Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
(a) On the axes, sketch the graph of y = 5(x+1)(3x-2)(x-2), stating the intercepts with the coordinate axes. [Figure 1.1] (b) Hence find the values...
Find $\int_3^5\left(\frac{1}{x-1}-\frac{1}{(x-1)^2}\right)\,\text{dx}$, giving your answer in the form a+lnb, where a and b are rational numbers.
The polynomial p(x) = ax³-9x²+bx-6, where a and b are constants, has a factor of x-2. The polynomial has a remainder of 66 when divided by x-3.
The first 3 terms in the expansion of $(a+x)^2(1-\frac{x}{3})^5$, in ascending powers of x, can be written in the form $27+ bx + cx^2$, where a, b ...
The functions f and g are defined as follows. f(x) = x² + 4x for x ∈ R g(x) = 1+e²x for x∈R
(a) (i) Find how many different 5-digit numbers can be formed using the digits 1, 3, 5, 6, 8 and 9. No digit may be used more than once in any 5-di...
(a) In this question, all lengths are in metres and time, t, is in seconds. [Figure 7.1] The diagram shows the displacement-time graph for a runner...
DO NOT USE A CALCULATOR IN THIS QUESTION. A curve has equation $y = (2-\sqrt{3})x^2+x-1$. The x-coordinate of a point A on the curve is $\frac{\sqr...
(a) (i) Write 6xy+3y+4x+2 in the form (ax+b)(cy+d), where a, b, c and d are positive integers. (ii) Hence solve the equation 6 sin $\theta$ cos $\t...
In this question all lengths are in centimetres. [Figure 10.1] The diagram shows a shaded shape. The arc AB is the major arc of a circle, centre O,...
Write (pqr)-23 / (p2r) q³ in the form paqor, where a, b and c are constants.
The diagram shows the quadrilateral OABC such that OA = a, OB = b and OC = c. The lines OB and AC intersect at the point P, such that AP : PC = 3:2...
A curve is such that d²y/dx² = (3x+2)^(-1/3). The curve has gradient 4 at the point (2, 6.2). Find the equation of the curve.
DO NOT USE A CALCULATOR IN THIS QUESTION. A curve has equation y = (3+√5)x²-8√5x+60.
A six-character password is to be made from the following eight characters. Digits: 1, 3, 5, 8, 9 Symbols: *, $, # No character may be used more th...
Variables x and y are such that y = Ax^b, where A and b are constants. When lgy is plotted against lgx, a straight line graph passing through the p...
The normal to the curve y = ln(x²+2)/(2x-3) at the point where x = 2 meets the y-axis at the point P. Find the coordinates of P.
Variables x and y are such that, when lny is plotted against lnx, a straight line graph passing through the points (6, 5) and (8, 9) is obtained. S...
The graph of y=a+2tanbx, where a and b are constants, passes through the point (0, -4) and has period 480°.
The curves y = x² and y² = 27x intersect at O(0, 0) and at the point A. Find the equation of the perpendicular bisector of the line OA.
Variables x and y are such that y = e2 + x cos 2x, where x is in radians. Use differentiation to find the approximate change in y as x increases fr...
Find the exact values of the constant k for which the line y=2x+1 is a tangent to the curve y = 4x² +kx+k-2.
In this question, a, b, c and d are positive constants.
A curve is such that d²y/dx² = sin(6x-π/2). Given that dy/dx = 1/2 at the point (π/4, 13π/12) on the curve, find the equation of the curve.
Relative to an origin O, the position vectors of the points A, B, C and D are OA = (6, -3), OB = (10, 3), OC = (x, y) and OD = (12, 7)
The 2nd, 8th and 44th terms of an arithmetic progression form the first three terms of a geometric progression. In the arithmetic progression, the ...
The diagram shows part of the curve y = (9-x)(x-3) and the line y = k−3, where k > 3. The line through the maximum point of the curve, parallel to ...
Using the binomial theorem, expand (1+e2x)4, simplifying each term.
On the axes, sketch the graph of y = 3 (x-3)(x-1)(x+2) stating the intercepts with the coordinate axes.
Find the values of the constant k for which (2k-1)x² +6x+k+1 = 0 has real roots.
The polynomial p(x) = mx³-29x2+39x+n, where m and n are constants, has a factor 3x-1, and remainder 6 when divided by x-1. Show that x-2 is a facto...
The function f is defined, for 0° ≤ x ≤ 810°, by f(x)=-2+cos(2x/3).
The points A(5, -4) and C(11, 6) are such that AC is the diagonal of a square, ABCD.
DAB is a sector of a circle, centre A, radius 18 cm. The lines CB and CD are tangents to the circle. Angle DAB is (7π/9) radians. [Figure 7.1]
A particle moves in a straight line so that, t seconds after passing through a fixed point O, its velocity, v ms⁻¹, is given by v = 3t² - 30t+72.
Solve the following simultaneous equations. 4x² + 3xy + y² = 8 xy+4 = 0
In this question all lengths are in centimetres. The volume and surface area of a sphere of radius r are (4/3)πr³ and 4πr² respectively. [Figure 11...
DO NOT USE A CALCULATOR IN THIS QUESTION. [Figure 12.1] The diagram shows part of the curve y=1/(2x+1) and part of the line 5y = x-1. The curve mee...
The functions f and g are defined, for x > 0, by f(x) = (2x²-1)/(3x) and g(x) = 1/x.
Find constants a, b and c such that √pq³r⁻³ / ((pq⁻¹)²)r⁻¹ = pᵃqᵇrᶜ.
A particle moves in a straight line such that its displacement, s metres, from a fixed point, at time t seconds, t≥ 0, is given by s = (1+3t)^(1/2).
A function f is such that f(x) = ln (2x+1), for x > -1/2. A function g is such that g(x) = 5x-7, for x ∈ R.
[Figure 4.a] The diagram shows the graph of y =|f(x)|, where f(x) is a cubic. Find the possible expressions for f(x).