Problems

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Solution: The tangent to the curve \(y = f(x)\) at \(x = a\) has the equation \(\frac{y-f(a)}{x-a} = f'(a) = g(a)+(a-p)g'(a)\). This passes through \((p,0)\) iff \begin{align*} && \frac{-f(a)}{p-a} &= g(a)+(a-p)g'(a) \\ \Leftrightarrow && -f(a) &= (p-a)g(a) -(a-p)^2g'(a) \\ \Leftrightarrow && -f(a) &= -f(a) -(a-p)^2g'(a) \\ \Leftrightarrow && 0 &= g'(a) \\ \end{align*}

1. In this case \(g(x) = A(x-q)(x-r) = A(x^2-(q+r)x+qr)\) and so we must have that \(g'(a) = 0\), ie \(A(2a-(q+r)) = 0 \Rightarrow 2a = q+r\) The gradient is \(g(a) +(a-p)g'(a) = g(a) = A(a-q)(a-r)\)
2. By the same reasoning, but with \(g(x) = A(x-p)(x-q)\) we have the gradient is \(A(c-p)(c-r)\). This is parallel iff \begin{align*} && (c-p)(c-r) &= (a-q)(a-r) \end{align*} The tangent at \(x = q\) is \(\frac{y-0}{x-q} = A(q-p)(q-r)\) or \( y = A(q-p)(q-r)(x-q)\)

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Solution:

Notice the total area is \(xf(x)\) and it is made up of the sum of the two integrals.

1. Suppose \((g(t))^3 + g(t) = t\). Notice that \((g(0))^3 + g(0) =0 \Rightarrow g(0)((g(0))^2 + 1) = 0 \Rightarrow g(0) = 0\). \begin{align*} && t &= (g(t))^3 + g(t) \\ \Rightarrow && 1 &= 3(g(t))^2 g'(t) + g'(t) \\ \Rightarrow && g'(t) &= \frac{1}{1 + 3(g(t))^2} > 0 \end{align*}
   

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   From our sketch, we can see we are interested in: \begin{align*} && \int_0^2 g(t) \d t &= 2 - \int_0^1 (x^3 + x) \d x \\ &&&= 2 - \frac14 - \frac12 = \frac54 \end{align*}
2. \(\,\)
   

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   From our second sketch, we can see that: \begin{align*} && \int_0^8 h(t) \d t &= 16 - \int_1^2 (x^3+x-2) \d x \\ &&&= 16 - \left ( \frac{8}{4} + \frac{2^2}{2} - 2 \cdot 2 \right)+ \left ( \frac{1}{4} + \frac{1}{2} - 2 \right) \\ &&&= \frac{59}{4} \end{align*}

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Solution: Claim: \(|x_1 + x_2| \leq |x_1| + |x_2|\) Proof: Case 1: \(x_1, x_2 \geq 0\). The inequality is equivalent to \(|x_1 + x_2| = x_1 + x_2 = |x_1|+|x_2|\) so it's an equality. Case 2: \(x_1, x_2 \leq 0\). The inequality is equivalent to \(|x_1+x_2| = -x_1-x_2 = |x_1|+|x_2\), so it's also an equality in this case. Case 3: (wlog) \(|x_1| \geq |x_2| > 0\) and \(x_1x_2 < 0\) then \(|x_1+x_2| = x_1-x_2 \leq x_1 \leq |x_1|+|x_2|\) We can prove this by induction, we've already proven the base case and: \(|x_1+x_2 + \cdots + x_n| \leq |x_1 + x_2 + \cdots x_{n-1}| + |x_n| \leq |x_1| + |x_2| + \cdots + |x_n|\)

1. \(\,\) \begin{align*} && |f(x) - 1| &= |a_1 x + a_2x^2 + \cdots + a_{n-1}x^{n-1} + x^n| \\ &&&\leq |a_1x| + |a_2x^2| + \cdots + |a_{n-1}x^{n-1}| + |x^n| \\ &&&\leq |a_1||x| + |a_2||x|^2 + \cdots + |a_{n-1}||x|^{n-1} + |x|^n \\ &&&\leq A|x| + A|x|^2 + \cdots + A|x|^{n-1} + |x|^n \\ &&&=A|x| \frac{1-|x|^{n-1}}{1-|x|} + |x|^n \\ &&&= \frac{A|x|-A|x|^{n}+|x|^{n+1}-|x|^n}{1-|x|} \\ &&&= \frac{A|x|-|x|^n(\underbrace{A-|x|+1}_{\geq0})}{1-|x|} \\ &&&\leq \frac{A|x|}{1-|x|} \end{align*}
2. If \(f(\omega) = 0\) then \begin{align*} && 1 & \leq \frac{A|\omega|}{1-|\omega|} \\ \Leftrightarrow && 1-|\omega| &\leq A |\omega| \\ \Leftrightarrow && 1 &\leq (1+A) |\omega| \\ \Leftrightarrow && \frac{1}{1+A} &\leq |\omega| \\ \end{align*} We also know \(\omega \leq 1 < 1 + A\)
3. If \(\omega\) is a root of \(f(x)\) then \(1/\omega\) is a root of \(1 + a_{n-1}x + a_{n-2}x^2 + \cdots + a_1x^{n-1}+x^n\) and so \(1/\omega\) satisfies that inequality, ie \begin{align*} && \frac{1}{1+A} && \leq &&|1/\omega| && \leq &&1 + A \\ \Leftrightarrow &&1+A && \geq&& |\omega| && \geq&& \frac{1}{1 + A} \end{align*}
4. First notice that it's equivalent to: \(0 = x^5 - 1x^4 - \frac{100}{135}x^3-\frac{91}{135}x^2-\frac{126}{135} + 1\) therefore all integer roots must be between \(-2,-1\) and \(1\) and \(2\). \(1\) doesn't work. \(-1\) works. Clearly \(2\) cannot work by parity argument, therefore the only integer root is \(-1\).

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Solution:

1. \begin{align*}\cos(\frac{\pi}{9}) \cos(\frac{2\pi}{9}) \cos(\frac{4\pi}{9}) &= \frac{\sin(\frac{2\pi}{9}) \cos(\frac{2\pi}{9}) \cos(\frac{4\pi}{9})}{2 \sin \frac{\pi}{9}} \\ &= \frac{\sin(\frac{4\pi}{9})\cos(\frac{4\pi}{9})}{4 \sin \frac{\pi}{9}} \\ &= \frac{\sin(\frac{8\pi}{9})}{8 \sin \frac{\pi}{9}} \\ &= \frac{1}{8} \end{align*}
2. Let \(\displaystyle P_n = \prod_{k=0}^{n} \cos\left(\frac{x}{2^k}\right)\). Claim: \(P_n = \frac{\sin 2x}{2^{n+1} \sin \l \frac{x}{2^n} \r}\). Proof: This is true for \(n = 0\), assume true for \(n-1\) \begin{align*} \sin\l \frac{x}{2^{n}} \r P_n &= P_{n-1} \cos\l \frac{x}{2^{n}} \r \sin\l \frac{x}{2^{n}} \r \\ &= P_{n-1} \frac{1}{2} \sin\l \frac{x}{2^{n-1}} \r \\ &= \frac{\sin 2x}{2^{n} \sin \l \frac{x}{2^{n-1}}\r} \frac{1}{2} \sin\l \frac{x}{2^{n}} \r \\ &= \frac{\sin 2x}{2^{n+1}} \end{align*} Hence \(P_n = \frac{\sin 2x}{2^{n+1} \sin \l \frac{x}{2^n}\r}\) Taking logs, we determine that: \begin{align*} && \sum_{k=0}^n \ln \cos \l \frac{x}{2^k} \r &= \ln \sin 2x - \ln \sin \l \frac{x}{2^n} \r - (n+1) \ln 2 \\ \Rightarrow && \sum_{k=0}^n \frac{1}{2^k} \tan \l \frac{x}{2^k} \r &= -2 \cot 2x + \frac{1}{2^n} \cot \l \frac{x}{2^n} \r - 0 \\ \end{align*} as required.
3. As \(n \to \infty\) \(\frac{x}{2^n} \to 0\), so \(\frac{\sin \frac{x}{2^n}}{\frac{x}{2^n}} = \frac{2^n \sin \frac{x}{2^n}}{x} \to 1\) \begin{align*}\prod_{k=1}^{\infty} \cos\left(\frac{x}{2^k}\right) &= \lim_{n \to \infty} \frac{\sin x}{2^n \sin \l \frac{x}{2^n} \r} \\ &= \lim_{n \to \infty} \frac{\sin x}{x \frac{2^n \sin \l \frac{x}{2^n} \r}{x} } \\ &= \lim_{n \to \infty} \frac{\sin x}{x} \\ \end{align*} \begin{align*} \sum_{j=2}^{\infty} \frac{1}{2^{j-2}} \tan\left(\frac{\pi}{2^j}\right) &= \sum_{j=0}^{\infty} \frac{1}{2^{j}} \tan\left(\frac{1}{2^j}\frac{\pi}{4}\right) \\ &= \lim_{n \to \infty} \l -2 \cot \frac{\pi}{2} + \frac{1}{2^n} \cot \l \frac{\pi}{4 \cdot 2^n} \r\r \\ &= \frac{4}{\pi} \lim_{n \to \infty} \l \frac{1}{2^n} \frac{\pi}{4} \cot \l \frac{\pi}{4 \cdot 2^n} \r\r \\ &\to \frac{\pi}{4} \end{align*}

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Solution:

1. If \(f(a) = a\) then the sequence is constant, ie \(a = p+a^2-pa \Rightarrow 0 = (a-p)(a-1)\). Therefore \(a = 1, p\) If there sequence has period \(2\) then there must be a solution to \(f(f(x)) = x\), ie \begin{align*} && x &= p+(f(x)-p)f(x) \\ &&&= p+(p+(x-p)x-p)(p+(x-p)x) \\ &&&= p + (x-p)x(p+(x-p)x) \\ &&&= p+(x^2-px)(x^2-px+p) \\ \Rightarrow && 0 &= x^4-2px^3+(p+p^2)x^2-(p^2+1)x+p \\ &&&= (x-1)(x-p)(x^2-(p-1)x+1) \end{align*} The first two roots (\(x = 1, p\)) are constant sequences, so we need the second quadratic to have a root, ie \((p-1)^2-4 \geq 0 \Rightarrow p \geq 3 , p \leq -1\). We also need this root not to be \(1\) or \(p\), ie \(1-(p-1)+1 = 3-p \neq 0\) and \(p^2-(p-1)p + 1 = 1+p \neq 0\) so \(p \neq -1, 3\). Therefore \(p > 3\) or \(p < -1\).
2. There exists a constant sequence iff there is a solution to \(f(x) = x\), ie \begin{align*} && x &= f(x) \\ &&&= q + (x-p)x \\ \Leftrightarrow && 0 &= x^2-(p+1)x + q \tag{has a solution} \\ \end{align*} But if it doesn't have a solution, clearly the RHS is always larger, and if it does have a solution then there is some point where the inequality doesn't hold. Suppose \(f(x) > x\) for all \(x\) then \(f(f(x)) > f(x) > x\) therefore there is no value where \(f(f(x)) = x\) which is required for any sequence of period 2. No, consider \(p = q = 0\) so \(f(x) = x^2\) then there cannot be a period \(2\) sequence by the first part, but also clearly \(u_n = 1\) is a valid constant sequence.

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Solution:

1. Looking for solution of the form \(y = mx+c\) we find that \(m = mx+c+x+1 \Rightarrow m = -1, c = -2\). At stationary points \(\frac{\d y}{\d x} = 0 \Rightarrow y = -x-1\). If \(y_3(0)= k < -2\) then the solution curve lies below \(y = -x-2\) and therefore it cannot cross \(y = -x -2\) to reach \(y = -x-1\) for a stationary point. Suppose \(Y = y+x\) then \(\frac{\d Y}{\d x} = \frac{\d y}{\d x} + 1=Y+2 \Rightarrow Y = Ae^x-2 \Rightarrow y= (k+2)e^x-x-2\)
   

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2. \(\,\) \begin{align*} && m &= x^2 + (mx+c)^2 -2x(mx+c) - 4x+4(mx+c) + 3 \\ &&0&= (m^2-2m+1)x^2+(2mc-2c-4+4m)x + (c^2+4c+3-m)\\ \Rightarrow && m &= 1 \\ \Rightarrow && 0 &= c^2+4c+2 \\ \Rightarrow &&&= (c+2)^2-2 \\ \Rightarrow && c &= -2 \pm \sqrt{2} \end{align*} Therefore the lines are \(y = x -2-\sqrt{2}\) and \(y = x -2+\sqrt{2}\). Any stationary point will satisfy \(y' = 0\), ie \(0 = x^2+y^2-2xy-4x+4y+3 = (x-y)^2-4(x-y)+3 = (x-y-3)(x-y-1)\) therefore they must lie on \(y = x-1\) or \(y = x-3\). Any point between these lines must have negative gradient (since one factor is positive and one factor is negative).
   

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Solution:

1. Given \(\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}\), we can form the following results: \begin{align*} && \begin{cases} \mathbf{a} \cdot (\mathbf{a} + \mathbf{b} + \mathbf{c}) &= 0 \\ \mathbf{b} \cdot (\mathbf{a} + \mathbf{b} + \mathbf{c}) &= 0 \\ \mathbf{c} \cdot (\mathbf{a} + \mathbf{b} + \mathbf{c}) &= 0 \\ \end{cases} \\ \Rightarrow && \begin{cases} \mathbf{a} \cdot \mathbf{a} + \mathbf{a} \cdot\mathbf{b} + \mathbf{a} \cdot\mathbf{c} &= 0 \\ \mathbf{b} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} &= 0 \\ \mathbf{c} \cdot \mathbf{a} + \mathbf{c} \cdot \mathbf{b} + \mathbf{c} \cdot \mathbf{c} &= 0 \\ \end{cases} \\ \Rightarrow && \begin{cases} \mathbf{a} \cdot\mathbf{b} + \mathbf{a} \cdot\mathbf{c} &= -1 \\ \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} &= -1 \\ \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c} &= -1 \\ \end{cases} \\ \Rightarrow && \begin{cases} \mathbf{a} \cdot\mathbf{b} + \mathbf{a} \cdot\mathbf{c} &= -1 \\ \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} &= -1 \\ \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c} &= -1 \\ \mathbf{a} \cdot \mathbf{b} +\mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c} &= -\frac12 \\ \end{cases} \\ \Rightarrow && \begin{cases} \mathbf{a} \cdot \mathbf{b} = -\frac12 \\ \mathbf{a} \cdot \mathbf{c} = -\frac12 \\ \mathbf{b} \cdot \mathbf{c} = -\frac12 \\ \end{cases} \end{align*} The triangle must be equilateral since the angles between each vertex are the same.
2. We have \(\displaystyle \sum_{i=1}^{4} \mathbf{a}_i = \mathbf{0}\) so \(\displaystyle \mathbf{a}_i \cdot \sum_{i=1}^{4} \mathbf{a}_i = 0\) or for each \(i\), \(\displaystyle \sum_{j \neq i} \mathbf{a}_i \cdot \mathbf{a}_j = -1\). \begin{align*} && \begin{cases} \mathbf{a}_1 \cdot \mathbf{a}_2 + \mathbf{a}_1 \cdot \mathbf{a}_3 + \mathbf{a}_1 \cdot \mathbf{a}_4 = -1 \\ \mathbf{a}_1 \cdot \mathbf{a}_2 + \mathbf{a}_2 \cdot \mathbf{a}_3 + \mathbf{a}_2 \cdot \mathbf{a}_4 = -1 \\ \mathbf{a}_1 \cdot \mathbf{a}_3 + \mathbf{a}_2 \cdot \mathbf{a}_3 + \mathbf{a}_3 \cdot \mathbf{a}_4 = -1 \\ \mathbf{a}_2 \cdot \mathbf{a}_4 + \mathbf{a}_2 \cdot \mathbf{a}_4 + \mathbf{a}_3 \cdot \mathbf{a}_4 = -1 \\ \end{cases} \\ && \text{adding the first two, subtracting the last two} \\ \Rightarrow && \begin{cases} \mathbf{a}_1 \cdot \mathbf{a}_2 +\cancel{\mathbf{a}_1 \cdot \mathbf{a}_3} + \cancel{\mathbf{a}_1 \cdot \mathbf{a}_4} = -1 \\ \mathbf{a}_1 \cdot \mathbf{a}_2 + \cancel{\mathbf{a}_2 \cdot \mathbf{a}_3} + \cancel{\mathbf{a}_2 \cdot \mathbf{a}_4} = -1 \\ \cancel{\mathbf{a}_1 \cdot \mathbf{a}_3} + \cancel{\mathbf{a}_2 \cdot \mathbf{a}_3} + \mathbf{a}_3 \cdot \mathbf{a}_4 = -1 \\ \cancel{\mathbf{a}_1 \cdot \mathbf{a}_4} + \cancel{\mathbf{a}_2 \cdot \mathbf{a}_4} + \mathbf{a}_3 \cdot \mathbf{a}_4 = -1 \\ \end{cases} \\ \Rightarrow && 2 (\mathbf{a}_1 \cdot \mathbf{a}_2) - 2(\mathbf{a}_3 \cdot \mathbf{a}_4) = 0 \end{align*} Rather than adding the first two and last two, we could have done any pair, resulting in the relations: \begin{align*} \mathbf{a}_1 \cdot \mathbf{a}_2 &= \mathbf{a}_3 \cdot \mathbf{a}_4 \\ \mathbf{a}_1 \cdot \mathbf{a}_3 &= \mathbf{a}_2 \cdot \mathbf{a}_4 \\ \mathbf{a}_1 \cdot \mathbf{a}_4 &= \mathbf{a}_2 \cdot \mathbf{a}_3 \end{align*}
   1. [(a)] The shape must be a parallelogram (from the angle requirement, but also cyclic quadrilateral (since all vectors are unit length), therefore it must be a rectangle
   2. [(b)] Given it's a regular tetrahedron, \(\mathbf{a}_i \cdot \mathbf{a}_j\) must be the same for all \(i \neq j\), ie \(-\frac13\). We are interested in \(|\mathbf{a}_i - \mathbf{a}_j|\) so consider, \begin{align*} |\mathbf{a}_i - \mathbf{a}_j|^2 &= (\mathbf{a}_i - \mathbf{a}_j) \cdot (\mathbf{a}_i - \mathbf{a}_j) \\ &= \mathbf{a}_i \cdot \mathbf{a}_i - 2 \mathbf{a}_i \cdot \mathbf{a}_j + \mathbf{a}_j \cdot \mathbf{a}_j \\ &= 1 - \frac23 + 1 \\ &= \frac43 \end{align*} Therefore the unit side lengths are \(\frac{2}{\sqrt{3}}\)

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Solution:

1. Consider \(f(M) = f(MI) = f(M)f(I)\). Since \(f(M) \neq 0\) we can divide by \(f(M)\) to obtain \(f(I) = 1\)
2. Let \(J = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), then \(J^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I\). Therefore \(1 = f(I) = f(J^2) = f(J)f(J) \Rightarrow f(J) = \pm 1 \Rightarrow f(J) = -1\) since \(f(J) \neq 1\). \begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix}J &= \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ &= \begin{pmatrix} b & a \\ d & c \end{pmatrix} \\ J\begin{pmatrix} a & b \\ c & d \end{pmatrix} &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix} \\ &= \begin{pmatrix} c & d \\ a & b \end{pmatrix} \\ J\begin{pmatrix} a & b \\ c & d \end{pmatrix}J &=\begin{pmatrix} d & c \\ b & a \end{pmatrix} \end{align*} Therefore \(f\left(\begin{pmatrix} c & d \\ a & b \end{pmatrix}\right) = f \left (J\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = f(J) f\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = -f\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right)\) and \(f\left(\begin{pmatrix} d & c \\ b & a \end{pmatrix}\right) = f\left(J\begin{pmatrix} a & b \\ c & d \end{pmatrix}J \right) = f(J)f\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right)f(J) = f\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right)\) as required.
3. First consider \(O\) the matrix of \(0\), then \begin{align*} && JO &= O \\ \Rightarrow && f(JO) &= f(O) \\ \Rightarrow && f(J)f(O) &= f(O) \\ \Rightarrow && -f(O) &= f(O) \\ \Rightarrow && f(O) &= 0 \end{align*} Now consider \(K_{k} = \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}\). Suppose \(A = \begin{pmatrix} a & b \\ ka & kb \end{pmatrix}\) then \begin{align*} K_{\frac1k}A &= \begin{pmatrix} 1 & 0 \\ 0 & \frac1k \end{pmatrix} \begin{pmatrix} a & b \\ ka & kb \end{pmatrix} \\ &= \begin{pmatrix} a & b \\ a & b \end{pmatrix} \end{align*} And so \(f(K_{\frac1k}A) = f\left ( \begin{pmatrix} a & b \\ a & b \end{pmatrix} \right) = - f \left ( \begin{pmatrix} a & b \\ a & b \end{pmatrix} \right) = 0\), therefore either \(f(K_{\frac1k}) = 0\) or \(f(A) = 0\), but we know that \(f(I) \neq 0\) therefore \(f(A) = 0\).
4. Let \(P = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\), then \(P^2 = \begin{pmatrix} 1 &2 \\ 0 & 1 \end{pmatrix}\), \(P^{-1} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}\), \(K_k^{-1}PK_k = K_k^{-1}\begin{pmatrix} 1 & k \\ 0 & k \end{pmatrix} = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}\). If \(A\) has an inverse then \(f(A) \neq 0\) since \(1 = f(I) = f(A)f(A^{-1})\), in particular, \(f(A)f(A^{-1}) = 1\). Using this for \(K_2\) we have: \(f(P)^2 = f(P^2) = f(K_2^{-1}PK_2) = f(P)\) therefore \(f(P) = 0, 1\), but since \(f(P)\) has an inverse, \(f(P) \neq 0\) so \(f(P) = 1\)

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Solution:

1. Notice that \(P = \begin{pmatrix} u \cos \alpha t\\ u \sin \alpha t - \frac12 g t^2 \end{pmatrix}\), so \begin{align*} && |OP|^2 &= u^2 \cos^2 \alpha t^2 + \left (u \sin \alpha t - \frac12 g t^2 \right)^2 \\ &&&= u^2 \cos^2 \alpha t^2 +u^2 \sin^2 \alpha t^2 - u \sin \alpha g t^3 +\frac14 g^2 t^4 \\ &&&= u^2 t^2 -u\sin \alpha g t^3 + \frac14g^2t^4 \\ && \frac{\d |OP|^2}{\d t} &= 2u^2 t - 3u \sin \alpha g t^2+g^2 t^3 \\ &&&= t \left (2u^2 - 3u \sin \alpha (gt)+(gt)^2\right) \\ && \Delta &= 9u^2 \sin^2 \alpha -4 \cdot 2u^2 \cdot 1 \\ &&&= u^2 (9\sin^2 \alpha -8) \\ \end{align*} Therefore if \(\sin \alpha < \frac{2\sqrt{2}}3\) the discriminant is negative, the quadratic factor is always positive and the distance \(|OP|\) is always increasing. Similarly, if \(\sin \alpha > \frac{2 \sqrt{2}}3\) then the derivative has a root. This means somewhere on its (possibly extended) trajectory \(OP\) is decreasing. This must be before it lands, since if it were after it 'landed' then both the \(x\) and \(y\) distances are increasing, therefore it cannot occur after it 'lands'.
2. Note that \(Q = \begin{pmatrix} -v t \\0 \end{pmatrix}\) \begin{align*} && |QP|^2 &= (u \cos \alpha t+vt)^2 + \left (u \sin \alpha t - \frac12 g t^2 \right)^2 \\ &&&= u^2 \cos^2 \alpha t^2+2u\cos \alpha v t^2 + v^2 t^2 +u^2 \sin^2 \alpha t^2 - u \sin \alpha g t^3 +\frac14 g^2 t^4 \\ &&&= (u^2+2u v \cos \alpha+v^2) t^2 - u \sin \alpha g t^3 + \frac14 g^2 t^4 \\ \\ \Rightarrow && \frac{\d |QP|^2}{\d t} &= 2(u^2+u v \cos \alpha+v^2) t - 3u \sin \alpha g t^2 + g^2 t^3 \\ &&&= t \left ( 2(u^2+2u v \cos \alpha+v^2) - 3u \sin \alpha (g t) + (g t)^2\right) \\ && \Delta &= 9u^2 \sin^2 \alpha - 8(u^2+2u v \cos \alpha+v^2) \\ &&&= (9 \sin^2 \alpha -8)u^2 - 16v \cos \alpha u - 8v^2 \\ &&&= \left (( \sin \alpha-2\sqrt{2}\cos \alpha)u-2\sqrt{2} v \right) \left ( ( \sin \alpha+2\sqrt{2}\cos \alpha)u+2\sqrt{2} v \right) \end{align*} Since the second bracket is clearly positive, the first bracket must be negative (for \(\Delta < 0\) and our derivative to be positive), ie \(2\sqrt{2} v > ( \sin \alpha-2\sqrt{2}\cos \alpha)u\)

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Solution:

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*}