Questions and Answers of Cambridge International AS & A Level Further Mathematics

Find the length of the curve y = 1/2ex + 1/2e-x from x = In 2 to x = In 3.
Show thatconverges. co In n CO Σ n=1 n
Given that show thatHence, determine the exact value of I6. 4 sec" xdx %3D
Find, to 1 decimal place, the surface area of revolution formed when y = (x - 2)3 is rotated about the x-axis from x = 3 to x = 4.
Determine whetherconverges or diverges. 00 Σ 1 e" オ=
Evaluate ∫32 1/√x2 – 4 dx.
You are given thata. Show thatb. Find the exact value of I4. 1, = x(1 – 2x*y"dx. 2.x) In
Given that x = 1/3t3, y = 1/2t2 , show that the arc length of the curve can be represented as Hence, determine, to 3 significant figures, the length of the curve for 1 ≤ t ≤ 2. (P+ f) dt.
Using the comparison test, or otherwise, show thatdiverges. 00 1 Σ n=1 Vn? + 3
Find the volume generated when y = 6/√9 – x2 is rotated about the x-axis from x = 1 to x = 2.
Show that the reduction formula for Hence, find I3, giving your answer in terms of e. In "e*dx is I,=e - In-1 3 for n>1 3
Determine whetherConverges or diverges. Σ In n ,3 n=1 77 8
Evaluate ∫10.75 coth 2x dx.
You are given that Hence, determine the value of I5, giving your answer correct to 4 decimal places. In ( In x)"dx. Show that I, = 2(In 2)" – nl-1 for n> 1
The polar curve C is given as r = θ. Find, to three decimal places, the length of the curve from θ = 0.1 to θ = 0.3.
DoesConverge or diverge? 3n? + 4 Σ 2n² 2n2 + 3n + 5 n=1
Determine the integral dx. V4 - 9x2
Find the exact volume generated when sin3x is rotated about the x-axis from x = 0 to x = π/2.
Evaluate whetherConverges or diverges. 00 Σ 3" 2" + 5" =1
Integrate x cosh x.
The curve y = sinh s is rotated about the x-axis from x = 0 to x = 1. Determine the volume generated.
Determine ∫sinh-1 x dx.
The curve C is given as y = 1/2x2.a. Show that the surface area of revolution of the curve about the x-axis from x = 0 to x = I isb. Using the substitution x= tan 0, or otherwise, show that the
A small smooth ball of mass m is travelling on a smooth horizontal floor with speed u. The ball then goes on to strike a fixed smooth vertical wall at an angle θ = 30° to the wall. The coefficient
Two smooth spheres A and B, of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere A has mass m and speed If and sphere B has mass am and
Two particles, P and Q, are resting on a smooth, horizontal surface. P is projected towards Q with velocity 2u and they collide directly. Given that the masses of P and Q are 2m and m, respectively,
A small smooth ball of mass 3m is travelling on a smooth horizontal floor with speed 2u. The ball then goes on to strike a fixed smooth vertical wall at an angle θ = 60° to the wall. The
Two particles, P and Q, are resting on a smooth surface. They are connected by a light string of length 4m. The particles are of mass 2m and 3m, respectively, and are resting 2m apart. Q is projected
Two perfectly elastic small smooth spheres A and B have masses 3m and m respectively. They lie at rest on a smooth horizontal plane with B at a distance a from a smooth vertical barrier. The line of
Three rail carriages are on a smooth, horizontal rail track. They are equally spaced apart. The carriages are labelled A, B and C, and their masses are 2m, 6m and 3m, respectively. Carriage A is
Three small spheres, A, B and C, of masses m, km and 6m respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with B between A and C. The coefficient
Two identical, 1 kg, smooth spheres, A and B, of equal radius, are travelling on a smooth surface. The velocity of A is 3ms-1 and the velocity of B is 2ms-1. Given that the spheres are moving in
A particle is travelling on a smooth, horizontal surface with velocity 3u. The particle collides with a smooth, vertical wall at an oblique angle of 30° to the wall. Find the angle between the path
A light, elastic string, of natural length 2m and modulus of elasticity 40N, has one end attached to a ceiling at the point O. A particle of mass 0.5 kg is attached to the other end of the string.
A particle, P, of mass 4m, is projected with velocity 2u towards another particle, Q, of mass 5m, which is at rest. Both particles are on a smooth, horizontal surface. The coefficient of restitution
Two smooth, identical spheres, P and Q, are resting on a smooth, horizontal surface. The masses of the spheres are 3m and m, respectively. P is projected towards Q with velocity u.a. Given that the
The diagram shows a lamina that consists of a uniform semicircular plate of radius a attached to a uniform rectangular plate of dimensions a and 2a.The density of the rectangular section is twice the
Two particles, P and Q, are resting on a smooth, horizontal surface. Their masses are 2 kg and 3 kg, respectively. Particle P is projected towards Q with velocity 5ms-1. It subsequently strikes Q and
Three particles are lying in a straight line on a smooth, horizontal surface. The particles are labelled A, B and C, with B between A and C. The masses of the particles are 5 kg, 3 kg and 2 kg,
A particle of mass in is dropped from a great height. The air resistance on the particle is given as inky N.a. Show that dv/dt = g – kv.b. Hence, show that v = g/k(1 – e-kt).c. State the speed of
Two particles, P and Q, are resting on a smooth, horizontal surface. P has mass 2m and Q has mass m. P is projected towards Q with velocity u and Q is projected towards P with velocity 3u. Given that
The diagram shows a smooth sphere, travelling horizontally, that is colliding with two smooth vertical walls at a corner point. The sphere has an initial velocity u, and given that the coefficient of
A smooth hemisphere, of radius 2a, is placed with its plane face on a horizontal surface. A particle, of mass 3m, is placed on the highest point of the hemisphere. The particle is then projected
Two smooth spheres of equal size are resting on a smooth, horizontal surface. Particle P, of mass m, is projected towards particle Q, of mass 2m, with velocity 3u. Particle Q is initially at rest.
Particles P and Q are resting on a smooth, horizontal surface. A smooth, vertical wall is at a distance 2r from Q, and the wall is perpendicular to the line through P and Q. The masses of P and Q are
A particle is projected from the top of a cliff that is 25 m above the sea below. The speed of the projection is 40ms-1 and the angle of elevation is 15°. The particle travels as a projectile and
Three identical smooth spheres, A, B and C, of masses 2m, M and m, respectively, are resting on a smooth, horizontal surface. The coefficient of restitution between A and B is 4/5, and the
Two particles, P and Q, of masses 2m and 3m, respectively, are resting on a smooth, horizontal surface. Particle P is projected towards Q with speed u. It strikes Q directly and the coefficient of
Two spheres, P and Q, which are of equal radii, are placed on a smooth, horizontal surface. Their masses are in and 2m, respectively. Initially, sphere Q is at rest and sphere P is projected with
A particle, P, of mass 3.5 kg is attached to one end of a light, elastic string, of natural length 1.2 m. The other end of the string is attached to a fixed point, 0. The particle is allowed to rest
A light, elastic string of natural length 2.4 m is stretched between two points, A and B, on a horizontal ceiling. The distance AB is 4m. The modulus of elasticity of the string is 60 N. A particle
In each of the following cases, integrate the acceleration expression using a = v v to obtain an expression dx for v.a. a = x2 + 3 b. a = e2x - xc. a = 3 -1/x2
A particle P starts from rest at a point O and travels in a straight line. The acceleration of P is (15 - 6x) ms-2 where xm is the displacement of P from O.i. Find the value of x for which P reaches
A particle P of mass 0.25 kg moves in a straight line on a smooth horizontal surface. P starts at the point O with speed 10ms-1 and moves towards a fixed point A on the line.At time is the
Sketch the curve in question 4.Question 4A Cartesian equation is given as x3 – y + y2x = 0. Find the polar equivalent.
For each of the tests in question 4a-c, state E(S) and Var(S) and hence the value of the test statistic, when approximating to the normal distribution.Question 4a-ca. H0: The population median is
A particle P is projected with speed 35ms-1 from a point O on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of P from O are xm and ym,
A light elastic string of natural length I and modulus of elasticity λ is stretched by a force T, causing the string to extend. In each of the following cases, work out the unknown value.a. T = 15N,
A light elastic string of natural length land modulus of elasticity 2 is stretched by an extension x, by means of a force T, causing the string to gain elastic potential energy (EPE). Find the
A particle of mass 0.4kg is hanging from a light elastic string. The string has natural length 2m and modulus of elasticity 100N. It is held at rest with the string extended by a total of 1.2m. If
A particle P of mass 0.28 kg is attached to the mid-point of a light elastic string of natural length 4m. The ends of the string are attached to fixed points A and B which are at the same horizontal
A light string of unknown natural length is fixed to a ceiling at one end, with a mass of 0.4 kg attached to the other end. The string is allowed to hang in equilibrium such that the length of the
Prove, using integration, that the elastic potential energy stored in a string of natural length land modulus of is elasticity λ is λx2/2I, where x is the extension of the string.
A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1.5m and modulus of elasticity 50N. The other end of the string is attached to a fixed ceiling and the
A particle P of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m. The ends of the string are attached to fixed points A and .8 which are 4.8m apart at the
A light string is attached to a ceiling at the point A. The natural length of the string is 1.3 m and the string has a mass of 3m attached to the free end. The string is allowed to rest in
A particle of mass 5 kg is attached to one end of a light elastic string of natural length 0.8 m. The other end of the string is attached to a fixed ceiling and the particle is allowed to rest,
A particle, P, of mass 3 kg is attached to one end of a light, elastic spring. The other end of the spring is attached to a ceiling at the point A. The spring has natural length 1.6m and modulus of
The ends of a light elastic string of natural length 0.8m and modulus of elasticity 2N are attached to fixed points A and B which are 1.2m apart at the same horizontal level. A particle of mass 0.3
A light, elastic string is attached to a ceiling at the point A. The natural length of the string is 1.6m and the string has a mass of 2m attached to the other end. The string is allowed to rest in
A particle of mass 3 kg is attached to one end of a light, elastic string of natural length 1.2m. The other end of the string is attached to a fixed ceiling and the particle is allowed to rest,
A light, elastic string, of natural length 2m and modulus of elasticity 50N, is attached to two points, A and B, by its opposite ends. The points A and B are at the same horizontal level and they are
Two points, A and B, lie 3m apart on a smooth horizontal surface. A light, elastic string, of natural length 0.8 m and modulus of elasticity 70N, is attached to A. Its other end is attached to a
The points A and B are 4.5m apart, with A vertically above B. Particle P, of mass 2 kg, is connected to A and B by means of two light, elastic springs. The spring attached to point A has natural
One end of a light, elastic string, of natural length 1.2m and modulus of elasticity 32N, is attached to a fixed point, B. A particle, P, of mass 1.5 kg, is then attached to the other end of the
Two fixed points, A and B, are such that A is 4m vertically above B. A light spring of natural length 1.5 m is attached to A, and at the other end it has a particle of mass 6 kg attached. A second
A light elastic string has one end attached to a ceiling. The other end is attached to a particle of mass 3m. The string has natural length 2a and modulus of elasticity 6mg. The particle rests in
A rough, inclined plane has a string attached to it at the point C. The string has natural length 1.4m and modulus of elasticity 80N. The string is then attached to a particle, P, of mass 4 kg. P is
A light, elastic string, of natural length 1.8 m and modulus of elasticity 45 N, is attached to a ceiling at point G. The lower end of the string is attached to a particle, P, of mass 1.8 kg. A
A particle, P, of mass 4 kg is resting on a rough, horizontal table. A light, elastic string, of natural length 2 in and modulus of elasticity 50 N, is attached to P. It is then passed over a smooth
A light, elastic string, of natural length 3/2 a and modulus of elasticity 2mg, is attached to a ceiling at the point A. The other end is attached to a particle, P, of mass 5m. A second light,
A particle, P, of mass m, is moving in a horizontal circle, having centre O, with angular speed √g/4a. The particle is attached to one end of a light, elastic string of natural length 3a and
A particle of mass 2 kg is being held in equilibrium on a smooth slope by a horizontal force, P, and a light, elastic spring. The spring has modulus of elasticity ION and is attached to the particle
A particle, P, of mass 3 kg, rests on a rough, horizontal table, where µ = 2/3. P is 3 attached to a light, elastic string of natural length 2in and modulus of elasticity 49 N. The string passes
A particle of mass km is placed on top of a light, vertical spring of natural length 2a. The modulus of elasticity of the spring is 3mg and the spring is standing upright so that it lies in a
In each case you are given the displacement function of a particle. Find the velocity and acceleration at the time given.a. x = 3t2 + 5t3, t = 2 b. x = e2t - 5t, t = 0 c. x = 5ln(t + 1) +
A force of magnitude 12N is applied to a particle of mass 0.3 kg, resting on a smooth horizontal surface, for two seconds. Find the speed of the particle after the two seconds.
A particle of mass 2kg is resting on a smooth horizontal surface. A force of magnitude 4xN is applied to the particle. Find the speed of the particle after it has travelled 6m.
A cyclist and his bicycle have a total mass of 81 kg. The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of 135 N and the motion is opposed by a resistance
A particle of mass 0.5 kg is travelling on a smooth horizontal surface with a constant speed of 4ms-1. A force of magnitude kt N is applied to the particle to slow it down. Given that the particle
A particle is travelling in a straight line such that its acceleration is 3x ms-2. Find v = f(x), given that when v = 5ms-2, x = 0m.
A particle is travelling with velocity function v = 1/t2 ms-1. Given that x = 2m when t = 1 s, find x = f(t). What does this tell you about the displacement of the particle?
A force of magnitude x/x + 3 N is used to drive a car of mass 1200 kg. The car starts from rest. Find the velocity of the car when x = 6m.
A force of magnitude (4t + 3) N is applied in the direction of motion of a particle of mass 5 kg. The particle travels in a straight line. Given that the particle is already travelling at 5ms-1 when
A particle moves along the x-axis with acceleration 3e-2x ms-2 directed towards the origin, O. Given that P passes through O with speed 5 ms-1, find an expression for the velocity in terms of x and,
An opposing force of magnitude 2t2N is applied to an object of mass 2 kg. At the time when the force begins to act, the object is already travelling at 3ms-1 and is passing through the point O. Find
A particle of mass 0.25 kg is travelling in a straight line at 6ms-1 when it passes through a point, O. An opposing force of magnitude x/x2 + 1 is then applied to the particle. Find the speed of the
A particle is travelling in a straight line with displacement function x = te-tm. Determine the time when the velocity is at a maximum.
A particle is travelling in a straight line with acceleration (3 + 2x) ms-2. As it passes through a point. O, its velocity is 2ms-1. Find v2 in terms of x.
A truck of mass 12000 kg is driving at a constant speed of 15 ms-1. The truck driver sees a red traffic light 100m ahead and applies the brakes. This produces a braking force of magnitude 300t2N.

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