Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
E [Figure: Venn Diagram with F and C circles, overlapping] There are 105 boys in a year group at a school. Some boys play football (F) and some pla...
A curve has equation y = x³ / sin 2x . Find
Solve
Solve the simultaneous equations 8^(p+1)/4^q = 2^11, 3^(2p+5)/27^q = 9^(3q)
Solve the quadratic equation (1-√3)x² + x + (1+√3) = 0, giving your answer in the form a+b√3, where a and b are constants.
y [Figure: Graph showing y = 2√x curve from O, a tangent/normal line AB. Point A(4,4) is on the curve. Point B is on the x-axis.] The diagram shows...
Two lines are tangents to the curve y = 12-4x-x². The equation of each tangent is of the form y = 2k+1-kx, where k is a constant.
The functions f and g are defined for real values of x ≥ 1 by f(x) = 4x-3, g(x) = (2x+1)/(3x-1).
A plane that can travel at 260km/h in still air heads due North. A wind with speed 40 km/h from a bearing of 310° blows the plane off course. Find ...
(i) On the axes below, sketch the graph of y = 2cos3x-1 for -90° ≤x≤ 90°. [Figure 1.1]
When lgy² is plotted against x, a straight line is obtained passing through the points (5, 12) and (3, 20). Find y in terms of x, giving your answe...
The first three terms in the expansion of (1-x/7)^14 (1-2x)^4 can be written as 1 + ax+bx². Find the value of each of the constants a and b.
(i) On the axes below, sketch the graph of y = |2x²-9x-5| showing the coordinates of the points where the graph meets the axes. [Figure 4.1]
(a) It is given that f : x ↦ √x for x ≥ 0, g : x ↦ x + 5 for x ≥ 0. Identify each of the following functions with one of f⁻¹, g⁻¹, fg, gf, f², g². ...
(a) Write √(p(qp²/r)²) / (p⁻¹(³√qr)) in the form p^a q^b r^c, where a, b and c are constants. (b) Solve log₇x + 2logₓ7 = 3.
It is given that y = (1+e^(x²))(x+5).
(a) Five teams took part in a competition in which each team played each of the other 4 teams. The following table represents the results after all...
A solid circular cylinder has a base radius of r cm and a height of h cm. The cylinder has a volume of 1200π cm³ and a total surface area of S cm².
The diagram shows a circle centre O, radius 10cm. The points A, B and C lie on the circumference of the circle such that AB = BC = 18 cm. [Figure 1...
A curve is such that d²y/dx² = 2(3x-1)^(-3/2). Given that the curve has a gradient of 6 at the point (3, 11), find the equation of the curve.
On each of the Venn diagrams below, shade the region indicated.
Given that y = 2 sin 3x + cos 3x, show that d²y/dx² + dy/dx + 3y = k sin 3x, where k is a constant to be determined.
A 5-digit code is formed using the following characters. Letters a c e i o u Numbers 1 2 3 4 5 6 Symbols @ * # No character can be repeated in a co...
Find the values of k for which the line y = kx+3 does not meet the curve y = x²+5x+12.
At the point where x = 1 on the curve y = k/(x+1)², the normal has a gradient of 1/3.
The diagram shows the points A (−3, 5) and B (5, −1). The mid-point of AB is M and the line PM is perpendicular to AB. The point P has coordinates ...
Do not use a calculator in this question. Solve the quadratic equation (√5-3)x² + 3x + (√5 +3) = 0, giving your answers in the form a+b√5, where a ...
The curve y = 2x²+k+4 intersects the straight line y = (k+4)x at two distinct points. Find the possible values of k.
[Figure 2.1] The diagram shows the graph of y = f(x), where f(x) is a cubic polynomial.
The 7th and 10th terms of an arithmetic progression are 158 and 149 respectively.
Find the coefficient of x² in the expansion of (x - 3/x) (x + 2/x)⁵.
f(x) = x²+2x-3 for x≥-1
A curve has equation y = ln(3x²-5) / (2x+1) for 3x² > 5.
A curve has equation y = (2x-1)√4x+3.
The polynomial p(x) = 6x³ +ax²+bx+2, where a and b are integers, has a factor of x-2.
In this question all lengths are in centimetres and all angles are in radians. [Figure 11.1] The diagram shows the rectangle ADEF, where AF = DE = ...
[Figure 12.1] The diagram shows the velocity-time graph of a particle P that travels 2775 m in 90s, reaching a final velocity of V ms⁻¹.
Solve the inequality (x-8)(x−10) > 35.
Find the value of x such that 4^(x+1) / 2^(x-1) = 32^(2/3) x 8^(1/3).
Solve the simultaneous equations. log3 (x+y) = 2 2log3 (x + 1) = log3 (y+2)
Find the exact value of ∫₄² (x+1)² / x² dx.