1.     (a)     Sketch the graph of $y=|x–2|.$ $$\begin{array}{}\text{[1]}& \end{array}$$
        (b)     Solve the inequality $|x–2|<3x-4.$ $$\begin{array}{}\text{[3]}& \end{array}$$ Check here for my solution

2.       Solve the equation $$\ln 3+\ln (2x+5)=2\ln (x+2).$$
Give your answer in a simplified exact form.
$$\begin{array}{}\text{[4]}& \end{array}$$

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3.     (a)     By sketching a suitable pair of graphs, show that the equation $\sec x=2–\frac{1}{2}x$ has exactly one root in the interval $0\le x<\frac{1}{2}\pi .$ $$\begin{array}{}\text{[2]}& \end{array}$$         (b)     Verify by calculation that this root lies between 0.8 and 1. $$\begin{array}{}\text{[2]}& \end{array}$$         (c)     Use the iterative formula $x_{n+1}={\cos }^{-1}(\frac{2}{4-x_n})$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $$\begin{array}{}\text{[3]}& \end{array}$$ Check here for my solution

4.       Find $${\displaystyle {\int }_{\frac{1}{6}\pi }^{\frac{1}{3}\pi }x{\sec }^{2}x\text{d}x.}$$
Give your answer in a simplified exact form.
$$\begin{array}{}\text{[7]}& \end{array}$$

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5.     (a)     Show that $$\frac{\cos 3x}{\sin x}+\frac{\sin 3x}{\cos x}=2\cot 2x$$
$$\begin{array}{}\text{[4]}& \end{array}$$

        (b)     Hence solve the equation $$\frac{\cos 3x}{\sin x}+\frac{\sin 3x}{\cos x}=4,$$

for $0<x<\pi .$ $$\begin{array}{}\text{[3]}& \end{array}$$ Check here for my solution

6. The variables $x$ and $y$ satisfy the differential equation
$$\frac{\text{d}y}{\text{d}x}=\frac{1+4y^2}{e^x}.$$
It is given that $y=0$ when $x=1$.

    (a)     Solve the differential equation, obtaining an expression for $y$ in terms of $x$.
$$\begin{array}{}\text{[7]}& \end{array}$$
      (b)     State what happens to the value of $y$ as $x$ tends to infinity.
$$\begin{array}{}\text{[1]}& \end{array}$$

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7. The equation of a curve is $$x^3+3x{y}^2–y^3=5.$$

    (a)     Show that $\frac{\text{d}y}{\text{d}x}=\frac{x^2+y^2}{y^2–2xy}.$
$$\begin{array}{}\text{[4]}& \end{array}$$
    (b)     Find the coordinates of the points on the curve where the tangent is parallel to the $y$-axis.
$$\begin{array}{}\text{[5]}& \end{array}$$

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8. In the diagram, $OABCDEFG$ is a cuboid in which $\overrightarrow{OA}=2$ units, $\overrightarrow{OC}=3$ units and $\overrightarrow{OD}=2$ units. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $\overrightarrow{OA},\overrightarrow{OC}$ and $\overrightarrow{OD}$ respectively. The point $M$ on $AB$ is such that $MB=2AM$. The midpoint of $FG$ is $N$.

    (a)     Express the vectors $\overrightarrow{OM}$ and $\overrightarrow{MN}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$.
$$\begin{array}{}\text{[3]}& \end{array}$$
      (b)     Find a vector equation for the line through $M$ and $N$.
$$\begin{array}{}\text{[2]}& \end{array}$$
      (c)     Find the position vector of $P$, the foot of the perpendicular from $D$ to the line through $M$ and $N$.
$$\begin{array}{}\text{[4]}& \end{array}$$

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9.       Let $f(x)=\frac{2+11x–10x^2}{(1+2x)(1–2x)(2+x)}.$

    (a)     Express $f(x)$ in partial fractions. $$\begin{array}{}\text{[5]}& \end{array}$$
    (b)     Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$. $$\begin{array}{}\text{[5]}& \end{array}$$

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10.     (a)   The complex numbers $v$ and $w$ satisfy the equations
$$v+iw=5\text{and}(1+2i)v–w=3i.$$

Solve the equations for $v$ and $w$, giving your answers in the form $x+iy$, where $x$ and $y$ are real.
$$\begin{array}{}\text{[6]}& \end{array}$$
    (b)   (i)   On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $|z–2–3i|=1.$
$$\begin{array}{}\text{[2]}& \end{array}$$
        (ii)   Calculate the least value of arg $z$ for points on this locus.
$$\begin{array}{}\text{[2]}& \end{array}$$

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