Cambridge Past Paper Questions

The matrix A is given by A = [[2, 2, -3], [2, 2, 3], [-3, 3, 3]] The matrix A has an eigenvector [[1], [-1], [1]]. Find the corresponding eigenvalu...

Answer only one of the following two alternatives. EITHER Show that the substitution v = 1/y reduces the differential equation (2/y³) (dy/dx)² + (1...

OR The lines l₁ and l₂ have equations r = 8i + 2j + 3k + λ(i – 2j) and r = 5i + 3j – 14k + μ(2j – 3k) respectively. The point P on l₁ and the point...

The roots of the cubic equation 2x³ + x² − 7 = 0 are α, β and γ. Using the substitution y = 1 + (1/x), or otherwise, find the cubic equation whose ...

Express 4/(r(r + 1)(r + 2)) in partial fractions and hence find sum from r=1 to n of 4/(r(r + 1)(r + 2)). Deduce the value of sum from r=1 to infin...

Prove by mathematical induction that, for all positive integers n, 10ⁿ + 3 × 4ⁿ⁺² + 5 is divisible by 9.

A curve C has polar equation r² = 8 cosec 2θ for 0 < θ < ½π. Find a cartesian equation of C. Sketch C. Determine the exact area of the sector bound...

Let In = ∫ (from 0 to (π/2)) cosⁿ x sin²x dx, for n ≥ 0. By differentiating cosⁿ⁻¹ x sin³ x with respect to x, prove that (n + 2)In = (n − 1)In−2 f...

Use de Moivre's theorem to express cot 7θ in terms of cot θ. Use the equation cot 7θ = 0 to show that the roots of the equation x⁶ − 21x⁴ + 35x² − ...

A curve C has equation y = x²/(x-2). Find the equations of the asymptotes of C. Show that there are no points on C for which 0 < y < 8. Sketch C, g...

Find a cartesian equation of the plane Π₁ passing through the points with coordinates (2, −1, 3), (4, 2, -5) and (−1, 3, −2). The plane Π₂ has cart...

Find the value of the constant k such that y = kx²e²ˣ is a particular integral of the differential equation d²y/dx² - 4 dy/dx + 4y = 4e²ˣ. (*) Henc...

Write down the eigenvalues of the matrix A, where A = [[-2, 1, -1], [0, -1, 2], [0, 0, 1]] and find corresponding eigenvectors. Find a matrix P and...

ANSWER ONLY ONE OF THE FOLLOWING TWO ALTERNATIVES. EITHER A curve C has parametric equations x = e²ᵗ cos 2t, y = e²ᵗ sin 2t, for -π/4 ≤ t ≤ π/4. Fi...

ANSWER ONLY ONE OF THE FOLLOWING TWO ALTERNATIVES. OR The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = [[1,-2,3,-4],...

It is given that ∑ (from r=1 to n) u_r = n²(2n + 3), where n is a positive integer.

Prove, by mathematical induction, that 5ⁿ + 3 is divisible by 4 for all non-negative integers n.

A curve C has equation tan y = x, for x > 0.

The matrix A, given by A = ( 1 2 -2 ) ( 6 4 -6 ) ( 6 5 -7 ) has eigenvalues 1, -1 and -2.

Let I_n = ∫ (from 0 to ½π) xⁿ sin x dx.

By finding a cubic equation whose roots are α, β and γ, solve the set of simultaneous equations α + β + γ = −1, α² + β² + γ² = 29, 1/α + 1/β + 1/γ ...

Let z = cos θ + i sin θ.

The curve C has equation y = (x² - 3x + 6) / (1 - x)

It is given that x = t½, where x > 0 and t > 0, and y is a function of x. (i) Show that dy/dx = 2t dy/dt and d²y/dx² = 2(d²y/dt²) + 4t(dy/dt)². (ii...

The curve C has polar equation r = a(1 + sin θ) for –π < θ < π, where a is a positive constant.

Answer only one of the following two alternatives. EITHER The curve C has equation y = ½(eˣ + e⁻ˣ) for 0 ≤ x ≤ 4.

The position vectors of the points A, B, C, D are i+j+3k, 3i – j + 5k, i − j + k, 5i – 5j + αk, respectively, where α is a positive integer. It is ...

The curve C is defined parametrically by x = eᵗ - t, y = 4e½ᵗ. Find the length of the arc of C from the point where t = 0 to the point where t = 3.

It is given that f(n) = 2³ⁿ + 8ⁿ⁻¹. By simplifying f(k) + f(k + 1), or otherwise, prove by mathematical induction that f(n) is divisible by 9 for e...

The curve C has polar equation r = cos 2θ, for -¼π ≤ θ ≤ ¼π.

It is given that the equation x³ – 21x² + kx – 216 = 0, where k is a constant, has real roots a, ar and ar⁻¹.

Let Sₙ = ∑ⁿᵣ₌₁(−1)ʳ⁻¹r².

The curve C has equation y = (x² + b)/(x + b), where b is a positive constant.

Find the particular solution of the differential equation 49(d²y/dx²) + 14(dy/dx) + y = 49x + 735, given that when x = 0, y = 0 and dy/dx = 0.

The linear transformation T : R⁴ → R³ is represented by the matrix M, where M = ⎛ 1 2 α -1 ⎞ ⎜ 2 6 -3 -3 ⎟ ⎝ 3 10 -6 -5 ⎠ and α is a co...

It is given that, for n ≥ 0, Iₙ = ∫⁰¹ᐟ⁴π secⁿx tan²x dx.

The line l₁ is parallel to the vector ai – j + k, where a is a constant, and passes through the point whose position vector is 9j + 2k. The line l₂...

OR It is given that e is an eigenvector of the matrix A, with corresponding eigenvalue λ. Let A = ⎛ 3 0 0 ⎞ ⎜ 2 7 0 ⎟ ⎝ 4 8 1 ⎠

A curve C has equation cos y = x, for −n < x < π.

Let un = 4 sin(n-½) sin ½ / (cos(2n-1) + cos 1)

The lines l₁ and l₂ have equations r = 6i + 2j + 7k + λ(i + j) and r = 4i + 4j + μ(−6j + k) respectively. The point P on l₁ and the point Q on l₂ a...

It is given that, for n ≥ 0, In = ∫0^1 xⁿe^(x³) dx.

A curve C is defined parametrically by x = 2/(eᵗ + e⁻ᵗ) and y = (eᵗ - e⁻ᵗ)/(eᵗ + e⁻ᵗ), for 0 ≤ t ≤ 1. The area of the surface generated when C is r...

The equation x³ − x + 1 = 0 has roots α, β, γ.

Find the particular solution of the differential equation 10d²x/dt² + 3dx/dt - x = t + 2, given that when t = 0, x = 0 and dx/dt = 0.

It is given that e is an eigenvector of the matrix A, with corresponding eigenvalue λ.

The curves C₁ and C₂ have equations y = ax/(x+5) and y = (x² + (a + 10)x + 5a + 26)/(x+5) respectively, where a is a constant and a > 2.

Answer only one of the following two alternatives. EITHER The curve C₁ has polar equation r² = 2θ, for 0 ≤ θ < ½π.