Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Last Saturday, Sarah recorded the colour and type of 160 cars in a car park. All the cars that were not red or silver in colour were grouped togeth...
The discrete random variable X has the following probability distribution. X|-2|0|1|3|4 P(X = x)|0.2|0.1|p|0.1|q
A summary of n values of x gave the following information: Σ(x – 20) = 136, Σ(x – 20)² = 2888. The mean of the n values of x is 24.25.
The digits 1, 3, 5, 6, 6, 6, 8 can be arranged to form many different 7-digit numbers.
The weights of packets of a certain type of biscuit are normally distributed with mean 400 grams and standard deviation σ grams.
The results of a survey at a certain large college show that the proportion of students who own a car is 1/4.
The coefficient of x³ in the expansion of (1 – px)⁵ is –2160. Find the value of the constant p.
A curve with equation y = f(x) passes through the points (0, 2) and (3, -1). It is given that f'(x) = kx² – 2x, where k is a constant. Find the val...
In the diagram, CXD is a semicircle of radius 7 cm with centre A and diameter CD. The straight line YABX is perpendicular to CD, and the arc CYD is...
A curve has equation y = (2x – 1)⁻¹ + 2x.
Two vectors, u and v, are such that u = (2) (q) (6) and v = (8) (q-1) (q²-7) , where q is a constant.
The diagram shows part of the curve with equation y = √(x³ + x²). The shaded region is bounded by the curve, the x-axis and the line x = 3. [Figure...
The diagram shows the curve with equation y = 4x^(1/2). [Figure X.X]
Solve the equation sec² θ + tan² θ = 5 tan θ + 4 for 0° < θ < 180°. Show all necessary working.
Given that x satisfies the equation |2x + 3| = |2x – 1|, find the value of |4x – 3| – |6x|.
The variables x and y satisfy the equation y = AePx+P, where A and p are constants. The graph of ln y against x is a straight line passing through ...
The equation of a curve is y = e^(2x) / (4x + 1) and the point P on the curve has y-coordinate 10.
The parametric equations of a curve are x = 2t - sin 2t, y = 5t + cos 2t, for 0 ≤ t ≤ ½π. At the point P on the curve, the gradient of the curve is 2.
The sequence of values given by the iterative formula x_n+1 = (2x_n^6 + 12x_n) / (3x_n^5 + 8) with initial value x₁ = 2, converges to α.
Show that ∫₁⁴ x^(-1/2) ln x dx = 2 – ln 4.
The variables x and y satisfy the relation sin y = tan x, where –½π < y < ½π. Show that dy/dx = 1 / (cos x √(cos 2x)).
The variables x and y satisfy the differential equation dy/dx = ky³e⁻ˣ, where k is a constant. It is given that y = 1 when x = 0, and that y = √e w...
Let f(x) = (12 + 12x - 4x²) / ((2 + x)(3 – 2x)).
Two planes have equations 2x + 3y – z = 1 and x – 2y + z = 3.
The diagram shows the curve y = sin³x √(cos x) for 0 ≤ x ≤ ½π, and its maximum point M. [Figure 10.1]
A small ring P of mass 0.03 kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an...
A particle is projected vertically upwards with speed 30 m s¯¹ from a point on horizontal ground.
Four coplanar forces of magnitudes FN, 5 N, 25 N and 15 N are acting at a point P in the directions shown in the diagram. Given that the forces are...
A car of mass 1500 kg is pulling a trailer of mass 300 kg along a straight horizontal road at a constant speed of 20 m s⁻¹. The system of the car a...
The velocity of a particle moving in a straight line is v m s⁻¹ at time t seconds after leaving a fixed point O. The diagram shows a velocity-time ...
A particle moves in a straight line. It starts from rest at a fixed point O on the line. Its acceleration at time t s after leaving O is a m s⁻², w...
The diagram shows the vertical cross-section PQR of a slide. The part PQ is a straight line of length 8 m inclined at angle α to the horizontal, wh...
A uniform object is made by joining together three solid cubes with edges 3m, 2m and 1m. The object has an axis of symmetry, with the cubes stacked...
A small ball is projected from a point O on horizontal ground. At time ts after projection the horizontal and vertically upwards displacements of t...
A particle P of mass 0.3 kg is attached to a fixed point A by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N. The pa...
A particle P of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N. The other end ...
Fig. 1 shows the cross-section of a solid cylinder through which a cylindrical hole has been drilled to make a uniform prism. The radius of the cyl...
A particle P is projected horizontally from a point O on a rough horizontal surface. The coefficient of friction between the particle and the surfa...
On each day that Tamar goes to work, he wears either a blue suit with probability 0.6 or a grey suit with probability 0.4. If he wears a blue suit ...
For 40 values of the variable x, it is given that ∑(x − c)² = 3099.2, where c is a constant. The standard deviation of these values of x is 3.2.
The times taken, in minutes, for trains to travel between Alphaton and Beeton are normally distributed with mean 140 and standard deviation 12.