Questions — Pre-U Pre-U 9795/1 (179 questions)

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Pre-U Pre-U 9795/1 2014 June Q13
8 marks Challenging +1.8
13 The complex number \(w\) has modulus 1. It is given that $$w ^ { 2 } - \frac { 2 } { w } + k \mathrm { i } = 0$$ where \(k\) is a positive real constant.
  1. Show that \(k = ( 3 - \sqrt { 3 } ) \sqrt { \frac { 1 } { 2 } \sqrt { 3 } }\).
  2. Prove that at least one of the remaining two roots of the equation \(z ^ { 2 } - \frac { 2 } { z } + k i = 0\) has modulus greater than 1 .
Pre-U Pre-U 9795/1 2016 June Q1
4 marks Moderate -0.5
1 Using standard summation results, show that \(\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } + r \right) \equiv \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 ) ^ { 2 }\).
Pre-U Pre-U 9795/1 2016 June Q2
6 marks Standard +0.3
2 Find a vector which is perpendicular to both of the lines $$\mathbf { r } = \left( \begin{array} { r } 11 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 1 \\ 7 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } - 6 \\ 1 \\ 4 \end{array} \right)$$ and hence find the shortest distance between them.
Pre-U Pre-U 9795/1 2016 June Q3
4 marks Challenging +1.2
3 A curve has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { 2 x - 3 }\).
  1. Show that the curve meets the line \(y = k\) when \(2 x ^ { 2 } - ( 2 k + 1 ) x + ( 3 k - 1 ) = 0\), and hence show that no part of the curve exists in the interval \(\frac { 1 } { 2 } < y < \frac { 9 } { 2 }\).
  2. Deduce the coordinates of the turning points of this curve.
Pre-U Pre-U 9795/1 2016 June Q4
6 marks Standard +0.8
4 A \(3 \times 3\) system of equations is given by the matrix equation \(\left( \begin{array} { r r r } - 1 & 3 & 1 \\ 5 & - 1 & 2 \\ - 1 & 1 & 0 \end{array} \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 1 \\ 16 \\ - 2 \end{array} \right)\).
  1. Show that this system of equations does not have a unique solution.
  2. Solve this system of equations and describe the geometrical significance of the solution.
Pre-U Pre-U 9795/1 2016 June Q5
8 marks Standard +0.8
5 Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 24 \mathrm { e } ^ { 2 x }\).
Pre-U Pre-U 9795/1 2016 June Q6
16 marks Challenging +1.2
6 The equation \(\sinh x + \sin x = 3 x\) has one positive root \(\alpha\).
  1. Show that \(2.5 < \alpha < 3\).
  2. By using the first two non-zero terms in the Maclaurin series for \(\sinh x + \sin x\), show that \(\alpha \approx \sqrt [ 4 ] { 60 }\).
  3. By taking the third non-zero term in this series, find a second approximation to \(\alpha\), giving your answer correct to 4 decimal places.
Pre-U Pre-U 9795/1 2016 June Q7
9 marks Standard +0.3
7
  1. Find all values of \(z\) for which \(z ^ { 3 } = 2 + 2 \mathrm { i }\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(\theta\) is an exact multiple of \(\pi\) in the interval \(0 < \theta < 2 \pi\).
  2. The vertices of a triangle in the Argand diagram correspond to the three roots of the equation \(z ^ { 3 } = 2 + 2 \mathrm { i }\). Sketch the triangle and determine its area.
Pre-U Pre-U 9795/1 2016 June Q8
12 marks Challenging +1.2
8
  1. \(S\) is the set \(\{ 1,2,4,8,16,32 \}\) and \(\times _ { 63 }\) is the operation of multiplication modulo 63 .
    1. Construct the multiplication table for \(\left( S , \times _ { 63 } \right)\).
    2. Show that \(\left( S , \times _ { 63 } \right)\) forms a group, \(G\). (You may assume that \(\times _ { 63 }\) is associative.)
    3. The group \(H\), also of order 6, has identity element \(e\) and contains two further elements \(x\) and \(y\) with the properties $$x ^ { 2 } = y ^ { 3 } = e \quad \text { and } \quad x y x = y ^ { 2 } .$$ (a) Construct the group table of \(H\).
      (b) List all the proper subgroups of \(H\).
    4. State, with justification, whether \(G\) and \(H\) are isomorphic.
Pre-U Pre-U 9795/1 2016 June Q9
10 marks Challenging +1.2
9 The cubic equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. State, in terms of \(a , b\) and \(c\), the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
  2. Find, in terms of \(a , b\) and \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
  3. Show that \(( \alpha - 2 \beta \gamma ) ( \beta - 2 \gamma \alpha ) ( \gamma - 2 \alpha \beta ) = c ( 2 a + 1 ) ^ { 2 } - 2 ( b + 2 c ) ^ { 2 }\).
  4. Deduce that one root of the equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) is twice the product of the other two roots if and only if \(c ( 2 a + 1 ) ^ { 2 } = 2 ( b + 2 c ) ^ { 2 }\).
Pre-U Pre-U 9795/1 2016 June Q10
10 marks Challenging +1.2
10
  1. Sketch the curve with polar equation \(r = \left| \frac { 1 } { 2 } + \sin \theta \right|\), for \(0 \leqslant \theta < 2 \pi\).
  2. Find in an exact form the total area enclosed by the curve.
Pre-U Pre-U 9795/1 2016 June Q11
5 marks Challenging +1.8
11
  1. The sequence of Fibonacci Numbers \(\left\{ F _ { n } \right\}\) is given by $$F _ { 1 } = 1 , \quad F _ { 2 } = 1 \quad \text { and } \quad F _ { n + 1 } = F _ { n } + F _ { n - 1 } \text { for } n \geqslant 2 .$$ Write down the values of \(F _ { 3 }\) to \(F _ { 6 }\).
  2. The sequence of functions \(\left\{ \mathrm { p } _ { n } ( x ) \right\}\) is given by $$\mathrm { p } _ { 1 } ( x ) = x + 1 \quad \text { and } \quad \mathrm { p } _ { n + 1 } ( x ) = 1 + \frac { 1 } { \mathrm { p } _ { n } ( x ) } \text { for } n \geqslant 1$$
    1. Find \(\mathrm { p } _ { 2 } ( x )\) and \(\mathrm { p } _ { 3 } ( x )\), giving each answer as a single algebraic fraction, and show that \(\mathrm { p } _ { 4 } ( x ) = \frac { 3 x + 5 } { 2 x + 3 }\).
    2. Conjecture an expression for \(\mathrm { p } _ { n } ( x )\) as a single algebraic fraction involving Fibonacci numbers, and prove it by induction for all integers \(n \geqslant 2\).
Pre-U Pre-U 9795/1 2016 June Q12
10 marks Challenging +1.8
12 The curve \(C\) has equation \(y = \ln \left( \tanh \frac { 1 } { 2 } x \right)\), for \(x > 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x\).
  2. For positive integers \(n\), the length of the arc of \(C\) between \(x = n\) and \(x = 2 n\) is \(L _ { n }\).
    1. Show by calculus that, when \(n\) is large, \(L _ { n } \approx n\).
    2. Explain how this result corresponds to the shape of \(C\).
Pre-U Pre-U 9795/1 2016 June Q13
17 marks Challenging +1.8
13
  1. (a) Given that \(x \geqslant 1\), show that \(\sec ^ { - 1 } x = \cos ^ { - 1 } \left( \frac { 1 } { x } \right)\), and deduce that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }\).
    (b) Use integration by parts to determine \(\int \sec ^ { - 1 } x \mathrm {~d} x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{5d526fd9-72f8-42b1-b156-fd4a0c764c82-4_670_1029_1073_596} The diagram shows the curve \(S\) with equation \(y = \sec ^ { - 1 } x\) for \(x \geqslant 1\). The line \(L\), with gradient \(\frac { 1 } { \sqrt { 2 } }\), is the tangent to \(S\) at the point \(P\) and cuts the \(x\)-axis at the point \(Q\). The point \(I\) has coordinates \(( 1,0 )\).
    (a) Determine the exact coordinates of \(P\) and \(Q\).
    (b) The region \(R\), shaded on the diagram, is bounded by the line segments \(P Q\) and \(Q I\) and the \(\operatorname { arc } I P\) of \(S\). Show that \(R\) has area $$\ln ( 1 + \sqrt { 2 } ) - \frac { \pi ( 8 - \pi ) \sqrt { 2 } } { 32 } .$$ {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9795/1 2016 Specimen Q6
7 marks Challenging +1.8
6 A group \(G\) has order 12.
  1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
    (1) \(x\) has order 6 ,
    (2) \(x ^ { 3 } = y ^ { 2 }\),
    (3) \(x y x = y\).
  2. Prove that \(y x ^ { 2 } y = x\).
  3. Prove that \(G\) is not a cyclic group.
Pre-U Pre-U 9795/1 2016 Specimen Q7
9 marks Challenging +1.2
7
  1. Use de Moivre's theorem to show that \(\tan 4 \theta = \frac { 4 t \left( 1 - t ^ { 2 } \right) } { 1 - 6 t ^ { 2 } + t ^ { 4 } }\), where \(\mathrm { t } = \tan \theta\).
  2. Given that \(\theta\) is the acute angle such that \(\tan \theta = \frac { 1 } { 5 }\), express \(\tan 4 \theta\) as a rational number in its simplest form, and verify that $$\frac { 1 } { 4 } \pi + \tan ^ { - 1 } \left( \frac { 1 } { 239 } \right) = 4 \tan ^ { - 1 } \left( \frac { 1 } { 5 } \right) .$$
Pre-U Pre-U 9795/1 2016 Specimen Q8
10 marks Standard +0.8
8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).
  1. Determine \(f ^ { \prime \prime } ( 1 )\).
  2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
  3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
  4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).
Pre-U Pre-U 9795/1 2016 Specimen Q9
13 marks Challenging +1.2
9
  1. Show that the substitution \(u = \frac { 1 } { y ^ { 3 } }\) transforms the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\) into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - 3 u = - 9 x .$$
  2. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\), given that \(y = \frac { 1 } { 2 }\) when \(x = 0\). Give your answer in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2016 Specimen Q10
12 marks Standard +0.3
10 The line \(L\) has equation \(\mathbf { r } = \left( \begin{array} { c } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 4 \\ 6 \end{array} \right)\) and the plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { c } 2 \\ - 6 \\ 3 \end{array} \right) = k\).
  1. Given that \(L\) lies in \(\Pi\), determine the value of \(k\).
  2. Find the coordinates of the point, \(Q\), in \(\Pi\) which is closest to \(P ( 10,2 , - 43 )\). Deduce the shortest distance from \(P\) to \(\Pi\).
  3. Find, in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers, an equation for the plane which contains both \(L\) and \(P\).
Pre-U Pre-U 9795/1 2016 Specimen Q13
6 marks Challenging +1.8
13 Define the repunit number, \(R _ { n }\), to be the positive integer which consists of a string of \(n 1 \mathrm {~s}\). Thus, $$R _ { 1 } = 1 , \quad R _ { 2 } = 11 , \quad R _ { 3 } = 111 , \quad \ldots , \quad R _ { 7 } = 1111111 , \quad \ldots , \text { etc. }$$ Use induction to prove that, for all integers \(n \geqslant 5\), the number $$13579 \times R _ { n }$$ contains a string of ( \(n - 4\) ) consecutive 7s.
Pre-U Pre-U 9795/1 2016 Specimen Q2
4 marks Standard +0.3
2 A curve has polar equation \(r = \sin \theta + \cos \theta\). Find the area enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\).
Pre-U Pre-U 9795/1 2016 Specimen Q3
Standard +0.3
3
  1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). 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 2016 Specimen Q4
6 marks Challenging +1.2
4
  1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
  2. Hence or otherwise find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).
Pre-U Pre-U 9795/1 2016 Specimen Q5
Standard +0.3
5 Use induction to prove that \(\sum _ { r = 1 } ^ { n } \left( \frac { 2 } { 4 r - 1 } \right) \left( \frac { 2 } { 4 r + 3 } \right) = \frac { 1 } { 3 } - \frac { 1 } { 4 n + 3 }\) for all positive integers \(n\).
Pre-U Pre-U 9795/1 2016 Specimen Q6
9 marks Standard +0.8
6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
  2. Deduce the coordinates of the turning points on \(C\).
  3. Sketch \(C\).