Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) Specimen

1. Show that \(\binom{n}{n-2} = \frac{n(n-1)}{2}\), where the positive integer \(n\) satisfies \(n \geqslant 2\). [1]
2. Solve the equation \(\binom{2n+1}{2n-1} - 2 \times \binom{n}{n-2} = 24\). [3]

Solve the simultaneous equations $$x - 2y = 5,$$ $$\frac{4}{x} - \frac{2}{y} = 5.$$ [4]

The equation of a curve is \(y = x^{\frac{3}{2}} \ln x\). Find the exact coordinates of the stationary point on the curve. [5]

A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).

1. Determine the area of the smaller segment. [4]

The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.

1. Find the equation of the line in its final position. [3]

1. Divide the quartic \(2x^4 - 5x^3 + 4x^2 + 2x - 3\) by the quadratic \(x^2 + x - 2\), identifying the quotient and the remainder. [4]
2. 1. Show that \((x - 1)\) is a factor of \(nx^{n+1} - (n + 1)x^n + 1\), where \(n\) is a positive integer. [1]
   2. Hence, or otherwise, find all the roots of \(3x^4 - 4x^3 + 1 = 0\). [4]

1. Express \(y^3 - 3y - 2\) in terms of \(x\), where \(x = y + 1\). [1]
2. Hence express $$\frac{2y + 5}{y^3 - 3y - 2}$$ in partial fractions. [5]
3. Find the exact value of $$\int_0^1 \frac{2y + 5}{y^3 - 3y - 2} dy.$$ [4]

1. Given that \(\cos \theta = \frac{7}{25}\), where \(\frac{3}{2}\pi < \theta < 2\pi\), determine the exact values of
   1. \(\sin \theta\), [3]
   2. \(\sin(\frac{1}{2}\theta)\), [3]
   3. \(\sec(\frac{1}{2}\theta)\). [1]
2. 1. Express \(4 \cos x - 3 \sin x\) in the form \(A \cos(x + \alpha)\), where \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [2]
   2. Hence find the greatest and least values of \(4 \cos x - 3 \sin x\) for \(0 \leqslant x \leqslant \pi\). [3]

1. 1. Find the general solution of the differential equation $$x \frac{dy}{dx} = y(1 + x \cot x),$$ expressing \(y\) in terms of \(x\). [5]
   2. Find the particular solution given that \(y = 1\) when \(x = \frac{1}{2}\pi\). [2]
2. The real variables \(x\) and \(y\) are related by \(x^2 - y^2 = 2ax - b\), where \(a\) and \(b\) are real constants.
   1. Show that \(\frac{dy}{dx} = 0\) can only be solved for \(x\) and \(y\) if \(b \geqslant a^2\). [5]
   2. Show that \(y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2\). [2]

Two curves are defined by \(y = x^k\) and \(y = x^{\frac{1}{k}}\), for \(x \geqslant 0\), where \(k > 0\).

1. Prove that, except for one value of \(k\), the curves intersect in exactly two points. [4]

The two curves enclose a finite region \(R\).

1. Find the area, \(A\), of \(R\), giving your answer in the form \(A = f(k)\) and distinguishing clearly between the cases \(k < 1\) and \(k > 1\). [4]
2. Determine the set of values of \(k\) for which \(A \leqslant 0.5\). [3]
3. The function \(f\) is given by \(f : x \mapsto A\) with \(k > 1\). Prove that \(f\) is one-one and determine its inverse. [4]

1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]

Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).

1. Find the speeds of the particles immediately after the collision. [4]
2. Determine the coefficient of restitution between the particles. [2]

A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.

1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]

The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.

1. Find the position vector of the particle \(3\) seconds later. [5]

A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]

After the particles have moved a distance of \(12\) m the string is cut.

1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
2. Find the value of \(t\) when \(P\) comes to rest. [3]

A gunner fires one shell from each of two guns on a stationary ship towards a vertical cliff \(AB\) of height \(100\) m whose foot \(A\) is at a horizontal distance \(600\) m from the point of projection.

1. Given that the shell from the first gun hits the cliff, travelling horizontally, at a point \(45\) m above \(A\), determine the initial velocity of the shell. Express your answer in the form \(a\mathbf{i} + b\mathbf{j}\), where \(a\) and \(b\) are integers. [6]
2. The shell from the second gun hits the cliff at its top point \(B\). Given that the initial speed of the shell is \(300\) m s\(^{-1}\), determine the possible angles of projection. [5]