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Exam-Style Questions on Hidden QuadraticProblems on Hidden Quadratic adapted from questions set in previous Mathematics exams. |
1. | GCSE Higher [807] |
Show your working as you find the value of \(w\) when \(y = 10\) and:
$$ y = 6x^4 + 7x^2 \text{ and } x = \sqrt{w + 1}. $$
2. | GCSE Higher [196] |
If \(y = 5x^4 + 3x^2\) and \(x=\sqrt{w+2}\), find \(w\) when \(y = 12\) showing each step of your working.
3. | IB Analysis and Approaches [577] |
(a) Show that the equation \( 2 \sin^2 x - 5 \cos x = -1\) may be written in the form \( 2 \cos^2 x + 5 \cos x = 3\)
(b) Hence, solve the equation \( 2 \sin^2 x - 5 \cos x = -1 \), \( 2\pi \lt x \lt 4\pi \).
4. | A-Level [808] |
(a) Show that the equation \[\sin^2\theta = 4\cos^2\theta - 3\sin\theta\cos\theta\] can be written as a quadratic equation in \(\tan\theta\).
(b) Hence, or otherwise, solve the equation in part (a) for \(0° \leqslant \theta \leqslant 180°\).
5. | IB Analysis and Approaches [710] |
Find the range of possible values of \(a\) such that \(5e^x - \ln{a} = e^{2x}\) has at least one real solution.
6. | A-Level [809] |
(a) Show that \(2\tan x - \cot x = 5\csc x\) may be written in the form
\[a\cos^2 x + b\cos x + c = 0\]stating the values of the constants \(a\), \(b\) and \(c\).
(b) Hence solve, for \(0 \leqslant x < 2\pi\), the equation
\[2\tan x - \cot x = 5\csc x\]giving your answers to 3 significant figures.
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