Algebra

Write $m\times m\times n\times n\times n$ using indices.

Write $p\times p\times p\times q\div5$ in conventional algebraic notation, and state its coefficient.

A rectangle has length $\frac{3x}{2}$ and width $y$. Write its area and its perimeter in conventional algebraic notation.

Work out $2x^2-5$ when $x=-3$.

The kinetic energy of an object is given by $E=\frac12mv^2$. Work out $E$ when $m=3.2\,\text{kg}$ and $v=5\,\text{m s}^{-1}$.

Use $E=mc^2$ to calculate $E$ when $m=4.2\times10^{-8}\,\text{kg}$ and $c=3\times10^8\,\text{m s}^{-1}$. Give your answer in standard form.

State whether $5x+7$ is an expression, equation or inequality.

For $8x^2-12x=4x(2x-3)$, state the number of terms on the left and name the two factors on the right.

Classify each statement as an equation, an identity or an inequality: $3(2x-1)=9$, $3(2x-1)\equiv6x-3$, and $3(2x-1)<9$.

Simplify $7a+3b-2a+5b$.

Factorise $6x^2+x-2$.

Simplify $\frac{x^2-9}{x^2+x-6}$, stating every value of $x$ excluded from the original expression.

Make $w$ the subject of $A=lw$.

Make $t$ the subject of $v=u+at$.

Make $a$ the subject of $P=\frac{a+b}{a-b}$.

Show that $4(n+2)-3(n-1)$ is equivalent to $n+11$.

Prove algebraically that the sum of two consecutive integers is odd.

An odd number is written as $2n+1$. Demonstrate algebraically that squaring it leaves remainder $1$ after division by $8$.

Given $f(x)=3x-4$, work out $f(6)$.

Given $f(x)=5x+2$, find $f^{-1}(x)$ and work out $f^{-1}(27)$.

Let $f(x)=2x+3$ and $g(x)=x^2-1$. Solve $fg(x)=19$, where $fg(x)=f(g(x))$.

State the quadrant containing the point $(-4,3)$.

Find the midpoint of the line segment joining $(-5,2)$ and $(3,-4)$.

The point $P$ divides the line segment from $A(-6,5)$ to $B(4,-5)$ in the ratio $AP:PB=3:2$. Find the coordinates of $P$.

Write the equation of the line with gradient $4$ and $y$-intercept $-7$.

Find the equation of the line through $(-2,5)$ and $(4,-1)$.

Find the equation of the line through $(3,-5)$ that is perpendicular to $2x-3y=6$. Give your answer in the form $y=mx+c$.

State the gradient and $y$-intercept of $y=-3x+8$.

A straight line crosses the axes at $(0,12)$ and $(6,0)$. Find its gradient and both intercepts.

A straight-line graph of water volume $V$ litres against time $t$ minutes passes through $(0,120)$ and $(8,72)$. Find and interpret its gradient and $V$-intercept, then write $V$ in terms of $t$.

Find the roots and the $y$-intercept of $y=x^2-5x+6$.

For $y=(x-4)^2-9$, state the turning point and axis of symmetry, and find the roots.

Complete the square for $y=2x^2+8x-10$. Hence state the turning point and find the roots.

A graph passes through $(0,1)$ and its $y$-value doubles whenever $x$ increases by $1$. Name the function $y=2^x$ as linear, quadratic, cubic, reciprocal or exponential.

For the graph $y=\frac6x$, state both asymptotes and the two quadrants containing its branches.

For $y=\cos x$ on $-180^\circ\leq x\leq360^\circ$, list the $x$-intercepts and the coordinates of every maximum and minimum needed for an accurate sketch.

Describe fully the transformation from $y=f(x)$ to $y=f(x)+4$.

The point $(3,-2)$ lies on $y=f(x)$. Find the corresponding point on $y=f(-x)$ and name the transformation.

The point $(-1,5)$ lies on $y=f(x)$. Find the corresponding point on $y=f(2-x)-3$, and describe the reflection and translations that produce the new graph.

The time $t$ hours for a fixed journey is modelled by $t=24/v$, where $v$ is the average speed in km/h. Work out the point on this graph when $v=16$ and interpret it.

Higher only: Plot $y=2^x$ and $y=9-2x$ for $1\le x\le3$. Use the intersection to estimate the solution of $2^x=9-2x$ to one decimal place.

For $0\le t\le5$, two moving objects have distances from a marker modelled by $d=3t^2+2$ and $d=14t$, with $d$ in metres and $t$ in seconds. Draw both graphs and estimate the later time when the objects are equally far from the marker.

A distance-time graph is a straight line from $(12,150)$ to $(32,510)$, where time is in seconds and distance is in metres. Calculate and interpret its gradient.

A velocity-time graph joins the points $(0,0)$, $(6,15)$, $(14,15)$ and $(20,3)$ with straight lines. Work out the distance travelled in the first $20$ seconds.

A curved velocity-time graph passes through the values $v=4,9,18,32$ m/s at $t=0,2,4,6$ seconds. Use three trapezia to estimate the distance travelled. A tangent at $t=4$ passes through $(2,11)$ and $(6,25)$; estimate the acceleration then.

Write down the equation of the circle with centre $(0,0)$ and radius $7$.

The point $P(5,12)$ lies on the circle $x^2+y^2=169$. Determine the tangent's equation there.

The tangent to $x^2+y^2=50$ at $Q(1,7)$ meets the positive coordinate axes. Find the exact area of the triangle enclosed by the tangent and the axes.

Solve $7x+5=33$.

Draw $y=4.5x-2$ and $y=9.2-x$ on the same axes. Use the intersection to solve $4.5x-2=9.2-x$, giving an estimate to one decimal place.

Solve $\frac{3(x-2)}{4}-\frac{x+1}{3}=\frac56$.

Solve $x^2+2x-35=0$ by factorising.

Higher only: Solve $x^2-8x+3=0$ by completing the square. Give exact answers.

Higher only: The curves $y=x^2$ and $y=5x+1$ intersect twice. Use the quadratic formula to find the exact $x$-coordinates, then give the values a graph should show to two decimal places.

Solve simultaneously $2x+y=11$ and $x-y=1$.

On the same axes draw $y=0.6x+1$ and $y=5-0.8x$. Use the graph to estimate their point of intersection to one decimal place.

Higher only: Solve simultaneously $y=x+1$ and $x^2+y^2=25$.

The iteration $x_{n+1}=\sqrt{10-x_n}$ starts with $x_0=3$. Work out $x_1$ and $x_2$, giving each to three decimal places.

Use $x_{n+1}=\sqrt{(18-x_n)/2}$ with $x_0=3$ to find $x_4$. Give the result to four decimal places.

Let $f(x)=x^3+x-12$. Show that $f(x)=0$ has a root between $2$ and $3$. Starting with $x_0=2.2$, use $x_{n+1}=\sqrt[3]{12-x_n}$ to find this root to four decimal places.

A rectangle has width $x$ cm and length $(x+3)$ cm. Its perimeter is $34$ cm. Form and solve an equation to find both dimensions.

A club sells $38$ tickets. Adult tickets cost £7 and junior tickets cost £4. The total received is £203. Form two equations and find how many tickets of each type were sold.

Mira is $4$ years older than Theo. In $3$ years, the product of their ages will be $192$. Form an equation and find their current ages.

Higher only: Solve $3x-7\le11$. Give the answer in set notation and describe its number-line representation.

Higher only: On coordinate axes, show the region satisfying both $y\ge2x-1$ and $x+y<5$. State the intersection of the boundary lines and identify which boundaries are included.

Higher only: Solve $(2x+3)(x-2)>0$. Give the solution in set notation and describe it on a number line.

A sequence has position-to-term rule $6n-2$. Write down its first four terms.

A sequence starts $2,5$. Each later term is one more than the sum of the previous two terms. Write down the next three terms.

The $n$th term of a sequence is $n^2-3n+4$. Generate the first six terms.

The sequence $1,4,9,16,\ldots$ is made from a named type of number. Name the type and write down the next two terms.

A Fibonacci-type sequence begins $3,7,10,17,27,\ldots$, with each term after the second equal to the sum of the previous two. Find the eighth term.

Higher only: A geometric progression begins $2,2\sqrt3,6,6\sqrt3,\ldots$. State the common ratio and find the eighth term in exact form.

Find an expression for the $n$th term of $7,11,15,19,\ldots$.

Higher only: Find an expression for the $n$th term of $2,7,14,23,34,\ldots$.

Higher only: A quadratic sequence starts $4,15,32,55$. Deduce its $n$th term and determine the position of the term equal to $207$.