Exam-Style Questions on Algebra

Solve the following linear equations showing clearly each step of your working.

(a) \( \frac{6}{x} = 24 \).

(b) \(11x - 5 = 9(x + 9) \).

Worked Solution

The diagram shows an isosceles triangle (not drawn to scale).

Angle ABC = angle ACB, AB = \(3x+8\) and AC = \(5x-7\).

Use an algebraic method to find the value of \(x\).

Worked Solution

The diagrams show a triangle and a rectangle (not drawn to scale).

The area of the rectangle is six times larger than the area of the triangle. Find the value of \( x \)

Worked Solution

The perimeter of the triangle is the same length as the perimeter of the square.

Find an expression for the length of one side of the square in terms of a. Give your answer in its simplest form.

Worked Solution

Multiply out and simplify:

Worked Solution

Aimee, Natasha and Ruby play hockey.

• Aimee has scored 6 more goals than Natasha.
• Ruby has scored 5 more goals than Aimee.

If altogether they have scored 35 goals, how many goals did they each score?

Worked Solution

Julie is on holiday and buys some fruit from the local market.

She buys \(x\) mangos, \((x+4)\) guavas and \((3x-5)\) bananas.

The total number of pieces of fruit Julie buys is 29.

(a) Find the number of mangos Julie buys.

(b) The cost of one mango is 35 baht.

The cost of one guava is 18 baht.

If the total cost of all the fruit is 455 baht, find the cost, in baht, of one banana.

(c) What is the equivalent cost of the fruit Julie has bought in British pounds if the exchange rate with Thai baht is currently:

£1 = 46 baht.

Worked Solution

(a) Simplify:

(b) Simplify:

(c) Solve:

Worked Solution

Luke, Leia and Han swim lengths of a swimming pool to raise money for charity.

• Luke swims 20 more lengths than Han
• Leia swims twice as many lengths as Han
• Altogether they swim 232 lengths.

How many lengths did each person swim?

Worked Solution

Simplify then find the square root of this expression:

Worked Solution

(a) The expression \( (x+1)(2x-3)(3x+4) \) can be written in the form \(ax^3 + bx^2 + cx + d \) where \(a, b, c\) and \(d\) are integers. Find the values of \(a, b, c\) and \(d\).

(b) Solve the following inequality:

Worked Solution

Two rectangles, not drawn to scale, are shown below. All measurements are in centimetres. Both rectangles have the same areas. Work out the perimeter of the rectangle on the left.

Worked Solution

The area of rectangle PQRS (not to scale) is 80cm².

(a) Show that \( x^2 + 14x = 40\).

(b) Find the value of \( x \) giving your answer correct to three significant figures.

Worked Solution

Expand and simplify the following expression:

Worked Solution

(a) Express the following as a single fraction in its simplest form.

(b) Expand and simplify the following:

Worked Solution

Show that you understand equations and inequalities by answering the following:

(a) Solve \(5x^2=80\)

(b) Solve \(8x + 2 \gt x + 7\)

(c) Write down the largest integer that satisfies \(8x - 2 \lt 25\)

(d) Solve the following pair of equations

Worked Solution

(a) Simplify \( 6(3a-2)-2(a-3) \)

(b) Simplify \( \frac{3x}{4} - \frac{2x}{3} \)

(c) Rearrange the formula to make m the subject:

(d) Solve \( x^8 = 5600 \)

(e) Solve \( |x+5| = 9 \)

(f) Solve by factorising:

Worked Solution

A red square has sides of length \(x\) cm. A blue rectangle has sides of length \(3x-1\) cm and \(x-2\) cm. The diagrams below are not drawn to scale.

The area of the rectangle is \(132\) cm² greater than the area of the square.

The difference between the perimeters of the two shapes is \(p\) cm.

Find the value of \(p\).

You must show all your working.

Worked Solution

A circle is drawn inside a square so that it touches all four sides of the square.

(a) If the sides of the square are each \(k\) mm in length and the area of the red shaded region is \(A\) mm² show that:

(b) Make \(k\) the subject of the formula \(4A=4k^2-\pi k^2\)

Worked Solution

One is added to the product of two consecutive positive even numbers. Show that the result is a square number.

Worked Solution

(a) Simplify the following expression.

(b) Make b the subject of the following formula.

Worked Solution

Rätselzeit school has an annual sports day in which the mathematics teachers are in charge of the scoring system.

The points scored for throwing the discus are worked out using the formula:

where \(p\) is the number of points scored when the discus is thrown a distance of \(d\) metres.

(a) How many points does Homer score for throwing the discus a distance of 52 metres?

(b) How far did Marge throw the discus if she scored 492 points?

The points scored for running 400 metres are worked out using the formula:

where \(p\) is the number of points scored when the time taken is \(t\) seconds.

(c) Lisa scores 1014 points for the 400 metres race. Work out the time, in seconds, it took Lisa.

The formula for the number of points scored in the 400 metres race should not be used for \(t\gt n\).

(d) State the value of \(n\) giving a reason for your answer.

Worked Solution

(a) Find the ratio \(a:b \) given the equation:

(b) Find the ratio \( c:d \) given the following equation where \( c \gt 0 \text{ and } d \gt 0 \).

Worked Solution

Betsy thinks that \((3x)^2\) is always greater than or equal to \(3x\).

Is she is correct?

Show your working to justify your decision

Worked Solution

Factorise the following expression

Worked Solution

Given that:

find the possible values of \(x\).

Worked Solution

Show that:

simplifies to \( \frac{a-x}{x+b}\) where \(a\) and \(b\) are integers.

Worked Solution

The expression below can be written as a single fraction in the form \( \dfrac{a-bx}{x^2-25} \) where \(a\) and \(b\) are integers.

Work out the value of \(a\) and the value of \(b\).

Worked Solution

If \(y = 5x^4 + 3x^2\) and \(x=\sqrt{w+2}\), find \(w\) when \(y = 12\) showing each step of your working.

Worked Solution