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

A particle has velocity v = kx sin x ms-1, where x is the distance from the point O. The particle travels in a straight line. Given that the particle has velocity v = 2m s-1 when x = 3π/2 m, find
A particle travels in a straight line with velocity v = 2 + sin tms-1. It passes through the point O when t = 0s. Find the displacement from O after four seconds.
A particle is travelling along a straight line with acceleration 4 cos x/4 ms-2. As it passes through the point where x = 0m its velocity is v = 7 ms-1.a. Find an expression for the velocity in terms
A ball of mass mkg is dropped from a very high tower. Due to air resistance the ball is subject to an opposing force of magnitude mkv N, where k is a constant and v is the velocity of the ball. Show
A particle of mass 2kg experiences a force of magnitude (3x – 1/x2) N being applied in the same direction as the particle's motion. When x = 1 m, is v = 4 ms-1. Find is = f(x).
A particle passes a point, O, with speed 12 ms-1, travelling in a straight line. The acceleration of the particle is -41ms-2. Find:a. The time taken for the particle to be at instantaneous restb. The
A toy rocket, of mass 1 kg, is modelled as a particle. It is launched from rest using its engines, which produce a force of size (20 - t) and have enough fuel for five seconds. After five seconds the
A discrete random variable, X, has the probability distribution function:a. Find the value of k and the probability generating function.b. Find E(X).c. Find Var(X). k P(X = x) = er x = 0, 1, 2, 3,
A uniform beam AB has length 2m and weight 70N. The beam is hinged at A to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached
Show that the centre of mass of this uniform lamina, in the shape of a trapezium, is given by: ah + 2bh a+ ab + b? 3a + 3b 3a + 36
In each of the following, use the relation v = rω to determine the unknown value.a. v = 4, r = 6. Find ω.b. v = 6, ω = 3, Find r.c. r = 5, ω = 0.8. Find v.
A particle of mass 0.9 kg is attached to a light inextensible string of length 2.4 m. The other end of the string is attached to a fixed point on the ceiling. The particle is describing horizontal
A particle of mass m is attached to the end of a light inelastic string of length 1.5a. The other end of the string is attached to a fixed point O. When the particle is resting in equilibrium, it is
One end of a light inextensible string is attached to a fixed point A and the other end of the string is attached to a particle P. The particle P moves with constant angular speed 5 rad s-1 in a
A particle of mass 0.2 kg is attached to a light inextensible string of length 1.5 m. The particle is moving in horizontal circles. Find the tension in the string if the speed of the particle is
A particle of mass 1.6 kg is attached to a light inextensible string of length 1.4 m. The other end of the string is attached to a fixed point on the ceiling. The particle is describing horizontal
A particle of mass in is attached to the end of a light inelastic string of length 1.2a. The other end of the string is attached to a fixed point O. When the particle is resting in equilibrium, it is
A smooth bead B of mass 0.3 kg is threaded on a light inextensible string of length 0.9m. One end of the string is attached to a fixed point A, and the other end of the string is attached to a fixed
A particle of mass 0.8 kg is attached to a light inextensible string of length 1.2m. The particle is moving in horizontal circles. Find the angular speed of the particle if the tension in the string
A particle of mass 2 kg is attached to a light, inextensible string of length 3 m. The other end of the string is attached to a fixed point on the ceiling. The particle is describing horizontal
A smooth hemisphere of radius 1.6a is placed plane face down and fixed onto a horizontal plane. A particle r of mass 2m is placed on the top of the hemisphere and projected with speed √1/4ga. As
A particle P of mass m is projected horizontally with speed √7/2ga from the lowest point of the inside of a fixed hollow smooth sphere of internal radius a and centre O. The angle between OP and
A particle is describing horizontal circles on a smooth, horizontal table. The particle is fixed to its circular path by an inelastic string of length 1.5 m. Given that the tension in the string is
A car is driving around a banked road inclined at 15° to the horizontal. The car has mass 1200 kg and the radius of the circular part of the road is 60m. The coefficient of friction between the road
A particle of mass 2m is attached to the end of a light, inelastic string of length a. The other end of the string is attached to a fixed point, O. When the particle is resting in equilibrium, it is
A particle is placed on a rough horizontal disc, 2a from the centre of the disc. The particle is of mass 3m.The disc then starts to spin at an angular speed of √g/4a. Given that the particle is on
A particle, P, of mass 3 kg is attached to two light, inextensible strings. One string is attached at its other end to a point, A. The other string has its other end attached to a point, B. A is 4m
A particle of mass m is attached to a light, inextensible string of length a. With the other end of the string attached to a fixed point, A, the particle rests in equilibrium. The particle is then
A car is driving on a horizontal circular section of road that has radius 50m. The car is of mass 800 kg and the coefficient of friction between the tyres and the road is 0.8. Find the maximum speed
A hemispherical bowl of radius a is resting in a fixed position where its rim is horizontal. A small ball of mass 2m is moving around the inside of the bowl such that the circle described by the ball
A particle of mass 3 kg is resting on the inside of a smooth, circular hoop of radius 2 m. From the bottom position, the particle is projected horizontally with speed 8ms-1. Find the greatest height
A particle of mass 1.5 kg is resting on a smooth horizontal table. The particle is attached to a light, inextensible string of length 2.5 m. This string is passed through a smooth hole in the table.
A light, inextensible string of length 4m is threaded through a smooth ring at the point O. One end of the string has a particle, P, of mass 4 kg, which is 1.2 m vertically below O. Particle Q, of
A smooth hemisphere, of radius I.5a, is placed plane face down and fixed onto a horizontal plane. A particle, of mass m, is placed on the top of the hemisphere and projected with speed √3/4ga. As
A particle is describing horizontal circles of radius a; the mass of the particle is m. Given that the particle can complete one circle in r seconds, and that the particle is held in its path by
A car is moving on a circular section of road where the road is banked at 25° to the horizontal. The radius of this section of road is 100m. The car has mass 1400 kg and is travelling at 30ms-1.
A particle of mass m is resting on the inside of a smooth, circular hoop of radius 3a. The particle is then projected from the lowest point, with horizontal speed u.a. State the minimum speed
A particle is placed on the inside of a rough, hollow cylinder of radius 2a. The cylinder is placed upright and 43 can rotate about an axis through its centre. The particle has mass in and the
A toy plane of mass 0.4kg is attached to one end of a light, inextensible string of length 6m. The other end of the string is attached to the point 0. The string is taut and makes an angle of 45°
A particle is held at rest on a smooth, circular track, which is an arc of radius a. The track is standing in a vertical plane, and it is fixed to a horizontal surface. The points O and A are such
A particle, P, is placed on a large, rough, horizontal disc. P is of mass 2.4 kg. The particle is then attached to a light, inextensible string of length 4 m. The string passes through a smooth hole
A string of length 2a is attached to a point O and has a particle of mass 3m attached to the other end. The particle is resting in equilibrium. The particle is then projected with horizontal speed
The diagram shows two uniform solid cylinders. The larger cylinder has density ρ and the smaller cylinder has density kρ. The cylinders are joined together by the faces of their planes, and their
A particle is projected from ground level with initial speed 30ms-1 at an angle of elevation of 25°. Find the horizontal distance travelled from the starting point when the height is 4m for the
A particle is projected from point A on horizontal ground with initial speed 25 ms-1 and angle of elevation 8. Given that the range of the particle is 60m, find the value of the angle θ.
A particle is projected from a point 5m above a horizontal plane. The angle of elevation is 10°. Given that the particle travels 75 m before hitting the ground, find the initial speed.
A football player kicks a football from a point on the ground with an angle of elevation of 15°. The ball must land on horizontal ground level between 10m and 20m from the player's feet. Find the
A particle is projected from a platform 4m above horizontal ground, with an initial speed of 20 ms-1 and an angle of elevation of 20°. Find the direction of the particle as it lands on the ground.
A basketball is thrown with speed 10ms-1 from a point 2m above the ground, at an angle of 45° above the horizontal. Find the speed of the basketball when it is at a height of 4m for the second time
A ball is kicked at a house window. The window is 4m up a vertical wall from a horizontal floor. The ball is kicked from a position that is 12 m distance from the foot of the wall. If the ball enters
A particle of mass 2 kg is fired up a smooth slope of length 4m, with initial speed 10ms-1, which is inclined at 30° above the horizontal. The bottom of the slope is at the same level as horizontal
A particle is projected from the top of an office building that is 35m tall. The initial speed is 14 ms-1 and the angle of depression is 20°. Find the height of the particle above the ground
A particle is projected horizontally from a point that is 12m above a horizontal surface, with a speed of 15ms-1. Find the horizontal distance travelled before the particle hits the surface below.
A particle is projected from a point on a horizontal surface with initial speed u and angle of elevation At any time during its motion, the particle is at the position (x, y), where x is the
A particle is projected across a horizontal area of land, with initial speed 30ms-1 and inclined at an angle of 40°. Find the duration of time for which the particle is at least 10 m above the
A particle is projected from a point on a horizontal plane with speed 20ms-1 at an angle of elevation of θ. Given that the particle passes through the point x = 20, y = 10, find the possible angles
A small stone is thrown from the top of a building that is 30m tall. The stone is given an initial speed of 5ms-1, and it is directed downwards with an angle of depression of 15°.a. Find the time
A particle is projected from a point on horizontal ground with initial speed u and angle of elevation α. Show that x2 + y2 = u2t2 - 10yt2 - 25t4, where (x, y) is the particle's position at time t.
A rod AB of length 3.1 m and mass 8 kg is balancing on a pivot at a point C, where AC is 1.2m. It is kept balanced by masses being placed at A and B. If the mass at A is 4kg, determine the mass at B.
The diagram shows a lamina that is formed by removing a small rectangle from a larger rectangle. Find the distance of the centre of mass from AB and from AC. B 4a 2a Sa За C
A uniform solid cylinder, of radius r and length 4r, has a uniform solid hemisphere of radius r, of the same material, attached to one of its plane faces. The plane faces of each solid coincide with
A uniform solid cylinder, of radius r and height 5r, is suspended from a point on the rim of its plane face. It is allowed to rest in equilibrium. Find the angle between the plane face of the
A rod AB of length 1.4m and mass 6 kg is balancing on a pivot at a point C, where AC is 0.5m. It is kept balanced by masses being placed at A and R. If the mass at A is 3 kg, determine the mass at B.
The diagram shows a uniform lamina in the shape of a trapezium. Find the distance of the centre of mass from edge AB and from edge AC. B 4.5 A 12 C
Two uniform cones with base radius r are joined together by their plane faces. Their lines of symmetry are aligned. The height of one cone is 6r and the height of the other cone is 2r.Given that the
A uniform solid cylinder, of radius 2r and height 7r, is resting on a sufficiently rough slope. The slope is inclined at an angle α. Find the maximum value of α such that the cylinder is on the
A uniform rod AB has weight 6N and length 0.8 m. The rod rests in limiting equilibrium with B in contact with a rough horizontal surface and AB inclined at 60° to the horizontal. Equilibrium is
A rod AB, of length 1.1 m and mass 4kg, is resting on a horizontal table with part of the rod hanging over the edge of the table. The rod is perpendicular to the edge of the table. Point A is in
The diagram shows two uniform laminas, each a right-angled triangle, that are joined together at one edge, AB. The smaller triangle is twice as dense as the larger triangle. Find the distance of the
Find, by using integration, the centre of mass of a solid hemisphere of radius 2r, measured from its plane face.
A ladder of length 4a is placed such that it rests against a smooth vertical wall and stands upon a rough horizontal floor. The angle between the ladder and the wall is 30°. The ladder has mass 2m.
The diagram shows the cross-section OABCDE through the centre of mass of a uniform prism on a rough inclined plane. The portion ADEO is a rectangle in which AD = OE = 0.6m and DE = AO = 0.8m; the
Find the moment about 0 of the forces shown, stating if it is clockwise or anticlockwise.a.b.c. 3 N 2m -3.4m 2N 5N
The image shows a uniform lamina that is formed by removing a square from a right-angled triangle. Find the coordinates of the centre of mass, as measured from the point O. 2a 2a 2a 4u
A uniform solid cylinder has length 4a and radius 2a at each end. Centred on the plane faces are points A and B, respectively, such that AB is 4a. At the plane face B, a hemisphere, of radius 2a and
A solid uniform cone, of base radius 2a and height 5a, is suspended by a point, B, on the rim of its circular base. The centre of the circular base is denoted by C. Find the angle BC makes with the
For each of the following rods, find the moment about the point P. a.b.c. A8N 2NA 2m 2.5m 3m ¥3.5N
The diagram shows a uniform square lamina of side 4r and density 2p attached to a uniform lamina in the shape of a quarter circle of radius 4r and density p. Find the distance of the centre of mass
A uniform solid cone, C1, of base radius 1.5r and height 4r, is connected to another uniform solid cone, C2, of base radius 1.5r and height r. Given that the cones are connected by the faces of their
The diagram shows a uniform lamina in the shape of a trapezium. The lamina is suspended from the point A. Find the angle between the vertical and the edge AB. За 2a A 4а
A rod, AB, of length 0.4m and mass 5 kg, is resting such that part of the rod is hanging over the edge of a horizontal table. The rod is positioned so that it is perpendicular to the edge of the
A piece of uniform wire is bent to form the letter D. This letter D consists of a straight edge of length 2m, and a semicircle of radius 1 m. The letter D is held upright with the straight edge in
A uniform ladder, of length 2a and mass in, is resting against a smooth vertical wall and a rough horizontal floor. The ladder is making an angle of 30° with the wall, and the coefficient of
A toy is constructed by joining a hemisphere to a cone by their plane faces. Both the hemisphere and cone have the same radius, r, and the cone has height. 10r. Given that the density of the cone is
A rod, AB, of length 5m and mass 12 kg is hanging over the edge of a cliff. The end A is 1.2m from the edge of the cliff, and the rod is assumed to be perpendicular to the cliff. A woman, of mass M,
A uniform lamina is made from a square of side 2a joined to a semicircle of radius 2a. This semicircle is then joined to another semicircle of radius 2a, which is in turn joined to a smaller
The diagram shows a cylinder with a hemisphere removed from one end, and a cone attached to the other end. Each part is solid and its mass uniformly distributed. The density of the cone is twice that
The diagram shows a uniform rod, of mass in and length 2a, smoothly hinged to a vertical wall. A light, inelastic string connects the rod to a point on the wall above the hinge. Find the magnitude
In each case, find the unknown such that the total moment about O is zero.a.b.c. 43° xN 7N 2m 2m o 1m SN 48
A solid uniform cone, of base radius r and height 6r, has a similar smaller cone of height 2r removed from the top to form a frustum. This frustum is placed, with its larger plane face, on a rough
A solid uniform hemisphere, of radius r, is placed onto a rough plane inclined at 45° to the horizontal. A force, P, parallel to and up the plane, is applied to the hemisphere at a point that is r/2
In a game a ball is kicked from the point 0 on horizontal ground, so that it lands on a scoring area which extends from 20m to 25m from O. If the angle of elevation when kicked is 35°, find the
A ball is thrown from a point 2m above horizontal ground. The initial speed is 20 ms-1 and the angle of elevation is 45°. Find the horizontal distance covered when the ball is 12m above the ground.
A particle is projected from horizontal ground with initial speed 15 m-1 and angle of elevation 60°. Find the horizontal distance travelled when the particle is first at a height of 8m above ground
A projectile is launched from a point on horizontal ground with speed U. Given that the horizontal distance travelled before hitting the ground again is 140m, and that the angle of projection is 25°
A particle P is projected from a point O with initial speed 10ms-1 at an angle of 45° above the horizontal. P subsequently passes through the point A, which is at an angle of elevation of 30° from
A projectile is launched from point 0 on horizontal ground, landing at the point A.a. OA = 205m, 0 = 40°. Find the speed u.b. θ = 30°, u = 32ms-1. Find the range OA.c. OA = 130m, u = 40m s-1. Find
A particle is projected from a point O on horizontal ground. The velocity of projection has magnitude 20ms-I and direction upwards at an angle θ to the horizontal. The particle passes through the
The Tax Bureau claims that people typically take 140 minutes to fill in a tax form. A researcher believes that this claim is incorrect and that, generally, it takes people longer to complete the
It has been found that 60% of the computer chips produced in a factory are faulty. As part of quality control, 100 samples of 4 chips are selected at random, and each chip is tested. The number of

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