Past Exam Questions
← Back9709 P33 - Jun 2013 - Q4
(i) Express \((\sqrt{3}) \cos x + \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving the exact values of \(R\) and \(\alpha\).
(ii) Hence show that
\(\int_{\frac{1}{6}\pi}^{\frac{1}{2}\pi} \frac{1}{((\sqrt{3}) \cos x + \sin x)^2} \, dx = \frac{1}{4}\sqrt{3}.\)
9709 P31 - Jun 2013 - Q9
(i) Express \(4 \cos \theta + 3 \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the value of \(\alpha\) correct to 4 decimal places.
(ii) Hence
(a) solve the equation \(4 \cos \theta + 3 \sin \theta = 2\) for \(0 < \theta < 2\pi\),
(b) find \(\int \frac{50}{(4 \cos \theta + 3 \sin \theta)^2} \, d\theta\).
9709 P33 - Nov 2012 - Q7
The diagram shows part of the curve \(y = \\sin^3 2x \\cos^3 2x\). The shaded region shown is bounded by the curve and the \(x\)-axis and its exact area is denoted by \(A\).
(i) Use the substitution \(u = \\sin 2x\) in a suitable integral to find the value of \(A\). [6]
(ii) Given that \(\\int_0^{k\\pi} |\\sin^3 2x \\cos^3 2x| \, dx = 40A\), find the value of the constant \(k\). [2]
9709 P31 - Nov 2012 - Q5
(i) By differentiating \(\frac{1}{\cos x}\), show that if \(y = \sec x\) then \(\frac{dy}{dx} = \sec x \tan x\).
(ii) Show that \(\frac{1}{\sec x - \tan x} \equiv \sec x + \tan x\).
(iii) Deduce that \(\frac{1}{(\sec x - \tan x)^2} \equiv 2 \sec^2 x - 1 + 2 \sec x \tan x\).
(iv) Hence show that \(\int_0^{\frac{1}{4}\pi} \frac{1}{(\sec x - \tan x)^2} \, dx = \frac{1}{4}(8\sqrt{2} - \pi)\).
9709 P33 - Nov 2011 - Q10
(i) Use the substitution \(u = \tan x\) to show that, for \(n \neq -1\),
\(\int_0^{\frac{\pi}{4}} (\tan^{n+2} x + \tan^n x) \, dx = \frac{1}{n+1}.\)
(ii) Hence find the exact value of
(a) \(\int_0^{\frac{\pi}{4}} (\sec^4 x - \sec^2 x) \, dx,\)
(b) \(\int_0^{\frac{\pi}{4}} (\tan^9 x + 5 \tan^7 x + 5 \tan^5 x + \tan^3 x) \, dx.\)
9709 P43 - Nov 2022 - Q5
Particles X and Y move in a straight line through points A and B. Particle X starts from rest at A and moves towards B. At the same instant, Y starts from rest at B.
At time t seconds after the particles start moving:
- the acceleration of X in the direction AB is given by \\((12t + 12) \, \text{ms}^{-2} \\\),
- the acceleration of Y in the direction AB is given by \\((24t - 8) \, \text{ms}^{-2} \\\).
(a) It is given that the velocities of X and Y are equal when they collide. Calculate the distance AB.
\((b) It is given instead that AB = 36 m. Verify that X and Y collide after 3 s.\)
9709 P41 - Jun 2015 - Q6
Two particles A and B start to move at the same instant from a point O. The particles move in the same direction along the same straight line. The acceleration of A at time t s after starting to move is a m/s2, where a = 0.05 - 0.0002t.
- Find A’s velocity when t = 200 and when t = 500.
- B moves with constant acceleration for the first 200 s and has the same velocity as A when t = 200. B moves with constant retardation from t = 200 to t = 500 and has the same velocity as A when t = 500.
- Find the distance between A and B when t = 500.
9709 P41 - Nov 2014 - Q4
Particles P and Q move on a straight line AOB. The particles leave O simultaneously, with P moving towards A and with Q moving towards B. The initial speed of P is 1.3 m s-1 and its acceleration in the direction OA is 0.1 m s-2. Q moves with acceleration in the direction OB of 0.016t m s-2, where t seconds is the time elapsed since the instant that P and Q started to move from O. When t = 20, particle P passes through A and particle Q passes through B.
- Given that the speed of Q at B is the same as the speed of P at A, find the speed of Q at time t = 0.
- Find the distance AB.
9709 P42 - Jun 2014 - Q2
A and B are two points which are 10 m apart on the same horizontal plane. A particle P starts to move from rest at A, directly towards B, with constant acceleration 0.5 m s-2. Another particle Q is moving directly towards A with constant speed 0.75 m s-1, and passes through B at the instant that P starts to move. At time T s after this instant, particles P and Q collide. Find
- the value of T,
- the speed of P immediately before the collision.
9709 P41 - Jun 2014 - Q7
Two cyclists P and Q travel along a straight road ABC, starting simultaneously at A and arriving simultaneously at C. Both cyclists pass through B 400 s after leaving A. Cyclist P starts with speed 3 m s-1 and increases this speed with constant acceleration 0.005 m s-2 until he reaches B.
(i) Show that the distance AB is 1600 m and find P's speed at B.
Cyclist Q travels from A to B with speed v m s-1 at time t seconds after leaving A, where
\(v = 0.04t - 0.0001t^2 + k,\)
and k is a constant.
(ii) Find the value of k and the maximum speed of Q before he has reached B.
Cyclist P travels from B to C, a distance of 1400 m, at the speed he had reached at B. Cyclist Q travels from B to C with constant acceleration a m s-2.
(iii) Find the time taken for the cyclists to travel from B to C and find the value of a.
9709 P41 - Nov 2012 - Q5
Particle P travels along a straight line from A to B with constant acceleration 0.05 m s-2. Its speed at A is 2 m s-1 and its speed at B is 5 m s-1.
(i) Find the time taken for P to travel from A to B, and find also the distance AB.
Particle Q also travels along the same straight line from A to B, starting from rest at A. At time t s after leaving A, the speed of Q is kt3 m s-1, where k is a constant. Q takes the same time to travel from A to B as P does.
(ii) Find the value of k and find Q's speed at B.
9709 P42 - Jun 2011 - Q7
A walker travels along a straight road passing through the points A and B on the road with speeds 0.9 m s-1 and 1.3 m s-1 respectively. The walker’s acceleration between A and B is constant and equal to 0.004 m s-2.
- Find the time taken by the walker to travel from A to B, and find the distance AB.
A cyclist leaves A at the same instant as the walker. She starts from rest and travels along the straight road, passing through B at the same instant as the walker. At time t s after leaving A the cyclist’s speed is kt3 m s-1, where k is a constant.
- Show that when t = 64.05 the speed of the walker and the speed of the cyclist are the same, correct to 3 significant figures.
- Find the cyclist’s acceleration at the instant she passes through B.
9709 P41 - Nov 2010 - Q4
A particle P starts from a fixed point O at time t = 0, where t is in seconds, and moves with constant acceleration in a straight line. The initial velocity of P is 1.5 m s-1 and its velocity when t = 10 is 3.5 m s-1.
- Find the displacement of P from O when t = 10.
Another particle Q also starts from O when t = 0 and moves along the same straight line as P. The acceleration of Q at time t is 0.03t m s-2.
- Given that Q has the same velocity as P when t = 10, show that it also has the same displacement from O as P when t = 10.
9709 P4 - Nov 2007 - Q6
(i) A man walks in a straight line from A to B with constant acceleration 0.004 m s-2. His speed at A is 1.8 m s-1 and his speed at B is 2.2 m s-1. Find the time taken for the man to walk from A to B, and find the distance AB.
(ii) A woman cyclist leaves A at the same instant as the man. She starts from rest and travels in a straight line to B, reaching B at the same instant as the man. At time t s after leaving A the cyclist’s speed is k(200t − t2) m s-1, where k is a constant. Find
- the value of k,
- the cyclist’s speed at B.
(iii) Sketch, using the same axes, the velocity-time graphs for the man’s motion and the woman’s motion from A to B.
9709 P4 - Jun 2007 - Q6
A particle P starts from rest at the point A and travels in a straight line, coming to rest again after 10 s. The velocity-time graph for P consists of two straight line segments (see diagram). A particle Q starts from rest at A at the same instant as P and travels along the same straight line as P. The velocity of Q is given by \(v = 3t - 0.3t^2\) for \(0 \leq t \leq 10\). The displacements from A of P and Q are the same when \(t = 10\).
- Show that the greatest velocity of P during its motion is 10 m s-1.
- Find the value of \(t\), in the interval \(0 < t < 5\), for which the acceleration of Q is the same as the acceleration of P.
9709 P4 - Nov 2004 - Q5
Particles P and Q start from points A and B respectively, at the same instant, and move towards each other in a horizontal straight line. The initial speeds of P and Q are 5 m s-1 and 3 m s-1 respectively. The accelerations of P and Q are constant and equal to 4 m s-2 and 2 m s-2 respectively (see diagram).
- Find the speed of P at the instant when the speed of P is 1.8 times the speed of Q.
- Given that AB = 51 m, find the time taken from the start until P and Q meet.
9709 P41 - Jun 2021 - Q4
Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point A.
Isabella accelerates for 5 s at a constant rate \(a \text{ m s}^{-2}\). She then travels at the constant speed she has reached for 10 s, before decelerating to rest at a constant rate over a period of 5 s.
Maria accelerates at a constant rate, reaching a speed of 5 \(\text{ m s}^{-1}\) in a distance of 27.5 m. She then maintains this speed for a period of 10 s, before decelerating to rest at a constant rate over a period of 5 s.
(a) Given that \(a = 1.1\), find which cyclist travels further.
(b) Find the value of \(a\) for which the two cyclists travel the same distance.
9709 P4 - Nov 2003 - Q7
A tractor A starts from rest and travels along a straight road for 500 seconds. The velocity-time graph for the journey is shown above. This graph consists of three straight line segments. Find
- the distance travelled by A,
- the initial acceleration of A.
Another tractor B starts from rest at the same instant as A, and travels along the same road for 500 seconds. Its velocity t seconds after starting is \((0.06t - 0.00012t^2)\) m s-1. Find
- how much greater B’s initial acceleration is than A’s,
- how much further B has travelled than A, at the instant when B’s velocity reaches its maximum.
9709 P41 - Nov 2019 - Q7
A particle moves in a straight line, starting from rest at a point O, and comes to instantaneous rest at a point P. The velocity of the particle at time t s after leaving O is v m s-1, where
\(v = 0.6t^2 - 0.12t^3\).
- Show that the distance OP is 6.25 m.
On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time t s after leaving O is s m, where
\(s = kt^3 + ct^5\).
\(It is given that the particle passes point P with velocity 1.25 m s-1 at time t = 5.\)
- Find the values of the constants k and c.
- Find the acceleration of the particle at time t = 5.
9709 P42 - Jun 2019 - Q7
Particles P and Q leave a fixed point A at the same time and travel in the same straight line. The velocity of P after t seconds is \(6t(t-3)\) m s-1 and the velocity of Q after t seconds is \((10 - 2t)\) m s-1.
- Sketch, on the same axes, velocity-time graphs for P and Q for \(0 \leq t \leq 5\).
- Verify that P and Q meet after 5 seconds.
- Find the greatest distance between P and Q for \(0 \leq t \leq 5\).