Exam-Style Questions.

1.	GCSE Higher

Multiply out and simplify:

$$(x-6)^2$$

Worked Solution

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2.	GCSE Higher

(a) Simplify:

$$ p^6 \times p^3$$

(b) Simplify:

$$ \dfrac{a^5b^7}{a^4b} $$

(c) Solve:

$$ \dfrac{2w}{9} \gt 10$$

Worked Solution

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3.	GCSE Higher

Find the highest common factor of the following two expressions:

$$ 8x^5y^3 $$ $$ 6x^2y $$

Worked Solution

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4.	GCSE Higher

Simplify then find the square root of this expression:

$$\frac{y}{(1-y)^2} - \frac{y}{1-y}$$

Worked Solution

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5.	GCSE Higher

(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:

$$(x-2)^2 \lt \frac{16}{49}$$

Worked Solution

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6.	GCSE Higher

(a) Simplify \( \left(\dfrac{3a}{a^3 - 3}\right)^0 \)

 

(b) Simplify \( \dfrac{9(2b-1)}{(2b-1)^2}\)

 

(c) Simplify \( (2c^3d^4)^5 \)

Worked Solution

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7.	GCSE Higher

$$ 9^{\frac{1}{3}} \times 3^n = 27^{\frac{4}{5}} $$

Work out the exact value of \(n\).

Worked Solution

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8.	GCSE Higher

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

$$ \dfrac{6}{x-3} - \dfrac{2}{x-1} $$

(b) Expand and simplify the following:

$$ (x+ 2)(3x-5)(5x+1) $$

Worked Solution

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9.	GCSE Higher

(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:

$$ w = \frac{m+n}{3-m} + 7 $$

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

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

(f) Solve by factorising:

$$ 9z^2 - 27z + 20 = 0 $$

Worked Solution

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10.	GCSE Higher

(a) The \(n\)th term of a sequence is \(2^n+2^{n+1}\)

Work out the 8th term of the sequence.

(b) The \(n\)th term of a different sequence is \(9(3^n + 3^{n+1})\)

Expand and express this expression as the sum of two powers of three.

Worked Solution

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11.	GCSE Higher

(a) Simplify the following expression.

$$ \frac{x^2 - 4}{3x^2 + 13x + 14}$$

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

$$ a = \frac{7(3b-c)}{b}$$

Worked Solution

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12.	GCSE Higher

(a) Without using a calculator, show that \(\sqrt{28}=2\sqrt7\)

(b) The point \(X\) is shown on the unit grid below. The point \(Y\) is \(\sqrt17\) units from \(X\) and lies on the intersection of two grid lines. Mark one possible position for \(Y\).

Worked Solution

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13.	GCSE Higher

If a, b and c are positive integers use the following statements to find the values of a, b and c.

$$ (ab^c)^3 = 27b^{21} $$ $$ b= 9a $$

Worked Solution

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14.	GCSE Higher

Factorise the following expression

$$6x^2-x-15$$

Worked Solution

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15.	GCSE Higher

Show that:

$$3 - (4x^2+11x+7) \div (x^2+4x+3) $$

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

Worked Solution

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16.	GCSE Higher

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.

$$ \frac{x-4}{x-5} - 2 + \frac{x+1}{x+5}$$

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

Worked Solution

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17.	GCSE Higher

\(y = a \times b^{x – 2}\) where \(a\) and \(b\) are numbers.

\(y = 5\) when \(x = 2\)

\(y = 0.005\) when \(x = 5\)

Work out the value of \(y\) when \(x = 4\)

Worked Solution

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