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 9794/2 2016 June Q10
10 marks Challenging +1.2
  1. Using the substitution \(u = \frac{1}{x}\), or otherwise, find \(\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [4]
  2. Evaluate \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) and \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [3]
  3. Show that, when \(n\) is a positive integer, the integral \(\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) takes one of the two values found in part (ii), distinguishing between the two cases. [3]
Pre-U Pre-U 9794/2 2016 June Q11
12 marks Standard +0.3
The function f is defined by \(f(x) = \sqrt{x}, x > 0\).
  1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]
The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).
    1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
    2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]
Pre-U Pre-U 9794/3 2016 June Q1
4 marks Moderate -0.8
The following data refer to the annual rate of inflation and the annual percentage pay increase measured on 10 randomly chosen occasions.
Inflation rate (\%)0.91.21.61.51.73.04.13.72.84.2
Pay increase (\%)4.84.73.84.45.65.52.40.40.61.7
Show that, for these data, the product moment correlation coefficient between the rate of inflation and the annual pay increase is \(-0.679\), correct to 3 significant figures. [4]
Pre-U Pre-U 9794/3 2016 June Q2
8 marks Moderate -0.8
The weights of pineapples on sale at a wholesaler are normally distributed with mean \(1.349\) kg and standard deviation \(0.236\) kg. Before going on sale the pineapples are classified as 'Small', 'Medium', 'Large' and 'Extra Large'.
  1. A pineapple is classified as 'Small' if it weighs less than \(1.100\) kg. Find the probability that a randomly chosen pineapple will be classified as 'Small'. [5]
  2. \(10\%\) of pineapples are classified as 'Extra Large'. Find the minimum weight required for a pineapple to be classified as 'Extra Large'. [3]
Pre-U Pre-U 9794/3 2016 June Q3
11 marks Moderate -0.3
Chris plays for his local hockey club. In his first 20 games for the club, the mean number of goals per game he has scored is \(0.7\), with a standard deviation of \(0.9\). In the next 5 games he scores \(0, 1, 0, 2, 1\) goals.
  1. Find the mean and standard deviation for the number of goals per game Chris has scored in all 25 games. [7]
  2. A sponsor pays Chris £65 each time he plays for the club and a further £25 for each goal he scores. Find the mean and standard deviation of the amount per game he earns from the sponsor for all 25 games. [4]
Pre-U Pre-U 9794/3 2016 June Q4
8 marks Moderate -0.3
A certain type of sweet is made in a variety of colours. \(20\%\) of the sweets made are blue. Sweets of the various colours are thoroughly mixed before being put into packets.
  1. In a packet that contains 10 sweets, find the probability that the packet contains
    1. at most 3 blue sweets, [1]
    2. exactly 3 blue sweets, [2]
    3. at least 1 blue sweet. [2]
  2. What is the smallest number of sweets that a packet should contain in order to be at least \(95\%\) certain of having at least 1 blue sweet? [3]
Pre-U Pre-U 9794/3 2016 June Q5
4 marks Moderate -0.3
The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]
Pre-U Pre-U 9794/3 2016 June Q6
5 marks Moderate -0.8
\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\). Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]
Pre-U Pre-U 9794/3 2016 June Q7
5 marks Moderate -0.8
A stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]
Pre-U Pre-U 9794/3 2016 June Q8
8 marks Moderate -0.3
A particle is projected with speed \(U \text{ m s}^{-1}\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac{12}{13}\), and reaches its maximum height after \(2.4\) seconds.
  1. Find \(U\) and the maximum height reached by the particle. [4]
  2. Find the horizontal range of the particle. [4]
Pre-U Pre-U 9794/3 2016 June Q9
8 marks Standard +0.8
A particle of mass \(0.01\) kg is projected vertically upwards from a point \(G\) at ground level with speed \(165 \text{ m s}^{-1}\) and reaches a maximum height of \(1237.5\) m. Throughout its motion it experiences a constant resistance.
  1. Find the acceleration of the particle as it ascends and hence the magnitude of the resistance. [4]
  2. During its descent back to \(G\) the particle experiences the same constant resistance. Find the time taken for the descent. [4]
Pre-U Pre-U 9794/3 2016 June Q10
7 marks Standard +0.3
  1. A particle \(A\) of mass \(m\) travelling with speed \(u\) on a smooth horizontal surface collides directly with a particle \(B\) of mass \(3m\) travelling with speed \(\frac{2u}{5}\) in the opposite direction. After the collision, \(A\) travels at speed \(\frac{2u}{5}\) and \(B\) travels at speed \(\frac{4u}{15}\), both in the same direction as \(B\) before the collision. Find \(A\) and the coefficient of restitution between the two particles. [4]
  2. A particle of mass 3 kg moving with velocity \((2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) \text{ m s}^{-1}\) receives an impulse of \((6\mathbf{i} - 6\mathbf{j} - 9\mathbf{k})\) N s. Find the velocity of the particle after the impulse. [3]
Pre-U Pre-U 9794/3 2016 June Q11
12 marks Standard +0.3
\includegraphics{figure_11} The diagram shows a particle, \(A\), of mass \(m_1\) at rest on a rough slope at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). Particle \(A\) is connected by a light inextensible string to another particle, \(B\), of mass \(m_2\). The string passes over a smooth peg at the top of the slope and particle \(B\) is hanging freely.
  1. In the case when \(m_2 = \frac{1}{4}m_1\), particle \(A\) is on the point of sliding down the slope.
    1. Draw a fully labelled diagram to show all the forces acting on the particles. [2]
    2. Find the coefficient of friction between \(A\) and the slope. [6]
  2. In the case when \(m_2 = m_1\), find the acceleration of the particles. [4]
Pre-U Pre-U 9795/1 2018 June Q1
5 marks Moderate -0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using the method of differences, prove that \(\sum_{r=1}^{n} \frac{3}{(3r-1)(3r+2)} = \frac{1}{2} - \frac{1}{3n+2}\). [2]
  3. Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(3r-1)(3r+2)}\). [1]
Pre-U Pre-U 9795/1 2018 June Q2
10 marks Standard +0.3
  1. Determine the asymptotes and turning points of the curve with equation \(y = \frac{x^2+3}{x+1}\). [7]
  2. Sketch the curve. [3]
Pre-U Pre-U 9795/1 2018 June Q3
7 marks Standard +0.3
The complex numbers \(z_1\) and \(z_2\) are such that \(|z_1| = 2\), \(\arg(z_1) = \frac{7}{12}\pi\), \(|z_2| = \sqrt{2}\) and \(\arg(z_2) = -\frac{1}{8}\pi\).
  1. Find, in exact form, the modulus and argument of \(\frac{z_1}{z_2}\). [3]
  2. Let \(z_3 = \left(\frac{z_1}{z_2}\right)^n\). It is given that \(n\) is the least positive integer for which \(z_3\) is a positive real number. Find this value of \(n\) and the exact value of \(z_3\). [4]
Pre-U Pre-U 9795/1 2018 June Q4
7 marks Challenging +1.2
A curve has polar equation \(r = \frac{3}{10}e^{3\theta}\) for \(\theta \geq 0\). The length of the arc of this curve between \(\theta = 0\) and \(\theta = \alpha\) is denoted by \(L(\alpha)\).
  1. Show that \(L(\alpha) = \frac{1}{3}(e^{3\alpha} - 1)\). [5]
  2. The point \(P\) on the curve corresponding to \(\theta = \beta\) is such that \(L(\beta) = OP\), where \(O\) is the pole. Find the value of \(\beta\). [2]
Pre-U Pre-U 9795/1 2018 June Q5
8 marks Standard +0.8
Find, in the form \(y = f(x)\), the solution of the differential equation \(\frac{dy}{dx} + y\tanh x = 2\cosh x\), given that \(y = \frac{3}{4}\) when \(x = \ln 2\). [8]
Pre-U Pre-U 9795/1 2018 June Q6
8 marks Challenging +1.8
The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).
  1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
  2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]
Pre-U Pre-U 9795/1 2018 June Q7
6 marks Challenging +1.2
The function \(y\) satisfies \(\frac{d^2y}{dx^2} + x^2y = x\), and is such that \(y = 1\) and \(\frac{dy}{dx} = 1\) when \(x = 1\).
  1. Using the given differential equation
    1. state the value of \(\frac{d^2y}{dx^2}\) when \(x = 1\), [1]
    2. find, by differentiation, the value of \(\frac{d^3y}{dx^3}\) when \(x = 1\). [2]
  2. Hence determine the Taylor series for \(y\) about \(x = 1\) up to and including the term in \((x-1)^3\) and deduce, correct to 4 decimal places, an approximation for \(y\) when \(x = 1.1\). [3]
Pre-U Pre-U 9795/1 2018 June Q8
8 marks Challenging +1.2
  1. Write down the values of the constants \(a\) and \(b\) for which \(m^3 = \frac{1}{6}m^3(am^2 + 2) - \frac{1}{12}m^2(bm)\). [1]
  2. Prove by induction that \(\sum_{r=1}^{n} r^5 = \frac{1}{6}n^3(n+1)^3 - \frac{1}{12}n^2(n+1)^2\) for all positive integers \(n\). [7]
Pre-U Pre-U 9795/1 2018 June Q9
8 marks Standard +0.3
  1. Use de Moivre's theorem to prove that \(\cos 3\theta = 4c^3 - 3c\), where \(c = \cos\theta\). [3]
  2. Solve the equation \(2\cos 3\theta - \sqrt{3} = 0\) for \(0 < \theta < \pi\), giving each answer in an exact form. [2]
  3. Deduce, in trigonometric form, the three roots of the equation \(x^3 - 3x - \sqrt{3} = 0\). [3]
Pre-U Pre-U 9795/1 2018 June Q10
10 marks Challenging +1.8
  1. Let \(G\) be a group of order 10. Write down the possible orders of the elements of \(G\) and justify your answer. [2]
  2. Let \(G_1\) be the cyclic group of order 10 and let \(g\) be a generator of \(G_1\) (that is, an element of order 10). List the ten elements of \(G_1\) in terms of \(g\) and state the order of each element. [4]
  3. The group \(G_2\) is defined as the set of ordered pairs \((x, y)\), where \(x \in \{0, 1\}\) and \(y \in \{0, 1, 2, 3, 4\}\), together with the binary operation \(\oplus\) defined by $$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$ where \(x_3 = x_1 + x_2\) modulo 2 and \(y_3 = y_1 + y_2\) modulo 5.
    1. List the elements of \(G_2\) and state the order of each element. [3]
    2. State, with justification, whether \(G_1\) and \(G_2\) are isomorphic. [1]
Pre-U Pre-U 9795/1 2018 June Q11
10 marks Challenging +1.3
Let \(\mathbf{A}\) be the matrix \(\begin{pmatrix} 17 & 12 \\ 12 & 10 \end{pmatrix}\).
    1. Determine the integer \(n\) for which \(27\mathbf{A} - \mathbf{A}^2 = n\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [2]
    2. Hence find \(\mathbf{A}^{-1}\) in the form \(p\mathbf{A} + q\mathbf{I}\) for rational numbers \(p\) and \(q\). [2]
  2. The plane transformation \(T\) is defined by \(T: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \mathbf{A} \begin{pmatrix} x \\ y \end{pmatrix}\). It is given that \(T\) is a stretch, with scale factor \(k\), parallel to the line \(y = mx\), where \(m > 0\).
    1. Find the value of \(k\). [2]
    2. By considering \(\mathbf{A} \begin{pmatrix} x \\ mx \end{pmatrix}\), or otherwise, determine the value of \(m\). [4]
Pre-U Pre-U 9795/1 2018 June Q12
15 marks Challenging +1.8
The curve \(C\) is given by \(y = \frac{1}{4}x^2 - \frac{1}{2}\ln x\) for \(2 \leq x \leq 8\).
  1. Find, in its simplest exact form, the length of \(C\). [5]
  2. When \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed. Show that the area of this surface is \(\pi(270 - 47\ln 2 - 2(\ln 2)^2)\). [10]