Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/2 Specimen Q7
12 marks Standard +0.3
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.
Pre-U Pre-U 9794/2 Specimen Q8
6 marks Standard +0.8
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 $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 + \sin x } \mathrm {~d} x = \sqrt { 3 } - 1$$
Pre-U Pre-U 9794/2 Specimen Q9
11 marks Standard +0.3
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.
Pre-U Pre-U 9794/2 Specimen Q10
12 marks Challenging +1.2
10
    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.
Pre-U Pre-U 9794/2 Specimen Q11
4 marks Challenging +1.2
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 \(L = 20 - 4 \sqrt { 21 }\).
Pre-U Pre-U 9794/2 Specimen Q12
5 marks Standard +0.3
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.
Pre-U Pre-U 9794/2 Specimen Q13
10 marks Standard +0.3
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.
Pre-U Pre-U 9794/2 Specimen Q14
13 marks Standard +0.8
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 $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$
Pre-U Pre-U 9795/1 Specimen Q1
4 marks Moderate -0.8
1 Using standard results given in MF20, show that $$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 2 r ^ { 2 } + 5 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right) ( 3 n + 8 )$$
Pre-U Pre-U 9795/1 Specimen Q2
5 marks Standard +0.3
2 The equation \(x ^ { 3 } - 14 x ^ { 2 } + 16 x + 21 = 0\) has roots \(\alpha , \beta , \gamma\). Determine the values of \(\alpha + \beta + \gamma\), \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
Pre-U Pre-U 9795/1 Specimen Q3
3 marks Standard +0.3
3
  1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { r r r } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
  2. State the value of \(k\) for which these three planes do not meet at a single point.
  3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).
Pre-U Pre-U 9795/1 Specimen Q4
6 marks Challenging +1.2
4 Two skew lines have equations \(\mathbf { r } = \left( \begin{array} { r } - 4 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 6 \\ 5 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 8 \\ 3 \end{array} \right)\). Find a vector which is perpendicular to both lines and determine the shortest distance between the two lines.
Pre-U Pre-U 9795/1 Specimen Q5
7 marks Challenging +1.2
5 The variables \(y\) and \(x\) are related by the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } , \quad - 2 < x < 2 .$$ By writing \(u = \frac { \mathrm { d } y } { \mathrm {~d} x }\), determine \(y\) explicitly in terms of \(x\), given that \(y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 }\) when \(x = 0\).
Pre-U Pre-U 9795/1 Specimen Q6
9 marks Standard +0.8
6 The set \(S\) consists of all real numbers except 1. The binary operation * is defined for all \(a , b\) in \(S\) by $$a * b = a + b - a b$$
  1. By considering the identity \(a + b - a b \equiv 1 - ( a - 1 ) ( b - 1 )\), or otherwise, show that \(S\) is closed under *.
  2. Show that * is associative on \(S\).
  3. Find the identity of \(S\) under \(*\), and the inverse of \(x\) for all \(x \in S\).
  4. The set \(S\), together with the binary operation *, forms a group \(G\). Find a subgroup of \(G\) of order 2 .
Pre-U Pre-U 9795/1 Specimen Q7
6 marks Standard +0.8
7 A curve has equation \(y = \frac { 4 x + 11 } { ( x + 3 ) ^ { 2 } }\).
  1. Show that the curve meets the line \(y = k\) if and only if \(k \leq 4\), and deduce the coordinates of the turning point on the curve.
  2. Sketch the curve, stating the coordinates of the points where it cuts the axes, and showing clearly its asymptotes and the turning point.
Pre-U Pre-U 9795/1 Specimen Q8
7 marks Standard +0.8
8 The curve \(C\) has polar equation \(r = \theta ^ { 2 } + 2 \theta\) for \(0 \leq \theta \leq 3\).
  1. Find the area of the region enclosed by \(C\) and the half-lines \(\theta = 0\) and \(\theta = 3\).
  2. Determine the length of \(C\).
Pre-U Pre-U 9795/1 Specimen Q9
16 marks Standard +0.8
9
  1. (a) Given that \(y = \tanh ^ { - 1 } x , - 1 < x < 1\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
    (b) Show that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
  2. Show that \(\int _ { 0 } ^ { 1 / \sqrt { 3 } } \frac { 2 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( 1 + \sqrt { 3 } ) - \frac { 1 } { 2 } \ln 2 + \frac { 1 } { 6 } \pi\).
Pre-U Pre-U 9795/1 Specimen Q10
24 marks Challenging +1.8
10
  1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
  2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
  3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leq \pi\).
    (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.
Pre-U Pre-U 9795/1 Specimen Q11
17 marks Challenging +1.2
11
    1. Write down the matrix which represents a rotation through an angle \(\alpha\) anticlockwise about the origin.
    2. Show that the plane transformation given by the matrix $$\left( \begin{array} { c c } \cos \theta + \sin \theta & - ( \sin \theta - \cos \theta ) \\ \sin \theta - \cos \theta & \cos \theta + \sin \theta \end{array} \right)$$ is the composition of a rotation, \(R\), and a second transformation, \(S\). Describe both \(R\) and \(S\) fully.
    1. Write down the matrix which represents a reflection in the line \(y = x \tan \frac { 1 } { 2 } \beta\). For \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), the plane transformation \(T\) is given by the matrix $$\left( \begin{array} { c c } 1 + \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & - 1 - \cos 2 \theta \end{array} \right)$$
    2. Show that \(T\) is the composition of a reflection and an enlargement, and describe these transformations in full.
    3. Find also the values of \(\theta\) for which \(T\) is an area-preserving transformation.
Pre-U Pre-U 9795/1 Specimen Q12
13 marks Challenging +1.2
12
  1. The sequence \(\left\{ u _ { n } \right\}\) is defined for all integers \(n \geq 0\) by $$u _ { 0 } = 1 \quad \text { and } \quad u _ { n } = n u _ { n - 1 } + 1 , \quad n \geq 1 .$$ Prove by induction that \(u _ { n } = n ! \sum _ { r = 0 } ^ { n } \frac { 1 } { r ! }\).
  2. (a) Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geq 0\), show that, for \(n \geq 1\), $$I _ { n } = n I _ { n - 1 } - \frac { 1 } { \mathrm { e } }$$ (b) Evaluate \(I _ { 0 }\) exactly and deduce the value of \(I _ { 1 }\).
    (c) Show that \(I _ { n } = n ! - \frac { u _ { n } } { \mathrm { e } }\) for all integers \(n \geq 1\).
Pre-U Pre-U 9795 Specimen Q1
Standard +0.3
1 The \(n\)th triangular number, \(T _ { n }\), is given by the formula \(T _ { n } = \frac { 1 } { 2 } n ( n + 1 )\).
  1. Express \(\frac { 1 } { T _ { n } }\) in terms of partial fractions.
  2. Hence, using the method of differences, show that \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).
Pre-U Pre-U 9795 Specimen Q2
Moderate -0.3
2
  1. On a single Argand diagram, sketch and clearly label each of the following loci:
    1. \(| z | = 4\),
    2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
    3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$
Pre-U Pre-U 9795 Specimen Q3
Standard +0.3
3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.
Pre-U Pre-U 9795 Specimen Q4
Standard +0.8
4 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Use de Moivre's theorem to prove that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
  2. Deduce the identity \(\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )\).
Pre-U Pre-U 9795 Specimen Q5
Standard +0.3
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 72 \mathrm { e } ^ { 3 x }$$