Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
10 (a) The first 3 terms of an arithmetic progression are 3 sin 2x, 5 sin 2x, 7 sin2x.
11 (a) Given that y = x²lnx, find dy/dx.
Use algebra to show that the equation 5x(x-3) = 5x-26 has no real solutions.
There are 3 women, 2 men and 4 children in a choir.
Variables x and y are such that y = cosx sin²x. Use differentiation to find the approximate change in y as x increases from 3 to 3+h, where h is sm...
It is given that y = mx² + x/2 + n, where m and n are non-zero constants. It is also given that 3(d²y/dx²) = (dy/dx)² - y for all values of x. Find...
The functions f and g are defined by f(x) = 3x²/(4x-1) for x < 0 and g(x) = 1/x² for x < 0.
In this question all lengths are in centimetres. The diagram shows a rectangle ABCD with BC = x. The area of the rectangle is 400 cm². Two identica...
The diagram shows a triangle OBC. OA : OB = 4 : 7 and OD : OC = 4 : 7. OB = b and OC = c. The point P is the point of intersection of AC and BD suc...
The polynomial p is such that p(x) = 6x³ +x² - 12x+5.
A curve has equation y = 5e^(2x−1)+e. The tangent to the curve at the point where x = 1 cuts the x-axis at the point P. Find the equation of the ta...
Find the number of different ways the 9 letters of the word POLYMATHS can be arranged when
An experiment was carried out and values of y for certain values of x were recorded. The table shows the values recorded. x 15 30 45 60 75 y...
The diagram shows part of the curve y = 32x-4x² - 48 and the line AB. The curve and the line AB meet the x-axis at A and meet again at the point B(...
The functions f and fg are defined by f(x) = e^(x²+3) for x < 0 fg(x) = e^(2x) for x > 3/2
In the binomial expansion of (2 + x/a)^n, the first three terms in increasing powers of x are b + abx + (9/8)abx². Find the values of the constants...
Calculators must not be used in this paper.
Point A has coordinates (3, -1). A circle has equation (x-4)² + (y+3)² = 5.
A curve has equation y = ((x²-1)/(x²+1))⁴.
Solutions to this question by accurate drawing will not be accepted.
The point A with x-coordinate 2 lies on the curve y = √4x+1. The diagram shows part of this curve and the tangent to the curve at A. Find the area ...
An arithmetic progression has common difference d. The 3rd term of this progression is 10.
In this question n ≥ 6.
The diagram shows the graph of y = |f(x)|, where f is a cubic polynomial. Find expressions for the two possible functions f(x). Write each expressi...
Solve the equation x³ + 1 = 6/x³.
A circle with centre C has the equation x²+y² - 10x-4y+24 = 0.
The polynomial p is such that p(x) = 3x³ − 7x² + ax + b, where a and b are integers. It is given that p'(-1) = 21 and that x-2 is a factor of p(x).
When ln y is plotted against x³, a straight line passing through the points (2,5) and (-8,25) is obtained.
A geometric progression has a 4th term of 8k⁶/27 and a 6th term of 32k¹⁰/243, where k is a constant. The common ratio of this geometric progression...
It is given that y = ln(3x² + 16) / (x+2).
It is given that f(x) = 2ln(3x-4), for x > a, and that f⁻¹ exists.
The diagram shows the shape OABCDEF. AOF is a straight line. OAB and OEF are sectors of a circle with centre O and radius r. Angle BOA = angle EOF....
The diagram shows the triangle OAB, where OA = a and OB = b. The point P lies on OA such that OP = (3/4)OA. The point Q lies on AB such that AQ = (...
A curve is such that its gradient at the point (x, y) is given by (5x-2)³ᐟ². The curve passes through the point (2, 32/5). Find the coordinates of ...
A curve has equation y = x²+2x-3.
In this question, k is a constant. It is given that 2x² + (3k-2)x+k = 0 has roots that are real and distinct. Find the set of possible values of k.
A sports club has the following members. 5 runners, 4 swimmers, 3 gymnasts