ZIMSEC A Level Pure Mathematics Paper 2 Past Papers

Topics in this paper

Complex Numbers Functions and Graphs Trigonometry Sequences and SeriesProjectile motionIntegrationContinuous random variablesDifferential equationsVectors, lines and planesKinematics

Explained questions

Solve the inequality $|x^2-2|<-x$ .
In solving $|x^2-2|<-x$ by cases, why is only $x=-2$ (not $x=1$ ) accepted from the case $x^2-2\ge0$ , and only $x=-1$ (not $x=2$ ) accepted from the case $x^2-2<0$ ?
The graph $y=f(x)$ has $y$ -intercept $(0,16)$ . State the $y$ -intercept of $y=f(x)-6$ .
The graph $y=f(x)$ has $x$ -intercepts $(-2,0)$ , $(2,0)$ and $(4,0)$ . State the $x$ -intercepts of $y=f(x+1)$ .
The graph $y=f(x)$ has $y$ -intercept $(0,16)$ . State the $y$ -intercept of $y=-\tfrac12f(x)$ .
The graph $y=f(x)$ has $x$ -intercepts $(-2,0)$ , $(2,0)$ and $(4,0)$ . State the $x$ -intercepts of $y=f(2x)$ .
Given $f(x)=\dfrac{2x-3}{x^2}$ , differentiate from first principles to find $f'(x)$ .
For $f(x)=\dfrac{2x-3}{x^2}$ , with $f'(x)=\dfrac{-2x+6}{x^3}$ , find the equation of the tangent to the curve at $x=2$ .
The complex number $u=\dfrac{5+4i}{3-2i}$ . Express $u$ in the form $x+iy$ .
For $u=\dfrac7{13}+\dfrac{22}{13}i$ , find the modulus $|u|$ .
For $u=\dfrac7{13}+\dfrac{22}{13}i$ , find the argument of $u$ in degrees.
The complex number $u\approx0.54+1.69i$ . What does the locus $|z-u|=2$ represent on an Argand diagram?
The function $f:x\mapsto\dfrac{m}{x}+n$ . State the domain of $f(x)$ .
The function $f:x\mapsto\dfrac{m}{x}+n$ satisfies $f(-1)=1\tfrac12$ and $f(2)=9$ . Find the values of $m$ and $n$ .
Given $f(x)=\dfrac5x+\dfrac{13}2$ , find an expression for $f^{-1}(x)$ .
Given $f^{-1}(x)=\dfrac{10}{2x-13}$ , evaluate $f^{-1}(4)$ .
Solve the inequality $\dfrac2{x-3}<\dfrac3{x-4}$ .
The point $(1,5)$ satisfies the simultaneous equations $n^2x+my=14$ and $ny-mx=8$ . Find one valid pair of values for $n$ and $m$ .
Find the equation of the curve which satisfies $\dfrac{dy}{dx}=-\dfrac{x}{y}$ and passes through $(-4,3)$ .
Express $2\cos\theta-\sqrt3\sin\theta$ in the form $R\sin(\theta-\alpha)$ , with $R>0$ and $0^\circ<\alpha<360^\circ$ .
Use the trapezium rule with 5 ordinates ( $h=0.25$ ) to find an approximate value of $\displaystyle\int_0^1x^2e^x\,dx$ , to 2 d.p.
Evaluate the exact value of $\displaystyle\int_0^1x^2e^x\,dx$ using integration by parts.
The trapezium rule gives $0.76$ for $\int_0^1x^2e^x\,dx$ , whose exact value is $e-2\approx0.718$ . Is this an over-estimation or under-estimation, and why?
Use the Newton-Raphson method, starting from $x_0=1$ , to find a root of $x^3+3x-7=0$ , correct to 2 d.p.
A region R is enclosed by the curve $y=\dfrac{a}{x}$ , where $a$ is a positive constant, the line $y=\dfrac12$ , the $y$ -axis and the line $y=2$ . Find the exact area of R, giving your answer in a form involving a single logarithm.
The region enclosed by $y=\dfrac{a}{x}$ , $y=\dfrac12$ , the $y$ -axis and $y=2$ has exact area $a\ln 4$ , where $a$ is a positive integer. Find the minimum value of $a$ for which this area exceeds 100.
The recurring decimal $5,4\dot{2}\dot{7}$ can be written as $5,4+0,027+0,00027+0,0000027+\ldots$ . What is the common ratio of the geometric series that follows the 5,4?
In a triangle ABC, $\hat{BAC}=\left(\dfrac{\pi}{2}+\theta\right)$ radians and $\hat{ACB}=\left(\theta-\dfrac{\pi}{2}\right)$ radians. Find $\hat{ABC}$ in radians.
In a proof by induction that $5^{2n}-3^{2n}$ is a multiple of 8 for every positive integer $n$ , the first step is to test $n=1$ . Evaluate $5^{2n}-3^{2n}$ when $n=1$ .
Angles $A$ and $B$ satisfy $A + B = 45^\circ$ , and it follows that $\tan A + \tan B + \tan A\tan B = 1$ . By putting $A = B = 22\tfrac12^\circ$ and writing $t = \tan 22\tfrac12^\circ$ , write down the resulting quadratic equation in $t$ .
Find $\int x^3\ln x\,dx$ .
Use the Newton-Raphson method to find, correct to 3 decimal places, the root of the equation $\sin x+\cos x=e^{x}-1$ that lies between $-4$ and $-3$ .
For $A=\begin{pmatrix}2&a\\0&1\end{pmatrix}$ it can be shown that $A^{n}=\begin{pmatrix}2^{n}&\left(2^{n}-1\right)a\\0&1\end{pmatrix}$ for every positive integer $n$ . State the top right entry of $A^{5}$ when $a=3$ .
A farmer has 56 m of mesh wire, which he uses as the complete perimeter of a rectangular pen for his goats. The width of the pen is $x$ m. Write down an expression, in terms of $x$ , for the area $A$ of the pen.
Given that $y=e^{x}\sin x$ , find $\dfrac{d^{2}y}{dx^{2}}$ , giving your answer in its simplest form.
Evaluate $\displaystyle\int_{\pi}^{\frac{3\pi}{2}}\frac{\cos x}{3+\cos^{2}x}\,dx$ , using the substitution $u=\sin x$ . Give the exact answer as a single natural logarithm.
Express $\dfrac{3x^{2}+6x+7}{x^{2}+2x+1}$ in the form $\dfrac{\mathrm{A}}{(x+\mathrm{B})^{2}}+\mathrm{C}$ , stating the values of the constants A, B and C.
For the statement $\displaystyle\sum_{r=1}^{n}(r+1)2^{r}=n2^{n+1}$ , evaluate the left hand side when $n=1$ .
ABCD is a square of side $x$ cm, drawn with B and C as the upper vertices and A and D as the lower vertices, and O is the midpoint of AD. Find angle AOB in radians, correct to 3 decimal places.
Simplify $\dfrac{3^{2x-3}\cdot81^{x+\frac12}}{27^x}$ , giving the answer as a single power of 3.
The extension $y$ of an elastic string varies directly as the magnitude of the force $F$ extending it. Given $y=0.45$ m when $F=6$ N, express $F$ as a function of $y$ .

Other papers in this subject

Paper 1

Track every paper you sit

Danho marks each sitting as sat, passed or still to try, and keeps the questions you got wrong in one place to come back to.

Get it on Google Play Download on the App Store

A Level Pure Mathematics Paper 2 Past Papers, 2008 to 2025

Sit the newest paper

Every sitting we hold