Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Find the cubic equation with roots α, β and γ such that α + β + γ = 3, α² + β² + γ² = 1, α³ + β³ + γ³ = −30, giving your answer in the form x³ + px...
Find a matrix A whose eigenvalues are -1, 1, 2 and for which corresponding eigenvectors are (1) (0), (0) (1) (1), (1) (0) (1) (1) respectively.
Using factorials, show that (n/(r-1)) + (n/r) = ((n+1)/r). Hence prove by mathematical induction that (a + x)ⁿ = (n/0)aⁿ + (n/1)aⁿ⁻¹x + ... + (n/r)...
The linear transformation T : R⁴ → R⁴ is represented by the matrix A, where A = (1 3 5 7) (2 8 7 9) (3 13 9 11) (6 24 21 27) Fin...
Find the general solution of the differential equation d²x/dt² + 7dx/dt + 10x = 116 sin 2t. State an approximate solution for large positive values...
The curve C has equation y = e⁻²ˣ. Find, giving your answers correct to 3 significant figures, (i) the mean value of dy/dx over the interval 0 ≤ x ...
A curve C has equation x² + 4xy – y² + 20 = 0. Show that, at stationary points on C, x = −2y. Find the coordinates of the stationary points on C, a...
Evaluate ∫₀^(π/2) x sin x dx. Given that I_n = ∫₀^(π/2) xⁿ sin x dx, prove that, for n > 1, I_n = n(π/2)ⁿ⁻¹ – n(n - 1)I_n₋₂. By first using the sub...
Let z = cos θ + i sin θ. Show that zⁿ + 1/zⁿ = 2 cos nθ and zⁿ - 1/zⁿ = 2i sin nθ. By considering (z - 1/z)⁴ (z + 1/z)², show that sin⁴ θ cos² θ = ...
Answer only one of the following two alternatives. EITHER The lines l₁ and l₂ have equations r = 6i – 3j + s(3i – 4j – 2k) and r = 2i – j – 4k + t(...
Answer only one of the following two alternatives. OR A curve C has parametric equations x = 1 − 3t², y = t(1 – 3t²), for 0 ≤ t ≤ 1/√3. Show that (...
Find ∑(4r – 3)(4r + 1), giving your answer in its simplest form.
Find the general solution of the differential equation d²x/dt² + 2dx/dt + 5x = 4 – 5t².
The cubic equation 2x³ – 3x² + 4x − 10 = 0 has roots α, β and γ.
The curve C has equation 2x³ + 3x²y – 3y³ – 16 = 0.
The points A, B and C have position vectors 2i – j + k, 3i + 4j – k and –i + 2j + 4k respectively.
The linear transformation T : R⁴ → R⁴ is represented by the matrix A, where A = [[1, -1, -2, 3], [5, -3, -4, 25], [6, -4, -6, 28], [7, -5, -8, 31]].
Let Iⁿ = ∫(from 0 to π/4) secⁿx dx for n > 0.
The curve C has equation y = (3x - 9) / ((x - 2)(x + 1)).
OR The polar equation of a curve C is r = a(1 + cos θ) for 0 ≤ θ < 2π, where a is a positive constant.
The roots of the cubic equation x³- 5x² + 13x – 4 = 0 are α, β, γ.
It is given that A= 2 3 1 0 -2 1 0 0 1
The curve C has polar equation r = a cos 3θ, for −π/6 < θ < π/6, where a is a positive constant.
The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = 3 2 0 1 6 5 -1 3 9 8 -2 5 -3 -2 0 -1
It is given that y = eˣu, where u is a function of x. The rth derivatives dʳy/dxʳ and dʳu/dxʳ are denoted by y⁽ʳ⁾ and u⁽ʳ⁾ respectively. Prove by m...
Let S_N = Σ(from r=1 to N) (3r+1)(3r+4) and T_N = Σ(from r=1 to N) 1/((3r+1)(3r+4)).
The curve C has equation y = (5x² + 5x + 1) / (x² + x + 1).
The position vectors of the points A, B, C, D are i + j + 3k, 3i + 4j + 5k, -i + 3k, mj + 4k, respectively, where m is a constant.
EITHER The curve C is defined parametrically by x = 18t - t² and y = 8t³, where 0 < t < 4.
OR Let I_n = ∫(from 1 to √2) (x² - 1)ⁿ dx.
The curve C has equation y = xº for 0 ≤ x ≤ 1, where a is a positive constant. Find, in terms of a, the coordinates of the centroid of the region e...
It is given that y = ln(ax + 1), where a is a positive constant. Prove by mathematical induction that, for every positive integer n, dry = (-1)-1 (...
The integral In, where n is a positive integer, is defined by In = ∫ from 1/2 to 1 of xⁿ sin πx dx.
The line y = 2x + 1 is an asymptote of the curve C with equation y = (x² + 1)/(ax + b)
Let SN = sum from r=1 to N of (5r + 1)(5r + 6) and TN = sum from r=1 to N of 1/((5r + 1)(5r + 6))
With O as the origin, the points A, B, C have position vectors i-j, 2i+j+7k, i-j+k respectively.
The equation x³ + 2x² + x + 7 = 0 has roots α, β, γ.
The matrix M is defined by M = [[2,m,1],[0,m,7],[0,0,1]] where m ≠ 0, 1, 2.
(i) Use de Moivre's theorem to show that sec 6θ = sec^6 θ / (32 - 48 sec² θ + 18 sec⁴ θ – sec^6 θ)
The matrix A is defined by A = [[1,5,1],[1,-2,-2],[2,3,θ]]
Answer only one of the following two alternatives. EITHER It is given that w = cos y and tan y d²y/dx² + (dy/dx)² + 2 tan y dy/dx = 1 + e^(-2x) sec y.
The cubic equation x³ + bx² + cx + d = 0, where b, c and d are constants, has roots α, β, γ. It is given that αβγ =-1.
Prove by mathematical induction that 7²ⁿ – 1 is divisible by 12 for every positive integer n.
The matrices A and B are given by A = (0 1; 1 0) and B = (1/2 -√3/2; √3/2 1/2)
The curve C has polar equation r = ln(1+π−θ), for 0 ≤ θ ≤ π.