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1. Solve the inequality $2|3x \ − \ 1| \lt |x \ + \ 1|$.

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2. Find the real root of the equation $ {\large \frac{ 2{\mathrm{e}}^{x} \ + \ \mathrm{e}^{−x} }{ 2 \ + \ \mathrm{e}^x } } = 3$, giving your answer correct to 3 decimal places.

Your working should show clearly that the equation has only one real root.

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3. (a) Given that $ \cos(x \ − \ {30}^{\circ}) = 2 \sin(x \ + \ {30}^{\circ})$, show that $ \tan x = {\large \frac{2 \ – \ \sqrt{3}}{1 \ – \ 2\sqrt{3}} }$.

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(b) Hence solve the equation $ \cos(x \ − \ {30}^{\circ}) = 2 \sin(x \ + \ {30}^{\circ})$, for $ \ {0}^{\circ} \lt x \lt {360}^{\circ} $.

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4. (a) Prove that ${\large \frac{1 \ – \ \cos 2\theta}{1 \ + \ \cos 2\theta} } \equiv {\tan}^{2} \theta$ .

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(b) Hence find the exact value of $ \displaystyle \int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} \frac{1 \ – \ \cos 2\theta}{1 \ + \ \cos 2\theta} \ \mathrm{d}\theta$.

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5. (a) Solve the equation $ z^2 \ − \ 2p\mathrm{i}z \ − \ q = 0$, where $p$ and $q$ are real constants.

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In an Argand diagram with origin $O$, the roots of this equation are represented by the distinct points $A$ and $B$.

(b) Given that $A$ and $B$ lie on the imaginary axis, find a relation between $p$ and $q$.

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(c) Given instead that triangle $OAB$ is equilateral, express $q$ in terms of $p$.

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6. The parametric equations of a curve are

$$ x = \ln(2 \ + \ 3t), \enspace y = \frac{t}{2 \ + \ 3t}.$$

(a) Show that the gradient of the curve is always positive.

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(b) Find the equation of the tangent to the curve at the point where it intersects the $y$-axis.

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7.

The diagram shows the curve $ y = {\large \frac {\tan^{−1}x} {\sqrt{x}} }$ and its maximum point $M$ where $x = a$.

(a) Show that $a$ satisfies the equation

$$ a = \tan \Big( \frac{2a}{1 \ + \ a^2} \Big).$$

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(b) Verify by calculation that $a$ lies between l.3 and 1.5.

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(c) Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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8. With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $ \ {\small \overrightarrow{OA}} \ = \ \begin{pmatrix}

1 \\[1pt]

2 \\[1pt]

1

\end{pmatrix} $ and $ \ {\small \overrightarrow{OB}} \ = \ \begin{pmatrix}

3 \\[1pt]

1 \\[1pt]

-2

\end{pmatrix} $. The line $l$ has equation $ \ {\small \mathbf{r}} \ = \ \begin{pmatrix}

2 \\[1pt]

3 \\[1pt]

1

\end{pmatrix} \ + \ \lambda \begin{pmatrix}

1 \\[1pt]

-2 \\[1pt]

1

\end{pmatrix} $.

(a) Find the acute angle between the directions of $AB$ and $l$.

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(b) Find the position vector of the point $P$ on $l$ such that $AP = BP$.

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9. The equation of a curve is $ y = x^{ – \frac{2}{ 3}} \ \ln x$ for $x \gt 0$. The curve has one stationary point.

(a) Find the exact coordinates of the stationary point.

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(b) Show that $ \displaystyle \int_{1}^{8} y \ \mathrm{d}x = 18 \ln 2 \ – \ 9 $.

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10. The variables $x$ and $t$ satisfy the differential equation $ {\large \frac{\mathrm{d}x} {\mathrm{d}t} } = {x}^{2}(1 \ + \ 2x)$, and $x = 1$ when $t = 0$.

Using partial fractions, solve the differential equation, obtaining an expression for $t$ in terms of $x$.

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