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Exam-Style Questions on Equations

Problems on Equations adapted from questions set in previous exams.

1.

GCSE Higher
Shapes

The trapezium and rectangle shown here have the same perimeters. The diagrams are not drawn to scale and the measurements are in centimetres. Calculate the area of the trapezium.


2.

GCSE Higher

Solve the simultaneous equations

$$5x+2y=0$$ $$5x-3y=15$$

3.

GCSE Higher

Solve the simultaneous equations:

$$3x-5y=18$$ $$4x+2y=11$$

4.

IGCSE Extended

(a) Show that the equation \(\frac{3}{x+1}+\frac{3x-9}{2}=1\) can be simplified to \(3x^2-8x-5=0\).

(b) Solve the equation \(3x^2-8x-5=0\) showing all of your working and giving answers to three significant figures.

(c) The total surface area of a cone with radius \(x\) and slant height \(8x\) is equal to the area of a circle with radius r. Show that \(r = 3x\).

[The curved surface area, \(A\), of a cone with radius \(r\) and slant height \(l\) is \(A=\pi rl\).]


5.

GCSE Higher

A rectangular sheet of paper can be cut into two identical rectangular pieces in two different ways, either by cutting along line A or by cutting along line B.

Rectangle

When the original sheet of paper is cut along line A, the perimeter of each of the two pieces is 56 cm.

When the original sheet of paper is cut along line B, the perimeter of each of the two pieces is 64 cm.

What is the perimeter of the original sheet of paper?


6.

IB Studies

A red rug has a width of \(x-3\) cm and a length of \(4x\) cm.

(a) Write down an expression for the area, A, in cm2, of the rug.

The area of the rug is 3240 cm2.

(b) Calculate the value of \(x\).

(c) Hence, write down the value of the length and of the width of the rug in centimetres.


7.

GCSE Higher

The diagram shows a circle with equation \(x^2+y^2=13\).

Diagram

A tangent to the circle touches the circle at the point J. The x-coordinate of J is 2. The tangent intersects the x-axis at K. Find the coordinates of the point K.


8.

GCSE Higher

(a) Find the coordinates of the point at which the curve \(y = k^x\) intersects the y-axis.

The equation of circle \(A\) is \(x^2+y^2=25\). This circle is translated by the vector \( \begin{pmatrix} 0 \\ 4 \\ \end{pmatrix} \) to give circle \(B\).

(b) Draw a sketch of circle \(B\) clearly labelling the points of intersection with the y-axis.


9.

GCSE Higher

The prices of two watches are in the ratio \(a:b\).

When the prices are both increased by £10, the ratio becomes \(5 : 7\).

When the prices are both reduced by £10, the ratio becomes \(1 : 3\).

Express the ratio \(a:b\) in its lowest terms.


10.

GCSE Higher

(a) The circumference of a circle is \(16 \pi \) cm and its centre is at the origin. Find the equation of the circle.

(b) The line \(12x+ ay = b\) is a tangent at the point (6, 5) to a different circle with centre at the origin. Find the values of a and b.


The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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