Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The diagram shows a triangle OAB where OA = OB = 10cm and angle AOB = 0.8 radians. Points C and D on OA and OB respectively are such that the arc C...
An arithmetic progression has first term 5 and common difference 6. For this progression, find the sum of all the terms that lie between 150 and 400.
The diagram shows a circle C of radius r, where x > 0 and y > 0 for all points on C. The least distance between any point on C and the x-axis is 8 ...
A geometric progression is such that its second term is – 120 and its sum to infinity is 160.
A curve is such that d²y/dx² = 6/x⁴ - 5/x³. It is given that the curve has a stationary point at (1/2, 9).
The diagram shows the curve with equation y = 4(3x+4)¹ᐟ² - 2x - 6 for values of x such that 0 ≤ x ≤ 7. The tangent to the curve at the point P (7,0...
Functions f and g are defined for all real values of x by f(x) = 4x² - c and g(x) = 2x+k, where c and k are positive constants. It is given that g⁻...
Solve the equation ln(3x+1) – ln(x - 5) = ln 7.
A curve passes through the point with coordinates (½π, 5) and is such that dy/dx = 4 sec²(½x). Find the equation of the curve.
The diagram shows the curves y = e^2x and y = 8e^-x. The shaded region is bounded by the two curves and the y-axis. [Figure]
A curve has equation y = 4 sinx / (3 + cos 2x) for values of x such that 0 ≤ x ≤ 2π.
A curve has equation 3e^2xy + 4e^3x + y³ = 18.
Solve the equation In(1-e-2x) + 3 = 0. Give your final answer correct to 4 decimal places.
The equation of a curve is xy² +ln(x+2y) = 1. Find the gradient of the curve at the point where x = 0.
The shaded region on the Argand diagram shows points representing complex numbers z defined by two inequalities. The shaded region is bounded by a ...
By first expressing the equation tan(x-60°) = 2cotx as a quadratic equation in tanx, solve the equation for 0° ≤ x ≤ 180°.
The square roots of -4+6√5i can be expressed in the Cartesian form x+iy, where x and y are real and exact. By first forming a quartic equation in x...
The variables x and θ satisfy the differential equation dx/dθ = (½x+1)sin²2θ, and x = 5 when θ = 0. Solve the differential equation and obtain an e...
The diagram shows the curve y = x³cos2x for 0 ≤ x ≤ ¼π. The curve has a maximum point at M, where x = p. [Figure 7.1]
Two lines have equations r = (-1 3 -4) + λ(2 3 -1) and r = (2 -3 -1) + μ(-1 -2 1).
The polynomial 6x³ + ax² + bx + 9 is denoted by p(x), where a and b are constants. It is given that (x-3) is a factor of p(x), and when the first d...
Let f(x) = (-7x²+2x-6)/((1+x)(4+x²))
Find the exact value of ∫[from 0 to π/2] x² cos(x/3)dx.
Three coplanar forces of magnitudes 40N, 30N and X N act at a point in the directions shown in the diagram. [Figure X.X]
A cyclist is travelling along a straight horizontal road at a speed of 4 ms⁻¹ when she passes a point O. She accelerates at a constant rate for a d...
An aeroplane is flying at a constant speed.
Two particles A and B have masses 0.3kg and 0.1kg respectively. The particles are attached to the ends of a light inextensible string. The string p...
Three particles P, Q and R, of masses 0.6kg, 0.4 kg and 0.8 kg respectively, are at rest in a straight line on a smooth horizontal plane. The dista...
A block of mass 12kg is placed on a rough plane inclined at an angle of α to the horizontal, where α = tan⁻¹0.5. A force of X N is applied to the b...
A particle moves in a straight line. The velocity v ms⁻¹ of the particle t s after leaving a fixed point O is given by v = k(20+pt-6t²), where k an...
Jacob throws three coins at the same time. The first coin is biased so that the probability of obtaining a head when it is thrown is 1/3. The secon...
Last year, an online store sold a large number of computers. 55% of the computers were made by company F, 30% were made by company G and 15% were m...
The lengths of 250 leaves of a certain type of plant are measured, correct to the nearest centimetre. The results are summarised in the table below...
Eddie has 16 toy cars, of which 8 are white, 5 are black and 3 are silver. He places all the cars in a bag and selects three of them at random, wit...
The mass of peaches sold per day in a supermarket is normally distributed with mean 65.8kg and standard deviation 9.6kg.
Alissa has 10 different books from the series Squares and Circles. The books look similar except for their colour. There are 3 blue books, 2 red bo...
The random variables X and Y have the independent distributions N(44, 16) and N(30, 9) respectively. Find P(X-Y < 15).
A researcher records the time, T seconds, taken by adults to complete a questionnaire. The results for a random sample of 60 adults who completed t...
The random variable X has the distribution Po(1.5).
The diagram shows the graph of the probability density function, f, of a random variable X. The graph is a straight line from (0, a) to (2, b), whe...
Amir believes that 20% of the students at his college are left-handed. His friend believes that the true proportion, p, is less than 20%. Amir plan...
Nikki is investigating the views of students at her school about the school sports facilities. She plans to give a survey to a sample of students. ...
(i) Show that the equation 3(2 sin x − cos x) = 2(sin x – 3 cos x) can be written in the form tan x = -¾. (ii) Solve the equation 3(2 sin x − cos x...
The diagram shows part of the curve y = a/x, where a is a positive constant. Given that the volume obtained when the shaded region is rotated throu...
The functions f and g are defined for x ∈ R by f : x → 4x - 2x², g : x → 5x + 3. (i) Find the range of f. (ii) Find the value of the constant k for...