Cambridge Past Paper Questions
Browse 23,507questions from 25 years of O-Level & A-Level exams. Click any question to practice.
[Figure 1] Figure 1 shows a sketch of part of the curve with equation $y = x(x-2)(x-4)$. The curve crosses the $x$-axis at the origin $O$ and at th...
Solve, for $0 \le \theta < 360^{\circ}$, the equation $4\sin^2\theta = 5 - 4\cos\theta$
The polynomial $P(x)$ is given by $P(x) = x^3 - 4x^2 - 7x + 10$. (a) Show that $(x-1)$ is a factor of $P(x)$. (b) Hence, or otherwise, factorise $P...
A curve C has equation y = 2x³ - 5x² - 4x + 1. The point P has x-coordinate 2.
Solve the equation 2 log₃(x) - log₃(x + 6) = 1.
A geometric series has first term a and common ratio r. The second term of the series is 12 and the sum to infinity is 64.
Which of the following expressions is equivalent to \frac{1}{2} \ln(9x^2)?
The function f is defined by f(x) = 3x^2 - 6x + 5, x \in \mathbb{R}, x \ge 1.
A curve C has equation y = (2x+3)e^{-x}.
The curve C has equation \cos(y) = x^2 \sin(x).
Using the substitution u = \tan x, find the exact value of \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 - \sin^2 x} dx.
Given that \theta is small and measured in radians, use the small angle approximations to show that \frac{\cos(3\theta) - 1}{\theta \sin(2\theta)} ...
The population of a species of insects in a laboratory is modelled by the equation P = (800e^(0.1t))/(1 + 4e^(0.1t)), t ≥ 0, where P is the populat...
Figure 1 shows a sketch of the curve with equation y = 2x ln x, x > 0.
The curve $C$ has equation $y = 2x + \cot x$, $0 < x < \pi$.
Water is leaking from a cylindrical tank. The depth of the water in the tank at time t minutes is h cm. The rate at which the depth of water is cha...
(a) Express (2x + 7) / ((x + 2)(2x + 1)) in partial fractions. (b) Hence find ∫ (2x + 7) / ((x + 2)(2x + 1)) dx.
The curve $C$ has equation $y = x^2(3-x)$. The point $P$ lies on $C$ and has $x$-coordinate $1$. (a) Find the equation of the tangent to $C$ at $P$...
A curve $C$ has equation $y = x^2-10x+26$. The line $L$ has equation $y = -3x+c$, where $c$ is a constant.
Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm{f}(x)$, where $\mathrm{f}(x) = (x+1)(x-2)(x-4)$. The curve $C$ crosses the $x$-...
Figure 2 shows a sketch of the curve $C$ with equation $y = \frac{1}{2}x^2-3x+8$. The points $P$ and $Q$ lie on $C$ such that the tangent to $C$ at...
I am unable to extract the questions, mark schemes, model answers, and examiner report traps because the Pearson Edexcel IAL Question Paper, Mark S...
A function f is defined by f(x) = (x^2 - 2x + 1) / (x - 3), x ∈ ℝ, x ≠ 3. (a) Express f(x) in the form Ax + B + C / (x - 3), where A, B and C are ...
(a) Show that 3cos(2θ) - sin(2θ) ≡ Rcos(2θ + α), where R > 0 and 0 < α < π/2, giving the value of R and α to 3 decimal places. (b) Hence solve, for...
A curve C has equation y = x e^(2x). (a) Find dy/dx. (b) Find the exact coordinates of the stationary point of C. (c) Determine the nature of this...
Using the substitution u = 1 + cos x, find the exact value of ∫(from π/2 to π) [sin(x) / (1 + cos x)^2] dx.
Solve, for 0 ≤ x < 360°, the equation 3 tan x = 5 sin x.
The curve C has equation x^2 + 2xy - 3y^2 + 16 = 0. (a) Show that dy/dx = (x + y)/(3y - x). (b) Find the equation of the normal to the curve C at t...
Given that the general solution of the differential equation (x + 1) dy/dx = y(x^2 - 1) is y = A(x - 1)e^(x^2/2), where A is an arbitrary constant....
A curve has parametric equations x = t^2 + 1, y = 2ln(t + 1), t > -1 (a) Find dy/dx in terms of t. (b) Find the equation of the normal to the curve...
A curve C has equation y = x³ - 6x² + 5x - 2.
The 5th term of an arithmetic series is 18 and the 10th term is 33. Find the first term and the common difference of the series.
The first three terms of an arithmetic series are 2x, x+4, and 2x-5 respectively.
The binomial expansion of (1 + px)ⁿ, where n is a positive integer, is 1 - 12x + 54x² + ...
Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the origin O and touches the x-axis at the point (3, 0). ...
Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the origin O and touches the x-axis at the point (3, 0). ...
A circle C has equation x^2 + y^2 - 4x + 6y - 12 = 0. (a) Find the coordinates of the centre of C and the radius of C.
A circle C has equation x^2 + y^2 - 4x + 6y - 12 = 0. (b) The point P(5, -7) lies on C. Find the equation of the tangent to C at the point P.
Find \int \left( 2x^3 - 3 + \frac{4}{\sqrt{x}} \right) \text{d}x, giving each term in its simplest form.
A curve has equation y = x^2(x - 2) + 3. (a) Find dy/dx.
A curve has equation y = x^2(x - 2) + 3. (b) Find the exact coordinates of the stationary points of the curve.
A rectangular box has an open top. The box has length $3x$ cm, width $x$ cm and height $h$ cm. The volume of the box is $2250 \mathrm{~cm}^3$.
A curve has equation y = x^3 - 6x^2 + 9x + 2.
The circle C has equation (x - 2)^2 + (y + 1)^2 = 25.
Given the function f(x) = x^2 - 6x + 13.
The equation e^(x) + 2x - 3 = 0 has a single root α. (a) Show that α lies between 0.5 and 0.6. (b) Using x₀ = 0.55 as a first approximation, apply...