Exam-Style Questions.

1.	GCSE Higher

Multiply out and simplify:

$$(x-6)^2$$

Worked Solution

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2.	GCSE Higher

Find the highest common factor of the following two expressions:

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

Worked Solution

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3.	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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4.	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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5.	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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6.	GCSE Higher

Factorise the following expression

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

Worked Solution

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7.	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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8.

A-Level

The function \(f\) is defined as \(f(x) = 12x^3 - 5x^2 -11x + 6 \).

(a) Use the Factor Theorem to show that \( (4x-3) \) is a factor of \(f(x)\)

(b) Express \(f(x)\) as a product of linear factors.

(c) The function \(g\) is defined as \( g( \theta )= 6 \cos \theta \cos 2\theta + 5 \sin^2 \theta - 5 \cos \theta + 1 \). Show that the function \( g( \theta ) \) can be written as \( f(x) \), where \( x = \cos \theta \).

(d) Hence solve the equation \( g(\theta ) = 0 \), giving your answers, in radians, in the interval \(0 \le \theta \le 2 \pi \).

Worked Solution

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