Questions — Pre-U (885 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Pre-U Pre-U 9794/1 2014 June Q8
4 marks Standard +0.3
8 The parametric equations of a curve are given by $$x = \mathrm { e } ^ { t } - 2 t , \quad y = \mathrm { e } ^ { t } - 5 t$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that \(t = - \ln 2\) at the point on the curve where the gradient is 3 .
Pre-U Pre-U 9794/1 2014 June Q9
9 marks Moderate -0.3
9 It is given that \(x , 6\) and \(x + 5\) are consecutive terms of a geometric progression.
  1. Show that \(x ^ { 2 } + 5 x - 36 = 0\) and find the possible values of \(x\).
  2. Hence find the possible values of the common ratio. Furthermore, \(x , 6\) and \(x + 5\) are the second, third and fourth terms of a geometric progression for which the sum to infinity exists.
  3. Find the first term and the sum to infinity.
Pre-U Pre-U 9794/1 2014 June Q10
4 marks Moderate -0.3
10
  1. Show that \(\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln \left( \frac { 3 } { \sqrt { 5 } } \right)\).
  2. Find \(\int x \sqrt { x - 2 } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2014 June Q11
11 marks Standard +0.3
11 A differential equation is given by \(2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ( 1 - y )\).
  1. Express \(\frac { 2 } { y ( 1 - y ) }\) in partial fractions.
  2. Hence show by integration that \(\frac { y ^ { 2 } } { ( 1 - y ) ^ { 2 } } = A \mathrm { e } ^ { x }\).
  3. Given that \(x = 0\) when \(y = 2\), find the value of \(A\) and express \(y\) in terms of \(x\).
Pre-U Pre-U 9794/1 2014 June Q12
10 marks Standard +0.8
12
  1. Use the identity \(\tan 2 x \equiv \frac { 2 \tan x } { 1 - \tan ^ { 2 } x }\) to show that \(\tan 4 x \equiv \frac { 4 \left( 1 - \tan ^ { 2 } x \right) \tan x } { 1 - 6 \tan ^ { 2 } x + \tan ^ { 4 } x }\).
  2. Hence, given that \(x = \frac { 1 } { 16 } \pi\) is a root of the equation \(\tan ^ { 4 } x + p \tan ^ { 3 } x - 6 \tan ^ { 2 } x - p \tan x + 1 = 0\) where \(p\) is a positive constant, find the value of \(p\).
Pre-U Pre-U 9794/2 2014 June Q1
2 marks Easy -1.2
1 The diagram shows the triangle \(A B C\). \(A B = 10 \mathrm {~cm} , A C = 7 \mathrm {~cm}\) and angle \(B A C = 100 ^ { \circ }\).
  1. Find the length \(B C\).
  2. Find the area of the triangle \(A B C\).
Pre-U Pre-U 9794/2 2014 June Q2
3 marks Moderate -0.8
2 Let \(\mathrm { f } ( x ) = x ^ { 2 } + k x + 4\), where \(k\) is a constant.
  1. Find an expression for the discriminant of f in terms of \(k\).
  2. Hence find the range of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has two distinct real roots.
Pre-U Pre-U 9794/2 2014 June Q3
4 marks Moderate -0.8
3 Given that \(\mathrm { f } ( x ) = x ^ { 3 }\), use differentiation from first principles to prove that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 }\).
Pre-U Pre-U 9794/2 2014 June Q4
7 marks Standard +0.3
4 The points \(A , B , C\) and \(D\) have coordinates \(( 2 , - 1,0 ) , ( 3,2,5 ) , ( 4,2,3 )\) and \(( - 1 , a , b )\) respectively, where \(a\) and \(b\) are constants.
  1. Find the angle \(A B C\).
  2. Given that the lines \(A B\) and \(C D\) are parallel, find the values of \(a\) and \(b\).
Pre-U Pre-U 9794/2 2014 June Q5
3 marks Easy -1.3
5 An arithmetic progression has first term 5 and common difference 7.
  1. Find the value of the 10th term.
  2. Find the sum of the first 15 terms. The terms of the progression are given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\).
  3. Evaluate \(\sum _ { n = 1 } ^ { 15 } \left( 2 x _ { n } + 1 \right)\).
Pre-U Pre-U 9794/2 2014 June Q6
5 marks Moderate -0.8
6 Given that the angle \(\theta\) is acute and \(\cos \theta = \frac { 3 } { 4 }\) find, without using a calculator, the exact value of \(\sin 2 \theta\) and of \(\cot \theta\).
Pre-U Pre-U 9794/2 2014 June Q7
2 marks Moderate -0.8
7
  1. Express \(z ^ { 4 } + 3 z ^ { 2 } - 4\) in the form \(\left( z ^ { 2 } + a \right) \left( z ^ { 2 } + b \right)\) where \(a\) and \(b\) are real constants to be found.
  2. Hence draw an Argand diagram showing the points that represent the roots of the equation \(z ^ { 4 } + 3 z ^ { 2 } - 4 = 0\).
Pre-U Pre-U 9794/2 2014 June Q8
6 marks Standard +0.3
8 Show that the graph of \(y = x ^ { 2 } - \ln x\) has only one stationary point and give the coordinates of that point in exact form.
Pre-U Pre-U 9794/2 2014 June Q9
7 marks Challenging +1.2
9 A new lake is stocked with fish. Let \(P _ { t }\) be the population of fish in the lake after \(t\) years. Two models using recurrence relations are proposed for \(P _ { t }\), with \(P _ { 0 } = 550\). $$\begin{aligned} & \text { Model } 1 : P _ { t } = 2 P _ { t - 1 } \mathrm { e } ^ { - 0.001 P _ { t - 1 } } \\ & \text { Model } 2 : P _ { t } = \frac { 1 } { 2 } P _ { t - 1 } \left( 7 - \frac { 1 } { 160 } P _ { t - 1 } \right) \end{aligned}$$
  1. Evaluate the population predicted by each model when \(t = 3\).
  2. Identify, with evidence, which one of the models predicts a stable population in the long term.
  3. Describe the long term behaviour of the population for the other model.
Pre-U Pre-U 9794/2 2014 June Q10
11 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 10 x ^ { 2 } + 28 x - 15\).
  1. Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x - 5\).
  3. Factorise \(\mathrm { f } ( x )\) fully and hence sketch the graph of \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2014 June Q11
12 marks Challenging +1.2
11 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 4 x - 7 = 0\) has a single root \(\alpha\), close to 1.9 , which can be found using an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\). Three possible functions that can be used for such an iteration are $$\mathrm { F } _ { 1 } ( x ) = \frac { 7 } { 4 } + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 } , \quad \mathrm {~F} _ { 2 } ( x ) = \sqrt [ 3 ] { 2 x ^ { 2 } - 4 x + 7 } , \quad \mathrm {~F} _ { 3 } ( x ) = \frac { 7 - 4 x } { x ^ { 2 } - 2 x }$$
  1. Differentiate each of these functions with respect to \(x\).
  2. Without performing any iterations, and using \(x = 1.9\), show that an iterative process based on only two of the given functions will converge. Determine which one will do so more rapidly. The sequence of errors, \(e _ { n }\), is such that \(e _ { n + 1 } \approx \mathrm {~F} ^ { \prime } ( \alpha ) e _ { n }\).
  3. Using the iteration from part (ii) with the most rapid convergence, estimate the number of iterations required to reduce the magnitude of the error from \(\left| e _ { 1 } \right|\) in the first term to less than \(10 ^ { - 10 } \left| e _ { 1 } \right|\).
Pre-U Pre-U 9794/2 2014 June Q12
9 marks Standard +0.8
12 A curve \(C\) is defined parametrically by $$x = \cos t ( 1 - 2 \sin t ) , \quad y = \sin t ( 1 - 3 \sin t ) , \quad 0 \leqslant t < 2 \pi$$
  1. Show that \(C\) intersects the \(y\)-axis at exactly three points, and state the values of \(t\) and \(y\) at these points.
  2. Find the range of values of \(t\) for which \(C\) lies above the \(x\)-axis.
Pre-U Pre-U 9795/1 2014 June Q1
4 marks Standard +0.8
1 The series \(S\) is given by \(S = \sum _ { r = 0 } ^ { N } ( N + r ) ^ { 2 }\).
  1. Write out the first three terms and the last three terms of the series for \(S\).
  2. Use the standard result \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } N ( N + 1 ) ( a N + 1 )\) for some positive integer \(a\) to be determined.
Pre-U Pre-U 9795/1 2014 June Q2
8 marks Standard +0.3
2
  1. Show that there is a value of \(t\) for which \(\mathbf { A B }\) is an integer multiple of the \(3 \times 3\) identity matrix \(\mathbf { I }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 2 & 1 \\ t & 1 & - t \\ 3 & 2 & 1 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { c r r } t - 2 & 0 & 5 \\ 12 & - 2 & - 6 \\ 3 t & 4 & 7 \end{array} \right) .$$
  2. Express the system of equations $$\begin{aligned} - 5 x + 5 z & = 8 \\ 12 x - 2 y - 6 z & = 12 \\ - 9 x + 4 y + 7 z & = 22 \end{aligned}$$ in the form \(\mathbf { C x } = \mathbf { u }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, and \(\mathbf { x }\) and \(\mathbf { u }\) are suitable column vectors.
  3. Use the result of part (i) to solve the system of equations given in part (ii).
Pre-U Pre-U 9795/1 2014 June Q3
5 marks Standard +0.3
3
  1. On a single copy of an Argand diagram, sketch the loci defined by $$| z + 2 | = 3 \quad \text { and } \quad \arg ( z - \mathrm { i } ) = - \frac { 1 } { 4 } \pi$$
  2. State the complex number \(z\) which corresponds to the point of intersection of these two loci.
Pre-U Pre-U 9795/1 2014 June Q4
5 marks Challenging +1.2
4 Let \(I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { 2 x + 1 } \mathrm {~d} x\) for \(n \geqslant 0\). Show that, for \(n \geqslant 1\), $$( 2 n + 3 ) I _ { n } = 27 \times 4 ^ { n } - n I _ { n - 1 }$$
Pre-U Pre-U 9795/1 2014 June Q5
6 marks Standard +0.3
5 The curve \(C\) has equation \(y = \frac { 12 ( x + 1 ) } { ( x - 2 ) ^ { 2 } }\).
  1. Determine the coordinates of any stationary points of \(C\).
  2. Sketch \(C\).
Pre-U Pre-U 9795/1 2014 June Q6
8 marks Standard +0.3
6 Solve the first-order differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 \ln x\) given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2014 June Q7
6 marks Standard +0.3
7 Let \(\mathrm { f } ( n ) = 11 ^ { 2 n - 1 } + 7 \times 4 ^ { n }\). Prove by induction that \(\mathrm { f } ( n )\) is divisible by 13 for all positive integers \(n\).
Pre-U Pre-U 9795/1 2014 June Q8
6 marks Standard +0.3
8
  1. Show that the line \(l\) with vector equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 5 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 5 \\ - 2 \\ 3 \end{array} \right)\) lies in the plane \(\Pi\) with cartesian equation \(x + 4 y + z + 11 = 0\).
  2. The plane \(\Pi\) is horizontal, and the point \(P ( 1,2 , k )\) is above it. Given that the point in \(\Pi\) which is directly beneath \(P\) is on the line \(l\), determine the value of \(k\).