Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Let a be a positive constant. The curve C₂ has equation y = ((x-a)/(x-2a))². The curve C3 has equation y = |(x-a)/(x-2a)|.
The points A, B, C have position vectors -2i+2j-k, -2i+j+2k, -2j+k, respectively, relative to the origin O.
Give full details of the geometrical transformation in the x-y plane represented by the matrix (6 0) (0 6) Let A = (3 4) (2 2)
It is given that y = xe^ax, where a is a constant. Prove by mathematical induction that, for all positive integers n, dⁿy/dxⁿ = (aⁿx + naⁿ⁻¹)e^ax.
Let S_n = Σ[from r=1 to n] ln( r(r+2) / (r+1)² ).
The cubic equation x³ + 2x² + 3x + 3 = 0 has roots α, β, γ.
The curve C has polar equation r = 3+2 sinθ, for -π < θ ≤ π. [Figure 5(a)]
The curve C has equation y = x² / (x-3).
The points A, B, C have position vectors 2i+2j, -j+k and 2i+j-7k respectively, relative to the origin O.
The equation x⁴ + 3x² + 2x+6 = 0 has roots α, β, γ, δ.
The matrix M is given by `M = \begin{pmatrix} 1 & 0 \\ k^2 & k \end{pmatrix}`, where k is a constant and `k \neq 0` or `1`.
The function f is such that f''(x) = f(x). Prove by mathematical induction that, for every positive integer n, `\frac{d^{2n-1}}{dx^{2n-1}}(xf(x)) =...
The curve C has polar equation `r = a\sec^2\theta`, where a is a positive constant and `0 \le \theta \le \frac{1}{4}\pi`.
The lines `l_1` and `l_2` have equations `r = 2i+k+\lambda(i-j+2k)` and `r = 2j+6k+\mu(i+2j-2k)` respectively. The point P on `l_1` and the point Q...
The curve C has equation `y = \frac{x^2-x}{x+1}`.
Prove by mathematical induction that, for all positive integers n, \frac{d^n}{dx^n}(x^2e^x) = (x^2+2nx+n(n-1))e^x.
The matrix M is given by M = \begin{pmatrix} k & 0 \\ 1 & 1 \end{pmatrix}, where k is a constant and k \neq 0 and k \neq 1.
The cubic equation 27x^3 +18x^2 + 6x-1 = 0 has roots α, β, γ.
The plane Π₁ has equation r = i− j − 2k + \lambda(i−2j − 3k) + \mu(3i− k).
The curve C has polar equation r = e^{-\theta} - e^{-\frac{1}{2}\pi}, where $0 \le \theta \le \frac{1}{2}\pi$.
The curve C has equation y = f(x), where f(x) = \frac{x^2}{x+1}.
The sequence u₁, u₂, u₃, ... is such that u₁ = 4 and u_n+1 = 3u_n - 2 for n ≥ 1.
The line l₁ has equation r = i+3j-k+λ(i−j−4k). The plane Π contains l₁ and is parallel to the vector 2i+5j-4k. The line l₂ is parallel to the vecto...
It is given that α + β + γ + δ = 2, α² + β² + γ² + δ² = 3, α³ + β³ + γ³ + δ³ = 4. It is given that α, β, γ, δ are the roots of the equation 6x⁴-12x...
The matrices A, B and C are given by A = (1 2 3) (2 1 3) (3 2 5) B = (0 -2) (-1 3) (0 0) C = (-2 -1 1) (1 1 3)
It is given that S_n = Σ_{r=1}^n u_r, where u_r = x^(f(r)) − x^(f(r+1)) and x > 0.
The curve C has equation y = (x²+3) / (x²+1).
The curve C₁ has polar equation r = a(cosθ + sinθ) for -¼π ≤ θ ≤ ¾π, where a is a positive constant. The curve C₂ with polar equation r = aθ inters...
Use standard results from the list of formulae (MF19) to find the following sums.
The cubic equation x³ +bx² +cx+d = 0, where b, c and d are constants, has roots α, β and γ. It is given that α+β + γ = 2, α² + β² + γ² = 3, α⁴ + β⁴...
The sequence of positive numbers u₁, u₂, u₃, ... is such that u₁ < 5 and, for n ≥ 1, u(n+1) = (6u(n) + 5) / (u(n) + 2)
Let k and m be non-zero constants. The matrices A, B and C are given by A = [[0,1],[-1,1],[1,1]], B = [[k,0],[0,m]], and C = [[2,-1],[1,2]].
The curve C has polar equation r² = tan 2θ, where 0 < θ < π/4.
The plane Π has equation x+3y+2z = 1. Relative to O, the points A, B, C have position vectors -j+2k, 2i-k, 2i-j-k, respectively.
The curve C has equation y = (x²+x+1)/(x+1).
From the study by Dement and Kleitman (sleep and dreams):
The study by Bandura et al. (aggression) used a sample of children.
From the Pepperberg study (parrot learning):
Two ethical guidelines are debriefing and informed consent. Suggest how ethical issues raised in the Piliavin et al. study (subway Samaritans) rela...
The study by Laney et al. is about false memories.
From the Canli et al. study (brain scans and emotions):
In the Schachter and Singer study (two factors in emotion), after each participant completed their session with the stooge they completed a questio...
Evaluate the Saavedra and Silverman study (button phobia) in terms of two strengths and two weaknesses. At least one of your evaluation points must...
One of the aims of the study by Schachter and Singer (two factors in emotion) was to find out whether the mood of a stooge affected the way partici...
Saavedra and Silverman studied a boy with a phobia of buttons and measured his distress using a 'feelings thermometer'.