Cambridge Past Paper Questions
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OR The linear transformation T : R⁴ → R4 is represented by the matrix M = (-1 2 3 4; 1 0 1 -1; 1 -2 -3 a; 1 2 5 2)
Let a be a positive constant.
The cubic equation 6x³ +px² -3x-5 = 0, where p is a constant, has roots α, β, γ.
The curve C has equation y = x2 2x + 1
The lines l₁ and l₂ have equations r = 3i+3k+λ(i+4j+4k) and r = 3i-5j-6k+µ(5j+6k) respectively.
Let A = 2 0 1 1 .
The curve C₁ has polar equation r = θcos θ, for 0 ≤ θ ≤ π/2. The point on C₁ furthest from the line θ = π/2 is denoted by P.
Prove by mathematical induction that 2⁴ⁿ + 31ⁿ – 2 is divisible by 15 for all positive integers n.
The equation x⁴ - 2x³ − 1 = 0 has roots α, β, γ, δ.
The matrix M represents the sequence of two transformations in the x-y plane given by a rotation of 60° anticlockwise about the origin followed by ...
The curve C has polar equation r = acot(⅙π − θ), where a is a positive constant and 0 ≤ θ ≤ ⅙π. It is given that the greatest distance of a point o...
Let t be a positive constant. The line l₁ passes through the point with position vector ti+j and is parallel to the vector -2i-j. The line l₂ passe...
The curve C has equation y = (x²+x+9)/(x+1).
The points A, B, C have position vectors 4i-4j+k, -4i+3j-4k, 4i-j-2k, respectively, relative to the origin O.
The sequence of positive numbers $u_1, u_2, u_3, \dots$ is such that $u_1 > 4$ and, for $n \ge 1$, $u_{n+1} = \frac{u_n^2+u_n+12}{2u_n}$.
The cubic equation $2x^3 + 5x^2 - 6 = 0$ has roots $\alpha, \beta, \gamma$.
The curve C has equation $y = \frac{2x^2-x-1}{x^2+x+1}$.
The curve C has polar equation $r^2 = \tan^{-1}\left(\frac{1}{2}\theta\right)$, where $0 \le \theta \le 2$.
The matrix A is given by $\mathbf{A} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{pmatrix}$.
Let A = ((3, 0), (1, 1))
The cubic equation x^3 + 4x^2 +6x+1 = 0 has roots α, β, γ.
The matrix M is given by M = ((a, b^2), (c^2, a)), where a, b, c are real constants and b ≠ 0.
The curve C has polar equation r^2 = 1/(θ^2+1), for 0 < θ < π.
The curve C has equation y = (x^2+2x-15) / (x-2)
The plane Π₁ has equation r = -4j −3k+ λ(i- j+k)+μ(i+j-k).
The cubic equation 2x³ +x² -px-5 = 0, where p is a positive constant, has roots α, β, γ.
The matrix M is given by M = ( (1/2) (-√3/2) / (√3/2) (1/2) ) ( (14) (0) / (0) (1) )
The points A, B, C have position vectors 2i+2j+4k, 2i+4j-k, -3i-3j+4k, respectively, relative to the origin O. The point D has position vector 2i+j...
The curve C has equation y = (x² + ax+1)/(x+2), where a > 5/2.
The curve C has polar equation r² = (π−θ) tan⁻¹ (π−θ), for 0 ≤ θ < π.
The cubic equation x³ +2x+1 = 0 has roots α, β, γ.
The sequence u₁, u₂, u₃, ... is such that u₁ = 5 and u_{n+1} = 6u_n + 5 for n ≥ 1.
The matrix M is given by M = ( (1, 2), (0, 1) ) ( (cosθ, -sinθ), (sinθ, cosθ) ) where 0 < θ < 2π.
The curve C has polar equation r = Ѳe^(Ѳ/2), for 0 ≤ θ≤ 2π.
The points A, B, C have position vectors i-2k, i+2j+2k, 2i-j-k, respectively.
The curve C has equation y = (2x² – 5x) / (2x² – 7x – 4).
The equation x³ + px + q = 0 has a repeated root. Prove that 4p³ + 27q² = 0.
The position vectors of points A, B, C, relative to the origin O, are a, b, c, where a = 3i + 2j – k, b = 4i – 3j + 2k, c = 3i – j – k.
Prove by mathematical induction that, for all positive integers n, `dⁿ/dxⁿ (eˣ sin x) = 2ⁿ/² eˣ sin(x + nπ/4)`.
The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = ( 3 4 2 5 ) ( 6 7 5 8 ) ( 9 9 9 9 ) ( 15 ...
The point P (2, 1) lies on the curve with equation `x³ - 2y³ = 3xy`.
Let `I_n = ∫₀¹ xⁿ(1-x)² dx`, for `n ≥ 0`. Show that, for `n ≥ 1`, `(3 + 2n)I_n = 2nI_n-₁`. Hence find the exact value of `I₃`.
The curve C has equation `y = (x² + px + 1)/(x - 2)`, where p is a constant. Given that C has two asymptotes, find the equation of each asymptote. ...
The vector e is an eigenvector of the matrix A, with corresponding eigenvalue λ, and is also an eigenvector of the matrix B, with corresponding eig...