A Level Pure Mathematics Practice Questions

Solve the inequality $|3^{-3x} - 54| < 27$.

In triangle $ABC$ the sides $BC$, $CA$ and $AB$ are $a$, $b$ and $c$ respectively, and $\cot C = \dfrac{a}{c}\operatorname{cosec} B - \cot B$. Given $a = 14.7$ cm, $c = 17.3$ cm and $B = 64.2°$, find angle $C$ correct to the nearest $0.1°$.

In triangle $ABC$, $a = 14.7$ cm, $c = 17.3$ cm, $B = 64.2°$ and $C = 65.3°$. Find angle $A$ correct to the nearest $0.1°$.

The diagram shows the graph of $y = f(x)$, passing through $A(-1, 0)$, $B(0, 1)$ and $C(1, 0)$. On the graph of $y = f(-x) + 3$, the image of $B$ is

A teacher earns $\$x$ in his first year and his salary increases each year by 10% of his first year's salary. His total salary after $n$ years is

A teacher's total salary after $n$ years is $\dfrac{nx(n+19)}{20}$, where $\$x$ is his first year's salary. Find the least value of $n$ for which his total salary exceeds 100 times his first salary.

What is the common difference of the salary arithmetic progression, in terms of $x$?

From $\dfrac{nx(n+19)}{20}>100x$, cancel $x$ and simplify to a quadratic inequality in $n$, in the form $n^2+bn+c>0$.

Find the set of values of $x$ which satisfy $|3x + 2| < -x + 1$.

In a circle of radius $r$ and centre $O$, the shaded region is bounded by the chord $AC$, the diameter $AB$ and the arc $BC$, with $\widehat{BAC} = \theta$ radians. Find an exact expression for the perimeter of the shaded region.

Given that $-4 < x < 2$ is the solution of the inequality $|x + a| < b$, find the values of $a$ and $b$.

In triangle $ABC$, $AB = m$ cm, $AC = 12$ cm and angle $B\hat{A}C = 60°$. Given that the area of the triangle is $\sqrt{675}$ cm$^2$, find the exact value of $m$.

Solve the simultaneous equations $2^x + 3^y = 5$ and $2^{x+2} - 3^{y+1} = 13$.

Solve the equation $5^{x-1} + 5^{x-2} = 30$.

Express $\dfrac{6}{3 + \sqrt{5} + \sqrt{14}}$ in the form $a + b\sqrt{c} + d\sqrt{e}$.

A polynomial $f(x)$ leaves remainder $-9$ when divided by $(x-3)$ and remainder $-6$ when divided by $(2x-1)$. Find the remainder when $f(x)$ is divided by $(x-3)(2x-1)$.

Find the inverse of $f(x) = ax + b$, where $a \ne 0$ and $b$ are constants.

The complex number $u = \dfrac{1}{-2+i}$. Find $|u|$.

Solve the inequality $|x-1| < x$.

Given $w = 1 + 2i$ and $u = 3 - i$, express $uw$ in the form $a + ib$.

Solve the equation $e^{2x} = 4e^{2-x}$, giving your answer in exact form.

The equation $x^3 - 2x^2 + 4x - 8 = 0$ has a root $x = 2i$. Find the other two roots.

Find, in terms of $k$, the equation of the perpendicular bisector of the line joining $A(2k, k)$ and $B(4k, 9k)$.

Solve the equation $3^{x-1} + 3^{x-2} = 12$.

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 $y$.