Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Solve 2sin(x + π/3) = -1 for 0 ≤ x ≤ 2π. Which of the following is the smaller value of x?
Solve 2sin(x + π/3) = -1 for 0 ≤ x ≤ 2π. Which of the following is the larger value of x?
Solve tany - 2 = coty for 0° ≤ y ≤ 180°. Which of the following is the smaller value of y, correct to 1 decimal place?
Solve tany - 2 = coty for 0° ≤ y ≤ 180°. Which of the following is the larger value of y, correct to 1 decimal place?
Find the y-intercept of the tangent to the curve y = x³ – 4x² + x + 1 at point A (2, -5).
The Venn diagram shows the universal set &, the set A and the set B. Given that n(B) = 5, n(A') = 10 and n(&) = 26, find
A 4-digit number is to be formed from the digits 1, 2, 5, 7, 8 and 9. Each digit may only be used once. Find the number of different 4-digit number...
Show that (1 – cos θ – sin θ)² – 2(1 – sin θ)(1 – cos θ) = 0.
Find the set of values of k for which the curve y = 2x² + kx + 2k – 6 lies above the x-axis for all values of x.
The line 3x + 4y = 15 cuts the curve 2xy = 9 at the points A and B. Find the length of the line AB.
The normal to the curve y + 2 = 3 tan x, at the point on the curve where x = 3π/4, cuts the y-axis at the point P. Find the coordinates of P.
It is given that f(x) = 6x³ – 5x² + ax + b has a factor of x + 2 and leaves a remainder of 27 when divided by x – 1.
(a) Given that the matrix A = (4 2 \ 3 -5), find
(i) Given that n is a positive integer, find the first 3 terms in the expansion of (1 + x/2)^n in ascending powers of x.
(a) (i) Find ∫√2x - 5dx.
(a) Solve cos 2x + 2sec 2x + 3 = 0 for 0° ≤ x ≤ 360°.
A particle P moves in a straight line such that, t s after leaving a point O, its velocity v m s⁻¹ is given by v = 36t –3t² for t ≥ 0.
Prove that ( (1 + sin θ)² / cos² θ ) + ( (1 - sin θ)² / cos² θ ) = 2 + 4 tan² θ.
The velocity-time graph represents the motion of a particle moving in a straight line. [Figure 2.1]
Variables x and y are related by the equation y = 10 − 4sin² x, where 0 ≤ x ≤ π/2. Given that x is increasing at a rate of 0.2 radians per second, ...
A piece of wire of length 96cm is formed into the rectangular shape PQRSTU shown in the diagram. It is given that PQ = TU = SR = xcm. It may be ass...
Find the equation of the normal to the curve y = (x² + 8) / (x - 2) at the point on the curve where x = 4.
The line y = 2x - 8 cuts the curve 2x² + y² - 5xy + 32 = 0 at the points A and B. Find the length of the line AB.
It is given that x ∈ R and that E = {x:-5 < x < 12}, S = {x: 5x + 24 > x²}, T= {x:2x + 7 > 15}.
A plane, whose speed in still air is 240kmh⁻¹, flies directly from A to B, where B is 500km from A on a bearing of 032°. There is a constant wind o...
A one-one function f is defined by f(x) = (x - 1)² – 5 for x ≥ k.
The function f(x) = x³ + x² + ax + b is divisible by x - 3 and leaves a remainder of 20 when divided by x + 1.
Variables x and y are such that when √y is plotted against x² a straight line graph passing through the points (1, 3) and (4, 18) is obtained. Expr...
The diagram shows the graph of a function y = f(x).
The position vectors of the points A and B, relative to an origin O, are 4i – 21j and 22i – 30j respectively. The point C lies on AB such that AB =...
Calculators must not be used in this question. The diagram shows a trapezium ABCD in which AD = 7cm and AB = (4+ √5)cm. AX is perpendicular to DC w...
The shaded region in the diagram is a segment of a circle with centre O and radius r cm. Angle AOB = π/3 radians.
Differentiate, with respect to x,
Solutions to this question by accurate drawing will not be accepted. The points A(-6, 2), B(2, 6) and C are the vertices of a triangle.
The function f is defined, for 0° < x < 360°, by f(x) = 1 + 3 cos 2x.
A curve has equation y = 3x + 1/(x-4)³.
Show that tan θ + cos θ / (1 + sin θ) = sec θ.
Vectors a, b and c are such that a = (4, 3), b = (2, 2) and c = (-5, 2)
Find the set of values of k for which the line y = k(4x – 3) does not intersect the curve y = 4x² + 8x - 8.
Matrices A and B are such that A = (-1 4; 7 6) and B = (2 1; 3 5)
A curve is such that dy/dx = 4x + 1/(x+1)² for x > 0. The curve passes through the point (1/2, 5/6).
The table shows values of variables V and p. V 10 50 100 200 p 95.0 8.5 3.0 1.1