Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The coordinates of points A, B, C and D are as follows. A(-4,3) B(6, -9) C(15,10) D(14, -1) The line L has equation y = 11x-75. The perpendicular b...
Given that y = x² tan(x/2), use calculus to find the approximate change in y as x increases from π/3 to π/3 + h, where h is small.
In this question, all lengths are in centimetres and all angles are in radians. [Figure showing a sector AOB of a circle and a cone formed from it]...
A particle P moves in a straight line. t seconds after passing a fixed point, O, the acceleration of P, a m s⁻², is given by a = t²-2 for 0 ≤ t ≤ 4...
A geometric progression has first term a and common ratio r, where r > 0. The sum of the 2nd and 3rd terms of the progression is 168. The sum of th...
Solve the inequality |5x+2|≥3.
In this question, all lengths are in metres and time is in seconds. A particle P moves in a straight line such that its displacement s from a fixed...
Functions f and g are such that f(x) = 3x / (x+4) for x > 0 g(x)=√(x+2) for x >-2. Solve the equation fg(x) = 1.
A 4-digit number is to be formed using the digits 0, 2, 4, 5, 6 and 8. The 4-digit number must not start with 0. Any digit may be used at most once...
The volume, V, of a sphere is increasing at the constant rate of 2π cm³ s⁻¹. Find the rate of change of the surface area, S, of this sphere when th...
The first three terms of an arithmetic progression can be written as 2ln(x³), 51n(x²), 2ln(x⁷).
The diagram shows part of the curve y = 15/x - 5/x². The curve meets the x-axis at the point A. The curve has a maximum at the point B. Find the ar...
The first three terms, in descending powers of x, in the expansion of (3x²-a)ⁿ (1+2/x²)² can be written as 729x¹² - 972x¹⁰ + bx⁸, where a, b and n ...
The equation of a curve is given by y = 2x² + ax + 14, where a is a constant. Given that this equation can also be written as y = 2(x – 3)² + b, wh...
Show that
Factorise completely the expression 2x³ – 11x² – 20x – 7.
A curve has the equation y = 2x sin x + π/3. The curve passes through the point P(π/2, a).
(a) Sets A and B are such that A = {x : sin x = 0.5 for 0° ≤ x ≤ 360°}, B = {x : cos (x − 30°) = − 0.5 for 0° ≤ x ≤ 360°}. Find the elements of
The figure shows a rectangular metal block of length 4x cm, with a cross-section which is a square of side xcm and area Acm². The block is heated a...
The diagram shows a circle, centre O, radius 4 cm, enclosed within a sector PBCDP of a circle, centre P. The circle centre O touches the sector at ...
Answer only one of the following two alternatives. EITHER
OR A curve has the equation `y = Ae^(2x) + Be⁻ˣ` where `x ≥ 0`. At the point where `x = 0`, `y = 50` and `dy/dx = -20`.
Show that COS X + COS X = 2 sec x. 1 - sin x 1 + sin x
A curve has the equation y = 2x sin x + π/3. The curve passes through the point P (π/2, a).
EITHER: A curve is such that dy/dx = 4x² – 9. The curve passes through the point (3, 1).
OR: A curve has the equation y = Ae²ˣ + Be⁻ˣ where x ≥ 0. At the point where x = 0, y = 50 and dy/dx = -20.
Solve the equation |2x + 10| = 7.
The expression x³ + ax² – 15x + b has a factor x – 2 and leaves a remainder of 75 when divided by x + 3. Find the value of a and of b.
A number, N₀, of fish of a particular species are introduced to a lake. The number, N, of these fish in the lake, t weeks after their introduction,...
Students take three multiple-choice tests, each with ten questions. A correct answer earns 5 marks. If no answer is given 1 mark is scored. An inco...
Find the set of values of m for which the line y = mx – 2 cuts the curve y = x² + 8x + 7 in two distinct points.
A 4-digit number is formed by using four of the seven digits 1, 3, 4, 5, 7, 8 and 9. No digit can be used more than once in any one number. Find ho...
A rectangular sheet of metal measures 60 cm by 45 cm. A scoop is made by cutting out squares, of side x cm, from two corners of the sheet and foldi...
Solve the equation
A plane, whose speed in still air is 250 kmh⁻¹, flies directly from A to B, where B is 500 km from A on a bearing of 060°. There is a constant wind...
Solutions to this question by accurate drawing will not be accepted. The diagram shows a quadrilateral ABCD in which A is the point (1, 4) and B is...
Solve the equation
Answer only one of the following two alternatives. EITHER The functions f and g are defined, for x > 1, by f(x) = (x + 1)² - 4, g(x) = (3x + 5) / (...
OR A particle moves in a straight line so that, at time t s after passing a fixed point O, its velocity is v m s⁻¹, where v = 6t + 4 cos 2t. Find
Solve the equation