Questions — AQA (3620 questions)

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AQA M2 2016 June Q3
9 marks Moderate -0.3
The diagram shows a uniform lamina \(ABCDEFGHIJKL\). \includegraphics{figure_3}
  1. Explain why the centre of mass of the lamina is \(35\) cm from \(AL\). [1 mark]
  2. Find the distance of the centre of mass from \(AF\). [4 marks]
  3. The lamina is freely suspended from \(A\). Find the angle between \(AB\) and the vertical when the lamina is in equilibrium. [3 marks]
  4. Explain, briefly, how you have used the fact that the lamina is uniform. [1 mark]
AQA M2 2016 June Q4
8 marks Standard +0.3
A particle \(P\), of mass \(6\) kg, is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point \(O\). A second particle \(Q\), of mass \(8\) kg, is attached to the other end of the string. The particle \(Q\) hangs at rest vertically below the ring, and the particle \(P\) moves with speed \(5 \text{ m s}^{-1}\) in a horizontal circle, as shown in the diagram. The angle between \(OP\) and the vertical is \(\theta\). \includegraphics{figure_4}
  1. Find the tension in the string. [1 mark]
  2. Find \(\theta\). [3 marks]
  3. Find the radius of the horizontal circle. [4 marks]
AQA M2 2016 June Q5
12 marks Standard +0.3
A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(ROT\) is \(30°\), as shown in the diagram. \includegraphics{figure_5}
  1. Find, in terms of \(g\), \(l\) and \(u\), the speed of the particle at the point \(T\). [3 marks]
  2. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle is at the point \(T\). [3 marks]
  3. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle returns to the point \(R\). [2 marks]
  4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\). [4 marks]
AQA M2 2016 June Q6
8 marks Standard +0.3
A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda mv\), where \(\lambda\) is a constant.
  1. Show that $$\frac{\text{d}v}{\text{d}t} = g - \lambda v$$ [2 marks]
  2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\). [6 marks]
AQA M2 2016 June Q7
9 marks Standard +0.8
A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2\mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.
  1. Draw a diagram to show the forces acting on the ladder. [2 marks]
  2. Find \(\tan \theta\) in terms of \(\mu\). [7 marks]
AQA M2 2016 June Q8
8 marks Challenging +1.8
A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]
AQA M3 2016 June Q1
4 marks Moderate -0.8
At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of \(400 \text{ m s}^{-1}\). The mass of the gun is \(1.5\) kg and the mass of the bullet is \(30\) grams.
  1. Find the speed of recoil of the gun. [2 marks]
  2. Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired. [2 marks]
AQA M3 2016 June Q2
6 marks Moderate -0.8
A lunar mapping satellite of mass \(m_1\) measured in kg is in an elliptic orbit around the moon, which has mass \(m_2\) measured in kg. The effective potential, \(E\), of the satellite is given by $$E = \frac{K^2}{2m_1r^2} - \frac{Gm_1m_2}{r}$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G\) Nm\(^2\)kg\(^{-2}\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:
  1. \(E\), [3 marks]
  2. \(K\). [3 marks]
AQA M3 2016 June Q3
12 marks Standard +0.8
A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}
  1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
  2. Find the distance \(PQ\). [7 marks]
AQA M3 2016 June Q4
14 marks Standard +0.3
A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4m\), has the same radius as \(A\) and is moving on the table with velocity \(u\). \includegraphics{figure_4} The sphere \(A\) collides directly with the sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision. [6 marks]
    2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined. [2 marks]
  2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{5}\). The sphere \(B\) collides with \(A\) again after rebounding from the wall. Show that \(e < b\), where \(b\) is a constant to be determined. [3 marks]
  3. Given that \(e = \frac{4}{7}\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall. [3 marks]
AQA M3 2016 June Q5
12 marks Challenging +1.8
A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics{figure_5}
    1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
    2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
  2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]
AQA M3 2016 June Q6
14 marks Challenging +1.2
In this question use \(\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}\). A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60°\) with the wall, as shown in the diagram. \includegraphics{figure_6} The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac{1}{4}u(1 + e)\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision. [7 marks]
  2. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac{1 + e}{7 - e}\). [7 marks]
AQA M3 2016 June Q7
13 marks Challenging +1.8
A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \text{ m s}^{-1}\), the truck is moving on a bearing of \(340°\). Relative to the car, which is travelling due east at a speed of \(15 \text{ m s}^{-1}\), the truck is moving on a bearing of \(300°\).
  1. Show that the speed of the truck is approximately \(24.7 \text{ m s}^{-1}\) and that it is moving on a bearing of \(318°\), correct to the nearest degree. [8 marks]
  2. At the instant when the truck is at a distance of \(400\) metres from the quad-bike, the bearing of the truck from the quad-bike is \(060°\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \text{ m s}^{-1}\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible. [5 marks]
AQA FP1 2014 June Q1
5 marks Moderate -0.8
A curve passes through the point \((9, 6)\) and satisfies the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$ Use a step-by-step method with a step length of \(0.25\) to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places. [5 marks]
AQA FP1 2014 June Q2
11 marks Standard +0.3
The quadratic equation $$2x^2 + 8x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
    1. Find the value of \(\alpha^2 + \beta^2\). [2 marks]
    2. Hence, or otherwise, show that \(\alpha^4 + \beta^4 = \frac{449}{2}\). [2 marks]
  3. Find a quadratic equation, with integer coefficients, which has roots $$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$ [5 marks]
AQA FP1 2014 June Q3
4 marks Standard +0.3
Use the formulae for \(\sum_{r=1}^{n} r^3\) and \(\sum_{r=1}^{n} r^2\) to find the value of $$\sum_{r=3}^{60} r^2(r - 6)$$ [4 marks]
AQA FP1 2014 June Q4
6 marks Standard +0.3
Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]
AQA FP1 2014 June Q5
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x(x + 3)\).
  1. Find the gradient of the line passing through the point \((-5, 10)\) and the point on \(C\) with \(x\)-coordinate \(-5 + h\). Give your answer in its simplest form. [3 marks]
  2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((-5, 10)\). State the value of this gradient. [2 marks]
AQA FP1 2014 June Q6
10 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).
  1. Write down the equations of all the asymptotes of \(C\). [2 marks]
  2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
    1. Find the \(y\)-coordinate of the stationary point. [1 mark]
    2. Sketch the curve \(C\). [2 marks]
  3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]
AQA FP1 2014 June Q7
10 marks Moderate -0.3
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a reflection in the line \(y = -x\); [1 mark]
    2. a stretch parallel to the \(y\)-axis of scale factor \(7\). [1 mark]
  2. Hence find the matrix corresponding to the combined transformation of a reflection in the line \(y = -x\) followed by a stretch parallel to the \(y\)-axis of scale factor \(7\). [2 marks]
  3. The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} -3 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{bmatrix}\).
    1. Show that \(\mathbf{A}^2 = k\mathbf{I}\), where \(k\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [1 mark]
    2. Show that the matrix \(\mathbf{A}\) corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form \(y = (\tan \theta)x\). [5 marks]
AQA FP1 2014 June Q8
9 marks Standard +0.3
  1. Find the general solution of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ giving your answer for \(x\) in terms of \(\pi\). [5 marks]
  2. Use your general solution to find the sum of all the solutions of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ that lie in the interval \(0 \leqslant x \leqslant 20\pi\). Give your answer in the form \(k\pi\), stating the exact value of \(k\). [4 marks]
AQA FP1 2014 June Q9
15 marks Standard +0.8
An ellipse \(E\) has equation $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$
  1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes. [2 marks]
  2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(-5 < k < 5\). [5 marks]
  3. The ellipse \(E\) is translated by the vector \(\begin{bmatrix} a \\ b \end{bmatrix}\) to form another ellipse whose equation is \(9x^2 + 16y^2 + 18x - 64y = c\). Find the values of the constants \(a\), \(b\) and \(c\). [5 marks]
  4. Hence find an equation for each of the two tangents to the ellipse \(9x^2 + 16y^2 + 18x - 64y = c\) that are parallel to the line \(y = x\). [3 marks]
AQA FP1 2016 June Q1
7 marks Moderate -0.3
The quadratic equation \(x^2 - 6x + 14 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
  2. Find a quadratic equation, with integer coefficients, which has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). [5 marks]
AQA FP1 2016 June Q2
5 marks Moderate -0.8
A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).
  1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
  2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]
AQA FP1 2016 June Q3
7 marks Moderate -0.3
The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).
  1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
  2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
    1. Find the gradient of the line. [1 mark]
    2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]