Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The equation of a curve is y = √(8x – x²). Find
The diagram shows a quadrilateral ABCD in which the point A is (-1, -1), the point B is (3, 6) and the point C is (9, 4). The diagonals AC and BD i...
(a) An arithmetic progression contains 25 terms and the first term is –15. The sum of all the terms in the progression is 525. Calculate (b) A coll...
In the expansion of (x² - a/x)⁷, the coefficient of x⁵ is –280. Find the value of the constant a.
A function f is such that f(x) = (√(x + 3) / 2) + 1, for x ≥ −3.
The diagram shows a plan for a rectangular park ABCD, in which AB = 40 m and AD = 60 m. Points X and Y lie on BC and CD respectively and AX, XY and...
The line y = x/k + k, where k is a constant, is a tangent to the curve 4y = x² at the point P.
The diagram shows a triangle ABC in which A has coordinates (1, 3), B has coordinates (5, 11) and angle ABC is 90°. The point X (4, 4) lies on AC. ...
The position vectors of the points A and B, relative to an origin O, are given by OA = (1, 0, 2) and OB = (k, -k, 2k), where k is a constant.
The diagram shows part of the curve y = 9/(2x + 3), crossing the y-axis at the point B (0, 3). The point A on the curve has coordinates (3, 1) and ...
A curve is defined for x > 0 and is such that dy/dx = x + 4/x². The point P (4, 8) lies on the curve.
The diagram shows a sector of a circle with centre O and radius 20 cm. A circle with centre C and radius x cm lies within the sector and touches it...
Given that cosx = p, where x is an acute angle in degrees, find, in terms of p,
Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6 cm and the height is 8 cm. The paper is cut from A to O and ope...
The equation of a curve is y = 2 / √(5x-6)
Relative to an origin O, the position vectors of points A and B are given by OA = i + 2j and OB = 4i + pk.
The diagram shows a rectangle ABCD in which point A is (0, 8) and point B is (4, 0). The diagonal AC has equation 8y + x = 64. Find, by calculation...
In the diagram, S is the point (0, 12) and T is the point (16, 0). The point Q lies on ST, between S and T, and has coordinates (x, y). The points ...
A function f is defined by f : x → 3 cos x − 2 for 0 ≤ x ≤ 2π.
The diagram shows part of the curve y = 8/x + 2x and three points A, B and C on the curve with x-coordinates 1, 2 and 5 respectively.
A curve has equation y = 2x² – 3x.
The diagram shows part of the curve y = x² + 1. Find the volume obtained when the shaded region is rotated through 360° about the y-axis. [Figure w...
The diagram shows a triangle AOB in which OA is 12 cm, OB is 5 cm and angle AOB is a right angle. Point P lies on AB and OP is an arc of a circle w...
A curve has equation y = 12 / (3-2x)
The equation of a curve is y = x³ + ax² + bx, where a and b are constants.
The diagram shows a pyramid OABCX. The horizontal square base OABC has side 8 units and the centre of the base is D. The top of the pyramid, X, is ...
The diagram shows a trapezium ABCD in which AB is parallel to DC and angle BAD is 90°. The coordinates of A, B and C are (2, 6), (5, −3) and (8, 3)...
A curve is such that d²y/dx² = 24/x³ - 4. The curve has a stationary point at P where x = 2.
The function f : x → 6 − 4 cos(½x) is defined for 0 ≤ x ≤ 2π.
Functions f and g are defined by f: x → 3x + 2, x ∈ R, g: x → 4x – 12, x ∈ R. Solve the equation f⁻¹(x) = gf(x).
In the expansion of (x + 2k)⁷, where k is a non-zero constant, the coefficients of x⁴ and x are equal. Find the value of k.
[Figure 1] shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF are identical isosceles triangles. Angle ABE = angl...
The diagram shows a metal plate OABC, consisting of a right-angled triangle OAB and a sector OBC of a circle with centre O. Angle AOB = 0.6 radians...
Points A, B and C have coordinates A (−3, 7), B (5, 1) and C (−1, k), where k is a constant.
Relative to an origin O, the position vectors of points A, B and C are given by `OA = (0, 2, -3)` `OB = (2, 5, -2)` `OC = (3, p, q)`
The function f is defined, for x ∈ R, by f : x → x² + ax + b, where a and b are constants.
The curve `y = f(x)` has a stationary point at `(2, 10)` and it is given that `f'(x) = 12/x³`.
The diagram shows part of the curve `y = √(9 – 2x²)`. The point P (2, 1) lies on the curve and the normal to the curve at P intersects the x-axis a...
A curve is such that dy/dx = 8/√(4x + 1). The point (2, 5) lies on the curve. Find the equation of the curve.
A curve has equation y = 2x² – 6x + 5.
In the expansion of (3 – 2x) (1 + x/2)^n, the coefficient of x is 7. Find the value of the constant n and hence find the coefficient of x².
The line x/a + y/b = 1, where a and b are positive constants, intersects the x- and y-axes at the points A and B respectively. The mid-point of AB ...