Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The variables x and y satisfy the differential equation (x²+1) dy/dx = kxe^(-2y), where k is a constant. It is given that y = 0 when x = 0 and that...
Let f(x) = (x²+2x-11) / ((3+x)(2+x²)).
With respect to the origin O, the points A, B, C and D have position vectors given by OA = (1 5 3), OB = (0 4 1), OC = (1 -3 1) and OD = (3 -5 4). ...
Solve the equation 3 × 2x+1 = 4 × 32x-3. Give your answer correct to 3 significant figures.
The shaded region in the Argand diagram, bounded by a line and a circle, represents the complex numbers z satisfying Rez < 2 and |z-(3+i)|≤2. The p...
Find the exact value of ∫(from 0 to 1) x tan⁻¹x dx.
The equation of a curve is 2y³ – 3x²y-x³ = 16.
Let f(x) = (x²+4ax+6a²) / ((x+2a)(x+3a)), where a is a positive constant.
The diagram shows a tank for holding water. The tank is in the shape of a cube of side 50 cm. At time t seconds, the depth of water in the tank is ...
Relative to the origin O, the position vectors of the points A, B and C are OA = 4i−2j, OB = 2i+8j+4k, and OC = -2i+6k. The midpoint of AB is M, as...
Solve the inequality | 3x+2|<3|2x-1|.
Find the quotient and the remainder when 3x⁴ – 2x² is divided by x + 1.
Solve the equation 2^(3x-4) = 3/5^x. Give your answer in the form (ln m)/(ln n), where m and n are integers.
Find the exact value of ∫(from 0 to π/2) x² sin 2x dx.
Solve the equation (5z)/(2-i) - zz* + 20+8i = 0. Give your answers in the form x+iy, where x and y are real.
The curve with equation y = e⁻⁵ˣln 5x has a stationary point at x = p.
The line l₁ passes through the point (3, 1, −6) and is parallel to the vector 2i+j+4k. The line l₂ passes through the point (−1, 3, −6) and is perp...
The diagram shows the graph of y = sec²x√(3+2 tanx) for -¼π ≤ x ≤ ¼π, and its minimum point M. [Figure 11.1]
A car of mass 900kg is moving along a straight horizontal road against a constant resistance to motion of 350N. At an instant when the car is movin...
Two particles, P and Q, of masses 3 kg and 5kg respectively, are at rest on a smooth horizontal plane. P is projected at a speed of 4 ms⁻¹ directly...
Coplanar forces of magnitudes 45N, 28N, 72N and 35N act at a point in the directions shown in the diagram. [Figure 3.1] Find, in either order, the ...
Two particles, A and B, of masses mkg and 3kg respectively, are connected by a light inextensible string. Particle B is on a fixed plane which is a...
A particle P of mass mkg is in equilibrium on a rough plane inclined at an angle θ° to the horizontal. The equilibrium of P is maintained by a forc...
A particle A of mass 2.5kg is released from rest from the top of a smooth plane, which makes an angle of sin⁻¹ 0.2 with the horizontal. The particl...
A particle P starts from a point O and moves in a straight line. The velocity v ms⁻¹ of P, at time t s after leaving O, is given by v = ¹/₃(2t-3)(t...
A block A of mass 2kg and a particle B of mass 0.5kg are connected by a light inextensible string inclined at 15° to the horizontal. They are pulle...
The diagram shows the displacement-time graph for the motion of a particle. The particle starts from rest at a point O and travels with constant ac...
A railway locomotive of mass 240000 kg is towing a coach of mass 36000 kg down a hill inclined at an angle of sin⁻¹ 0.04 to the horizontal. The dri...
A car of mass 1400 kg travelling on a straight horizontal road accelerates uniformly from a speed of 15 m s⁻¹ to 20 m s⁻¹ over a distance of 175 m.
A particle P of mass 0.1 kg is projected vertically upwards with speed 30 m s⁻¹ from horizontal ground. At the same instant a particle Q of mass 0....
A block of mass 5 kg is being pulled straight down a line of greatest slope of a rough plane by a force of magnitude 20 N. The plane is inclined at...
A particle starts from rest at a point O. The acceleration of the particle at time t s after leaving O is a m s⁻², where a = 2(t+1)⁻½ - 1 for t ≥ 0...
Four coplanar forces of magnitudes 20 N, 30 N, 15 N and 25 N act at a point P in the directions shown in Fig. 7.1. The forces act in a vertical pla...
The diagram shows the velocity-time graph for the motion of an athlete. The athlete runs in a straight line from point A to point B, runs back to p...
The engine of a motorcycle can generate a maximum power of 105kW. The mass of the motorcycle and its rider is 400kg. The total resistance to the mo...
A block of mass 4kg is pulled along a rough horizontal road by a constant force of magnitude 25 N acting at an angle of 36° above the horizontal. T...
A particle P moves in a straight line. At time ts after passing through a point O on the line, the displacement of P from O is sm, where s = 0.01t³...
Particles A and B, of masses 2kg and 6kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fix...
Particle P of mass m kg and particle Q of mass 2 kg are free to move on a smooth horizontal plane. P and Q are moving directly towards each other w...
A particle P of mass 6 kg lies on a rough plane inclined at an acute angle θ to the horizontal. A horizontal force of magnitude 12N acts on P as sh...
The random variable X takes the value x with probability kx², where k is a constant and x takes the values -2, 1, 2, 3 only.
A fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly until a 6 is obtained.
The back-to-back stem-and-leaf diagram shows the annual salaries, in dollars, of 27 employees at each of two companies, Browns and Greens. Browns ...