where \(m>0,\) express \(S_{T+m}\) in terms of \(c,d,k,T\) and \(m.\) Show that, if \(d/c>(1+k)^{T}-1,\) the firm will eventually pay off the debt.

The definition of the derivative \(f'\) of a (differentiable) function f is $$f'(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h}. \quad (*)$$

1. The function f has derivative \(f'\) and satisfies $$f(x + y) = f(x)f(y)$$ for all \(x\) and \(y\), and \(f'(0) = k\) where \(k \neq 0\). Show that \(f(0) = 1\). Using \((*)\), show that \(f'(x) = kf(x)\) and find \(f(x)\) in terms of \(x\) and \(k\).
2. The function g has derivative \(g'\) and satisfies $$g(x + y) = \frac{g(x) + g(y)}{1 + g(x)g(y)}$$ for all \(x\) and \(y\), \(|g(x)| < 1\) for all \(x\), and \(g'(0) = k\) where \(k \neq 0\). Find \(g'(x)\) in terms of \(g(x)\) and \(k\), and hence find \(g(x)\) in terms of \(x\) and \(k\).

Let \(f(x) = (x-p)g(x)\), where g is a polynomial. Show that the tangent to the curve \(y = f(x)\) at the point with \(x = a\), where \(a \neq p\), passes through the point \((p, 0)\) if and only if \(g'(a) = 0\). The curve \(C\) has equation $$y = A(x - p)(x - q)(x - r),$$ where \(p\), \(q\) and \(r\) are constants with \(p < q < r\), and \(A\) is a non-zero constant.

1. The tangent to \(C\) at the point with \(x = a\), where \(a \neq p\), passes through the point \((p, 0)\). Show that \(2a = q + r\) and find an expression for the gradient of this tangent in terms of \(A\), \(q\) and \(r\).
2. The tangent to \(C\) at the point with \(x = c\), where \(c \neq r\), passes through the point \((r, 0)\). Show that this tangent is parallel to the tangent in part (i) if and only if the tangent to \(C\) at the point with \(x = q\) does not meet the curve again.

The real numbers \(a_1\), \(a_2\), \(a_3\), \(\ldots\) are all positive. For each positive integer \(n\), \(A_n\) and \(G_n\) are defined by \[ A_n = \frac{a_1+a_2 + \cdots + a_n}n \ \ \ \ \ \text{and } \ \ \ \ \ G_n = \big( a_1a_2\cdots a_n\big) ^{1/n} \,. \]

1. Show that, for any given positive integer \(k\), \[ (k+1) ( A_{k+1} - G_{k+1}) \ge k (A_k-G_k) \] if and only if \[\lambda^{k+1}_k -(k+1)\lambda_{{k}} +k \ge 0\,, \] where \( \lambda_{{k}} = \left(\dfrac{a_{k+1}}{G_{k}}\right)^{\frac1 {k+1}}\,\).
2. Let \[ \f(x)=x^{k+1} -(k+1)x +k \,, \] where \(x > 0\) and \(k\) is a positive integer. Show that \(\f(x)\ge0\) and that \(\f(x)=0\) if and only if \(x = 1\,\).
3. Deduce that:
   1. \(A_n \ge G_n\) for all \(n\); \\
   2. if \(A_n=G_n\) for some \(n\), then \(a_1=a_2 = \cdots = a_n\,\).

The line \(y=a^2 x\) and the curve \(y=x(b-x)^2\), where \(0 < a < b\,\), intersect at the origin \(O\) and at points \(P\) and \(Q \). The \(x\)-coordinate of \(P\) is less than the \(x\)-coordinate of \(Q\). Find the coordinates of \(P\) and \(Q\), and sketch the line and the curve on the same axes. Show that the equation of the tangent to the curve at \(P\) is \[ y = a(3a-2b)x + 2a(b-a)^2 . \] This tangent meets the \(y\)-axis at \(R\). The area of the region between the curve and the line segment \(OP\) is denoted by \(S\). Show that \[ S= \frac1{12}(b-a)^3(3a+b)\,. \] The area of triangle \(OPR\) is denoted by~\(T\). Show that \(S>\frac{1}{3}T\,\).

The points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), where \(p>0\) and \(q<0\), lie on the curve \(C\) with equation $$y^2= 4ax\,, $$ where \(a>0\,\). Show that the equation of the tangent to \(C\) at \(P\) is $$y= \frac 1 p \, x +ap\,.$$ The tangents to the curve at \(P\) and at \(Q \) meet at \(R\). These tangents meet the \(y\)-axis at \(S\) and \(T\) respectively, and \(O\) is the origin. Prove that the area of triangle \(OPQ\) is twice the area of triangle \(RST\).

Let \[ \f(x) = 3ax^2 - 6x^3\, \] and, for each real number \(a\), let \({\rm M}(a)\) be the greatest value of \(\f(x)\) in the interval \(-\frac13 \le x \le 1\). Determine \({\rm M} (a)\) for \(a\ge0\). [The formula for \({\rm M} (a)\) is different in different ranges of \(a\); you will need to identify three ranges.]

Let \(L_a\) denote the line joining the points \((a,0)\) and \((0, 1-a)\), where \(0< a < 1\). The line~\(L_b\) is defined similarly.

1. Determine the point of intersection of \(L_a\) and \(L_b\), where \(a\ne b\).
2. Show that this point of intersection, in the limit as $b\to a\(, lies on the curve~\)C$ given by \[ y=(1-\sqrt x)^2\, \ \ \ \ (0< x < 1)\,. \]
3. Show that every tangent to~\(C\) is of the form~\(L_a\) for some~\(a\).

1. A function \(\f(x)\) satisfies \(\f(x) = \f(1-x)\) for all \(x\). Show, by differentiating with respect to \(x\), that \(\f'(\frac12) =0\,\). If, in addition, \(\f(x) = \f(\frac1x)\) for all (non-zero) \(x\), show that \(\f'(-1)=0\) and that \(\f'(2)=0\).
2. The function \(\f\) is defined, for \(x\ne0\) and \(x\ne1\), by \[ \f(x) = \frac {(x^2-x+1)^3}{(x^2-x)^2} \,. \] Show that \(\f(x)= \f(\frac 1 x)\) and \(\f(x) = \f(1-x)\). Given that it has exactly three stationary points, sketch the curve \(y=\f(x)\).
3. Hence, or otherwise, find all the roots of the equation \(\f(x) = \dfrac {27} 4\,\) and state the ranges of values of \(x\) for which \(\f(x) > \dfrac{27} 4\,\). Find also all the roots of the equation \(\f(x) = \dfrac{343}{36}\,\) and state the ranges of values of \(x\) for which \(\f(x) > \dfrac{343}{36}\).

1. Sketch the curve \(y= x^4-6x^2+9\) giving the coordinates of the stationary points. Let \(n\) be the number of distinct real values of \(x\) for which \[ x^4-6x^2 +b=0. \] State the values of \(b\), if any, for which
   1. \(n=0\,\);
   2. \(n=1\,\);
   3. \(n=2\,\);
   4. \(n=3\,\);
   5. \(n=4\,\).
2. For which values of \(a\) does the curve \(y= x^4-6x^2 +ax +b\) have a point at which both \(\dfrac{\d y}{\d x}=0\) and \(\dfrac{\d^2y}{\d x^2}=0\,\)? For these values of \(a\), find the number of distinct real values of \(x\) for which \(\vphantom{\dfrac{A}{B}}\) \[ x^4-6x^2 +ax +b=0\,, \] in the different cases that arise according to the value of \(b\).
3. Sketch the curve \(y= x^4-6x^2 +ax\) in the case \(a>8\,\).

1. Find the coordinates of the turning points of the curve \(y=27x^3-27x^2+4\). Sketch the curve and deduce that \(x^2(1-x)\le 4/27\) for all \(x\ge0\,\). Given that each of the numbers \(a\), \(b\) and \(c\) lies between \(0\) and \(1\), prove by contradiction that at least one of the numbers \(bc(1-a)\), \(ca(1-b)\) and \(ab(1-c)\) is less than or equal to \(4/27\).
2. Given that each of the numbers \(p\) and \(q\) lies between \(0\) and \(1\), prove that at least one of the numbers \(p(1-q)\) and \(q(1-p)\) is less than or equal to \(1/4\).

A small goat is tethered by a rope to a point at ground level on a side of a square barn which stands in a large horizontal field of grass. The sides of the barn are of length \(2a\) and the rope is of length \(4a\). Let \(A\) be the area of the grass that the goat can graze. Prove that \(A\le14\pi a^2\) and determine the minimum value of \(A\).

Show Solution

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The areas are \(8\pi a^2 + \frac14 \pi (4a-x)^2 + \frac14 \pi (2a-x)^2 + \frac14\pi(2a+x)^2+\frac14 \pi x^2\) ie \begin{align*} A &= \frac{\pi}{4} \left ( x^2 \left (1 + 1 + 1 + 1 \right) + x \left (4a-4a-8a \right)+\left (32a^2+16a^2+4a^2+4a^2 \right)\right) \\ &= \frac{\pi}{4} \left (4x^2-8ax+56a^2 \right) \\ &= \pi(x^2-2ax+14a^2) \\ &= \pi ((x-a)^2+13a^2) \end{align*} Since \(x \in [0, 2a]\) we have \(13\pi a^2 \leq A \leq 14 \pi a^2\)

The point \(P\) has coordinates \(\l p^2 , 2p \r\) and the point \(Q\) has coordinates \(\l q^2 , 2q \r\), where \(p\) and~\(q\) are non-zero and \(p \neq q\). The curve \(C\) is given by \(y^2 = 4x\,\). The point \(R\) is the intersection of the tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\). Show that \(R\) has coordinates \(\l pq , p+q \r\). The point \(S\) is the intersection of the normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\). If \(p\) and \(q\) are such that \(\l 1 , 0 \r\) lies on the line \(PQ\), show that \(S\) has coordinates \(\l p^2 + q^2 + 1 , \, p+q \r\), and that the quadrilateral \(PSQR\) is a rectangle.

Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a square. The result changes if, instead of maximising the sum of lengths of sides of the rectangle, we seek to maximise the sum of \(n\)th powers of the lengths of those sides for \(n\geqslant 2\). What happens if \(n=2\)? What happens if \(n=3\)? Justify your answers.

1. By considering \((1+x+x^{2}+\cdots+x^{n})(1-x)\) show that, if \(x\neq1\), \[ 1+x+x^{2}+\cdots+x^{n}=\frac{1-x^{n+1}}{1-x}. \]
2. By differentiating both sides and setting \(x=-1\) show that \[ 1-2+3-4+\cdots+(-1)^{n-1}n \] takes the value \(-n/2\) is \(n\) is even and the value \((n+1)/2\) if \(n\) is odd.
3. Show that \[ 1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n-1}n^{2}=(-1)^{n-1}(An^{2}+Bn) \] where the constants \(A\) and \(B\) are to be determined.

\(\lozenge\) is an operation which take polynomials in \(x\) to polynomials in \(x\); that is, given a polynomial \(\mathrm{h}(x)\) there is another polynomial called \(\lozenge\mathrm{h}(x)\). It is given that, if \(\mathrm{f}(x)\) and \(\mathrm{g}(x)\) are any two polynomials in \(x\), the following are always true:

1. \(\lozenge(\mathrm{f}(x)\mathrm{g}(x))=\mathrm{g}(x)\lozenge\mathrm{f}(x)+\mathrm{f}(x)\lozenge\mathrm{g}(x),\)
2. \(\lozenge(\mathrm{f}(x)+\mathrm{g}(x))=\lozenge\mathrm{f}(x)+\lozenge\mathrm{g}(x),\)
3. \(\lozenge x=1\)
4. if \(\lambda\) is a constant then \(\lozenge(\lambda\mathrm{f}(x))=\lambda\lozenge\mathrm{f}(x).\)

Given a curve described by \(y=\mathrm{f}(x)\), and such that \(y\geqslant0\), a push-off of the curve is a new curve obtained as follows: for each point \((x,\mathrm{f}(x))\) with position vector \(\mathbf{r}\) on the original curve, there is a point with position vector \(\mathbf{s}\) on the new curve such that \(\mathbf{s-r}=\mathrm{p}(x)\mathbf{n},\) where \(\mathrm{p}\) is a given function and \(\mathbf{n}\) is the downward-pointing unit normal to the original curve at \(\mathbf{r}\).

1. For the curve \(y=x^{k},\) where \(x>0\) and \(k\) is a positive integer, obtain the function \(\mathrm{p}\) for which the push-off is the positive \(x\)-axis, and find the value of \(k\) such that, for all points on the original curve, \(\left|\mathbf{r}\right|=\left|\mathbf{r-s}\right|\).
2. Suppose that the original curve is \(y=x^{2}\) and \(\mathrm{p}\) is such that the gradient of the curves at the points with position vectors \(\mathbf{r}\) and \(\mathbf{s}\) are equal (for every point on the original curve). By writing \(\mathrm{p}(x)=\mathrm{q}(x)\sqrt{1+4x^{2}},\) where \(\mathrm{q}\) is to be determined, or otherwise, find the form of \(\mathrm{p}\).

The function \(\mathrm{f}\) is defined by \[ \mathrm{f}(x)=ax^{2}+bx+c. \] Show that \[ \mathrm{f}'(x)=\mathrm{f}(1)\left(x+\tfrac{1}{2}\right)+\mathrm{f}(-1)\left(x-\tfrac{1}{2}\right)-2\mathrm{f}(0)x. \] If \(a,b\) and \(c\) are real and such that \(\left|\mathrm{f}(x)\right|\leqslant1\) for \(\left|x\right|\leqslant1\), show that \(\left|\mathrm{f}'(x)\right|\leqslant4\) for \(\left|x\right|\leqslant1\). Find particular values of \(a,b\) and \(c\) such that, for the corresponding function \(\mathrm{f}\) of the above form \(\left|\mathrm{f}(x)\right|\leqslant1\) for all \(x\) with \(\left|x\right|\leqslant1\) and \(\mathrm{f}'(x)=4\) for some \(x\) satisfying \(\left|x\right|\leqslant1\).

In this question, you may assume that, if a continuous function takes both positive and negative values in an interval, then it takes the value \(0\) at some point in that interval.

1. The function \(\f\) is continuous and \(\f(x)\) is non-zero for some value of \(x\) in the interval \(0\le x \le 1\). Prove by contradiction, or otherwise, that if \[ \int_0^1 \f(x) \d x = 0\,, \] then \(\f(x)\) takes both positive and negative values in the interval \(0\le x\le 1\).
2. The function \(\g\) is continuous and \[ \int_0^1 \g(x) \, \d x = 1\,, \quad \int_0^1 x\g(x) \, \d x = \alpha\, , \quad \int_0^1 x^2\g(x) \, \d x = \alpha^2\,. \tag{\(*\)} \] Show, by considering \[ \int_0^1 (x - \alpha)^2 \g(x) \, \d x \,, \] that \(\g(x)=0\) for some value of \(x\) in the interval \(0\le x\le 1\). Find a function of the form \(\g(x) = a+bx\) that satisfies the conditions \((*)\) and verify that \(\g(x)=0\) for some value of \(x\) in the interval \(0\le x \le 1\).
3. The function \(\h\) has a continuous derivative \(\h'\) and \[ \h(0) = 0\,, \quad \h(1) = 1\,, \quad \int_0^1 \h(x) \, \d x = \beta\,, \quad \int_0^1 x \h(x) \, \d x = \ts \frac{1}{2} \ds \beta (2 - \beta) \,. \] Use the result in part \bf (ii) \rm to show that \(\h^\prime(x)=0\) for some value of \(x\) in the interval \(0\le x\le 1\).

A spherical loaf of bread is cut into parallel slices of equal thickness. Show that, after any number of the slices have been eaten, the area of crust remaining is proportional to the number of slices remaining. A European ruling decrees that a parallel-sliced spherical loaf can only be referred to as `crusty' if the ratio of volume \(V\) (in cubic metres) of bread remaining to area \(A\) (in square metres) of crust remaining after any number of slices have been eaten satisfies \(V/A<1\). Show that the radius of a crusty parallel-sliced spherical loaf must be less than \(2\frac23\) metres. [{\sl The area \(A\) and volume \(V\) formed by rotating a curve in the \(x\)--\(y\) plane round the \(x\)-axis from \(x=-a\) to \(x=-a+t\) are given by \[ A= 2\pi\int_{-a}^{-a+t} { y}\left( 1+ \Big(\frac{\d {y}}{\d x}\Big)^2\right)^{\frac12} \d x\;, \ \ \ \ \ \ \ \ \ \ \ V= \pi \int_{-a}^{-a+t} {y}^2 \d x \;. \ \ ] \] }

For any number \(x\), the largest integer less than or equal to \(x\) is denoted by \([x]\). For example, \([3.7]=3\) and \([4]=4\). Sketch the graph of \(y=[x]\) for \(0\le x<5\) and evaluate \[ \int_0^5 [x]\;\d x. \] Sketch the graph of \(y=[\e^{x}]\) for \(0\le x< \ln n\), where \(n\) is an integer, and show that \[ \int_{0}^{\ln n}[\e^{x}]\, \d x =n\ln n - \ln (n!). \]

Show Solution

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\begin{align*} \int_0^5 [x]\;\d x &= 0 \cdot 1 + 1 \cdot 1 + 2 \cdot 1 + 3 \cdot 1 + 4 \cdot 1 \\ &= 10 \end{align*}

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\begin{align*} \int_{0}^{\ln n}[\e^{x}]\, \d x &= \sum_{k=1}^{n-1} \int_{\ln k}^{\ln (k+1)}[\e^{x}]\, \d x \\ &= \sum_{k=1}^{n-1} k \l \ln (k+1) - \ln (k) \r\\ &= \sum_{k=1}^{n-1} \l( (k+1) \l \ln (k+1) - \ln (k) \r - \ln(k+1) \r \\ &= \sum_{k=1}^{n-1} (k+1) \ln (k+1) - \sum_{k=1}^{n-1} k \ln (k) - \sum_{k=1}^{n-1} \ln (k+1) \\ &= n \ln n - 1 \ln 1 - \sum_{k=1}^{n-1} \ln (k+1) \\ &= n \ln n - \ln \l \prod_{k=1}^{n-1} (k+1)\r \\ &= n \ln n - \ln (n!) \end{align*}

If \({\rm f}(t)\ge {\rm g}(t)\) for \(a\le t\le b\), explain very briefly why \(\displaystyle \int_a^b {\rm f}(t) \d t \ge \int_a^b {\rm g}(t) \d t\). Prove that if \(p>q>0\) and \(x\ge1\) then $$\frac{x^p-1}{ p}\ge\frac{x^q-1}{ q}.$$ Show that this inequality also holds when \(p>q>0\) and \(0\le x\le1\). Prove that, if \(p>q>0\) and \(x\ge0\), then $$\frac{1}{ p}\left(\frac{x^p}{ p+1}-1\right)\ge \frac{1}{q}\left(\frac{x^q}{ q+1}-1\right).$$

Find constants \(a_{1}\), \(a_{2}\), \(u_{1}\) and \(u_{2}\) such that, whenever \({\mathrm P}\) is a cubic polynomial, \[\int_{-1}^{1}{\mathrm P}(t)\,{\mathrm d}t =a_{1}{\mathrm P}(u_{1})+a_{2}{\mathrm P}(u_{2}).\]

Explain diagrammatically, or otherwise, why \[ \frac{\mathrm{d}}{\mathrm{d}x}\int_{a}^{x}\mathrm{f}(t)\,\mathrm{d}t=\mathrm{f}(x). \] Show that, if \[ \mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t+1, \] then \(\mathrm{f}(x)=\mathrm{e}^{x}.\) What is the solution of \[ \mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t? \] Given that \[ \int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t=\int_{x}^{1}t^{2}\mathrm{f}(t)\,\mathrm{d}t+x-\frac{x^{5}}{5}+C, \] find \(\mathrm{f}(x)\) and show that \(C=-2/15.\)

Criticise each step of the following arguments. You should correct the arguments where necessary and possible, and say (with justification) whether you think the conclusion are true even though the argument is incorrect.

1. The function \(g\) defined by \[ \mathrm{g}(x)=\frac{2x^{3}+3}{x^{4}+4} \] satisfies \(\mathrm{g}'(x)=0\) only for \(x=0\) or \(x=\pm1.\) Hence the stationary values are given by \(x=0\), \(\mathrm{g}(x)=\frac{3}{4}\) and \(x=\pm1,\) \(\mathrm{g}(x)=1.\) Since \(\frac{3}{4}<1,\) there is a minimum at \(x=0\) and maxima at \(x=\pm1.\) Thus we must have \(\frac{3}{4}\leqslant\mathrm{g}(x)\leqslant1\) for all \(x\).
2. \({\displaystyle \int(1-x)^{-3}\,\mathrm{d}x=-3(1-x)^{-4}}\quad\) and so \(\quad{\displaystyle \int_{-1}^{3}(1-x)^{-3}\,\mathrm{d}x=0.}\)

1. Prove that \[ \sum_{r=1}^{n}r(r+1)(r+2)(r+3)(r+4)=\tfrac{1}{6}n(n+1)(n+2)(n+3)(n+4)(n+5) \] and deduce that \[ \sum_{r=1}^{n}r^{5}<\tfrac{1}{6}n(n+1)(n+2)(n+3)(n+4)(n+5). \]
2. Prove that, if \(n>1,\) \[ \sum_{r=0}^{n-1}r^{5}>\tfrac{1}{6}(n-5)(n-4)(n-3)(n-2)(n-1)n. \]
3. Let \(\mathrm{f}\) be an increasing function. If the limits \[ \lim_{n\rightarrow\infty}\sum_{r=0}^{n-1}\frac{a}{n}\mathrm{f}\left(\frac{ra}{n}\right)\qquad\mbox{ and }\qquad\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\frac{a}{n}\mathrm{f}\left(\frac{ra}{n}\right) \] both exist and are equal, the definite integral \({\displaystyle \int_{0}^{a}\mathrm{f}(x)\,\mathrm{d}x}\) is defined to be their common value. Using this definition, prove that \[ \int_{0}^{a}x^{5}\,\mathrm{d}x=\tfrac{1}{6}a^6. \]

The function \(\mathrm{f}\) satisfies the condition \(\mathrm{f}'(x)>0\) for \(a\leqslant x\leqslant b\), and \(\mathrm{g}\) is the inverse of \(\mathrm{f}.\) By making a suitable change of variable, prove that \[ \int_{a}^{b}\mathrm{f}(x)\,\mathrm{d}x=b\beta-a\alpha-\int_{\alpha}^{\beta}\mathrm{g}(y)\,\mathrm{d}y, \] where \(\alpha=\mathrm{f}(a)\) and \(\beta=\mathrm{f}(b)\). Interpret this formula geometrically, in the case where \(\alpha\) and \(a\) are both positive. Prove similarly and interpret (for \(\alpha>0\) and \(a>0\)) the formula \[ 2\pi\int_{a}^{b}x\mathrm{f}(x)\,\mathrm{d}x=\pi(b^{2}\beta-a^{2}\alpha)-\pi\int_{\alpha}^{\beta}\left[\mathrm{g}(y)\right]^{2}\,\mathrm{d}y. \]

Show Solution

Let \(u = f(x)\) then \(\frac{\d u}{\d x} = f'(x)\) and \begin{align*} \int_a^b f(x) \d x &\underbrace{=}_{\text{IBP}} \left [ xf(x) \right]_a^b - \int_a^b x f'(x) \d x \\ &\underbrace{=}_{u = f(x)} b \beta - a \alpha - \int_{u = f(a) = \alpha}^{u = f(b) = \beta} g(u) \d u \\ &= b \beta - a \alpha - \int_{\alpha}^{\beta} g(u) \d u \end{align*}

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\[ \underbrace{\int_{a}^{b}\mathrm{f}(x)\,\mathrm{d}x}_{\text{red area}}=\underbrace{b\beta}_{\text{whole area}}-\underbrace{a\alpha}_{\text{area in green}}-\underbrace{\int_{\alpha}^{\beta}\mathrm{g}(y)\,\mathrm{d}y}_{\text{area in blue}}, \] \begin{align*} 2\pi \int_a^b x f(x) \d x &\underbrace{=}_{\text{IBP}}\pi \left [ x^2 f(x) \right]_a^b - \pi \int_a^b x^2 f'(x) \d x \\ &\underbrace{=}_{x = g(u)} \pi (b^2 \beta - a^2 \alpha) - \pi \int_{u = f(a) = \alpha}^{u = f(b) = \beta} [g(u)]^2 \d u \\ &= \pi(b^2 \beta - a^2 \alpha) - \pi \int_\alpha^\beta [g(u)]^2 \d u \end{align*} This is the volume outside the function in the volume of revolution about the \(y\) axis between \( \alpha\) and \(\beta\).

Three points, \(A\), \(B\) and \(C\), lie in a horizontal plane, but are not collinear. The point \(O\) lies above the plane. Let \(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(P\) is a point with \(\overrightarrow{OP} = \alpha\mathbf{a} + \beta\mathbf{b} + \gamma\mathbf{c}\), where \(\alpha\), \(\beta\) and \(\gamma\) are all positive and \(\alpha + \beta + \gamma < 1\). Let \(k = 1 - (\alpha + \beta + \gamma)\).

1. The point \(L\) is on \(OA\), the point \(X\) is on \(BC\) and \(LX\) passes through \(P\). Determine \(\overrightarrow{OX}\) in terms of \(\beta\), \(\gamma\), \(\mathbf{b}\) and \(\mathbf{c}\) and show that \(\overrightarrow{OL} = \frac{\alpha}{k+\alpha}\mathbf{a}\).
2. Let \(M\) and \(Y\) be the unique pair of points on \(OB\) and \(CA\) respectively such that \(MY\) passes through \(P\), and let \(N\) and \(Z\) be the unique pair of points on \(OC\) and \(AB\) respectively such that \(NZ\) passes through \(P\). Show that the plane \(LMN\) is also horizontal if and only if \(OP\) intersects plane \(ABC\) at the point \(G\), where \(\overrightarrow{OG} = \frac{1}{3}(\mathbf{a} + \mathbf{b} + \mathbf{c})\). Where do points \(X\), \(Y\) and \(Z\) lie in this case?
3. State what the condition \(\alpha + \beta + \gamma < 1\) tells you about the position of \(P\) relative to the tetrahedron \(OABC\).

1. The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), respectively. Each of these vectors is a unit vector (so \(\mathbf{a} \cdot \mathbf{a} = 1\), for example) and $$\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.$$ Show that \(\mathbf{a} \cdot \mathbf{b} = -\frac{1}{2}\). What can be said about the triangle ABC? You should justify your answer.
2. The four distinct points \(A_i\) (\(i = 1, 2, 3, 4\)) have unit position vectors \(\mathbf{a}_i\) and $$\sum_{i=1}^{4} \mathbf{a}_i = \mathbf{0}.$$ Show that \(\mathbf{a}_1 \cdot \mathbf{a}_2 = \mathbf{a}_3 \cdot \mathbf{a}_4\).
   1. Given that the four points lie in a plane, determine the shape of the quadrilateral with vertices \(A_1\), \(A_2\), \(A_3\) and \(A_4\).
   2. Given instead that the four points are the vertices of a regular tetrahedron, find the length of the sides of this tetrahedron.

Let \(\mathbf{r}\) be the position vector of a point in three-dimensional space. Describe fully the locus of the point whose position vector is \(\mathbf{r}\) in each of the following four cases:

1. \(\left(\mathbf{a-b}\right) \cdot \mathbf{r}=\frac{1}{2}(\left|\mathbf{a}\right|^{2}-\left|\mathbf{b}\right|^{2});\)
2. \(\left(\mathbf{a-r}\right)\cdot\left(\mathbf{b-r}\right)=0;\)
3. \(\left|\mathbf{r-a}\right|^{2}=\frac{1}{2}\left|\mathbf{a-b}\right|^{2};\)
4. \(\left|\mathbf{r-b}\right|^{2}=\frac{1}{2}\left|\mathbf{a-b}\right|^{2}.\)

\(ABC\) is a triangle whose vertices have position vectors \(\mathbf{a,b,c}\)brespectively, relative to an origin in the plane \(ABC\). Show that an arbitrary point \(P\) on the segment \(AB\) has position vector \[ \rho\mathbf{a}+\sigma\mathbf{b}, \] where \(\rho\geqslant0\), \(\sigma\geqslant0\) and \(\rho+\sigma=1\). Give a similar expression for an arbitrary point on the segment \(PC\), and deduce that any point inside \(ABC\) has position vector \[ \lambda\mathbf{a}+\mu\mathbf{b}+\nu\mathbf{c}, \] where \(\lambda\geqslant0\), \(\mu\geqslant0\), \(\nu\geqslant0\) and \(\lambda+\mu+\nu=1\). Sketch the region of the plane in which the point \(\lambda\mathbf{a}+\mu\mathbf{b}+\nu\mathbf{c}\) lies in each of the following cases:

1. \(\lambda+\mu+\nu=-1\), \(\lambda\leqslant0\), \(\mu\leqslant0\), \(\nu\leqslant0\);
2. \(\lambda+\mu+\nu=1\), \(\mu\leqslant0\), \(\nu\leqslant0\).

Show Solution

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Suppose \(P\) is a fraction \(0 \leq k\leq 1\) along \(AB\), then \(\overrightarrow{OP} = \overrightarrow{OA} +\overrightarrow{AP} = \overrightarrow{OA} +k\overrightarrow{AB} = \mathbf{a} + k(\mathbf{b} - \mathbf{a}) = \lambda \mathbf{b} + (1-k) \mathbf{a}\), ie an arbitrary point on \(AB\) has the position vector where \((1-k) = \rho \geq 0\) and \(k= \sigma \geq 0\) and \((1-\lambda) + \lambda = 1\). Any point on the segment \(PC\) will be of the form \(l\mathbf{c} + (1-l) (k \mathbf{b} + (1-k) \mathbf{a})\) which has the form \(\lambda \mathbf{a} + \mu \mathbf{b} + \nu \mathbf{c}\) where \(\lambda + \mu + \nu = (1-l)(1-k) + (1-l)k + l = 1\) and all coefficients are positive.

1. This is equivalent to the area inside the triangle where every point (\(\mathbf{a}, \mathbf{b}, \mathbf{c}\)) has been send to it's negative (\(-\mathbf{a}, -\mathbf{b}, -\mathbf{c}\)), ie
   

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2. Writing points as: \begin{align*} \lambda\mathbf{a}+\mu\mathbf{b}+\nu\mathbf{c} &= (1-\mu - \nu)\mathbf{a}+\mu\mathbf{b}+\nu\mathbf{c} \\ &= \mathbf{a} + (-\mu)(\mathbf{a}-\mathbf{b}) + (-\nu)(\mathbf{a} - \mathbf{c}) \\ &= \mathbf{a} + (-\mu)(\mathbf{a}-\mathbf{b}) + (-\nu)(\mathbf{a} - \mathbf{c}) + (1+\mu+\nu)\mathbf{0}\\ \end{align*} So this is a translation of \(\mathbf{a}\) of the triangle with vertices at \(\mathbf{0}, \mathbf{a-b}, \mathbf{a-c}\), or a triangle with vertices \(\mathbf{a}, 2\mathbf{a-b}, 2\mathbf{a-c}\).
   

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The vertices of a plane quadrilateral are labelled \(A\), \(B\), \(A'\) and \(B'\), in clockwise order. A point \(O\) lies in the same plane and within the quadrilateral. The angles \(AOB\) and \(A'OB'\) are right angles, and \(OA=OB\) and \(OA'=OB'\). Use position vectors relative to \(O\) to show that the midpoints of \(AB\), \(BA'\), \(A'B'\) and \(B'A\) are the vertices of a square. Given that the lengths of \(OA\) and \(OA'\) are fixed (and the conditions of the first paragraph still hold), find the value of angle \(BOA'\) for which the area of the square is greatest.

Three non-collinear points \(A\), \(B\) and \(C\) lie in a horizontal ceiling. A particle \(P\) of weight \(W\) is suspended from this ceiling by means of three light inextensible strings \(AP\), \(BP\) and \(CP\), as shown in the diagram. The point \(O\) lies vertically above \(P\) in the ceiling.

1. Show that the unit vector in the direction \(PB\) can be written in the form \[ -\frac13\, {\bf i} - \frac{\sqrt2\,}3\, {\bf j} + \frac{\sqrt2\, }{\sqrt3 \,} \,{\bf k} \,,\] where \(\bf i\,\), \(\, \bf j\) and \(\bf k\) are the usual mutually perpendicular unit vectors with \(\bf j\) parallel to \(OA\) and \(\bf k\) vertically upwards.
2. Find expressions in vector form for the forces acting on \(P\).
3. Show that \(U=\sqrt6 V\) and find \(T\), \(U\) and \(V\) in terms of \(W\).

Note: a regular octahedron is a polyhedron with eight faces each of which is an equilateral triangle.

1. Show that the angle between any two faces of a regular octahedron is \(\arccos \left( -{\frac1 3} \right)\).
2. Find the ratio of the volume of a regular octahedron to the volume of the cube whose vertices are the centres of the faces of the octahedron.

By considering a suitable scalar product, prove that \[ (ax+by+cz)^2 \le (a^2+b^2+c^2)(x^2+y^2+z^2) \] for any real numbers \(a\), \(b\), \(c\), \(x\), \(y\) and \(z\). Deduce a necessary and sufficient condition on \(a\), \(b\), \(c\), \(x\), \(y\) and \(z\) for the following equation to hold: \[ (ax+by+cz)^2 = (a^2+b^2+c^2)(x^2+y^2+z^2) \,. \]

1. Show that \((x+2y+2z)^2 \le 9(x^2+y^2+z^2)\) for all real numbers \(x\), \(y\) and \(z\).
2. Find real numbers \(p\), \(q\) and \(r\) that satisfy both \[ p^2+4q^2+9r^2 = 729 \text{ and } 8p+8q+3r = 243\,. \]

In 3-dimensional space, the lines \(m_1\) and \(m_2\) pass through the origin and have directions \(\bf i + j\) and \(\bf i +k \), respectively. Find the directions of the two lines \(m_3\) and \(m_4\) that pass through the origin and make angles of \(\pi/4\) with both \(m_1\) and \(m_2\). Find also the cosine of the acute angle between \(m_3\) and \(m_4\). The points \(A\) and \(B\) lie on \(m_1\) and \(m_2\) respectively, and are each at distance \(\lambda \surd2\) units from~\(O\). The points \(P\) and \(Q\) lie on \(m_3\) and \(m_4\) respectively, and are each at distance \(1\) unit from~\(O\). If all the coordinates (with respect to axes \(\bf i\), \(\bf j\) and \(\bf k\)) of \(A\), \(B\), \(P\) and \(Q\) are non-negative, prove that:

1. there are only two values of \(\lambda\) for which \(AQ\) is perpendicular to \(BP\,\);
2. there are no non-zero values of \(\lambda\) for which \(AQ\) and \(BP\) intersect.

Three ships \(A\), \(B\) and \(C\) move with velocities \({\bf v}_1\), \({\bf v}_2\) and \(\bf u\) respectively. The velocities of \(A\) and \(B\) relative to \(C\) are equal in magnitude and perpendicular. Write down conditions that \(\bf u\), \({\bf v}_1\) and \({\bf v}_2\) must satisfy and show that \[ \left| {\bf u} -{\textstyle\frac12} \l {\bf v}_1 + {\bf v}_2 \r \right|^2 = \left|{\textstyle\frac12} \l {\bf v}_1 - {\bf v}_2 \r \right|^2 \] and \[ \l {\bf u} -{\textstyle\frac12} \l {\bf v}_1 + {\bf v}_2 \r \r \cdot \l {\bf v}_1 - {\bf v}_2 \r = 0 \;. \] Explain why these equations determine, for given \({\bf v}_1\) and \({\bf v}_2\), two possible velocities for \(C\,\), provided \({\bf v}_1 \ne {\bf v}_2 \,\). If \({\bf v}_1\) and \({\bf v}_2\) are equal in magnitude and perpendicular, show that if \({\bf u} \ne {\bf 0}\) then \({\bf u} = {\bf v}_1 + {\bf v}_2\,\).

The plane \[ {x \over a} + {y \over b} +{z \over c} = 1 \] meets the co-ordinate axes at the points \(A\), \(B\) and \(C\,\). The point \(M\) has coordinates \(\left( \frac12 a, \frac12 b, \frac 12 c \right)\) and \(O\) is the origin. Show that \(OM\) meets the plane at the centroid \(\left( \frac13 a, \frac13 b, \frac 13 c \right)\) of triangle \(ABC\). Show also that the perpendiculars to the plane from \(O\) and from \(M\) meet the plane at the orthocentre and at the circumcentre of triangle \(ABC\) respectively. Hence prove that the centroid of a triangle lies on the line segment joining its orthocentre and circumcentre, and that it divides this line segment in the ratio \(2 : 1\,\). [The orthocentre of a triangle is the point at which the three altitudes intersect; the circumcentre of a triangle is the point equidistant from the three vertices.]

The cuboid \(ABCDEFGH\) is such \(AE\), \(BF\), \(CG\), \(DH\) are perpendicular to the opposite faces \(ABCD\) and \(EFGH\), and \(AB =2, BC=1, AE={\lambda}\). Show that if \(\alpha\) is the acute angle between the diagonals \(AG\) and \(BH\) then $$\cos {\alpha} = |\frac {3-{\lambda}^2} {5+{\lambda}^2} |$$ Let \(R\) be the ratio of the volume of the cuboid to its surface area. Show that \(R<\frac{1}{3}\) for all possible values of \(\lambda\). Prove that, if \(R\ge \frac{1}{4}\), then \(\alpha \le \arccos \frac{1}{9}\).

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Set \(A\) to be the origin, then \(B = \langle 2, 0, 0 \rangle, G = \langle 2, 1, \lambda \rangle, H = \langle 0, 1, \lambda \rangle\), in particular \begin{align*} && AG&= \langle 2, 1, \lambda \rangle \\ && BH &= \langle -2, 1, \lambda \rangle \\ \Rightarrow && \cos \alpha &= |\frac{-4+1+\lambda^2}{\sqrt{2^2+1^2+\lambda^2}\sqrt{(-2)^2+1^2+\lambda^2}}| \\ &&&= |\frac{-3+\lambda^2}{5+\lambda^2}| \end{align*} \begin{align*} && \text{Volume} &= 2\lambda \\ && \text{Surface area} &= 2\cdot2\lambda + 2\cdot\lambda + 2\cdot2 \\ \Rightarrow && R&= \frac{\lambda}{3\lambda + 2} < \frac{1}{3} \\ && \frac14 &\leq R \\ \Rightarrow && 3\lambda +2 &\leq 4\lambda \\ \Rightarrow &&2 & \leq \lambda \end{align*} Then \(\frac{\lambda^2-3}{5+\lambda^2}\) is increasing as \(\lambda\) increases, in particularly the smallest value is \(\frac{1}{9}\).

A ship sails at \(20\) kilometres/hour in a straight line which is, at its closest, 1 kilometre from a port. A tug-boat with maximum speed 12 kilometres/hour leaves the port and intercepts the ship, leaving the port at the latest possible time for which the interception is still possible. How far does the tug-boat travel?

1. Show that four vertices of a cube, no two of which are adjacent, form the vertices of a regular tetrahedron. Hence, or otherwise, find the volume of a regular tetrahedron whose edges are of unit length.
2. Find the volume of a regular octahedron whose edges are of unit length.
3. Show that the centres of the faces of a cube form the vertices of a regular octahedron. Show that its volume is half that of the tetrahedron whose vertices are the vertices of the cube.

Points \(\mathbf{A},\mathbf{B},\mathbf{C}\) in three dimensions have coordinate vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\), respectively. Show that the lines joining the vertices of the triangle \(ABC\) to the mid-points of the opposite sides meet at a point \(R\). \(P\) is a point which is {\bf not} in the plane \(ABC\). Lines are drawn through the mid-points of \(BC\), \(CA\) and \(AB\) parallel to \(PA\), \(PB\) and \(PC\) respectively. Write down the vector equations of the lines and show by inspection that these lines meet at a common point \(Q\). Prove further that the line \(PQ\) meets the plane \(ABC\) at \(R\).

A single stream of cars, each of width \(a\) and exactly in line, is passing along a straight road of breadth \(b\) with speed \(V\). The distance between the successive cars is \(c\).

Four rigid rods \(AB\), \(BC\), \(CD\) and \(DA\) are freely jointed together to form a quadrilateral in the plane. Show that if \(P\), \(Q\), \(R\), \(S\) are the mid-points of the sides \(AB\), \(BC\), \(CD\), \(DA\), respectively, then \[|AB|^{2}+|CD|^{2}+2|PR|^{2}=|AD|^{2}+|BC|^{2}+2|QS|^{2}.\] Deduce that \(|PR|^{2}-|QS|^{2}\) remains constant however the vertices move. (Here \(|PR|\) denotes the length of \(PR\).)

A ship is sailing due west at \(V\) knots while a plane, with an airspeed of \(kV\) knots, where \(k>\sqrt{2},\) patrols so that it is always to the north west of the ship. If the wind in the area is blowing from north to south at \(V\) knots and the pilot is instructed to return to the ship every thirty minutes, how long will her outward flight last? Assume that the maximum distance of the plane from the ship during the above patrol was \(d_{w}\) miles. If the air now becomes dead calm, and the pilot's orders are maintained, show that the ratio \(d_{w}/d_{c}\) of \(d_{w}\) to the new maximum distance, \(d_{c}\) miles, of the plane from the ship is \[ \frac{k^{2}-2}{2k(k^{2}-1)}\sqrt{4k^{2}-2}. \]

Let \(A,B,C\) be three non-collinear points in the plane. Explain briefly why it is possible to choose an origin equidistant from the three points. Let \(O\) be such an origin, let \(G\) be the centroid of the triangle \(ABC,\) let \(Q\) be a point such that \(\overrightarrow{GQ}=2\overrightarrow{OG},\) and let \(N\) be the midpoint of \(OQ.\)

1. Show that \(\overrightarrow{AQ}\) is perpendicular to \(\overrightarrow{BC}\) and deduce that the three altitudes of \(\triangle ABC\) are concurrent.
2. Show that the midpoints of \(AQ,BQ\) and \(CQ\), and the midpoints of the sides of \(\triangle ABC\) are all equidistant from \(N\).

The island of Gammaland is totally flat and subject to a constant wind of \(w\) kh\(^{-1},\) blowing from the West. Its southernmost shore stretches almost indefinitely, due east and west, from the coastal city of Alphabet. A novice pilot is making her first solo flight from Alphaport to the town of Betaville which lies north-east of Alphaport. Her instructor has given her the correct heading to reach Betaville, flying at the plane's recommended airspeed of \(v\) kh\(^{-1},\) where \(v>w.\) On reaching Betaport the pilot returns with the opposite heading to that of the outward flight and, so featureless is Gammaland, that she only realises her error as she crosses the coast with Alphaport nowhere in sight. Assuming that she then turns West along the coast, and that her outward flight took \(t\) hours, show that her return flight takes \[ \left(\frac{v+w}{v-w}\right)t\ \text{hours.} \] If Betaville is \(d\) kilometres from Alphaport, show that, with the correct heading, the return flight should have taken \[ t+\frac{\sqrt{2}wd}{v^{2}-w^{2}}\ \text{hours.} \]

Describe geometrically the possible intersections of a plane with a sphere. Let \(P_{1}\) and \(P_{2}\) be the planes with equations \begin{alignat*}{1} 3x-y-1 & =0,\\ x-y+1 & =0, \end{alignat*} respectively, and let \(S_{1}\) and \(S_{2}\) be the spheres with equations \begin{alignat*}{1} x^{2}+y^{2}+z^{2} & =7,\\ x^{2}+y^{2}+z^{2}-6y-4z+10 & =0, \end{alignat*} respectively. Let \(C_{1}\) be the intersection of \(P_{1}\) and \(S_{1},\) let \(C_{2}\) be the intersection of \(P_{2}\) and \(S_{2}\) and let \(L\) be the intersection of \(P_{1}\) and \(P_{2}.\) Find the points where \(L\) meets each of \(S_{1}\) and \(S_{2}.\) Determine, giving your reasons, whether the circles \(C_{1}\) and \(C_{2}\) are linked.

A square pyramid has its base vertices at the points \(A\) \((a,0,0)\), \(B\) \((0,a,0)\), \(C\) \((-a,0,0)\) and \(D\) \((0,-a,0)\), and its vertex at \(E\) \((0,0,a)\). The point \(P\) lies on \(AE\) with \(x\)-coordinate \(\lambda a\), where \(0<\lambda<1\), and the point \(Q\) lies on \(CE\) with \(x\)-coordinate \(-\mu a\), where \(0<\mu<1\). The plane \(BPQ\) cuts \(DE\) at \(R\) and the \(y\)-coordinate of \(R\) is \(-\gamma a\). Prove that $$ \gamma = {\lambda \mu \over \lambda + \mu - \lambda \mu}. $$ Show that the quadrilateral \(BPRQ\) cannot be a parallelogram.

The points \(P\) and \(R\) lie on the sides \(AB\) and \(AD,\) respectively, of the parallelogram \(ABCD.\) The point \(Q\) is the fourth vertex of the parallelogram \(APQR.\) Prove that \(BR,CQ\) and \(DP\) meet in a point.

The tetrahedron \(ABCD\) has \(A\) at the point \((0,4,-2)\). It is symmetrical about the plane \(y+z=2,\) which passes through \(A\) and \(D\). The mid-point of \(BC\) is \(N\). The centre, \(Y\), of the sphere \(ABCD\) is at the point \((3,-2,4)\) and lies on \(AN\) such that \(\overrightarrow{AY}=3\overrightarrow{YN}.\) Show that \(BN=6\sqrt{2}\) and find the coordinates of \(B\) and \(C\). The angle \(AYD\) is \(\cos^{-1}\frac{1}{3}.\) Find the coordinates of \(D\). [There are two alternative answers for each point.]

Show Solution

Since \(B\) and \(C\) are reflections of each other in the plane \(y+z=2\) (since that's what it means to be symmetrical), we must have that \(N\) also lies on the plane \(y+z=2\). Since \(\overrightarrow{AY}=3\overrightarrow{YN}.\) we must have \(\overrightarrow{AN}=\overrightarrow{AY}+\overrightarrow{YN} = \frac43\overrightarrow{AY} = \frac43\begin{pmatrix} 3\\-6\\6\end{pmatrix} = \begin{pmatrix} 4\\-8\\8\end{pmatrix}\) and \(N\) is the point \((4,-4,6)\) (which fortunately is on our plane as expected). \(Y\) is the point \((3,-2,4)\) \(|\overrightarrow{AY}| = \sqrt{3^2+(-6)^2+6^2} = 3\sqrt{1+4+4} = 9\)

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Notice that \(BN^2 + 3^2 = 9^2 \Rightarrow BN^2 = 3\sqrt{3^2-1} = 6\sqrt2\). Therefore \(\overrightarrow{NB} = \pm 6\sqrt{2} \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ 1\end{pmatrix}\) and \(\{ B, C\} =\{ (4, 2, 12), (4, -10, 0)\}\). Suppose \(D = (x,y,z)\) then \begin{align*} && \begin{pmatrix} -1 \\ 2 \\ -2\end{pmatrix} \cdot \begin{pmatrix} x- 3 \\ y+2 \\ z-4\end{pmatrix} &= 3 \cdot 9 \cdot \frac13 = 9\\ \Rightarrow && 9 &= 3-x+2(y+2)-2(z-4) \\ &&&= -x+2y-2z+15 \\ \Rightarrow && 6 &= x-2y+2z \\ && 2 &= x -4y \\ \\ \Rightarrow && 81 &= (4y+2-3)^2+(y+2)^2+(2-y-4)^2 \\ &&&= (4y-1)^2+2(y+2)^2 \\ &&&= 16y^2-8y+1+2y^2+8y+8 \\ &&&= 18y^2+9 \\ \Rightarrow && y^2 &= 2 \\ \Rightarrow && y &= \pm 2 \end{align*} Therefore \(\displaystyle D \in \left \{ (10, 2, 0), (-6, -2, 4) \right \}\)

A straight stick of length \(h\) stands vertically. On a sunny day, the stick casts a shadow on flat horizontal ground. In cartesian axes based on the centre of the Earth, the position of the Sun may be taken to be \(R(\cos\theta,\sin\theta,0)\) where \(\theta\) varies but \(R\) is constant. The positions of the base and tip of the stick are \(a(0,\cos\phi,\sin\phi)\) and \(b(0,\cos\phi,\sin\phi)\), respectively, where \(b-a=h\). Show that the displacement vector from the base of the stick to the tip of the shadow is \[ Rh(R\cos\phi\sin\theta-b)^{-1}\begin{pmatrix}-\cos\theta\\ -\sin^{2}\phi\sin\theta\\ \cos\phi\sin\phi\sin\theta \end{pmatrix}. \] {[}`Stands vertically' means that the centre of the Earth, the base of the stick and the tip of the stick are collinear, `horizontal' means perpendicular to the stick.

A bus has the shape of a cuboid of length \(a\) and height \(h\). It is travelling northwards on a journey of fixed distance at constant speed \(u\) (chosen by the driver). The maximum speed of the bus is \(w\). Rain is falling from the southerly direction at speed \(v\) in straight lines inclined to the horizontal at angle \(\theta\), where \(0<\theta<\frac12\pi\). By considering first the case \(u=0\), show that for \(u>0\) the total amount of rain that hits the roof and the back or front of the bus in unit time is proportional to \[ h\big \vert v\cos\theta - u \big\vert + av\sin\theta \,. \] Show that, in order to encounter as little rain as possible on the journey, the driver should choose \( u=w\) if either \(w< v\cos\theta\) or \( a\sin\theta > h\cos\theta\). How should the speed be chosen if \(w>v\cos\theta\) and \( a\sin\theta < h\cos\theta\)? Comment on the case \( a\sin\theta = h\cos\theta\). How should the driver choose \(u\) on the return journey?

I stand at the top of a vertical well. The depth of the well, from the top to the surface of the water, is \(D\). I drop a stone from the top of the well and measure the time that elapses between the release of the stone and the moment when I hear the splash of the stone entering the water. In order to gauge the depth of the well, I climb a distance \(\delta\) down into the well and drop a stone from my new position. The time until I hear the splash is \(t\) less than the previous time. Show that \[ t = \sqrt{\frac{2D}g} - \sqrt{\frac{2(D-\delta)}g} + \frac \delta u\,, \] where \(u\) is the (constant) speed of sound. Hence show that \[ D = \tfrac12 gT^2\,, \] where \(T= \dfrac12 \beta + \dfrac \delta{\beta g}\) and \(\beta = t - \dfrac \delta u\,\). Taking \(u=300\,\)m\,s\(^{-1}\) and \(g=10\,\)m\,s\(^{-2}\), show that if \(t= \frac 15\,\)s and \(\delta=10\,\)m, the well is approximately \(185\,\)m deep.

On the (flat) planet Zog, the acceleration due to gravity is \(g\) up to height \(h\) above the surface and \(g'\) at greater heights. A particle is projected from the surface at speed \(V\) and at an angle \(\alpha\) to the surface, where \(V^2 \sin^2\alpha > 2 gh\,\). Sketch, on the same axes, the trajectories in the cases \(g'=g\) and \(g' < g\). Show that the particle lands a distance \(d\) from the point of projection given by \[ d = \left(\frac {V-V'} g + \frac {V'}{ g'} \right) V\sin2\alpha\,, \] where \(V' = \sqrt{V^2-2gh\,\rm{cosec}^2\alpha\,}\,\).

The Norman army is advancing with constant speed \(u\) towards the Saxon army, which is at rest. When the armies are \(d\) apart, a Saxon horseman rides from the Saxon army directly towards the Norman army at constant speed \(x\). Simultaneously a Norman horseman rides from the Norman army directly towards the Saxon army at constant speed \(y\), where $y > u$. The horsemen ride their horses so that \(y - 2x < u < 2y - x\). When each horseman reaches the opposing army, he immediately rides straight back to his own army without changing his speed. Represent this information on a displacement-time graph, and show that the two horsemen pass each other at distances \[ \frac{xd }{ x + y} \;\; \mbox{and} \;\; \frac{xd(2y -x-u)} {(u+x ) ( x + y )} \] from the Saxon army. Explain briefly what will happen in the cases (i) \(u > 2y - x\) and (ii) \(u < y - 2x\).

A particle is travelling in a straight line. It accelerates from its initial velocity \(u\) to velocity \(v\), where \(v > \vert u \vert > 0\,\), travelling a distance \(d_1\) with uniform acceleration of magnitude \(3a\,\). It then comes to rest after travelling a further distance \(d_2\,\) with uniform deceleration of magnitude \(a\,\). Show that

1. if \(u>0\) then \(3d_1 < d_2\,\);
2. if \(u<0\) then \(d_2 < 3d_1 < 2d_2\,\).

Point \(B\) is a distance \(d\) due south of point \(A\) on a horizontal plane. Particle \(P\) is at rest at \(B\) at \(t=0\), when it begins to move with constant acceleration \(a\) in a straight line with fixed bearing~\(\beta\,\). Particle \(Q\) is projected from point \(A\) at \(t=0\) and moves in a straight line with constant speed \(v\,\). Show that if the direction of projection of \(Q\) can be chosen so that \(Q\) strikes \(P\), then \[ v^2 \ge ad \l 1 - \cos \beta \r\;. \] Show further that if \(v^2 >ad(1-\cos\beta)\) then the direction of projection of \(Q\) can be chosen so that \(Q\) strikes \(P\) before \(P\) has moved a distance \(d\,\).

A competitor in a Marathon of \(42 \frac38\) km runs the first \(t\) hours of the race at a constant speed of 13 km h\(^{-1}\) and the remainder at a constant speed of \(14 + 2t/T\) km h\(^{-1}\), where \(T\) hours is her time for the race. Show that the minimum possible value of \(T\) over all possible values of \(t\) is 3. The speed of another competitor decreases linearly with respect to time from 16~km~h\(^{-1}\) at the start of the race. If both of these competitors have a run time of 3 hours, find the maximum distance between them at any stage of the race.

A tortoise and a hare have a race to the vegetable patch, a distance \(X\) kilometres from the starting post, and back. The tortoise sets off immediately, at a steady \(v\) kilometers per hour. The hare goes to sleep for half an hour and then sets off at a steady speed \(V\) kilometres per hour. The hare overtakes the tortoise half a kilometre from the starting post, and continues on to the vegetable patch, where she has another half an hour's sleep before setting off for the return journey at her previous pace. One and quarter kilometres from the vegetable patch, she passes the tortoise, still plodding gallantly and steadily towards the vegetable patch. Show that \[ V= \frac{10}{4X-9} \] and find \(v\) in terms of \(X\). Find \(X\) if the hare arrives back at the starting post one and a half hours after the start of the race.

A particle moves so that \({\bf r}\), its displacement from a fixed origin at time \(t\), is given by \[{\bf r} = \l \sin{2t} \r {\bf i} + \l 2\cos t \r \bf{j}\,,\] where \(0 \le t < 2\pi\).

1. Show that the particle passes through the origin exactly twice.
2. Determine the times when the velocity of the particle is perpendicular to its displacement.
3. Show that, when the particle is not at the origin, its velocity is never parallel to its displacement.
4. Determine the maximum distance of the particle from the origin, and sketch the path of the particle.

A train is made up of two engines, each of mass \(M\), and \(n\) carriages, each of mass \(m\). One of the engines is at the front of the train, and the other is coupled between the \(k\)th and \((k+1)\)th carriages. When the train is accelerating along a straight, horizontal track, the resistance to the motion of each carriage is \(R\) and the driving force on each engine is \(D\), where \(2D >nR\,\). The tension in the coupling between the engine at the front and the first carriage is \(T\).

1. Show that \[ T = \frac{n(mD+MR)}{nm+2M}\,. \]
2. Show that \(T\) is greater than the tension in any other coupling provided that \(k> \frac12n\,\).
3. Show also that, if \(k> \frac12n\,\), then at least one of the couplings is in compression (that is, there is a negative tension in the coupling).

A particle of mass \(m\) is pulled along the floor of a room in a straight line by a light string which is pulled at constant speed \(V\) through a hole in the ceiling. The floor is smooth and horizontal, and the height of the room is \(h\). Find, in terms of \(V\) and \(\theta\), the speed of the particle when the string makes an angle of \(\theta\) with the vertical (and the particle is still in contact with the floor). Find also the acceleration, in terms of \(V\), \(h\) and \(\theta\). Find the tension in the string and hence show that the particle will leave the floor when \[ \tan^4\theta = \frac{V^2}{gh}\,. \]

A light smoothly jointed planar framework in the form of a regular hexagon \(ABCDEF\) is suspended smoothly from \(A\) and a weight 1kg is suspended from \(C\). The framework is kept rigid by three light rods \(BD\), \(BE\) and \(BF\). What is the direction and magnitude of the supporting force which must be exerted on the framework at \(A\)? Indicate on a labelled diagram which rods are in thrust (compression) and which are in tension. Find the magnitude of the force in \(BE\).

A projectile of mass \(m\) is fired horizontally from a toy cannon of mass \(M\) which slides freely on a horizontal floor. The length of the barrel is \(l\) and the force exerted on the projectile has the constant value \(P\) for so long as the projectile remains in the barrel. Initially the cannon is at rest. Show that the projectile emerges from the barrel at a speed relative to the ground of \[ \sqrt{\frac{2PMl}{m(M+m)}}. \]

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A librarian wishes to pick up a row of identical books from a shelf, by pressing her hands on the outer covers of the two outermost books and lifting the whole row together. The covers of the books are all in parallel vertical planes, and the weight of each book is \(W\). With each arm, the librarian can exert a maximum force of \(P\) in the vertical direction, and, independently, a maximum force of \(Q\) in the horizontal direction. The coefficient of friction between each pair of books and also between each hand and a book is \(\mu.\) Derive an expression for the maximum number of books that can be picked up without slipping, using this method. {[}You may assume that the books are thin enough for the rotational effect of the couple on each book to be ignored.{]}

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The force acting vertically on each of the outer books must be (by symmetry) \(\frac{nW}{2}\). The force acting horizontally on the outer books (and between each book in the horizontal direction) will be the same (we might as well say \(Q\) since increasing this force doesn't make any task less achievable. Looking at an end book, it will have force \(\frac{nW}{2}\) acting on one side, but it this force needs to not slip, ie \(\frac{nW}{2} \leq \mu Q\) \begin{align*} && \frac{nW}{2} &\leq \mu Q \\ \Rightarrow && n &\leq \frac{2\mu Q}{W} \\ && \frac{nW}{2} & \leq P \\ && n & \leq \frac{2P}{W} \\ \Rightarrow && n &\leq \frac2{W}\min \left (P, \mu Q \right) \end{align*}

One end of a thin uniform inextensible, but perfectly flexible, string of length \(l\) and uniform mass per unit length is held at a point on a smooth table a distance \(d(< l)\) away from a small vertical hole in the surface of the table. The string passes through the hole so that a length \(l-d\) of the string hangs vertically. The string is released from rest. Assuming that the height of the table is greater than \(l\), find the time taken for the end of the string to reach the top of the hole.

The diagrams below show two separate systems of particles, strings and pulleys. In both systems, the pulleys are smooth and light, the strings are light and inextensible, the particles move vertically and the pulleys labelled with \(P\) are fixed. The masses of the particles are as indicated on the diagrams.

1. For system I show that the acceleration, \(a_1\), of the particle of mass~\(M\), measured in the downwards direction, is given by \[ a_1= \frac{M-m}{M+m} \, g \,, \] where \(g\) is the acceleration due to gravity. Give an expression for the force on the pulley due to the tension in the string.
2. For system II show that the acceleration, \(a_2\), of the particle of mass \(M\), measured in the downwards direction, is given by \[ a_2= \frac{ M - 4\mu}{M+4\mu}\,g \,, \] where \(\mu = \dfrac{m_1m_2}{m_1+m_2}\). In the case \(m= m_1+m_2\), show that \(a_1= a_2\) if and only if \(m_1=m_2\).

The diagram shows two particles, \(A\) of mass \(5m\) and \(B\) of mass \(3m\), connected by a light inextensible string which passes over two smooth, light, fixed pulleys, \(Q\) and \(R\), and under a smooth pulley \(P\) which has mass \(M\) and is free to move vertically. Particles \(A\) and \(B\) lie on fixed rough planes inclined to the horizontal at angles of \(\arctan \frac 7{24}\) and \(\arctan\frac43\) respectively. The segments \(AQ\) and \(RB\) of the string are parallel to their respective planes, and segments \(QP\) and \(PR\) are vertical. The coefficient of friction between each particle and its plane is \(\mu\).

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1. Given that the system is in equilibrium, with both \(A\) and \(B\) on the point of moving up their planes, determine the value of \(\mu\) and show that \(M = 6m\).
2. In the case when \(M = 9m\), determine the initial accelerations of \(A\), \(B\) and \(P\) in terms of \(g\).

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First note our triangles are 7-24-25 and 3-4-5 triangles, so we can easily calculate \(\sin\) and \(\cos\) of our angles. \begin{questionparts} \item \begin{align*} \text{N2}(\uparrow, P): && 2T - Mg &= 0 \\ \\ \text{N2}(\perp AQ): && R_A - 5mg \cdot \frac{24}{25} &= 0 \\ \Rightarrow && R_A &= \frac{24}{5}mg \\ \text{N2}(\parallel AQ): && T - \mu R_A - 5mg \cdot \frac{7}{25} &= 0 \\ \Rightarrow && T &= \frac15 mg \l 7+24 \mu \r \\ \text{N2}(\perp BR): && R_B - 3mg \cdot \frac{3}{5}&= 0 \\ \Rightarrow && R_B &= \frac{9}{5}mg \\ \text{N2}(\parallel AQ): && T - \mu R_B - 3mg \cdot \frac{4}{5} &= 0 \\ \Rightarrow && T &= \frac15 mg \l 12+9 \mu \r \\ \\ \Rightarrow && 12 + 9 \mu &= 7 + 24 \mu \\ \Rightarrow && \mu &= \frac{5}{15} = \frac13 \\ \\ \Rightarrow && Mg &= 2 \cdot \frac15 \cdot mg \cdot (7 + 24 \cdot \frac13) \\ &&&= 6mg \\ \Rightarrow && M &= 6m \end{align*} \item Assuming \(\mu = \frac13\) \begin{align*} &&9m \ddot{p} &= 9mg - 2T \\ &&5m \ddot{a} &= T - 3mg \\ &&3m \ddot{b} &= T - 3mg \\ &&2\ddot{p} &= \ddot{a}+\ddot{b} \\ \Rightarrow &&30m\ddot{p} &= 8T - 24mg \\ &&9m\ddot{p} &= 9mg - 2T \\ \Rightarrow && 66m \ddot{p} &=12mg \\ \Rightarrow && \ddot{p} &= \frac{2}{11}g \\ && T &= 9mg - 9m \frac{2}{11} g = \frac{9^2}{11}mg\\ && \ddot{a} &= \frac{3}{22} g \\ && \ddot{b} &= \frac{5}{22}g \end{align*}

A plane is inclined at an angle \(\arctan \frac34\) to the horizontal and a small, smooth, light pulley~\(P\) is fixed to the top of the plane. A string, \(APB\), passes over the pulley. A particle of mass~\(m_1\) is attached to the string at \(A\) and rests on the inclined plane with \(AP\) parallel to a line of greatest slope in the plane. A particle of mass \(m_2\), where \(m_2>m_1\), is attached to the string at \(B\) and hangs freely with \(BP\) vertical. The coefficient of friction between the particle at \(A\) and the plane is \( \frac{1}{2}\). The system is released from rest with the string taut. Show that the acceleration of the particles is \(\ds \frac{m_2-m_1}{m_2+m_1}g\). At a time \(T\) after release, the string breaks. Given that the particle at \(A\) does not reach the pulley at any point in its motion, find an expression in terms of \(T\) for the time after release at which the particle at \(A\) reaches its maximum height. It is found that, regardless of when the string broke, this time is equal to the time taken by the particle at \(A\) to descend from its point of maximum height to the point at which it was released. Find the ratio \(m_1 : m_2\). \noindent [Note that \(\arctan \frac34\) is another notation for \(\tan^{-1} \frac34\,\).]

A long light inextensible string passes over a fixed smooth light pulley. A particle of mass 4~kg is attached to one end \(A\) of this string and the other end is attached to a second smooth light pulley. A long light inextensible string \(BC\) passes over the second pulley and has a particle of mass 2 kg attached at \(B\) and a particle of mass of 1 kg attached at \(C\). The system is held in equilibrium in a vertical plane. The string \(BC\) is then released from rest. Find the accelerations of the two moving particles. After \(T\) seconds, the end \(A\) is released so that all three particles are now moving in a vertical plane. Find the accelerations of \(A\), \(B\) and \(C\) in this second phase of the motion. Find also, in terms of \(g\) and \(T\), the speed of \(A\) when \(B\) has moved through a total distance of \(0.6gT^{2}\)~metres.

Hank's Gold Mine has a very long vertical shaft of height \(l\). A light chain of length \(l\) passes over a small smooth light fixed pulley at the top of the shaft. To one end of the chain is attached a bucket \(A\) of negligible mass and to the other a bucket \(B\) of mass \(m\). The system is used to raise ore from the mine as follows. When bucket \(A\) is at the top it is filled with mass \(2m\) of water and bucket \(B\) is filled with mass \(\lambda m\) of ore, where \(0<\lambda<1\). The buckets are then released, so that bucket \(A\) descends and bucket \(B\) ascends. When bucket \(B\) reaches the top both buckets are emptied and released, so that bucket \(B\) descends and bucket \(A\) ascends. The time to fill and empty the buckets is negligible. Find the time taken from the moment bucket \(A\) is released at the top until the first time it reaches the top again. This process goes on for a very long time. Show that, if the greatest amount of ore is to be raised in that time, then \(\lambda\) must satisfy the condition \(\mathrm{f}'(\lambda)=0\) where \[\mathrm{f}(\lambda)=\frac{\lambda(1-\lambda)^{1/2}} {(1-\lambda)^{1/2}+(3+\lambda)^{1/2}}.\]

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A small smooth ring \(R\) of mass \(m\) is free to slide on a fixed smooth horizontal rail. A light inextensible string of length~\(L\) is attached to one end,~\(O\), of the rail. The string passes through the ring, and a particle~\(P\) of mass~\(km\) (where \(k>0\)) is attached to its other end; this part of the string hangs at an acute angle \(\alpha\) to the vertical and it is given that \(\alpha\) is constant in the motion. Let \(x\) be the distance between \(O\) and the ring. Taking the \(y\)-axis to be vertically upwards, write down the Cartesian coordinates of~\(P\) relative to~\(O\) in terms of \(x\), \(L\) and~\(\alpha\).

1. By considering the vertical component of the equation of motion of \(P\), show that \[ km\ddot x \cos\alpha = T \cos\alpha - kmg\,, \] where \(T\) is the tension in the string. Obtain two similar equations relating to the horizontal components of the equations of motion of \(P\) and \(R\).
2. Show that \(\dfrac {\sin\alpha}{(1-\sin\alpha)^2_{\vphantom|}} = k\), and deduce, by means of a sketch or otherwise, that motion with \(\alpha\) constant is possible for all values of~\(k\).
3. Show that \(\ddot x = -g\tan\alpha\,\).

A triangular wedge is fixed to a horizontal surface. The base angles of the wedge are \(\alpha\) and \(\frac\pi 2-\alpha\). Two particles, of masses \(M\) and \(m\), lie on different faces of the wedge, and are connected by a light inextensible string which passes over a smooth pulley at the apex of the wedge, as shown in the diagram. The contacts between the particles and the wedge are smooth.

1. Show that if \(\tan \alpha> \dfrac m M \) the particle of mass \(M\) will slide down the face of the wedge.
2. Given that \(\tan \alpha = \dfrac{2m}M\), show that the magnitude of the acceleration of the particles is \[ \frac{g\sin\alpha}{\tan\alpha +2} \] and that this is maximised at \(4m^3=M^3\,\).

A block of mass \(4\,\)kg is at rest on a smooth, horizontal table. A smooth pulley \(P\) is fixed to one edge of the table and a smooth pulley \(Q\) is fixed to the opposite edge. The two pulleys and the block lie in a straight line. Two horizontal strings are attached to the block. One string runs over pulley \(P\); a particle of mass \(x\,\)kg hangs at the end of this string. The other string runs over pulley \(Q\); a particle of mass \(y\,\)kg hangs at the end of this string, where \(x > y\) and \(x + y = 6\,\). The system is released from rest with the strings taut. When the \(4\,\)kg block has moved a distance \(d\), the string connecting it to the particle of mass \(x\,\)kg is cut. Show that the time taken by the block from the start of the motion until it first returns to rest (assuming that it does not reach the edge of the table) is \(\sqrt{d/(5g)\,} \,\f(y)\), where \[ \f(y)= \frac{10}{ \sqrt{6-2y}}+ \left(1 + \frac{4}{ y} \right) \sqrt {6 -2y}. \] Calculate the value of \(y\) for which \(\f'(y)=0\).

A wedge of mass \(M\) rests on a smooth horizontal surface. The face of the wedge is a smooth plane inclined at an angle \(\alpha\) to the horizontal. A particle of mass \(m\) slides down the face of the wedge, starting from rest. At a later time \(t\), the speed \(V\) of the wedge, the speed \(v\) of the particle and the angle \(\beta\) of the velocity of the particle below the horizontal are as shown in the diagram.

1. \(V\sin\alpha=v\sin(\beta-\alpha)\);
2. \(\tan\beta=(1+m/M)\tan\alpha\);
3. \(2gy=v^2(M+m\cos^2\beta)/M\).

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1. Find the condition which \(m,M\) and \(\theta\) must satisfy to ensure that the ball will fall to the ground. Assuming that this condition is satisfied, show that the velocity \(v\) of the ball when it hits the ground satisfies \[ v^{2}=\frac{2g(M-m\sin\theta)d\sin\theta}{M+m}. \]
2. Find the condition which \(m,M\) and \(\theta\) must satisfy if the wagon is not to collide with the pulley at the top of the incline.

A horizontal rail is fixed parallel to a vertical wall and at a distance \(d\) from the wall. A~uniform rod \(AB\) of length \(2a\) rests in equilibrium on the rail with the end \(A\) in contact with the wall. The rod lies in a vertical plane perpendicular to the wall. It is inclined at an angle~\(\theta\) to the vertical (where \(0<\theta<\frac12\pi\)) and \(a\sin\theta < d\), as shown in the diagram.

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1. In the case \(W> T\sin(\alpha+\beta)\), show that the block will remain at rest provided \[ W\sin\lambda \ge T\cos(\alpha+\beta- \lambda)\,, \] where \(\lambda\) is the acute angle such that \(\tan\lambda = \mu\).
2. In the case \(W=T\tan\phi\), where \(2\phi =\alpha+\beta\), show that the block will start to move in a direction that makes an angle \(\phi\) with the horizontal.

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1. Assuming the block is at rest we must have: \begin{align*} \text{N2}(\uparrow): && 0 &= T \sin \beta\cos \alpha + T \cos \beta \sin \alpha +R -W \\ \Rightarrow && W &> T \sin \beta\cos \alpha + T \cos \beta \sin \alpha \\ &&&= T\sin(\alpha+\beta) \\ \Rightarrow && R &= W-T\sin(\alpha+\beta)\\ \\ \text{N2}(\rightarrow): && 0 &= T \cos \beta \cos \alpha - T \sin \beta \sin \alpha - F \\ \Rightarrow && T \cos(\alpha+\beta) &= F \\ &&&\leq \mu (W-T\sin(\alpha+\beta)) \\ \Rightarrow && W \sin \lambda &\geq T \cos (\alpha+\beta)\cos \lambda +T \sin (\alpha+\beta) \sin \lambda \\ &&&= T\cos(\alpha+\beta-\lambda) \end{align*}
2. If \(W = T\tan \phi\) where \(2\phi = \alpha + \beta\) then \begin{align*} \text{N2}(\uparrow): && ma_y &= T\sin(\alpha+\beta) - W \\ &&&= T \sin(\alpha+\beta) - T \tan \left ( \frac{\alpha+\beta}{2} \right ) \\ &&&= T \tan \left ( \frac{\alpha+\beta}{2} \right ) \left ( 2 \cos^2 \left ( \frac{\alpha+\beta}{2} \right ) -1\right) \\ &&&= T \tan \phi \cos \left ( \alpha+\beta\right ) \tag{notice this is positive so \(R=F=0\)} \\ \text{N2}(\rightarrow): && ma_x &= T \cos(\alpha+\beta) \\ \Rightarrow && \frac{a_y}{a_x} &= \tan \phi \end{align*} Therefore we are accelerating at an angle \(\phi\) to the horizontal

A hollow circular cylinder of internal radius \(r\) is held fixed with its axis horizontal. A uniform rod of length \(2a\) (where \(a < r\)) rests in equilibrium inside the cylinder inclined at an angle of \(\theta\) to the horizontal, where \(\theta\ne0\). The vertical plane containing the rod is perpendicular to the axis of the cylinder. The coefficient of friction between the cylinder and each end of the rod is \(\mu\), where \(\mu > 0\). Show that, if the rod is on the point of slipping, then the normal reactions \(R_1\) and \(R_2\) of the lower and higher ends of the rod, respectively, on the cylinder are related by \[ \mu(R_1+R_2) = (R_1-R_2)\tan\phi \] where \(\phi\) is the angle between the rod and the radius to an end of the rod. Show further that \[ \tan\theta = \frac {\mu r^2}{r^2 -a^2(1+\mu^2)}\,. \] Deduce that \(\lambda <\phi \), where \(\tan\lambda =\mu\).

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Let \(M\) be the midpoint of \(AB\), then \begin{align*} \overset{\curvearrowright}{M}: && R_1 \sin \phi-\mu R_1 \cos \phi &= R_2 \sin \phi+\mu R_2 \cos \phi \\ \Rightarrow && (R_1-R_2) \tan \phi &= \mu(R_1+R_2) \end{align*} As required. \begin{align*} && \cos \phi = \frac{a}{r} &,\,\, \sin \phi = \frac{\sqrt{r^2-a^2}}{r} \\ \text{N2}(\rightarrow): && R_1\cos(\phi + \theta)+\mu R_1 \sin(\phi + \theta) &= R_2 \cos(\theta - \phi) + \mu R_2 \sin(\theta - \phi) \\ \Rightarrow && R_1(\cos \theta \cos \phi - \sin \theta \sin \phi)+ \mu R_1 (\sin \theta \cos \phi + \cos \theta \sin \phi) &= R_2 (\cos\theta \cos \phi + \sin \theta \sin \phi)+ \mu R_2 (\sin \theta \cos \phi - \cos \theta \sin \phi) \\ && R_1 (1 - \tan \theta \tan \phi)+\mu R_1 (\tan \theta + \tan \phi) &= R_2(1 + \tan \theta \tan \phi) +\mu R_2 (\tan \theta - \tan \phi) \\ && 0 &= (R_1-R_2)(1+\mu \tan \theta)+(R_1+R_2)(-\tan \theta \tan\phi+\mu \tan \phi) \\ \Rightarrow && \frac{R_1+R_2}{R_1-R_2} &= \frac{1+\mu \tan \theta}{\tan \phi (\tan \theta - \mu))} \\ \Rightarrow && \frac{\tan \phi}{\mu} &= \frac{1+\mu \tan \theta}{\tan \phi (\tan \theta - \mu))} \\ \Rightarrow && \tan^2 \phi &= \frac{\mu(1+\mu \tan \theta)}{\tan \theta - \mu} \\ \Rightarrow && \frac{r^2-a^2}{a^2} &= \frac{\mu(1+\mu \tan \theta)}{\tan \theta - \mu} \\ \Rightarrow && \tan \theta (r^2-a^2-a^2\mu^2) &= \mu a^2+\mu(r^2-a^2) \\ \Rightarrow && \tan \theta &= \frac{\mu r^2}{r^2-(1+\mu^2)a^2} \end{align*} Since \(\mu r^2 > 0\) we must also have \(r^2 > a^2(1+\mu^2)\) ie \(\\sec^2 \phi > 1 + \mu^2 = \sec^2 \lambda\) and the result follows.

A wedge of mass \(km\) has the shape (in cross-section) of a right-angled triangle. It stands on a smooth horizontal surface with one face vertical. The inclined face makes an angle \(\theta\) with the horizontal surface. A particle \(P\), of mass \(m\), is placed on the inclined face and released from rest. The horizontal face of the wedge is smooth, but the inclined face is rough and the coefficient of friction between \(P\) and this face is \(\mu\).

1. When \(P\) is released, it slides down the inclined plane at an acceleration \(a\) relative to the wedge. Show that the acceleration of the wedge is \[ \frac {a \cos\theta}{k+1}\,. \] To a stationary observer, \(P\) appears to descend along a straight line inclined at an angle~\(45^\circ\) to the horizontal. Show that \[ \tan\theta = \frac k {k+1}\,. \] In the case \(k=3\), find an expression for \(a\) in terms of \(g\) and \(\mu\).
2. What happens when \(P\) is released if \(\tan\theta \le \mu\)?

A straight uniform rod has mass \(m\). Its ends \(P_1\) and \(P_2\) are attached to small light rings that are constrained to move on a rough rigid circular wire with centre \(O\) fixed in a vertical plane, and the angle \(P_1OP_2\) is a right angle. The rod rests with \(P_1\) lower than \(P_2\), and with both ends lower than \(O\). The coefficient of friction between each of the rings and the wire is \(\mu\). Given that the rod is in limiting equilibrium (i.e. \ on the point of slipping at both ends), show that \[ \tan \alpha = \frac{1-2\mu -\mu^2}{1+2\mu -\mu^2}\,, \] where \(\alpha\) is the angle between \(P_1O\) and the vertical (\(0<\alpha<45^\circ\)). Let \(\theta\) be the acute angle between the rod and the horizontal. Show that \(\theta =2\lambda\), where \(\lambda \) is defined by \(\tan \lambda= \mu\) and \(0<\lambda<22.5^\circ\).

A particle of weight \(W\) is placed on a rough plane inclined at an angle of \(\theta\) to the horizontal. The coefficient of friction between the particle and the plane is \(\mu\). A horizontal force \(X\) acting on the particle is just sufficient to prevent the particle from sliding down the plane; when a horizontal force \(kX\) acts on the particle, the particle is about to slide up the plane. Both horizontal forces act in the vertical plane containing the line of greatest slope. Prove that \[ \left( k-1 \right) \left( 1 + \mu^2 \right) \sin \theta \cos \theta = \mu \left( k + 1 \right) \] and hence that $\displaystyle k \ge \frac{ \left( 1+ \mu \right)^2} { \left( 1 - \mu \right)^2}$ .

Two particles, \(A\) and \(B\), of masses \(m\) and \(2m\), respectively, are placed on a line of greatest slope, \(\ell\), of a rough inclined plane which makes an angle of \(30^{\circ}\) with the horizontal. The coefficient of friction between \(A\) and the plane is \(\frac16\sqrt{3}\) and the coefficient of friction between \(B\) and the plane is \(\frac13 \sqrt{3}\). The particles are at rest with \(B\) higher up \(\ell\) than \(A\) and are connected by a light inextensible string which is taut. A force \(P\) is applied to \(B\).

1. Show that the least magnitude of \(P\) for which the two particles move upwards along \(\ell\) is \(\frac{11}8 \sqrt{3}\, mg\) and give, in this case, the direction in which \(P\) acts.
2. Find the least magnitude of \(P\) for which the particles do not slip downwards along~\(\ell\).

A bead \(B\) of mass \(m\) can slide along a rough horizontal wire. A light inextensible string of length \(2\ell\) has one end attached to a fixed point \(A\) of the wire and the other to \(B\,\). A particle \(P\) of mass \(3m\) is attached to the mid-point of the string and \(B\) is held at a distance \(\ell\) from~\(A\,\). The bead is released from rest. Let \(a_1\) and \(a_2\) be the magnitudes of the horizontal and vertical components of the initial acceleration of \(P\,\). Show by considering the motion of \(P\) relative to \(A\,\), or otherwise, that \(a_1= \sqrt 3 a_2\,\). Show also that the magnitude of the initial acceleration of \(B\) is \(2a_1\,\). Given that the frictional force opposing the motion of \(B\) is equal to \(({\sqrt{3}}/6)R\), where \(R\) is the normal reaction between \(B\) and the wire, show that the magnitude of the initial acceleration of \(P\) is~\(g/18\,\).

A rod \(AB\) of length 0.81 m and mass 5 kg is in equilibrium with the end \(A\) on a rough floor and the end \(B\) against a very rough vertical wall. The rod is in a vertical plane perpendicular to the wall and is inclined at \(45^{\circ}\) to the horizontal. The centre of gravity of the rod is at \(G\), where \(AG = 0.21\) m. The coefficient of friction between the rod and the floor is 0.2, and the coefficient of friction between the rod and the wall is 1.0. Show that the friction cannot be limiting at both \(A\) and \(B\). A mass of 5 kg is attached to the rod at the point \(P\) such that now the friction is limiting at both \(A\) and \(B\). Determine the length of \(AP\).

A small bullet of mass \(m\) is fired into a block of wood of mass \(M\) which is at rest. The speed of the bullet on entering the block is \(u\). Its trajectory within the block is a horizontal straight line and the resistance to the bullet's motion is \(R\), which is constant.

1. The block is fixed. The bullet travels a distance \(a\) inside the block before coming to rest. Find an expression for \(a\) in terms of \(m\), \(u\) and \(R\).
2. Instead, the block is free to move on a smooth horizontal table. The bullet travels a distance \(b\) inside the block before coming to rest relative to the block, at which time the block has moved a distance \(c\) on the table. Find expressions for \(b\) and \(c\) in terms of \(M\), \(m\) and \(a\).

A small light ring is attached to the end \(A\) of a uniform rod \(AB\) of weight \(W\) and length \(2a\). The ring can slide on a rough horizontal rail. One end of a light inextensible string of length \(2a\) is attached to the rod at \(B\) and the other end is attached to a point \(C\) on the rail so that the rod makes an angle of \(\theta\) with the rail, where \(0 < \theta < 90^{\circ}\). The rod hangs in the same vertical plane as the rail. A force of \(kW\) acts vertically downwards on the rod at \(B\) and the rod is in equilibrium.

1. You are given that the string will break if the tension \(T\) is greater than \(W\). Show that (assuming that the ring does not slip) the string will break if $$2k + 1 > 4 \sin \theta.$$
2. Show that (assuming that the string does not break) the ring will slip if $$2k + 1 > (2k + 3)\mu \tan \theta,$$ where \(\mu\) is the coefficient of friction between the rail and the ring.
3. You are now given that \(\mu \tan \theta < 1\). Show that, when \(k\) is increased gradually from zero, the ring will slip before the string breaks if $$\mu < \frac{2 \cos \theta}{1 + 2 \sin \theta}.$$

Show Solution

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1. \(\,\) \begin{align*} \overset{\curvearrowright}{A}:&& W \cos \theta \cdot a + kW \cos \theta \cdot 2a - T \cos \theta \sin \theta \cdot 2a - T \sin \theta \cos \theta \cdot 2a &= 0 \\ && (2k+1) \cos \theta W &= T \cos \theta \cdot 4 \sin \theta \\ \Rightarrow && T &= \frac{2k+1}{4 \sin \theta} W \\ \Rightarrow && \text{breaks if }\quad \quad 2k+1 &> 4 \sin \theta \end{align*}
2. \(\,\) \begin{align*} \text{N2}(\uparrow): && R - W - kW - T \sin \theta &= 0 \\ \Rightarrow && R &= (k+1)W - T \sin \theta \\ &&&= (k+1)W - \frac{2k+1}{4} W \\ &&&= \frac{2k+3}{4}W \\ \text{N2}(\leftarrow): && F_A - T \cos \theta &= 0 \\ \Rightarrow && F_A &= \frac{2k+1}{4 }\cot \theta \\ \Rightarrow && \text{slips if }\quad \quad\quad \quad\quad \quad F_A &> \mu R \\ \Rightarrow && \text{slips if }\quad \quad \frac{2k+1}{4 }\cot \theta &> \mu \frac{2k+3}{4}W \\ \Rightarrow && 2k+1 &> (2k+3) \mu \tan \theta \end{align*}
3. The condition for breaking is \(k > 2\sin \theta -\frac12\). The condition for slipping is equivalent to: \begin{align*} && 2k+1 &> (2k+3) \mu \tan \theta \\ \Leftrightarrow && 2k(1- \mu \tan \theta) &> 3 \mu \tan \theta-1 \\ \Leftrightarrow && k &> \frac{3 \mu \tan \theta-1}{2(1- \mu \tan \theta)} \end{align*} Therefore we will slip first if: \begin{align*} && \frac{3 \mu \tan \theta-1}{2(1- \mu \tan \theta)} &< 2 \sin \theta - \frac12 \\ \Leftrightarrow && 3 \mu \tan \theta-1 &< 4 \sin \theta (1- \mu \tan \theta) - (1- \mu \tan \theta) \\ &&&=4 \sin \theta - 1 + \mu \tan \theta (1-4 \sin \theta) \\ \Leftrightarrow && 3 \mu \tan \theta &< 4 \sin \theta + \mu \tan \theta (1- 4 \sin \theta) \\ \Leftrightarrow && \mu \tan \theta(3-1+4\sin \theta) &< 4 \sin \theta \\ \Leftrightarrow && \mu &< \frac{2 \cos \theta}{1+2 \sin \theta} \end{align*}

Two identical rough cylinders of radius \(r\) and weight \(W\) rest, not touching each other but a negligible distance apart, on a horizontal floor. A thin flat rough plank of width \(2a\), where \(a < r\), and weight \(kW\) rests symmetrically and horizontally on the cylinders, with its length parallel to the axes of the cylinders and its faces horizontal. A vertical cross-section is shown in the diagram below. \vspace{1.1cm} \hspace{5.0cm} \begin{pspicture}(9.3,-5.00 ) \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25} \psline(-2,-3)(7,-3) \pscircle(1,-1.5){1.50} \pscircle(4.02,-1.5 ){1.50} \psline[linewidth=3pt](1.45,-0.06)(3.58,-0.06) \end{pspicture} \vspace{-1.5cm} The coefficient of friction at all four contacts is \(\frac12\). The system is in equilibrium.

1. Let \(F\) be the frictional force between one cylinder and the floor, and let \(R\) be the normal reaction between the plank and one cylinder. Show that \[ R\sin\theta = F(1+\cos\theta)\,, \] where \(\theta\) is the acute angle between the plank and the tangent to the cylinder at the point of contact. Deduce that \(2\sin\theta \le 1+\cos\theta\,\).
2. Show that \[ N= \left( 1+\frac2 k\right)\left(\frac{1+\cos\theta}{\sin\theta} \right) F \,, \] where \(N\) is the normal reaction between the floor and one cylinder. Write down the condition that the cylinder does not slip on the floor and show that it is satisfied with no extra restrictions on \(\theta\).
3. Show that \(\sin\theta\le\frac45\,\) and hence that \(r\le5a\,\).

Two long circular cylinders of equal radius lie in equilibrium on an inclined plane, in \mbox{contact} with one another and with their axes horizontal. The weights of the upper and lower \mbox{cylinders} are \(W_1\) and \(W_2\), respectively, where \(W_1>W_2\)\,. The coefficients of friction \mbox{between} the \mbox{inclined} plane and the upper and lower cylinders are \(\mu_1\) and \(\mu_2\), respectively, and the \mbox{coefficient} of friction \mbox{between} the two cylinders is \(\mu\). The angle of inclination of the plane is~\(\alpha\) (which is positive).

1. Let \(F\) be the magnitude of the frictional force between the two cylinders, and let \(F_1\) and \(F_2\) be the magnitudes of the frictional forces between the upper cylinder and the plane, and the lower cylinder and the plane, respectively. Show that \(F=F_1=F_2\,\).
2. Show that \[ \mu \ge \dfrac{W_1+W_2}{W_1-W_2} \,,\] and that \[ \tan\alpha \le \frac{ 2 \mu_1 W_1}{(1+\mu_1)(W_1+ W_2)}\,. \]

The diagram shows three identical discs in equilibrium in a vertical plane. Two discs rest, not in contact with each other, on a horizontal surface and the third disc rests on the other two. The angle at the upper vertex of the triangle joining the centres of the discs is \(2\theta\).

1. Show that the normal reaction between the horizontal surface and a disc in contact with the surface is \(\frac32 W\,\).
2. Find the normal reaction between two discs in contact and show that the magnitude of the frictional force between two discs in contact is \(\dfrac{W\sin\theta}{2(1+\cos\theta)}\,\).
3. Show that if \(\mu <2- \surd3\,\) there is no value of \(\theta\) for which equilibrium is possible.

A thin non-uniform bar \(AB\) of length \(7d\) has centre of mass at a point \(G\), where \(AG=3d\). A light inextensible string has one end attached to \(A\) and the other end attached to \(B\). The string is hung over a smooth peg \(P\) and the bar hangs freely in equilibrium with \(B\) lower than~\(A\). Show that \[ 3\sin\alpha = 4\sin\beta\,, \] where \(\alpha\) and \(\beta\) are the angles \(PAB\) and \(PBA\), respectively. Given that \(\cos\beta=\frac45\) and that \(\alpha\) is acute, find in terms of \(d\) the length of the string and show that the angle of inclination of the bar to the horizontal is \(\arctan \frac17\,\).

A uniform rod \(AB\) of length \(4L \) and weight \(W\) is inclined at an angle \(\theta\) to the horizontal. Its lower end \(A\) rests on a fixed support and the rod is held in equilibrium by a string attached to the rod at a point \(C\) which is \(3L \) from \(A\). The reaction of the support on the rod acts in a direction \(\alpha\) to \(AC\) and the string is inclined at an angle \(\beta\) to \(CA\). Show that \[ \cot\alpha = 3\tan \theta + 2 \cot \beta\,. \] Given that \(\theta =30^\circ\) and \(\beta = 45^\circ\), show that \(\alpha= 15^\circ\).

A painter of weight \(kW\) uses a ladder to reach the guttering on the outside wall of a house. The wall is vertical and the ground is horizontal. The ladder is modelled as a uniform rod of weight \(W\) and length \(6a\). The ladder is not long enough, so the painter stands the ladder on a uniform table. The table has weight \(2W\) and a square top of side \(\frac12 a\) with a leg of length \(a\) at each corner. The foot of the ladder is at the centre of the table top and the ladder is inclined at an angle \(\arctan 2\) to the horizontal. The edge of the table nearest the wall is parallel to the wall. The coefficient of friction between the foot of the ladder and the table top is \(\frac12\). The contact between the ladder and the wall is sufficiently smooth for the effects of friction to be ignored.

1. Show that, if the legs of the table are fixed to the ground, the ladder does not slip on the table however high the painter stands on the ladder.
2. It is given that \(k=9\) and that the coefficient of friction between each table leg and the ground is \(\frac13\). If the legs of the table are not fixed to the ground, so that the table can tilt or slip, determine which occurs first when the painter slowly climbs the ladder.

\(AB\) is a uniform rod of weight \(W\,\). The point \(C\) on \(AB\) is such that \(AC>CB\,\). The rod is in contact with a rough horizontal floor at \(A\,\) and with a cylinder at \(C\,\). The cylinder is fixed to the floor with its axis horizontal. The rod makes an angle \({\alpha}\) with the horizontal and lies in a vertical plane perpendicular to the axis of the cylinder. The coefficient of friction between the rod and the floor is \(\tan \lambda_1\) and the coefficient of friction between the rod and the cylinder is \(\tan \lambda_2\,\). Show that if friction is limiting both at \(A\) and at \(C\), and \({\alpha} \ne {\lambda}_2 - {\lambda}_1\,\), then the frictional force acting on the rod at \(A\) has magnitude $$ \frac{ W\sin {\lambda}_1 \, \sin({\alpha}-{\lambda}_2)} {\sin ({\alpha}+{\lambda}_1-{\lambda}_2)} \;.$$ %and that %$$ %p=\frac{\cos{\alpha} \, \sin({\alpha}+{\lambda}_1-{\lambda}_2)} %{2\cos{\lambda}_1 \, \sin {\lambda}_2}\;. %$$

A rigid straight beam \(AB\) has length \(l\) and weight \(W\). Its weight per unit length at a distance \(x\) from \(B\) is \(\alpha Wl^{-1} (x/l)^{\alpha-1}\,\), where \(\alpha\) is a positive constant. Show that the centre of mass of the beam is at a distance \(\alpha l/(\alpha+1)\) from \(B\). The beam is placed with the end \(A\) on a rough horizontal floor and the end \(B\) resting against a rough vertical wall. The beam is in a vertical plane at right angles to the plane of the wall and makes an angle of \(\theta\) with the floor. The coefficient of friction between the floor and the beam is \(\mu\) and the coefficient of friction between the wall and the beam is also \(\mu\,\). Show that, if the equilibrium is limiting at both \(A\) and \(B\), then \[ \tan\theta = \frac{1-\alpha \mu^2}{(1+\alpha)\mu}\;. \] Given that \(\alpha =3/2\,\) and given also that the beam slides for any \(\theta<\pi/4\,\) find the greatest possible value of \(\mu\,\).

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1. If the normal reactions on the lower and higher wheels of the lorry are equal, show that the sum of the frictional forces between the wheels and the ground is zero.
2. If \(P\) is such that the lorry does not tip over (but the normal reactions on the lower and higher wheels of the lorry need not be equal), show that \[ W\tan(\alpha - \beta) \le P \le W\tan(\alpha+\beta)\;, \] where \(\tan\beta = d/h\,\).

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Two identical uniform cylinders, each of mass \(m,\) lie in contact with one another on a horizontal plane and a third identical cylinder rests symmetrically on them in such a way that the axes of the three cylinders are parallel. Assuming that all the surfaces in contact are equally rough, show that the minimum possible coefficient of friction is \(2-\sqrt{3}.\)

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First observe that many forces are equal by symmetry. Also notice that \(A\) and \(B\) are trying to roll in opposite directions, therefore there is no friction between \(A\) and \(B\). Considering the system as a whole \(R_1 = \frac32 mg\). \begin{align*} \text{N2}(\uparrow,C): && 0 &= -mg +2R_3\cos 30^{\circ} + 2F_{CA} \cos 60^{\circ} \\ \Rightarrow && mg &= \sqrt{3}R_3 + F_{CA} \\ \\ \text{N2}(\uparrow, A): && 0 &= -mg + \frac32mg-R_3\cos 30^{\circ} -F_{AC} \cos 60^\circ \\ \Rightarrow && mg &= \sqrt{3}R_3+F_{AC} \\ \text{N2}(\rightarrow, A): && 0 &= F_{AC}\cos 30^{\circ}+F_1-R_3 \cos 60^\circ -R_2 \\ \Rightarrow && 0 &=F_{AC}\sqrt{3}+2F_1-R_3-2R_2 \\ \overset{\curvearrowleft}{A}: && 0 &= F_1 - F_{AC} \end{align*} Since \(F_1 = F_{AC} = F_{CA}\) we can rewrite everything in terms of \(F= F_1\) so. \begin{align*} && mg &= \sqrt{3}R_3 + F \\ && 0 &= (2+\sqrt{3})F -R_3-2R_2 \\ \Rightarrow && (2+\sqrt3)F &\geq R_3 \\ \Rightarrow && F &\geq (2-\sqrt{3})R_3 \\ \Rightarrow && \mu & \geq 2 - \sqrt{3} \end{align*}

A light rod of length \(2a\) is hung from a point \(O\) by two light inextensible strings \(OA\) and \(OB\) each of length \(b\) and each fixed at \(O\). A particle of mass \(m\) is attached to the end \(A\) and a particle of mass \(2m\) is attached to the end \(B.\) Show that, in equilibrium, the angle \(\theta\) that the rod makes the horizontal satisfies the equation \[ \tan\theta=\frac{a}{3\sqrt{b^{2}-a^{2}}}. \] Express the tension in the string \(AO\) in terms of \(m,g,a\) and \(b\).

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The centre of mass of the rod will be at a point \(G\) which divides the rod in a ratio \(1:2\). Let \(M\) be the midpoint of \(AB\), so \(|AM| = a\) To be in equilibrium \(G\) must lie directly below \(O\). Note that \(OM^2 +a^2 = b^2 \Rightarrow OM = \sqrt{b^2-a^2}\) Notice that \(AG = \frac{4}{3}a\) and \(AM = a\), so \(|MG| = \frac13 a\). Therefore \(\displaystyle \tan \theta = \frac{\tfrac{a}{3}}{\sqrt{b^2-a^2}} \Rightarrow \tan \theta = \frac{a}{3\sqrt{b^2-a^2}}\).

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Notice that \(\frac{\sin \beta}{\frac43a} = \frac{\sin \angle OGA}{b} = \frac{\sin \alpha}{\frac23 a} \Rightarrow \sin \beta = 2 \sin \alpha\) \begin{align*} \text{N2}(\rightarrow, A): && C\cos \theta - T_A \sin \beta&= 0 \\ \text{N2}(\rightarrow, B): && T_B \sin \alpha - C\cos \theta &= 0 \\ \Rightarrow && T_B \sin \alpha &= T_A\sin \beta \\ \Rightarrow && T_B &= 2T_A \\ \text{N2}(\uparrow, A): && T_A \cos \beta+C\sin \theta-mg &= 0 \\ \text{N2}(\uparrow, B): && T_B \cos \alpha- C\sin \theta-2mg &= 0 \\ \Rightarrow && T_A \cos \beta+2T_A \cos \alpha&=3mg \\ \end{align*} Using the cosine rule: \((\frac23a)^2 = b^2 + OG^2 - 2b|OG|\cos \alpha\) and \((\frac43a)^2 = b^2 + OG^2-2b|OG|\cos \beta\). \(|OG|^2 = b^2 + (\frac43a)^2-\frac83ab \cos \angle A = b^2 +\frac{16}{9}a^2-\frac83a^2 = b^2-\frac89a^2\). Therefore \(\cos \alpha = \frac{2b^2-\frac43a^2}{2b |OG|}\), \(\cos \beta = \frac{2b^2-\frac83a^2}{2b|OG|}\) Therefore \(\cos \beta + 2 \cos \alpha = \frac{18b^2-16a^2}{6b|OG|} = \frac{9b^2-8a^2}{b\sqrt{9b^2-a^2}} = \frac{\sqrt{9b^2-a^2}}b\) Therefore \(\displaystyle T_A = \frac{3bmg}{\sqrt{9b^2-8a^2}}\)

Show that the point \(T\) with coordinates \[ \left( \frac{a(1-t^2)}{1+t^2} \; , \; \frac{2bt}{1+t^2}\right) \tag{\(*\)} \] (where \(a\) and \(b\) are non-zero) lies on the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 \,. \]

1. The line \(L\) is the tangent to the ellipse at \(T\). The point \((X,Y)\) lies on \(L\), and \(X^2\ne a^2\). Show that \[ (a+X)bt^2 -2aYt +b(a-X) =0 \,. \] Deduce that if \(a^2Y^2>(a^2-X^2)b^2\), then there are two distinct lines through \((X,Y)\) that are tangents to the ellipse. Interpret this result geometrically. Show, by means of a sketch, that the result holds also if \(X^2=a^2\,\).
2. The distinct points \(P\) and \(Q\) are given by \((*)\), with \(t=p\) and \(t=q\), respectively. The tangents to the ellipse at \(P\) and \(Q\) meet at the point with coordinates \((X,Y)\), where \(X^2\ne a^2\,\). Show that \[ (a+X)pq = a-X\] and find an expression for \(p+q\) in terms of \(a\), \(b\), \(X\) and \(Y\). Given that the tangents meet the \(y\)-axis at points \((0,y_1)\) and \((0,y_2)\), where \(y_1+y_2 = 2b\,\), show that \[ \frac{X^2}{a^2} +\frac{Y}{b}= 1 \,. \]

A curve \(C\) is determined by the parametric equations \[ x=at^2 \, , \; y = 2at\,, \] where \(a>0\)\,.

1. Show that the normal to \(C\) at a point \(P\), with non-zero parameter \(p\), meets \(C\) again at a point \(N\), with parameter \(n\), where \[ n= - \left( p + \frac{2}{p} \right). \]
2. Show that the distance \(\left| PN \right|\) is given by \[ \vert PN\vert^2 = 16a^2\frac{(p^2+1)^3}{p^4} \] and that this is minimised when \(p^2=2\,\).
3. The point \(Q\), with parameter \(q\), is the point at which the circle with diameter \(PN\) cuts~\(C\) again. By considering the gradients of \(QP\) and \(QN\), show that \[ 2 = p^2-q^2 + \frac{2q}p \,. \] Deduce that \(\left| PN \right|\) is at its minimum when \(Q\) is at the origin.

The curve \(C_1\) has parametric equations \(x=t^2\), \(y= t^3\), where \(-\infty < t < \infty\,\). Let \(O\) denote the point \((0,0)\). The points \(P\) and \(Q\) on \(C_1\) are such that \(\angle POQ\) is a right angle. Show that the tangents to \(C_1\) at \(P\) and \(Q\) intersect on the curve \(C_2\) with equation \(4y^2=3x-1\). Determine whether \(C_1\) and \(C_2\) meet, and sketch the two curves on the same axes.

The midpoint of a rod of length \(2b\) slides on the curve \(y =\frac14 x^2\), \(x\ge0\), in such a way that the rod is always tangent, at its midpoint, to the curve. Show that the curve traced out by one end of the rod can be written in the form \begin{align*} x& = 2 \tan\theta - b \cos\theta \\ y& = \tan^2\theta - b \sin\theta \end{align*} for some suitably chosen angle \(\theta\) which satisfies \(0\le \theta < \frac12\pi\,\). When one end of the rod is at a point \(A\) on the \(y\)-axis, the midpoint is at point \(P\) and \(\theta = \alpha\). Let \(R\) be the region bounded by the following: \hspace{2cm} the curve \(y=\frac14x^2\) between the origin and \(P\); \hspace{2cm} the \(y\)-axis between \(A\) and the origin; \hspace{2cm} the half-rod \(AP\). \noindent Show that the area of \(R\) is \(\frac 23 \tan^3 \alpha\).

The curve \(C\) has equation \(xy = \frac12\). The tangents to \(C\) at the distinct points \(P\big(p, \frac1{2p}\big)\) and \(Q\big(q, \frac1{2q}\big)\) where \(p\) and \(q\) are positive, intersect at \(T\) and the normals to \(C\) at these points intersect at \(N\). Show that \(T\) is the point \[ \left( \frac{2pq}{p+q}\,,\, \frac 1 {p+q}\right)\!. \] In the case \(pq=\frac12\), find the coordinates of \(N\). Show (in this case) that \(T\) and \(N\) lie on the line \(y=x\) and are such that the product of their distances from the origin is constant.

The distinct points \(P\) and \(Q\), with coordinates \((ap^2,2ap)\) and \((aq^2,2aq)\) respectively, lie on the curve \(y^2=4ax\). The tangents to the curve at \(P\) and \(Q\) meet at the point \(T\). Show that \(T\) has coordinates \(\big(apq, a(p+q)\big)\). You may assume that \(p\ne0\) and \(q\ne0\). The point \(F\) has coordinates \((a,0)\) and \(\phi\) is the angle \(TFP\). Show that \[ \cos\phi = \frac{pq+1}{\sqrt{(p^2+1)(q^2+1)}\ } \] and deduce that the line \(FT\) bisects the angle \(PFQ\).

A curve is given parametrically by \begin{align*} x&= a\big( \cos t +\ln \tan \tfrac12 t\big)\,,\\ y&= a\sin t\,, \end{align*} where \(0 < t < \frac12 \pi\) and \(a\) is a positive constant. Show that \(\ds \frac{\d y}{\d x} = \tan t\) and sketch the curve. Let \(P\) be the point with parameter \(t\) and let \(Q\) be the point where the tangent to the curve at \(P\) meets the \(x\)-axis. Show that \(PQ=a\). The {\sl radius of curvature}, \(\rho\), at \(P\) is defined by \[ \rho= \frac {\big(\dot x ^2+\dot y^2\big)^{\frac32}} {\vert \dot x \ddot y - \dot y \ddot x\vert \ \ } \,, \] where the dots denote differentiation with respect to \(t\). Show that \(\rho =a\cot t\). The point \(C\) lies on the normal to the curve at \(P\), a distance \(\rho\) from \(P\) and above the curve. Show that \(CQ\) is parallel to the \(y\)-axis.

An ellipse has equation $\dfrac{x^2}{a^2} +\dfrac {y^2}{b^2} = 1$. Show that the equation of the tangent at the point \((a\cos\alpha, b\sin\alpha)\) is \[ y=- \frac {b \cot \alpha} a \, x + b\, {\rm cosec\,}\alpha\,. \] The point \(A\) has coordinates \((-a,-b)\), where \(a\) and \(b\) are positive. The point \(E\) has coordinates \((-a,0)\) and the point \(P\) has coordinates \((a,kb)\), where \(0 < k < 1\). The line through \(E\) parallel to \(AP\) meets the line \(y=b\) at the point \(Q\). Show that the line \(PQ\) is tangent to the above ellipse at the point given by \(\tan(\alpha/2)=k\). Determine by means of sketches, or otherwise, whether this result holds also for \(k=0\) and \(k=1\).

A curve is defined parametrically by \[ x=t^2 \;, \ \ \ y=t (1 + t^2 ) \;. \] The tangent at the point with parameter \(t\), where \(t\ne0\,\), meets the curve again at the point with parameter \(T\), where \(T\ne t\,\). Show that \[ T = \frac{1 - t^2 }{2t} \mbox { \ \ \ and \ \ \ } 3t^2\ne 1\;. \] Given a point \(P_0\,\) on the curve, with parameter \(t_0\,\), a sequence of points \(P_0 \, , \; P_1 \, , \; P_2 \, , \ldots\) on the curve is constructed such that the tangent at \(P_i\) meets the curve again at \(P_{i+1}\). If \(t_0 = \tan \frac{ 7 } {18}\pi\,\), show that \(P_3 = P_0\) but \(P_1\ne P_0\,\). Find a second value of \(t_0\,\), with \(t_0>0\,\), for which \(P_3 = P_0\) but \(P_1\ne P_0\,\).

Show that the equation \(x^3 + px + q=0\) has exactly one real solution if \(p \ge 0\,\). A parabola \(C\) is given parametrically by \[ x = at^2, \: \ \ y = 2at \: \: \: \ \ \ \ \ \ \l a > 0 \r \;. \] Find an equation which must be satisfied by \(t\) at points on \(C\) at which the normal passes through the point \(\l h , \; k \r\,\). Hence show that, if \(h \le 2a \,\), exactly one normal to \(C\) will pass through \(\l h , \; k \r \, \). Find, in Cartesian form, the equation of the locus of the points from which exactly two normals can be drawn to \(C\,\). Sketch the locus.

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If \(p \geq 0\) then the derivative is \(x^2+p \geq 0\) and in particular the function is increasing. Therefore it will have exactly \(1\) real root (as for very large negative \(x\) it is negative, and vice-versa fo positive \(x\)). \begin{align*} && \frac{\d y}{\d x} &= \frac{\dot{y}}{\dot{x}} \\ &&&= \frac{2a}{2at} \\ &&&= \frac{1}{t} \\ \text{eq of normal} && \frac{k-2at}{h-at^2} &= -t \\ \Rightarrow && k-2at &= at^3-th \\ && 0 &= at^3+(2a-h)t-k \end{align*} Since \(a > 0\) this is the same constraint as the first part, in particular \(2a-h \geq 0 \Leftrightarrow 2a \geq h\). If exactly two normals can be drawn to \(C\) we must have that our equation has a repeated root, ie \begin{align*} && 0 &= at^3+(2a-h)t-k\\ && 0 &= 3at^2+2a-h\\ \Rightarrow && 0 &= 3at^3+ 3(2a-h)t-3k \\ && 0 &= 3at^3+(2a-h)t \\ \Rightarrow && 0 &= 2(2a-h)t-3k \\ \Rightarrow && t &= \frac{3k}{2(2a-h)} \\ \Rightarrow && 0 &= 3a \left (\frac{3k}{2(2a-h)} \right)^2+2a-h \\ && 0 &= 27ak^2+4(2a-h)^3 \end{align*}

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A closed curve is given by the equation $$ x^{2/n} + y^{2/n} = a^{2/n} \eqno(*) $$ where \(n\) is an odd integer and \(a\) is a positive constant. Find a parametrization \(x=x(t)\), \(y=y(t)\) which describes the curve anticlockwise as \(t\) ranges from \(0\) to \(2\pi\). Sketch the curve in the case \(n=3\), justifying the main features of your sketch. The area \(A\) enclosed by such a curve is given by the formula $$ A= {1\over 2} \int_0^{2\pi} \left[ x(t) {\d y(t)\over \d t} - y(t) {\d x(t)\over \d t} \right] \,\d t \,. $$ Use this result to find the area enclosed by (\(*\)) for \(n=3\).

Two curves are given parametrically by \[ x_{1}=(\theta+\sin\theta),\qquad y_{1}=(1+\cos\theta),\tag{1} \]and \[ x_{2}=(\theta-\sin\theta),\qquad y_{1}=-(1+\cos\theta),\tag{2} \] Find the gradients of the tangents to the curves at the points where \(\theta= \pi/2\) and \(\theta=3\pi/2\). Sketch, using the same axes, the curves for \(0\le\theta \le 2\pi\). Find the equation of the normal to the curve (1) at the point with parameter \(\theta\). Show that this normal is a tangent to the curve (2).

Sketch the curve \(C_{1}\) whose parametric equations are \(x=t^{2},\) \(y=t^{3}.\) The circle \(C_{2}\) passes through the origin \(O\). The points \(R\) and \(S\) with real non-zero parameters \(r\) and \(s\) respectively are other intersections of \(C_{1}\) and \(C_{2}.\) Show that \(r\) and \(s\) are roots of an equation of the form \[ t^{4}+t^{2}+at+b=0, \] where \(a\) and \(b\) are real constants. By obtaining a quadratic equation, with coefficients expressed in terms of \(r\) and \(s\), whose roots would be the parameters of any further intersections of \(C_{1}\) and \(C_{2},\) or otherwise, show that \(O\), \(R\) and \(S\) are the only real intersections of \(C_{1}\) and \(C_{2}.\)

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Suppose the circle has centre \((c,d)\), then \begin{align*} && c^2+d^2 &= (t^2-c)^2+(t^3-d)^2 \\ \Rightarrow && 0 &= t^4-2ct^2+t^6-2t^3d \\ \Rightarrow && 0 &= t^4+t^2-2td-2c \end{align*} So by setting \(a = -2d\) and \(b = -2c\) we have the desired equation. By matching the coefficients of \(t^4, t^3, t^2\) we must have: \begin{align*} && 0 &= (t^2-(r+s)t+rs)(t^2+t(r+s)-rs+(r+s)^2+1) \\ \Rightarrow && 0 &= t^2+(r+s)t-rs+(r+s)^2+1 \\ && \Delta &= (r+s)^2 -4(1-rs+(r+s)^2) \\ &&&= -4+4rs-3(r+s)^2 \\ &&&=-4-2(r+s)^2-(r-s)^2 < 0 \end{align*} Therefore there are no further (real) solutions. Hence \(O, R, S\) are the only solutions.

A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(u\) and angle of projection \(\alpha\), where \(0 < \alpha < \frac{1}{2}\pi\).

1. Show that if \(\sin \alpha < \frac{2\sqrt{2}}{3}\), then the distance \(OP\) is increasing throughout the flight. Show also that if \(\sin \alpha > \frac{2\sqrt{2}}{3}\), then \(OP\) will be decreasing at some time before the particle lands.
2. At the same time as \(P\) is projected, a particle \(Q\) is projected horizontally from \(O\) with speed \(v\) along the ground in the opposite direction from the trajectory of \(P\). The ground is smooth. Show that if $$2\sqrt{2}v > (\sin \alpha - 2\sqrt{2} \cos \alpha)u,$$ then \(QP\) is increasing throughout the flight of \(P\).

In this question, the \(x\)-axis is horizontal and the positive \(y\)-axis is vertically upwards. A particle is projected from the origin with speed \(u\) at an angle \(\alpha\) to the vertical. The particle passes through the fixed point \((h \tan \beta, h)\), where \(0 < \beta < 90^{\circ}\) and \(h > 0\).

1. Show that $$c^2 - ck \cot \beta + 1 + k \cot^2 \beta = 0, \quad (*)$$ where \(c = \cot \alpha\) and \(k = \frac{2u^2}{gh}\). You are given that there are two distinct values of \(\alpha\) that satisfy equation \((*)\). Let \(\alpha_1\) and \(\alpha_2\) be these values.
   1. Show that $$\cot \alpha_1 + \cot \alpha_2 = k \cot \beta.$$ Show also that $$\alpha_1 + \alpha_2 = \beta.$$
   2. Show that $$k > 2(1 + \sec \beta).$$
2. By considering the greatest height attained by the particle, show that \(k \geq 4 \sec^2 \alpha\).

Two thin vertical parallel walls, each of height \(2a\), stand a distance \(a\) apart on horizontal ground. The projectiles in this question move in a plane perpendicular to the walls.

1. A particle is projected with speed \(\sqrt{5ag}\) towards the two walls from a point \( A\) at ground level. It just clears the first wall. By considering the energy of the particle, find its speed when it passes over the first wall. Given that it just clears the second wall, show that the angle its trajectory makes with the horizontal when it passes over the first wall is \(45^\circ\,\). Find the distance of \(A\) from the foot of the first wall.
2. A second particle is projected with speed \(\sqrt{5ag}\) from a point \(B\) at ground level towards the two walls. It passes a distance \(h\) above the first wall, where \(h>0\). Show that it does not clear the second wall.

A particle is projected at speed \(u\) from a point \(O\) on a horizontal plane. It passes through a fixed point \(P\) which is at a horizontal distance \(d\) from \(O\) and at a height \(d \tan \beta\) above the plane, where \(d>0\) and \(\beta \) is an acute angle. The angle of projection \(\alpha\) is chosen so that \(u\) is as small as possible.

1. Show that \(u^2 = gd \tan \alpha\) and \(2\alpha = \beta + 90^\circ\,\).
2. At what angle to the horizontal is the particle travelling when it passes through~\(P\)? Express your answer in terms of \(\alpha\) in its simplest form.

1. Two particles move on a smooth horizontal surface. The positions, in Cartesian coordinates, of the particles at time \(t\) are \((a+ut\cos\alpha \,,\, ut\sin\alpha)\) and \((vt\cos\beta\,,\, b+vt\sin\beta )\), where \(a\), \(b\), \(u\) and \(v\) are positive constants, \(\alpha\) and \(\beta\) are constant acute angles, and \(t\ge0\). Given that the two particles collide, show that \[ u \sin(\theta+\alpha) = v\sin(\theta +\beta)\,, \] where \(\theta \) is the acute angle satisfying \(\tan\theta = \dfrac b a\).
2. A gun is placed on the top of a vertical tower of height \(b\) which stands on horizontal ground. The gun fires a bullet with speed \(v\) and (acute) angle of elevation \(\beta\). Simultaneously, a target is projected from a point on the ground a horizontal distance \(a\) from the foot of the tower. The target is projected with speed \(u\) and (acute) angle of elevation \(\alpha\), in a direction directly away from the tower. Given that the target is hit before it reaches the ground, show that \[ 2u\sin\alpha (u\sin\alpha - v\sin\beta)>bg\,. \] Explain, with reference to part (i), why the target can only be hit if \(\alpha>\beta\).

The point \(O\) is at the top of a vertical tower of height \(h\) which stands in the middle of a large horizontal plain. A projectile \(P\) is fired from \(O\) at a fixed speed \(u\) and at an angle \(\alpha\) above the horizontal. Show that the distance \(x\) from the base of the tower when \(P\) hits the plain satisfies \[ \frac{gx^2}{u^2} = h(1+\cos 2\alpha) + x \sin 2\alpha \,. \] Show that the greatest value of \(x\) as \(\alpha\) varies occurs when \(x=h\tan2\alpha\) and find the corresponding value of \(\cos 2\alpha\) in terms of \(g\), \(h\) and \(u\). Show further that the greatest achievable distance between \(O\) and the landing point is \(\dfrac {u^2}g +h\,\).

A short-barrelled machine gun stands on horizontal ground. The gun fires bullets, from ground level, at speed \(u\) continuously from \(t=0\) to \(t= \dfrac{\pi}{ 6\lambda}\), where \(\lambda\) is a positive constant, but does not fire outside this time period. During this time period, the angle of elevation \(\alpha\) of the barrel decreases from \(\frac13\pi\) to \(\frac16\pi\) and is given at time \(t\) by \[ \alpha =\tfrac13 \pi - \lambda t\,. \] Let \(k = \dfrac{g}{2\lambda u}\). Show that, in the case \(\frac12 \le k \le \frac12 \sqrt3\), the last bullet to hit the ground does so\\[2pt] at a distance \[ \frac{ 2 k u^2 \sqrt{1-k^2}}{g} \] from the gun. What is the corresponding result if \(k<\frac12\)?

A particle is projected from a point \(O\) on horizontal ground with initial speed \(u\) and at an angle of \(\theta\) above the ground. The motion takes place in the \(x\)-\(y\) plane, where the \(x\)-axis is horizontal, the \(y\)-axis is vertical and the origin is \(O\). Obtain the Cartesian equation of the particle's trajectory in terms of \(u\), \(g\) and~\(\lambda\), where \(\lambda=\tan\theta\). Now consider the trajectories for different values of \(\theta\) with \(u\)~fixed. Show that for a given value of~\(x\), the coordinate~\(y\) can take all values up to a maximum value,~\(Y\), which you should determine as a function of \(x\), \(u\) and~\(g\). Sketch a graph of \(Y\) against \(x\) and indicate on your graph the set of points that can be reached by a particle projected from \(O\) with speed \(u\). Hence find the furthest distance from \(O\) that can be achieved by such a projectile.

A particle of mass \(m\) is projected due east at speed \(U\) from a point on horizontal ground at an angle \(\theta\) above the horizontal, where $0<\theta< 90^\circ\(. In addition to the gravitational force \)mg$, it experiences a horizontal force of magnitude \(mkg\), where \(k\) is a positive constant, acting due west in the plane of motion of the particle. Determine expressions in terms of \(U\), \(\theta\) and~\(g\) for the time, \(T_H\), at which the particle reaches its greatest height and the time, \(T_L \), at which it lands. Let \(T = U\cos\theta /(kg)\). By considering the relative magnitudes of \(T_H\), \(T_L \) and \(T\), or otherwise, sketch the trajectory of the particle in the cases \(k\tan\theta<\frac12\), \ \ \(\frac12 < k\tan\theta<1\), and \(k\tan\theta>1\). What happens when \(k\tan\theta =1\)?

A particle is projected at an angle of elevation \(\alpha\) (where \(\alpha>0\)) from a point \(A\) on horizontal ground. At a general point in its trajectory the angle of elevation of the particle from \(A\) is \(\theta\) and its direction of motion is at an angle \(\phi\) above the horizontal (with \(\phi\ge0\) for the first half of the trajectory and \(\phi\le0\) for the second half). Let \(B\) denote the point on the trajectory at which \(\theta = \frac12 \alpha\) and let \(C\) denote the point on the trajectory at which \(\phi = -\frac12\alpha\).

1. Show that, at a general point on the trajectory, \(2\tan\theta = \tan \alpha + \tan\phi\,\).
2. Show that, if \(B\) and \(C\) are the same point, then \( \alpha = 60^\circ\,\).
3. Given that \(\alpha < 60^\circ\,\), determine whether the particle reaches the point \(B\) first or the point \(C\) first.

Two particles, \(A\) and \(B\), are projected simultaneously towards each other from two points which are a distance \(d\) apart in a horizontal plane. Particle \(A\) has mass \(m\) and is projected at speed \(u\) at angle \(\alpha\) above the horizontal. Particle \(B\) has mass \(M\) and is projected at speed \(v\) at angle \(\beta\) above the horizontal. The trajectories of the two particles lie in the same vertical plane. The particles collide directly when each is at its point of greatest height above the plane. Given that both \(A\) and \(B\) return to their starting points, and that momentum is conserved in the collision, show that \[ m\cot \alpha = M \cot \beta\,. \] Show further that the collision occurs at a point which is a horizontal distance \(b\) from the point of projection of \(A\) where \[ b= \frac{Md}{m+M}\, , \] and find, in terms of \(b\) and \(\alpha\), the height above the horizontal plane at which the collision occurs.

A tennis ball is projected from a height of \(2h\) above horizontal ground with speed \(u\) and at an angle of \(\alpha\) below the horizontal. It travels in a plane perpendicular to a vertical net of height \(h\) which is a horizontal distance of \(a\) from the point of projection. Given that the ball passes over the net, show that \[ \frac 1{u^2}< \frac {2(h-a\tan\alpha)}{ga^2\sec^2\alpha}\,. \] The ball lands before it has travelled a horizontal distance of \(b\) from the point of projection. Show that \[ \sqrt{u^2\sin^2\alpha +4gh \ } < \frac{bg}{u\cos\alpha} + u \sin\alpha\,. \] Hence show that \[ \tan\alpha < \frac{h(b^2-2a^2)}{ab(b-a)}\,. \]

A tall shot-putter projects a small shot from a point \(2.5\,\)m above the ground, which is horizontal. The speed of projection is \(10\,\text{ms}^{- 1}\) and the angle of projection is \(\theta\) above the horizontal. Taking the acceleration due to gravity to be \(10\,\text{ms}^{-2}\), show that the time, in seconds, that elapses before the shot hits the ground is \[ \frac1{\sqrt2}\left ( \sqrt{1-c}+ \sqrt{2-c}\right), \] where \(c = \cos2\theta\). Find an expression for the range in terms of \(c\) and show that it is greatest when \(c= \frac15\,\). Show that the extra distance attained by projecting the shot at this angle rather than at an angle of \(45^\circ\) is \(5(\sqrt6 -\sqrt2 -1)\,\)m.

A particle is projected from a point on a horizontal plane, at speed \(u\) and at an angle~\(\theta\) above the horizontal. Let \(H\) be the maximum height of the particle above the plane. Derive an expression for \(H\) in terms of \(u\), \(g\) and \(\theta\). A particle \(P\) is projected from a point \(O\) on a smooth horizontal plane, at speed \(u\) and at an angle~\(\theta\) above the horizontal. At the same instant, a second particle \(R\) is projected horizontally from \(O\) in such a way that \(R\) is vertically below \(P\) in the ensuing motion. A light inextensible string of length \(\frac12 H\) connects \(P\) and \(R\). Show that the time that elapses before the string becomes taut is \[ (\sqrt2 -1)\sqrt{H/g\,}\,. \] When the string becomes taut, \(R\) leaves the plane, the string remaining taut. Given that \(P\) and \(R\) have equal masses, determine the total horizontal distance, \(D\), travelled by \(R\) from the moment its motion begins to the moment it lands on the plane again, giving your answer in terms of \(u\), \(g\) and \(\theta\). Given that \(D=H\), find the value of \(\tan\theta\).

A particle is projected at an angle \(\theta\) above the horizontal from a point on a horizontal plane. The particle just passes over two walls that are at horizontal distances \(d_1\) and \(d_2\) from the point of projection and are of heights \(d_2\) and \(d_1\), respectively. Show that \[ \tan\theta = \frac{d_1^2+d_\subone d_\subtwo +d_2^2}{d_\subone d_\subtwo}\,. \] Find (and simplify) an expression in terms of \(d_1\) and \(d_2\) only for the range of the particle.

Two points \(A\) and \(B\) lie on horizontal ground. A particle \(P_1\) is projected from \(A\) towards \(B\) at an acute angle of elevation \(\alpha\) and simultaneously a particle \(P_2\) is projected from \(B\) towards \(A\) at an acute angle of elevation \(\beta\). Given that the two particles collide in the air a horizontal distance \(b\) from \(B\), and that the collision occurs after \(P_1\) has attained its maximum height \(h\), show that \[ 2h \cot\beta < b < 4h \cot\beta \hphantom{\,,} \] and \[ 2h \cot\alpha < a < 4h \cot\alpha \,, \] where \(a\) is the horizontal distance from \(A\) to the point of collision.

A particle is projected under gravity from a point \(P\) and passes through a point \(Q\). The angles of the trajectory with the positive horizontal direction at \(P\) and at \(Q\) are \(\theta\) and \(\phi\), respectively. The angle of elevation of \(Q\) from \(P\) is \(\alpha\).

1. Show that \(\tan\theta +\tan\phi = 2\tan\alpha\).
2. It is given that there is a second trajectory from \(P\) to \(Q\) with the same speed of projection. The angles of this trajectory with the positive horizontal direction at \(P\) and at \(Q\) are \(\theta'\) and \(\phi'\), respectively. By considering a quadratic equation satisfied by \(\tan\theta\), show that \(\tan(\theta+\theta') = -\cot\alpha\). Show also that \(\theta+\theta'=\pi+\phi+\phi'\,\).

Two particles \(P\) and \(Q\) are projected simultaneously from points \(O\) and \(D\), respectively, where~\(D\) is a distance \(d\) directly above \(O\). The initial speed of \(P\) is \(V\) and its angle of projection {\em above} the horizontal is \(\alpha\). The initial speed of \(Q\) is \(kV\), where \(k>1\), and its angle of projection {\em below} the horizontal is \(\beta\). The particles collide at time \(T\) after projection. Show that \(\cos\alpha = k\cos\beta\) and that \(T\) satisfies the equation \[ (k^2-1)V^2T^2 +2dVT\sin\alpha -d^2 =0\,. \] Given that the particles collide when \(P\) reaches its maximum height, find an expression for~\(\sin^2\alpha\) in terms of \(g\), \(d\), \(k\) and \(V\), and deduce that \[ gd\le (1+k)V^2\,. \]

In this question, use \(g=10\,\)m\,s\(^{-2}\). In cricket, a fast bowler projects a ball at \(40\,\)m\,s\(^{-1}\) from a point \(h\,\)m above the ground, which is horizontal, and at an angle \(\alpha\) above the horizontal. The trajectory is such that the ball will strike the stumps at ground level a horizontal distance of \(20\,\)m from the point of projection.

1. Determine, in terms of \(h\), the two possible values of \(\tan\alpha\). Explain which of these two values is the more appropriate one, and deduce that the ball hits the stumps after approximately half a second.
2. State the range of values of \(h\) for which the bowler projects the ball below the horizontal.
3. In the case \(h=2.5\), give an approximate value in degrees, correct to two significant figures, for \(\alpha\). You need not justify the accuracy of your approximation.

A particle is projected from a point on a plane that is inclined at an angle~\(\phi\) to the horizontal. The position of the particle at time \(t\) after it is projected is \((x,y)\), where \((0,0)\) is the point of projection, \(x\) measures distance up the line of greatest slope and \(y\) measures perpendicular distance from the plane. Initially, the velocity of the particle is given by \((\dot x, \dot y) = (V\cos\theta, V\sin\theta)\), where \(V>0\) and \(\phi+\theta<\pi/2\,\). Write down expressions for \(x\) and \(y\). The particle bounces on the plane and returns along the same path to the point of projection. Show that \[2\tan\phi\tan\theta =1\] and that \[ R= \frac{V^2\cos^2\theta}{2g\sin\phi}\,, \] where \(R\) is the range along the plane. Show further that \[ \frac{2V^2}{gR} = 3\sin\phi + {\rm cosec}\,\phi \] and deduce that the largest possible value of \(R\) is \(V^2/ (\sqrt{3}\,g)\,\).

{\sl In this question take the acceleration due to gravity to be \(10\,{\rm m \,s}^{-2}\) and neglect air resistance.} The point \(O\) lies in a horizontal field. The point \(B\) lies \(50\,\)m east of \(O\). A particle is projected from \(B\) at speed \(25\,{\rm m\,s}^{-1}\) at an angle \(\arctan \frac12\) above the horizontal and in a direction that makes an angle \(60^\circ\) with \(OB\); it passes to the north of \(O\).

1. Taking unit vectors \(\mathbf i\), \(\mathbf j\) and \(\mathbf k\) in the directions east, north and vertically upwards, respectively, find the position vector of the particle relative to \(O\) at time \(t\)~seconds after the particle was projected, and show that its distance from \(O\) is \[ 5(t^2- \sqrt5 t +10)\, {\rm m}. \] When this distance is shortest, the particle is at point \(P\). Find the position vector of \(P\) and its horizontal bearing from \(O\).
2. Show that the particle reaches its maximum height at \(P\).
3. When the particle is at \(P\), a marksman fires a bullet from \(O\) directly at \(P\). The initial speed of the bullet is \(350\,{\rm m\,s}^{-1}\). Ignoring the effect of gravity on the bullet show that, when it passes through \(P\), the distance between \(P\) and the particle is approximately~\(3\,\)m.

A solid right circular cone, of mass \(M\), has semi-vertical angle \(\alpha\) and smooth surfaces. It stands with its base on a smooth horizontal table. A particle of mass \(m\) is projected so that it strikes the curved surface of the cone at speed \(u\). The coefficient of restitution between the particle and the cone is \(e\). The impact has no rotational effect on the cone and the cone has no vertical velocity after the impact.

1. The particle strikes the cone in the direction of the normal at the point of impact. Explain why the trajectory of the particle immediately after the impact is parallel to the normal to the surface of the cone. Find an expression, in terms of \(M\), \(m\), \(\alpha\), \(e\) and \(u\), for the speed at which the cone slides along the table immediately after impact.
2. If instead the particle falls vertically onto the cone, show that the speed \(w\) at which the cone slides along the table immediately after impact is given by \[ w= \frac{mu(1+e)\sin\alpha\cos\alpha}{M+m\cos^2\alpha}\,. \] Show also that the value of \(\alpha\) for which \(w\) is greatest is given by \[ \cos \alpha = \sqrt{ \frac{M}{2M+m}}\ . \]

A smooth, straight, narrow tube of length \(L\) is fixed at an angle of \(30^\circ\) to the horizontal. A~particle is fired up the tube, from the lower end, with initial velocity \(u\). When the particle reaches the upper end of the tube, it continues its motion until it returns to the same level as the lower end of the tube, having travelled a horizontal distance \(D\) after leaving the tube. Show that \(D\) satisfies the equation \[ 4gD^2 - 2 \sqrt{3} \left( u^2 - Lg \right)D - 3L \left( u^2 - gL \right) = 0 \] and hence that \[ \frac{{\rm d}D}{ {\rm d}L} = - \frac{ 2\sqrt{3}gD - 3(u^2-2gL)} { 8gD - 2 \sqrt{3} \left(u^2 - gL \right)}. \] The final horizontal displacement of the particle from the lower end of the tube is \(R\). Show that \(\dfrac{\d R}{\d L} = 0\) when \(2D = L \sqrt 3\), and determine, in terms of \(u\) and \(g\), the corresponding value of \(R\).

A projectile of unit mass is fired in a northerly direction from a point on a horizontal plain at speed \(u\) and an angle \(\theta\) above the horizontal. It lands at a point \(A\) on the plain. In flight, the projectile experiences two forces: gravity, of magnitude \(g\); and a horizontal force of constant magnitude \(f\) due to a wind blowing from North to South. Derive an expression, in terms of \(u\), \(g\), \(f\) and \(\theta\) for the distance \(OA\).

1. Determine the angle \(\alpha\) such that, for all \(\theta>\alpha\), the wind starts to blow the projectile back towards \(O\) before it lands at \(A\).
2. An identical projectile, which experiences the same forces, is fired from \(O\) in a northerly direction at speed \(u\) and angle \(45^\circ\) above the horizontal and lands at a point \(B\) on the plain. Given that \(\theta\) is chosen to maximise \(OA\), show that \[ \frac{OB}{OA} = \frac{ g-f}{\; \sqrt{g^2+f^2\;}- f \;\;}\;. \] Describe carefully the motion of the second projectile when \(f=g\).

A particle \(P\) is projected in the \(x\)-\(y\) plane, where the \(y\)-axis is vertical and the \(x\)-axis is horizontal. The particle is projected with speed \(V\) from the origin at an angle of \(45 ^\circ\) above the positive \(x\)-axis. Determine the equation of the trajectory of \(P\). The point of projection (the origin) is on the floor of a barn. The roof of the barn is given by the equation \(y= x \tan \alpha +b\,\), where \(b>0\) and \(\alpha\) is an acute angle. Show that, if the particle just touches the roof, then \(V(-1+ \tan\alpha) =-2 \sqrt{bg}\); you should justify the choice of the negative root. If this condition is satisfied, find, in terms of \(\alpha\), \(V\) and \(g\), the time after projection at which touching takes place. A particle \(Q\) can slide along a smooth rail fixed, in the \(x\)-\(y\) plane, to the under-side of the roof. It is projected from the point \((0,b)\) with speed \(U\) at the same time as \(P\) is projected from the origin. Given that the particles just touch in the course of their motions, show that \[ 2 \sqrt 2 \, U \cos \alpha = V \big(2 + \sin\alpha\cos\alpha -\sin^2\alpha) . \]