Cambridge Past Paper Questions

The curve C has equation `y = ½(eˣ + e⁻ˣ)` for `0 ≤ x ≤ ln 5`. Find

The curve C has polar equation `r = 3 + 2 cos θ`, for `-π < θ < π`. The straight line l has polar equation `r cos θ = 2`. Sketch both C and l on a ...

Answer only one of the following two alternatives. **EITHER** Let `ω = cos(π/5) + i sin(π/5)`. Show that `ω⁵ + 1 = 0` and deduce that `ω⁴ – ω³ + ω...

The curve C has equation y = 2x² for 0 ≤ x ≤ 4. Find

Let f(r) = r(r + 1)(r + 2).

Let I_n denote ∫ (from 0 to ∞) xⁿe⁻²ˣ dx.

The curve C has equation y = λx + x/(x – 2), where λ is a non-zero constant.

The curve C has parametric equations x = ⅓t³ – ln t, y = ⅔t³, for 1 ≤ t ≤ 3.

The plane Π has equation r = 2i + 3j – k + λ(i – 2j + 2k) + μ(3i + j – 2k). The line l, which does not lie in Π, has equation r = 3i + 6j + 12k + t...

Write down the eigenvalues of the matrix A, where A = [[1, 4, -16], [0, 2, 3], [0, 0, 3]]

Answer only one of the following two alternatives. EITHER The roots of the equation x⁴ – 3x² + 5x − 2 = 0 are α, β, γ, δ, and αⁿ + βⁿ + γⁿ + δⁿ is ...

Answer only one of the following two alternatives. OR The linear transformation T : R⁴ → R³ is represented by the matrix M, where M = [[2, 1, -1, 4...

The curve C has polar equation r = 2e^θ, for π/6 < θ < π/3. Find

The cubic equation x³ – px − q = 0, where p and q are constants, has roots α, β, γ. Show that

It is given that S_n = ∑_{r=1}^n u_r = 2n² + n. Write down the values of S1, S2, S3, S4. Express u_r in terms of r, justifying your answer. Find ∑_...

It is given that I_n = ∫_0^1 x^n / √(1 + 2x) dx. Show that, for n ≥ 1, (2n + 1)I_n = √3 - nI_{n-1}. Show that I_3 = 2/35 (√3 + 1).

It is given that y = (1 + x)² ln(1 + x). Find d³y/dx³. Prove by mathematical induction that, for every integer n ≥ 3, dⁿy/dxⁿ = (-1)^{n-1} 2(n-3)! ...

The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = [[1, -3, -1, 2], [4, -10, 0, 2], [1, -1, 3, -4], [5, -12, 1, 1]]. F...

The square matrix A has λ as an eigenvalue with e as a corresponding eigenvector. Show that e is an eigenvector of A² and state the corresponding e...

The plane Π₁ has equation r = [[2], [3], [-2]] + s[[1], [0], [-1]] + t[[0], [-1], [-2]]. Find a cartesian equation of Π₁. The plane Π₂ has equation...

The curve C has parametric equations x = t², y=t-t³, for 0 ≤ t ≤ 1. Find the surface area generated when C is rotated through 2π radians about the ...

The curve C has equation y = (px² + 4x + 1) / (x + 1), where p is a positive constant and p ≠ 3.

Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form cos θ + i sin θ, where –π < θ ≤ π. Simplify (x...

Answer only one of the following two alternatives. OR Given that 1/y d²y/dx² - 6/y² (dy/dx)² + 2/y³ (dy/dx) + 3y = 25e^{-2x} and that v = y³, show ...

Given that Uk = 1/√(2k-1) - 1/√(2k+1), express ∑(from k=1 to n) Uk in terms of n. Deduce the value of ∑(from k=1 to ∞) Uk

A curve C has parametric equations x = eᵗ cos t, y = eᵗ sin t, for 0 ≤ t ≤ π. Find the arc length of C.

It is given that u_r = r × r! for r = 1, 2, 3, ... . Let S_n = u_1 + u_2 + u_3 + ... + u_n. Write down the values of 2! - S_1, 3! - S_2, 4! – S_3, ...

A curve C has equation y = (2x² + x - 1) / (x-1). Find the equations of the asymptotes of C. Show that there is no point on C for which 1 < y < 9.

Find the value of a for which the system of equations x - y + 2z = 4, x + ay - 3z = b, x - y + 7z = 13, where a and b are constants, has no unique ...

Use de Moivre's theorem to show that cos 5θ = cos θ(16 sin⁴ θ – 12 sin² θ + 1). By considering the equation cos 5θ = 0, show that the exact value o...

Let I_n = ∫₀¹ (1-x)ⁿeˣ dx. Show that, for all positive integers n, I_n = nI_(n-1) - 1. Find the exact value of I_4. By considering the area of the ...

A circle has polar equation r = a, for 0 ≤ θ < 2π, and a cardioid has polar equation r = a(1 – cos θ), for 0 ≤ θ < 2π, where a is a positive consta...

Given that x(d²y/dx²) + (2x+2)(dy/dx) + (2-3x)y = 10e^(2x) and that v = xy, show that d²v/dx² + 2(dv/dx) – 3v = 10e^(2x). Find the general solution...

The line l₁ is parallel to the vector i – 2j – 3k and passes through the point A, whose position vector is 3i + 3j - 4k. The line l₂ is parallel to...

Answer only one of the following two alternatives. EITHER The roots of the quartic equation x⁴ + 4x³ + 2x² – 4x + 1 = 0 are α, β, γ and δ. Find the...

The curve C is defined parametrically by x = 2 cos³t and y = 2 sin³t, for 0 < t < π. Show that, at the point with parameter t, d²y/dx² = ¹⁄₆ sec⁴t ...

Find the general solution of the differential equation d²x/dt² + 4dx/dt + 4x = 7 - 2t².

Given that a is a constant, prove by mathematical induction that, for every positive integer n, dⁿ/dxⁿ(xeᵃˣ) = naⁿ⁻¹eᵃˣ + aⁿxeᵃˣ.

The sequence a₁, a₂, a₃, ... is such that, for all positive integers n, a_n = (n+5)/√(n²-n+1) - (n+6)/√(n²+n+1). The sum ∑ᴺ_n=1 a_n is denoted by S...

The cubic equation x³ + px² + qx + r = 0, where p, q and r are integers, has roots α, β and γ, such that α + β + γ = 15, α² + β² + γ² = 83. Write d...

The matrix A, where A = (1 0 0 10 -7 10 7 -5 8) has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that (0 2 1)ᵀ is an eigenvec...

The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = (1 -2 -3 1 3 -5 -7 7 5 -9 -13 9 7 -13 -19 11) Find the rank of M a...

The curve C has equation y = (2x² + kx)/(x+1), where k is a constant. Find the set of values of k for which C has no stationary points. For the cas...

It is given that I_n = ∫ᵉ₁ (ln x)ⁿ dx for n ≥ 0. Show that I_n = (n − 1)[I_{n-2} − I_{n-1}] for n ≥ 2. Hence find, in an exact form, the mean value...

Using de Moivre's theorem, show that tan 5θ = (5 tan θ - 10 tan³ θ + tan⁵ θ) / (1 - 10 tan² θ + 5 tan⁴ θ). Hence show that the equation x² – 10x + ...

Answer only one of the following two alternatives. EITHER The points A, B and C have position vectors i, 2j and 4k respectively, relative to an or...

Use the method of differences to find Σr=1^n 1 / ((2r)² - 1). Deduce the value of Σr=1^∞ 1 / ((2r)² - 1).