Problems

1. Sketch the curve \(y=\sqrt{1-x} + \sqrt{3+x}\;\). Use your sketch to show that only one real value of \(x\) satisfies \[ \sqrt{1-x} + \sqrt{3+x} = x+1\,, \] and give this value.
2. Determine graphically the number of real values of \(x\) that satisfy \[ 2\sqrt{1-x} = \sqrt{3+x} + \sqrt{3-x}\;. \] Solve this equation.

Write down the cubes of the integers \(1, 2, \ldots , 10\). The positive integers \(x\), \(y\) and \(z\), where \(x < y\), satisfy \[ x^3+y^3 = kz^3\,, \tag{\(*\)} \] where \(k\) is a given positive integer.

1. In the case \(x+y =k\), show that \[ z^3 = k^2 -3kx+3x^2\,. \] Deduce that \((4z^3 - k^2)/3\) is a perfect square and that \(\frac14 {k^2} \le z^3 < k^2\,\). Use these results to find a solution of \((*)\) when \(k=20\).
2. By considering the case \(x+y = z^2\), find two solutions of \((*)\) when \(k=19\).

In this question, you may assume without proof that any function \(\f\) for which \(\f'(x)\ge 0\) is increasing; that is, \(\f(x_2)\ge \f(x_1)\) if \(x_2\ge x_1\,\).

1. 1. Let \(\f(x) =\sin x -x\cos x\). Show that \(\f(x)\) is increasing for \(0\le x \le \frac12\pi\,\) and deduce that \(\f(x)\ge 0\,\) for \(0\le x \le \frac12\pi\,\).
   2. Given that \(\dfrac{\d}{\d x} (\arcsin x) \ge1\) for \(0\le x< 1\), show that \[ \arcsin x\ge x \quad (0\le x < 1). \]
   3. Let \(\g(x)= x\cosec x\, \text{ for }0< x < \frac12\pi\). Show that \(\g\) is increasing and deduce that \[ ({\arcsin x})\, x^{-1} \ge x\,{\cosec x} \quad (0 < x < 1). \]
2. Given that $\dfrac{\d}{\d x} (\arctan x)\le 1\text{ for }x\ge 0$, show by considering the function \(x^{-1} \tan x\) that \[ (\tan x)( \arctan x) \ge x^2 \quad (0< x < \tfrac12\pi). \]

1. Find all the values of \(\theta\), in the range \(0^\circ <\theta<180^\circ\), for which \(\cos\theta=\sin 4\theta\). Hence show that \[ \sin 18^\circ = \frac14\left( \sqrt 5 -1\right). \]
2. Given that \[ 4\sin^2 x + 1 = 4\sin^2 2x \,, \] find all possible values of \(\sin x\,\), giving your answers in the form \(p+q\sqrt5\) where \(p\) and \(q\) are rational numbers.
3. Hence find two values of \(\alpha\) with \(0^\circ <\alpha<90^\circ\) for which \[ \sin^23\alpha + \sin^25\alpha = \sin^2 6\alpha\,. \]

The points \(A\) and \(B\) have position vectors \(\bf a \) and \(\bf b\) with respect to an origin \(O\), and \(O\), \(A\)~and~\(B\) are non-collinear. The point \(C\), with position vector \(\bf c\), is the reflection of \(B\) in the line through \(O\) and \(A\). Show that \(\bf c\) can be written in the form \[ \bf c = \lambda \bf a -\bf b \] where \(\displaystyle \lambda = \frac{2\,{\bf a .b}}{{\bf a.a}}\). The point \(D\), with position vector \(\bf d\), is the reflection of \(C\) in the line through \(O\) and \(B\). Show that \(\bf d\) can be written in the form \[ \bf d = \mu\bf b - \lambda \bf a \] for some scalar \(\mu\) to be determined. Given that \(A\), \(B\) and \(D\) are collinear, find the relationship between \(\lambda\) and \(\mu\). In the case \(\lambda = -\frac12\), determine the cosine of \(\angle AOB\) and describe the relative positions of \(A\), \(B\) and \(D\).

For any given function \(\f\), let \[ I = \int [\f'(x)]^2 \,[\f(x)]^n \d x\,, \tag{\(*\)} \] where \(n\) is a positive integer. Show that, if \(\f(x)\) satisfies \(\f''(x) =k \f(x)\f'(x)\) for some constant \(k\), then (\(*\)) can be integrated to obtain an expression for \(I\) in terms of \(\f(x)\), \(\f'(x)\), \(k\) and \(n\).

1. Verify your result in the case \(\f(x) = \tan x\,\). Hence find \[ \displaystyle \int \frac{\sin^4x}{\cos^{8}x} \, \d x\;. \]
2. Find \[ \displaystyle \int \sec^2x\, (\sec x + \tan x)^6\,\d x\;. \]

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.

The end \(A\) of an inextensible string \(AB\) of length \(\pi\) is attached to a point on the circumference of a fixed circle of unit radius and centre \(O\). Initially the string is straight and tangent to the circle. The string is then wrapped round the circle until the end \(B\) comes into contact with the circle. The string remains taut during the motion, so that a section of the string is in contact with the circumference and the remaining section is straight. Taking \(O\) to be the origin of cartesian coordinates with \(A\) at \((-1,0)\) and \(B\) initially at \((-1, \pi)\), show that the curve described by \(B\) is given parametrically by \[ x= \cos t + t\sin t\,, \ \ \ \ \ \ y= \sin t - t\cos t\,, \] where \(t\) is the angle shown in the diagram.

Two particles, \(A\) of mass \(2m\) and \(B\) of mass \(m\), are moving towards each other in a straight line on a smooth horizontal plane, with speeds \(2u\) and \(u\) respectively. They collide directly. Given that the coefficient of restitution between the particles is \(e\), where \(0 < e \le 1\), determine the speeds of the particles after the collision. After the collision, \(B\) collides directly with a smooth vertical wall, rebounding and then colliding directly with \(A\) for a second time. The coefficient of restitution between \(B\) and the wall is \(f\), where \(0 < f \le 1\). Show that the velocity of \(B\) after its second collision with \(A\) is \[ \tfrac23 (1-e^2)u - \tfrac13(1-4e^2)fu \] towards the wall and that \(B\) moves towards (not away from) the wall for all values of \(e\) and \(f\).

Show Solution

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Solution:

+ - ↺

Since the coefficient of restitution is \(e\) and the speed of approach is \(3u\), \(v_B = v_A + 3eu\), \begin{align*} \text{COM}: && 2m\cdot2u + m \cdot (-u) &= 2m v_A + m(v_A + 3eu) \\ \Rightarrow && 3u &= 3v_A + 3eu \\ \Rightarrow && v_A &= (1-e)u \\ \Rightarrow && v_B &= (1+2e)u \end{align*} After rebounding from the wall, the velocity of \(B\) will be \(-fv_B\). So for the second collision (between the particles) we will have:

+ - ↺

\begin{align*} \text{NEL}: && w_B - w_A &= e((1-e)u+(1+2e)fu) \\ \Rightarrow && w_B - w_A &= (1-e+f+2ef)eu \tag{1} \\ \text{COM}: && 2m w_A + w_B &= 2m(1-e)u -m(1+2e)fu \\ \Rightarrow && 2w_A + w_B &= (2-2e -f-2ef)u \tag{2} \\ (2) + 2\times(1): && 3w_B &= (2-2e -f-2ef)u +2(1-e+f+2ef)eu \\ &&&= (2-2e-f-2ef)u+(2e-2e^2+2ef+4e^2f)u \\ &&&= (2-2e^2-f+4e^2f)u \\ &&&= 2(1-e^2)-f(1-4e^2)u \\ \Rightarrow && w_B &= \frac23 (1-e^2)u-\frac13(1-4e^2)fu \end{align*} Since we've always taken towards the wall as positive, the question is whether or not this is positive for all values of \(e\) and \(f\). The first term is clearly positive, so in order to have a chance of being negative, we must have that \(1-4e^2 > 0\) and \(f\) is as large as possible, so wlog \(f = 1\). \begin{align*} 2-2e^2-1+4e^2 = 1+2e^2 > 0 \end{align*} \end{align*}

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 \(\tan\theta\).