Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The curve C has equation \( y = x^{3/2} \).
Let \( I_n = \int_0^{\pi/2} \cos^n x \, dx \), where \( n \ge 0 \).
Answer only one of the following two alternatives. EITHER
Answer only one of the following two alternatives. OR The curve C has equation \( y = \frac{x^2 + \lambda x - 6\lambda^2}{x + 3} \) where \( \lambd...
The roots of the cubic equation x³ – 7x² + 2x – 3 = 0 are α, β, γ. Find the values of
Prove, by mathematical induction, that, for integers n ≥ 2, 4ⁿ > 2n + 3ⁿ.
Given that f(r) = 1 / ((r + 1)(r + 2)), show that
The curve C has polar equation r = 2 + 2 cos θ, for 0 ≤ θ ≤ π. Sketch the graph of C. Find the area of the region R enclosed by C and the initial l...
A matrix A has eigenvalues –1, 1 and 2, with corresponding eigenvectors [1, -1, -2]ᵀ, [-1, -3, 5]ᵀ and [2, 5, -3]ᵀ respectively. Find A.
Write down the values of θ, in the interval 0 ≤ θ < 2π, for which cos θ + i sin θ is a fifth root of unity. By writing the equation (z + 1)⁵ = z⁵ i...
The linear transformations T₁ : R⁴ → R⁴ and T₂ : R⁴ → R⁴ are represented by the matrices M₁ = [[1, 1, 1, 4], [2, 1, 4, 11], [3, 4, 1, 9], [4, -...
Find the particular solution of the differential equation d²y/dx² + 2dy/dx + 5y = 10e⁻²ˣ, given that y = 5 and dy/dx = 1 when x = 0.
The curve C has equation y = (2x² + 2x + 3) / (x² + 2) Show that, for all x, 1 ≤ y ≤ 2. Find the coordinates of the turning points on C. Find the e...
The curve C has equation y = 2(x/3)^(3/2), where 0 ≤ x ≤ 3. Show that the arc length of C is 2(2√2 – 1). Find the coordinates of the centroid of th...
Answer only one of the following two alternatives. EITHER Show that ∫ (from 0 to π) eˣ sin x dx = (1 + e^π) / 2. Given that Iₙ = ∫ (from 0 to π) e...
Answer only one of the following two alternatives. OR The position vectors of the points A, B, C, D are 2i + 4j – 3k, –2i + 5j – 4k, i + 4j + k, i...
Find the area of the region enclosed by the curve with polar equation r = 2(1 + cos θ), for 0 ≤ θ < 2π.
Prove by mathematical induction that 5^(2n) – 1 is divisible by 8 for every positive integer n.
The cubic equation x³ – 2x² – 3x + 4 = 0 has roots α, β, γ. Given that c = α + β + γ, state the value of c. Use the substitution y = c − x to find ...
Let I_n = ∫₀¹ 1/(1 + x²)^n dx. Prove that, for every positive integer n, 2nI_{n+1} = 2⁻ⁿ + (2n − 1)I_n. Given that I₁ = π/4, find the exact value o...
Use the method of differences to show that Σ_{r=1}^N 1/((2r + 1)(2r + 3)) = 1/6 - 1/(2(2N + 3)). Deduce that Σ_{r=N+1}^2N 1/((2r + 1)(2r + 3)) < 1/...
The matrix A is given by A = [[4, -5, 3], [3, -4, 3], [1, -1, 2]] Show that e = [[1], [1], [1]] is an eigenvector of A and state the corresponding ...
By considering the binomial expansion of (z - 1/z)^6, where z = cos θ + i sin θ, express sin⁶ θ in the form 1/32 (p + q cos 2θ + r cos 4θ + s cos 6...
The linear transformations T₁ : R⁴ → R⁴ and T₂ : R⁴ → R⁴ are represented by the matrices M₁ and M₂ respectively, where M₁ = [[1, -2, 3, 5], [3, -4,...
Find x in terms of t given that d²x/dt² + 4dx/dt + x = 6e⁻²ᵗ, and that, when t = 0, x = 5/3 and dx/dt = 7/6. State lim_{t→∞} x.
The curve C has equation y = (2x²-3x-2)/(x²-2x+1). State the equations of the asymptotes of C. Show that y ≤ 25/12 at all points of C. Find the coo...
Answer only one of the following two alternatives. EITHER. The curve C has equation y = 2 sec x, for 0 ≤ x ≤ ¼π. Show that the arc length s of C is...
Answer only one of the following two alternatives. OR. The points A, B, C and D have coordinates as follows: A (2, 1, -2), B (4, 1, -1), C (3, -2, ...
The equation x³ + px + q = 0, where p and q are constants, with q ≠ 0, has one root which is the reciprocal of another root. Prove that p + q² = 1.
Expand and simplify (r + 1)⁴ – r⁴. Use the method of differences together with the standard results for Σr and Σr² to show that Σ(from r=1 to n) r³...
Prove by mathematical induction that, for all non-negative integers n, 11^(2n) + 25^n + 22 is divisible by 24.
Obtain the general solution of the differential equation d²x/dt² – 6dx/dt + 25x = 195 sin 2t.
The curve C has polar equation r = a(1 + sin θ), where a is a positive constant and 0 ≤ θ < 2π. Draw a sketch of C. Find the exact value of the are...
The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = [ [2, -1, 1, 3], [2, 0, 0, 5], [6, -2, 2, 11], [10, -3, 3, 19] ]. (...
Use de Moivre's theorem to show that tan 5θ = (5t - 10t³ + t⁵) / (1 - 10t² + 5t⁴), where t = tan θ. Deduce that the roots of the equation t⁴ – 10t²...
The curve C has parametric equations x = t², y = t - ⅓ t³, for 0 ≤ t ≤ 1. Find (i) the arc length of C, (ii) the surface area generated when C is r...
The matrix M, where M = [ [-2, 2, 2], [2, 1, 2], [-3, -6, -7] ], has an eigenvector (0, 1, -1)^T. Find the corresponding eigenvalue. It is given th...
It is given that I_n = ∫(from 0 to π/4) (sin^(2n+2) x) / (cos x) dx, where n ≥ 0. Show that I_n – I_(n+1) = (2^(−(n+2))) / (2n + 1). [5] Hence show...
The line l₁ passes through the points A (2, 3, −5) and B (8, 7, −13). The line l₂ passes through the points C (-2, 1, 8) and D (3, −1, 4). Find the...
Answer only one of the following two alternatives. EITHER The curve C has parametric equations x = t², y = (2 - t)³⁄², for 0 ≤ t ≤ 2. Find (i) d²y...
Use the List of Formulae (MF10) to show that ∑_{r=1}^{13} (3r² – 5r + 1) and ∑_{r=0}^{9} (r³ – 1) have the same numerical value.
Find the value of the constant k for which the system of equations 2x - 3y + 4z = 1, 3x - y = 2, x + 2y + kz = 1, does not have a unique solution. ...
The sequence a1, a2, a3, ... is such that a₁ > 5 and an+1 = 4an/5 + 5/an for every positive integer n. Prove by mathematical induction that a„ > 5 ...
The roots of the cubic equation x³ – 7x² + 2x − 3 = 0 are α, β and γ. Find the values of Deduce a cubic equation, with integer coefficients, having...
The curves C₁ and C₂ have polar equations C₁: r = 1/√2, for 0 < θ < 2π, C₂: r = √(sin10), for 0 ≤ θ≤ π. Find the polar coordinates of the point of ...
A curve has equation x² – 6xy + 25y² = 16. Show that dy/dx = 0 at the point (3, 1). By finding the value of d²y/dx² at the point (3, 1), determine ...
Let In = ∫₀^π xⁿ sin x dx, where n is a non-negative integer. Show that In = n(π)ⁿ⁻¹ – n(n − 1)In-2, for n ≥ 2. Find the exact value of I₄.
By considering ∑_{r=1}^{n} z²ʳ⁻¹, where z = cos θ + i sin θ, show that, if sin θ ≠ 0, ∑_{r=1}^{n} sin (2r − 1)θ = sin² nθ / sin θ Deduce that ∑_{r=...
The curve C has parametric equations x = 4t + 2t³, y = 4t - 2t², for 0 ≤ t ≤ 4. Find the arc length of C, giving your answer correct to 3 significa...