Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The number of insects in a population t weeks after the start of observations is denoted by N. The population is decreasing at a rate proportional ...
The curve with equation y = e^2x ln(x – 1) has a stationary point when x = p. (i) Show that p satisfies the equation x = 1 + exp exp(2(x-1)) where...
COS X (i) By differentiating -, show that if y sin x 10 dy = - cosec²x. [2] = cot x then dy dx (ii) Show that 2 π ∫ x cosec²x dx = 4(π + 1n 4). [6] π
Two lines l and m have equations r = ai + 2j + 3k + λ(i – 2j + 3k) and r = 2i + j + 2k + μ(2i – j + k) respectively, where a is a constant. It is g...
Let f(x) = x² + x + 6 x²(x + 2) (i) Express f(x) in partial fractions. (ii) Hence, showing full working, show that the exact value of S 4 f(x) dx...
(i) By first expanding cos(2x + x), show that cos 3x = 4 cos³ x − 3 cos x. 3 [4] (ii) Hence solve the equation cos 3x + 3 cos x + 1 = 0, for 0 ≤ x...
10 (a) The complex number u is given by u = −3 – (2√10)i. Showing all necessary working and without using a calculator, find the square roots of u....
Solve the equation 5 ln(4 – 3ˣ) = 6. Show all necessary working and give the answer correct to 3 decimal places.
The curve with equation y = e⁻²ˣ / (1-x²) has a stationary point in the interval −1 < x < 1. Find dy/dx and hence find the x-coordinate of this sta...
The polynomial x⁴ + 3x³ + ax + b, where a and b are constants, is denoted by p(x). When p(x) is divided by x² + x − 1 the remainder is 2x + 3. Find...
The context for the parts is: Express (√6) sin x + cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°. Hence solve an equation.
The equation of a curve is 2x²y – xy² = a³, where a is a positive constant. Show that there is only one point on the curve at which the tangent is ...
The variables x and θ satisfy the differential equation sin(θ/2) dx/dθ = (x + 2) cos(θ/2) for 0 < θ < π. It is given that x = 1 when θ = π/3. Solve...
The context for the parts involves complex numbers.
Let f(x) = (2x² + x + 8) / ((2x−1)(x² + 2)).
It is given that ∫(from 0 to a) x cos(x/3) dx = 3, where the constant a is such that 0 < a < (3/2)π.
The line l has equation r = i + 3j – 2k + λ(i – 2j + 3k). The plane p has equation 2x + y − 3z = 5.
Solve the inequality 2|x + 2| > |3x – 1|.
The polynomial 6x³ + ax² + bx − 2, where a and b are constants, is denoted by p(x). It is given that (2x + 1) is a factor of p(x) and that when p(x...
Showing all necessary working, solve the equation (3^(2x) + 3^(-x)) / (3^(2x) - 3^(-x)) = 4. Give your answer correct to 3 decimal places.
Throughout this question the use of a calculator is not permitted. The complex number with modulus 1 and argument 1/3π is denoted by w. The complex...
The plane m has equation x + 4y – 8z = 2. The plane n is parallel to m and passes through the point P with coordinates (5, 2, −2).
The diagram shows the graph of y = sec x for 0 ≤ x < 1/2π.
The variables x and t satisfy the differential equation 5 dx/dt = (20−x)(40 – x). It is given that x = 10 when t = 0.
The diagram shows the graph of y = e^(cosx) sin³x for 0 ≤ x ≤ π, and its maximum point M. The shaded region R is bounded by the curve and the x-axis.
A crane is lifting a load of 1250 kg vertically at a constant speed Vm s⁻¹. Given that the power of the crane is a constant 20 kW, find the value o...
The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her ...
A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of...
Two blocks A and B of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an...
A small ring P is threaded on a fixed smooth horizontal rod AB. Three horizontal forces of magnitudes 4.5 N, 7.5 N and F N act on P [Figure X.X].
A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of ...
A particle moves in a straight line, starting from rest at a point O, and comes to instantaneous rest at a point P. The velocity of the particle at...
A particle moves in a straight line. The displacement of the particle at time ts is sm, where s = t³ – 6t² + 4t. Find the velocity of the particle ...
The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes...
Differentiate with respect to x
Find the set of values of k for which the equation x2 + (k – 2)x + (2k – 4) = 0 has real roots.
Solve the equation 3x(x² + 6) = 8 – 17x2.
Given that log8p=x and log8q = y, express in terms of x and/or y
The function f is defined by f(x) = (2x + 1)² - 3 for x = -1/2.
Solve
Answer only one of the following two alternatives. EITHER A curve has equation y = ln x / x², where x > 0.
OR A curve is such that dy/dx = 6 cos(2x + π/2) for -π/4 <= x <= 5π/4. The curve passes through the point (π/4, 5).
Find the coordinates of the points of intersection of the curve y² + y = 10x - 8x² and the line y + 4x + 1 = 0.
The expression 6x³ + ax² − (a + 1)x + b has a remainder of 15 when divided by x + 2 and a remainder of 24 when divided by x + 1. Show that a = 8 an...