Questions (33218 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
OCR H240/03 2023 June Q10
7 marks Standard +0.3
A particle \(P\) of mass \(m \text{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j}) \text{N}\) and \((a\mathbf{i} + b\mathbf{j}) \text{N}\). The resultant of these two forces is \(\mathbf{R} \text{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).
  1. Show that \(3a + b = 10\). [3]
It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10} \text{ms}^{-2}\).
  1. Determine the value of \(m\). [4]
OCR H240/03 2023 June Q11
8 marks Standard +0.3
\includegraphics{figure_11} A uniform rod \(AB\), of weight \(20 \text{N}\) and length \(2.8 \text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55°\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30°\) to the horizontal (see diagram).
  1. Show that the magnitude of the force acting on the rod at \(B\) is \(9.56 \text{N}\), correct to 3 significant figures. [3]
  2. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to 3 significant figures. [5]
OCR H240/03 2023 June Q12
13 marks Standard +0.8
In this question you should take the acceleration due to gravity to be \(10 \text{ms^{-2}\).} \includegraphics{figure_12} A small ball \(P\) is projected from a point \(A\) with speed \(39 \text{ms}^{-1}\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac{5}{13}\) and \(\cos \theta = \frac{12}{13}\). Point \(A\) is \(20 \text{m}\) vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.
  1. Find the maximum height of \(P\) above the ground during its motion. [3]
The time taken for \(P\) to travel from \(A\) to \(C\) is \(7\) seconds.
  1. Determine the value of \(T\). [3]
  2. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). [1]
At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geq 0\), the velocity \(v \text{ms}^{-1}\) of \(Q\) is given by $$v = kt^3 + 6t^2 + \frac{3}{2}t,$$ where \(k\) is a positive constant.
  1. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision. [6]
OCR H240/03 2023 June Q13
12 marks Challenging +1.2
\includegraphics{figure_13} The diagram shows a small block \(B\), of mass \(2 \text{kg}\), and a particle \(P\), of mass \(4 \text{kg}\), which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60°\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.
  1. Determine, in terms of \(g\), the tension in the string. [7]
When \(P\) is moving at \(2 \text{ms}^{-1}\) the string breaks. In the \(0.5\) seconds after the string breaks \(P\) moves \(1.9 \text{m}\) down the plane.
  1. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures. [5]
AQA AS Paper 1 2018 June Q1
1 marks Easy -1.8
Three of the following points lie on the same straight line. Which point does not lie on this line? Tick one box. [1 mark] \((-2, 14)\) \((-1, 8)\) \((1, -1)\) \((2, -6)\)
AQA AS Paper 1 2018 June Q2
1 marks Easy -1.2
A circle has equation \((x - 2)^2 + (y + 3)^2 = 13\) Find the gradient of the tangent to this circle at the origin. Circle your answer. [1 mark] \(-\frac{3}{2}\) \quad \(-\frac{2}{3}\) \quad \(\frac{2}{3}\) \quad \(\frac{3}{2}\)
AQA AS Paper 1 2018 June Q3
2 marks Easy -1.2
State the interval for which \(\sin x\) is a decreasing function for \(0° \leq x \leq 360°\) [2 marks]
AQA AS Paper 1 2018 June Q4
5 marks Moderate -0.8
  1. Find the first three terms in the expansion of \((1 - 3x)^4\) in ascending powers of \(x\). [3 marks]
  2. Using your expansion, approximate \((0.994)^4\) to six decimal places. [2 marks]
AQA AS Paper 1 2018 June Q5
5 marks Standard +0.3
Point \(C\) has coordinates \((c, 2)\) and point \(D\) has coordinates \((6, d)\). The line \(y + 4x = 11\) is the perpendicular bisector of \(CD\). Find \(c\) and \(d\). [5 marks]
AQA AS Paper 1 2018 June Q6
7 marks Standard +0.8
\(ABC\) is a right-angled triangle. \includegraphics{figure_6} \(D\) is the point on hypotenuse \(AC\) such that \(AD = AB\). The area of \(\triangle ABD\) is equal to half that of \(\triangle ABC\).
  1. Show that \(\tan A = 2 \sin A\) [4 marks]
    1. Show that the equation given in part (a) has two solutions for \(0° \leq A \leq 90°\) [2 marks]
    2. State the solution which is appropriate in this context. [1 mark]
AQA AS Paper 1 2018 June Q7
5 marks Standard +0.8
Prove that \(n\) is a prime number greater than \(5 \Rightarrow n^4\) has final digit \(1\) [5 marks]
AQA AS Paper 1 2018 June Q8
8 marks Moderate -0.3
Maxine measures the pressure, \(P\) kilopascals, and the volume, \(V\) litres, in a fixed quantity of gas. Maxine believes that the pressure and volume are connected by the equation $$P = cV^d$$ where \(c\) and \(d\) are constants. Using four experimental results, Maxine plots \(\log_{10} P\) against \(\log_{10} V\), as shown in the graph below. \includegraphics{figure_8}
  1. Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph. [2 marks]
  2. Calculate the value of each of the constants \(c\) and \(d\). [4 marks]
  3. Estimate the pressure of the gas when the volume is \(2\) litres. [2 marks]
AQA AS Paper 1 2018 June Q9
8 marks Moderate -0.3
Craig is investigating the gradient of chords of the curve with equation \(\mathrm{f}(x) = x - x^2\) Each chord joins the point \((3, -6)\) to the point \((3 + h, \mathrm{f}(3 + h))\) The table shows some of Craig's results.
\(x\)\(\mathrm{f}(x)\)\(h\)\(x + h\)\(\mathrm{f}(x + h)\)Gradient
\(3\)\(-6\)\(1\)\(4\)\(-12\)\(-6\)
\(3\)\(-6\)\(0.1\)\(3.1\)\(-6.51\)\(-5.1\)
\(3\)\(-6\)\(0.01\)
\(3\)\(-6\)\(0.001\)
\(3\)\(-6\)\(0.0001\)
  1. Show how the value \(-5.1\) has been calculated. [1 mark]
  2. Complete the third row of the table above. [2 marks]
  3. State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to \(0\) [1 mark]
  4. Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks]
AQA AS Paper 1 2018 June Q10
11 marks Standard +0.3
A curve has equation \(y = 2x^2 - 8x\sqrt{x} + 8x + 1\) for \(x \geq 0\)
  1. Prove that the curve has a maximum point at \((1, 3)\) Fully justify your answer. [9 marks]
  2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]
AQA AS Paper 1 2018 June Q11
1 marks Easy -2.0
In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) A ball, initially at rest, is dropped from a height of \(40\,\mathrm{m}\) above the ground. Calculate the speed of the ball when it reaches the ground. Circle your answer. [1 mark] \(-28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(-780\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(780\,\mathrm{m}\,\mathrm{s}^{-1}\)
AQA AS Paper 1 2018 June Q12
1 marks Easy -1.8
An object of mass \(5\,\mathrm{kg}\) is moving in a straight line. As a result of experiencing a forward force of \(F\) newtons and a resistant force of \(R\) newtons it accelerates at \(0.6\,\mathrm{m}\,\mathrm{s}^{-2}\) Which one of the following equations is correct? Circle your answer. [1 mark] \(F - R = 0\) \quad \(F - R = 5\) \quad \(F - R = 3\) \quad \(F - R = 0.6\)
AQA AS Paper 1 2018 June Q13
6 marks Moderate -0.8
A vehicle, which begins at rest at point \(P\), is travelling in a straight line. For the first \(4\) seconds the vehicle moves with a constant acceleration of \(0.75\,\mathrm{m}\,\mathrm{s}^{-2}\) For the next \(5\) seconds the vehicle moves with a constant acceleration of \(-1.2\,\mathrm{m}\,\mathrm{s}^{-2}\) The vehicle then immediately stops accelerating, and travels a further \(33\,\mathrm{m}\) at constant speed.
  1. Draw a velocity-time graph for this journey on the grid below. [3 marks] \includegraphics{figure_13}
  2. Find the distance of the car from \(P\) after \(20\) seconds. [3 marks]
AQA AS Paper 1 2018 June Q14
6 marks Moderate -0.8
In this question use \(g = 9.81\,\mathrm{m}\,\mathrm{s}^{-2}\) Two particles, of mass \(1.8\,\mathrm{kg}\) and \(1.2\,\mathrm{kg}\), are connected by a light, inextensible string over a smooth peg. \includegraphics{figure_14}
  1. Initially the particles are held at rest \(1.5\,\mathrm{m}\) above horizontal ground and the string between them is taut. The particles are released from rest. Find the time taken for the \(1.8\,\mathrm{kg}\) particle to reach the ground. [5 marks]
  2. State one assumption you have made in answering part (a). [1 mark]
AQA AS Paper 1 2018 June Q15
6 marks Moderate -0.3
A cyclist, Laura, is travelling in a straight line on a horizontal road at a constant speed of \(25\,\mathrm{km}\,\mathrm{h}^{-1}\) A second cyclist, Jason, is riding closely and directly behind Laura. He is also moving with a constant speed of \(25\,\mathrm{km}\,\mathrm{h}^{-1}\)
  1. The driving force applied by Jason is likely to be less than the driving force applied by Laura. Explain why. [1 mark]
  2. Jason has a problem and stops, but Laura continues at the same constant speed. Laura sees an accident \(40\,\mathrm{m}\) ahead, so she stops pedalling and applies the brakes. She experiences a total resistance force of \(40\,\mathrm{N}\) Laura and her cycle have a combined mass of \(64\,\mathrm{kg}\)
    1. Determine whether Laura stops before reaching the accident. Fully justify your answer. [4 marks]
    2. State one assumption you have made that could affect your answer to part (b)(i). [1 mark]
AQA AS Paper 1 2018 June Q16
7 marks Moderate -0.8
A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement \(3\) metres from \(A\). Its velocity, \(v\,\mathrm{m}\,\mathrm{s}^{-1}\), at time \(t\) seconds is defined by $$v = 0.06(2 + t - t^2)$$
  1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds. [4 marks]
  2. In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43\,\mathrm{m}\,\mathrm{s}^{-1}\) Find the time taken for the ball to reach its highest point. [3 marks]
AQA AS Paper 1 2019 June Q1
1 marks Easy -1.8
State the number of solutions to the equation \(\tan 4\theta = 1\) for \(0° < \theta < 180°\) Circle your answer. [1 mark] 1 2 4 8
AQA AS Paper 1 2019 June Q2
1 marks Easy -1.2
Dan believes that for every positive integer \(n\), at least one of \(2^n - 1\) and \(2^n + 1\) is prime. Which value of \(n\) shown below is a counter example to Dan's belief? Circle your answer. [1 mark] \(n = 3\) \(n = 4\) \(n = 5\) \(n = 6\)
AQA AS Paper 1 2019 June Q3
5 marks Moderate -0.3
It is given that \((x + 1)\) and \((x - 3)\) are two factors of \(f(x)\), where $$f(x) = px^3 - 3x^2 - 8x + q$$
  1. Find the values of \(p\) and \(q\). [3 marks]
  2. Fully factorise \(f(x)\). [2 marks]
AQA AS Paper 1 2019 June Q4
4 marks Moderate -0.8
Show that \(\frac{\sqrt{6}}{\sqrt{3} - \sqrt{2}}\) can be expressed in the form \(m\sqrt{n} + n\sqrt{m}\), where \(m\) and \(n\) are integers. Fully justify your answer. [4 marks]
AQA AS Paper 1 2019 June Q5
5 marks Moderate -0.8
  1. Sketch the curve \(y = g(x)\) where $$g(x) = (x + 2)(x - 1)^2$$ [3 marks]
  2. Hence, solve \(g(x) \leq 0\) [2 marks]