Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
In the diagram, A is the point (−1, 3) and B is the point (3, 1). The line L₁ passes through A and is parallel to OB. The line L₂ passes through B ...
Relative to an origin O, the position vectors of the points A and B are given by OA = (-2/3) and OB = (4/p) (i) Find the value of p for which OA is...
(i) Find the first 3 terms in the expansion of (1 + ax)⁵ in ascending powers of x. (ii) Given that there is no term in x in the expansion of (1 – 2...
(a) Find the sum of all the multiples of 5 between 100 and 300 inclusive. (b) A geometric progression has a common ratio of -½ and the sum of the f...
A solid rectangular block has a square base of side x cm. The height of the block is h cm and the total surface area of the block is 96 cm². (i) Ex...
The diagram shows the curve y = (x – 2)² and the line y + 2x = 7, which intersect at points A and B. Find the area of the shaded region.
The equation of a curve is y = ½(2x – 3)³ – 4x. (i) Find dy/dx. (ii) Find the equation of the tangent to the curve at the point where the curve int...
The function f: x → 4-3 sin x is defined for the domain 0 ≤ x ≤ 2π. (i) Solve the equation f(x) = 2. (ii) Sketch the graph of y = f(x). (iii) Find ...
Find ∫(x³ + 1/x²) dx.
The equation x² + px + q = 0, where p and q are constants, has roots –3 and 5.
A curve has equation y = 4/(3x-4) and P (2, 2) is a point on the curve.
The function f is defined by f : x → (x+3)/(2x-1), x ∈ R, x ≠ 1/2.
The line L₁ passes through the points A (2, 5) and B (10, 9). The line L₂ is parallel to L₁ and passes through the origin. The point C lies on L₂ s...
Relative to the origin O, the position vectors of the points A, B and C are given by `OA = (2 5)`, `OB = (4 3)`, `OC = (10 6)`. The point D is such...
The function f is such that f(x) = 3 – 4 cosᵏx, for 0 ≤ x ≤ π, where k is a constant.
The diagram shows part of the curve y = 4√x – x. The curve has a maximum point at M and meets the x-axis at O and A. [Figure X.X]
The diagram shows the region enclosed by the curve y = 6/(2x-3), the x-axis and the lines x = 2 and x = 3. Find, in terms of π, the volume obtained...
The equation of a curve is y = 4√x + 2/√x
The coefficient of x³ in the expansion of (a + x)⁵ + (2 – x)⁶ is 90. Find the value of the positive constant a.
The point A has coordinates (−1, −5) and the point B has coordinates (7, 1). The perpendicular bisector of AB meets the x-axis at C and the y-axis ...
The diagram shows a metal plate made by removing a segment from a circle with centre O and radius 8 cm. The line AB is a chord of the circle and an...
The vector OA has a magnitude of 15 units and is in the same direction as the vector 3i – 4k. The vector OB has a magnitude of 14 units and is in t...
The diagram shows part of the curve y = −x² + 8x – 10 which passes through the points A and B. The curve has a maximum point at A and the gradient ...
Functions f and g are defined by f: x → 2x + 5 for x ∈ R, g: x → 8/(x-3) for x ∈ R, x ≠ 3.
A curve is such that dy/dx = 6/x² and (2, 9) is a point on the curve. Find the equation of the curve.
Find the coefficient of x² in the expansion of
The straight line y = mx + 14 is a tangent to the curve y = 12/x + 2 at the point P. Find the value of the constant m and the coordinates of P.
The diagram shows a square ABCD of side 10 cm. The mid-point of AD is O and BXC is an arc of a circle with centre O. [Figure 4.1]
It is given that a = sin θ – 3 cos θ and b = 3 sin θ + cos θ, where 0° ≤ θ ≤ 360°.
Relative to an origin O, the position vectors of points A and B are given by OA = i - 2j + 2k and OB = 3i + pj + qk, where p and q are constants.
The point R is the reflection of the point (−1, 3) in the line 3y + 2x = 33. Find by calculation the coordinates of R.
The volume of a solid circular cylinder of radius r cm is 250π cm³.
A function f is defined by f(x) = 5/(1-3x) for x ≥ 1.
The diagram shows the curve y = √(1 + 4x), which intersects the x-axis at A and the y-axis at B. The normal to the curve at B meets the x-axis at C...
Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to (10, 3) meets the x-axis.
Find the coefficient of x² in the expansion of (1 + x²)((x/2) - (4/x))⁶
The reflex angle θ is such that cos θ = k, where 0 < k < 1.
The diagram shows a sector of a circle with radius r cm and centre O. The chord AB divides the sector into a triangle AOB and a segment AXB. Angle ...
The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of ea...
The diagram shows a trapezium ABCD in which BA is parallel to CD. The position vectors of A, B and C relative to an origin O are given by O A = (3/...
The equation of a curve is such that d²y/dx² = 2x – 1. Given that the curve has a minimum point at (3, –10), find the coordinates of the maximum po...
The diagram shows part of the curve y = 8 – √(4 – x) and the tangent to the curve at P (3, 7). [Figure 9.1]
Functions f and g are defined by f : x → 2x – 3, x ∈ R, g : x → x² + 4x, x ∈ R. Function h is defined by h : x → x² + 4x for x ≥ k, and it is given...