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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.



(a) Solve the following trigonometric equation for \(–360° \lt x \lt 360°\):

$$ 5 \sin^2 x + 2\sin x + 3 = 7 \cos^2 x $$

giving your answers to the nearest integer.

(b) Hence find the smallest positive solution of the equation

$$ 5 \sin^2(3\theta + 20°) + 2\sin (3\theta + 20°) + 3 = 7 \cos^2 (3\theta + 20°) $$

giving your answer to 2 decimal places.



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

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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