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 9795/1 2014 June Q9
2 marks Hard +2.3
9
  1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
  2. Prove that any group in which every non-identity element is self-inverse is abelian.
  3. Simon believes that if \(x\) and \(y\) are two distinct self-inverse elements of a group, then the element \(x y\) is also self-inverse. By considering the group of the six permutations of \(\left( \begin{array} { l l } 1 & 2 \end{array} \right)\), produce a counter-example to prove him wrong.
  4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct self-inverse elements of \(G\). By considering the set \(H = \{ i , x , y , x y \}\), prove by contradiction that \(G\) cannot contain all self-inverse elements.
Pre-U Pre-U 9795/1 2014 June Q10
13 marks Challenging +1.8
10
  1. Use de Moivre's theorem to show that \(2 \cos 6 \theta \equiv 64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 2\).
  2. Hence find, in exact trigonometric form, the six roots of the equation $$x ^ { 6 } - 6 x ^ { 4 } + 9 x ^ { 2 } - 3 = 0$$
  3. By considering the product of these six roots, determine the exact value of $$\cos \left( \frac { 1 } { 18 } \pi \right) \cos \left( \frac { 5 } { 18 } \pi \right) \cos \left( \frac { 7 } { 18 } \pi \right) .$$
Pre-U Pre-U 9795/1 2014 June Q11
9 marks Challenging +1.2
11 A curve has polar equation \(r = \mathrm { e } ^ { \sin \theta }\) for \(- \pi < \theta \leqslant \pi\).
  1. State the polar coordinates of the point where the curve crosses the initial line.
  2. State also the polar coordinates of the points where \(r\) takes its least and greatest values.
  3. Sketch the curve.
  4. By deriving a suitable Maclaurin series up to and including the term in \(\theta ^ { 2 }\), find an approximation, to 3 decimal places, for the area of the region enclosed by the curve, the initial line and the line \(\theta = 0.3\).
Pre-U Pre-U 9795/1 2014 June Q12
10 marks Challenging +1.8
12
  1. (a) Show that \(\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }\).
    (b) Hence, or otherwise, show that, if \(\tanh x = \frac { 1 } { k }\) for \(k > 1\), then \(x = \frac { 1 } { 2 } \ln \left( \frac { k + 1 } { k - 1 } \right)\) and find an expression in terms of \(k\) for \(\sinh 2 x\).
  2. A curve has equation \(y = \frac { 1 } { 2 } \ln ( \tanh x )\) for \(\alpha \leqslant x \leqslant \beta\), where \(\tanh \alpha = \frac { 1 } { 3 }\) and \(\tanh \beta = \frac { 1 } { 2 }\). Find, in its simplest exact form, the arc length of this curve.
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 9794/1 2014 June Q4
2 marks Moderate -0.8
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Moderate -0.3
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.3
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Moderate -0.8
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-3_648_684_342_731}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2015 June Q1
3 marks Easy -1.2
1 Find the set of values of \(x\) for which \(x ^ { 2 } - x - 12 < 0\).
Pre-U Pre-U 9794/1 2015 June Q2
5 marks Moderate -0.8
2 Solve the following simultaneous equations. $$x ^ { 2 } + 2 y ^ { 2 } = 36 \quad x + 2 y = 10$$
Pre-U Pre-U 9794/1 2015 June Q3
3 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_305_825_630_660} The diagram shows a triangle \(A B C\) in which angle \(B = 39 ^ { \circ }\), angle \(C = 28 ^ { \circ } , A B = x \mathrm {~cm}\) and \(A C = ( 2 x - 1 ) \mathrm { cm }\). Find the value of \(x\).
Pre-U Pre-U 9794/1 2015 June Q4
6 marks Moderate -0.8
4 A population, \(P\), is modelled by the equation \(P = a \mathrm { e } ^ { b t }\) where \(t\) is time in years, and \(a\) and \(b\) are constants.
  1. By considering logarithms, show that a graph of \(\ln P\) against \(t\) is a straight line. State the intercept on the vertical axis and the gradient.
  2. Use the graph below to obtain values for \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_657_750_1530_740}
Pre-U Pre-U 9794/1 2015 June Q5
9 marks Moderate -0.8
5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).
  1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
  2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
  3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).
Pre-U Pre-U 9794/1 2015 June Q6
6 marks Moderate -0.3
6 The functions f and g are given by \(\mathrm { f } ( x ) = \frac { 3 } { x - 1 }\) for all \(x \neq 1\) and \(\mathrm { g } ( x ) = x + 2\) for all real \(x\).
  1. Find gf, stating its domain and range.
  2. Find \(( \mathrm { gf } ) ^ { - 1 }\), stating any values of \(x\) for which \(( \mathrm { gf } ) ^ { - 1 }\) is not defined.
Pre-U Pre-U 9794/1 2015 June Q7
9 marks Standard +0.3
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following vector equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$
  1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of their point of intersection.
  2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
Pre-U Pre-U 9794/1 2015 June Q8
11 marks Moderate -0.3
8 The complex numbers \(w\) and \(z\) are given by \(w = 3 - \mathrm { i }\) and \(z = 1 + \mathrm { i }\).
  1. Express \(\frac { z } { w }\) in the form \(p + \mathrm { i } q\) where \(p\) and \(q\) are real numbers.
  2. On the same Argand diagram, mark the points representing \(z , w\) and \(\frac { z } { w }\).
  3. Find the value in radians of \(\arg w\).
  4. Show that \(z + \frac { 2 } { z }\) is a real number.
Pre-U Pre-U 9794/1 2015 June Q9
7 marks Standard +0.3
9 A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x }\). Find the exact coordinates of the stationary points of the curve.
Pre-U Pre-U 9794/1 2015 June Q10
11 marks Standard +0.3
10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
  2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).
Pre-U Pre-U 9794/1 2015 June Q11
10 marks Standard +0.8
11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that $$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$ where \(A\) is an arbitrary constant and \(x > 0\).
Pre-U Pre-U 9794/2 2015 June Q1
3 marks Easy -1.8
1 Show that \(\frac { 31 } { 6 - \sqrt { 5 } } = 6 + \sqrt { 5 }\).
Pre-U Pre-U 9794/2 2015 June Q2
4 marks Easy -1.2
2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 2\). The curve passes through the point \(( 1,3 )\). Find the equation of the curve.
Pre-U Pre-U 9794/2 2015 June Q3
4 marks Easy -1.2
3 The function f is given by \(\mathrm { f } ( x ) = | x - 2 | + 3\) for \(- 5 \leqslant x \leqslant 5\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Explain why f is not one-one.
Pre-U Pre-U 9794/2 2015 June Q4
4 marks Moderate -0.8
4 Find the volume of the solid generated when the region bounded by the \(x\)-axis, \(x = 1 , x = 2\) and the curve given by \(y = x ^ { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Pre-U Pre-U 9794/2 2015 June Q5
8 marks Moderate -0.3
5
  1. Show that the equation \(\sin x - x + 1 = 0\) has a root between 1.5 and 2 .
  2. Use the iteration \(x _ { n + 1 } = 1 + \sin x _ { n }\), with a suitable starting value, to find that root correct to 2 decimal places.
  3. Sketch the graphs of \(y = \sin x\) and \(y = x - 1\), on the same set of axes, for \(0 \leqslant x \leqslant \pi\).
  4. Explain why the equation \(\sin x - x + 1 = 0\) has no root other than the one found in part (ii). [1]