Edexcel GCSE Maths exam readiness check

Question 1

Work out $(3.2 	imes 10^{5}) 	imes (4 	imes 10^{-2})$.

Give your answer in standard form.

Accepted Answer

Multiply the numbers: $3.2 	imes 4 = 12.8$.

Add the indices: $10^{5} 	imes 10^{-2} = 10^{3}$.

So the product is $12.8 	imes 10^{3}$, which is not yet standard form.

$12.8 	imes 10^{3} = 1.28 	imes 10^{4}$.

Question 2

Solve $3x^{2} - 7x - 5 = 0$.

Give your solutions correct to 3 significant figures.

Accepted Answer

$a = 3$, $b = -7$, $c = -5$.

$x = \dfrac{7 \pm \sqrt{49 + 60}}{6} = \dfrac{7 \pm \sqrt{109}}{6}$.

$\sqrt{109} = 10.4403...$

$x = \dfrac{17.4403}{6} = 2.9067... = 2.91$ (3 s.f.)

$x = \dfrac{-3.4403}{6} = -0.5734... = -0.573$ (3 s.f.)

Question 3

The value of a car falls by 18% in its first year and by a further 12% in its second year.

At the end of two years the car is worth AED 108 240.

Work out the value of the car when it was new.

Accepted Answer

Falling by 18% means multiplying by $0.82$; falling by a further 12% means multiplying by $0.88$.

Combined multiplier: $0.82 	imes 0.88 = 0.7216$.

So $	ext{new value} 	imes 0.7216 = 108\,240$.

$	ext{new value} = 108\,240 \div 0.7216 = 150\,000$.

The car was worth **AED 150 000** when new.

Question 4

A right-angled triangle has legs of length $\sqrt{12}$ cm and $\sqrt{27}$ cm.

(a) Show that the area of the triangle is $9$ cm².

(b) Work out the length of the hypotenuse, giving your answer in the form $a\sqrt{b}$ where $b$ is as small as possible.

Accepted Answer

$\sqrt{12} = \sqrt{4 	imes 3} = 2\sqrt{3}$ and $\sqrt{27} = \sqrt{9 	imes 3} = 3\sqrt{3}$.

(a) Area $= 	frac{1}{2} 	imes 2\sqrt{3} 	imes 3\sqrt{3} = 	frac{1}{2} 	imes 6 	imes 3 = 9$ cm².

(b) $h^{2} = 12 + 27 = 39$, so $h = \sqrt{39}$ cm.

Question 5

Two mathematically similar bottles have heights 12 cm and 18 cm.

The smaller bottle has a surface area of 240 cm² and holds 400 ml.

Work out the surface area and the capacity of the larger bottle.

Accepted Answer

Length scale factor $k = \dfrac{18}{12} = 1.5$.

Area scale factor $= k^{2} = 2.25$, so surface area $= 240 	imes 2.25 = 540$ cm².

Volume scale factor $= k^{3} = 3.375$, so capacity $= 400 	imes 3.375 = 1350$ ml.

Edexcel GCSE Maths — free exam readiness check