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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.

1.

GCSE Higher

Multiply out and simplify:

$$(x-6)^2$$

2.

GCSE Higher

Find the highest common factor of the following two expressions:

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

3.

GCSE Higher

Simplify then find the square root of this expression:

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

4.

GCSE Higher

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

 

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

 

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


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.

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

6.

GCSE Higher

Factorise the following expression

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

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 \(b\).


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 \(0 \le \theta \le 2 \pi \).


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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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