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

1 Solve the equation $$x \sqrt { 32 } - \sqrt { 24 } = ( 3 \sqrt { 3 } - 5 ) ( \sqrt { 6 } + x \sqrt { 2 } )$$

2 You are given that \(\ln ( 12 ) = 2.484907\) and \(\ln ( 3 ) = 1.098612\), correct to 6 decimal places. Use the laws of logarithms to obtain the values of \(\ln ( 36 )\) and \(\ln ( 0.5 )\), correct to 4 decimal places. You must show your numerical working.

3 Show that $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \pi - x ) \cos 2 x \mathrm {~d} x = \frac { 1 } { 4 } + \frac { 3 } { 8 } \pi$$

4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).

5 A circle \(S\) has centre at the point \(( 3,1 )\) and passes through the point \(( 0,5 )\).

1. Find the radius of \(S\) and hence write down its cartesian equation.
2. (a) Determine the two points on \(S\) where the \(y\)-coordinate is twice the \(x\)-coordinate.
   (b) Calculate the length of the minor arc joining these two points.

6

1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
2. Find an analogous result for three numbers in geometric progression.
3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.

7 A cubic polynomial is given by $$\mathrm { P } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.

1. If \(\mathrm { P } ( x )\) is exactly divisible by \(x - 1\), and has a local maximum at \(x = - 1\), determine the values of \(a\) and \(b\).
2. Sketch the curve \(y = \mathrm { P } ( x )\), marking the intercepts and the \(x\)-coordinates of the stationary points.
3. Expand and simplify \(\mathrm { P } ( 1 + x )\), and deduce that \(\mathrm { P } ( 1 + x ) = - \mathrm { P } ( 1 - x )\). Interpret this result graphically.

8

1. Show that $$\tan x = \frac { 2 t } { 1 - t ^ { 2 } } \text { for } 0 \leq t < 1 , \text { where } t = \tan \frac { 1 } { 2 } x$$ and deduce that $$\sin x = \frac { 2 t } { 1 + t ^ { 2 } }$$
2. Using the substitution \(t = \tan \frac { 1 } { 2 } x\), show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 + \sin x } \mathrm {~d} x = \sqrt { 3 } - 1$$

9 A curve has equation $$y = \mathrm { e } ^ { 3 x } - 5 \mathrm { e } ^ { 2 x } + 8 \mathrm { e } ^ { x }$$

1. Find the exact coordinates of the stationary points of \(y\).
2. Determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } > 0$$
3. Determine the nature of the stationary points on the curve.

10

1. 1. By writing \(\sec x = \frac { 1 } { \cos x }\), prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x .$$
   2. Deduce that \(y = \sec x\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \sqrt { y ^ { 2 } - 1 } , \quad 0 \leq x < \frac { 1 } { 2 } \pi .$$
2. A curve lies in the first quadrant of the cartesian plane with origin \(O\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-4_707_698_845_685} The normal to the curve at the point \(P ( x , y )\) meets the \(x\)-axis at the point \(Q\). The angle between \(O P\) and the \(x\)-axis is \(u\), and the angle between \(Q P\) and the \(x\)-axis is \(v\).
   1. If $$\tan v = \tan ^ { 2 } u$$ obtain a differential equation satisfied by the curve.
   2. The curve passes through the point \(( 2,1 )\). By solving the differential equation, find an equation for the curve in the implicit form $$\mathrm { F } ( x , y ) = C ,$$ where \(C\) is a constant that should be determined.

11 Three light inextensible strings \(A C , C D\) and \(D B\), each of length 10 cm , are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-5_300_670_475_699} The ends \(A\) and \(B\) are fixed to points 20 cm apart on the same horizontal level. Two heavy particles, each of mass 2 kg , are attached at \(C\) and \(D\). The system remains in a vertical plane.

1. Determine the tension in each string.
2. The string \(C D\) is replaced by one of length \(L \mathrm {~cm}\), made of the same material. If the tension in \(A C\) is 50 N , show that \(L = 20 - 4 \sqrt { 21 }\).

12

1. Whilst a helicopter is hovering, the floor of its cargo hold maintains an angle of \(30 ^ { \circ }\) to the horizontal. There is a box of mass 20 kg on the floor. If the box is just on the point of sliding, show by resolving forces that the coefficient of friction between the box and the floor is \(\frac { 1 } { \sqrt { 3 } }\).
2. The helicopter ascends at a constant acceleration 0.5 g . If the cargo hold is now maintained at \(10 ^ { \circ }\) to the horizontal, determine the frictional force and the normal reaction between the box and the floor.

13 Professor Oldham wishes to illustrate and test Newton's experimental law of impacts. A ball is dropped from rest from a height \(h\) above a rigid horizontal board and rebounds to a height \(H\). The time taken to reach the height \(H\) after the first impact is \(T\). These quantities are recorded using very accurate measuring devices.

1. Show that $$H = e ^ { 2 } h \quad \text { and } \quad T = e \sqrt { \frac { 2 h } { g } }$$ are predicted by Newton's law, where \(e\) is the coefficient of restitution between the ball and the board.
2. If \(h = 180 \mathrm {~cm}\) and \(H = 45 \mathrm {~cm}\), determine \(T\) from these formulae. The experiment is repeated for initial heights \(h , 2 h , 3 h , \ldots , 15 h\) where \(h = 180 \mathrm {~cm}\). The corresponding rebound heights and times taken to reach that height after the first impact are recorded. The mean of the 15 rebound heights is found to be 3.3 m .
3. Find the mean of the rebound heights predicted by Newton's law and give one reason why this differs from the experimental value. Professor Oldham is able to repeat the experiment on the surface of the moon using the same experimental set-up inside a laboratory.
4. The mean of the rebound heights is unchanged, but the mean of the rebound times is substantially increased. Comment on these findings.

14 A particle \(P\) is projected from the point \(O\), at the top of a vertical wall of height \(H\) above a horizontal plane, with initial speed \(V\) at an angle \(\alpha\) above the horizontal. At time \(t\) the coordinates of the particle are \(( x , y )\) referred to horizontal and vertical axes at \(O\).

1. Express \(x\) and \(y\) as functions of \(t\). Let \(\theta\) be the angle \(O P\) makes with the horizontal at time \(t\).
2. (a) Show that $$\tan \theta = \tan \alpha - \frac { g } { 2 V \cos \alpha } t$$ (b) Show that when the particle attains its greatest height above the point of projection, where \(O P\) makes an angle \(\beta\) with the horizontal, $$\tan \beta = \frac { 1 } { 2 } \tan \alpha .$$ (c) If the particle strikes the ground where \(O P\) makes an angle \(\beta\) below the horizontal, show that $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$