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1. Use logarithms to solve the equation $5^{3−2x} = 4(7^{x})$, giving your answer correct to 3 decimal places.

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2. Show that $ \displaystyle \int_{0}^{\frac{1}{4}\pi} x^{2} \cos 2x \textrm{d}x = \frac{1}{32}(\pi^{2} \ – \ 8)$.

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3. Let $\displaystyle f(\theta) = \frac{1 \ – \ \cos 2\theta + \sin 2\theta}{1 + \cos 2\theta + \sin 2\theta}$.

(i) Show that $f(\theta) = \tan \theta$.

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(ii) Hence show that $ \displaystyle \int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} f(\theta) \textrm{d}\theta = \frac{1}{2}\ln \frac{3}{2}.$

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4. The equation of a curve is $\displaystyle y = \frac{1 + \textrm{e}^{-x}}{1 \ – \ \textrm{e}^{-x} }$, for $x \gt 0$.

(i) Show that $\frac{\textrm{d}y}{\textrm{d}x}$ is always negative.

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(ii) The gradient of the curve is equal to −1 when $x = a$. Show that $ \ a \ $ satisfies the equation $e^{2a} − 4e^{a} + 1 = 0$. Hence find the exact value of $a$.

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5. The variables x and y satisfy the differential equation

$$\displaystyle (x+1)y \frac{\textrm{d}y}{\textrm{d}x}= y^2 + 5. $$

It is given that $y = 2$ when $x = 0$. Solve the differential equation obtaining an expression for $y^{2}$ in terms of $x$.

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6. The diagram shows the curve $y = x^4 − 2x^3 − 7x − 6$. The curve intersects the $x$-axis at the points $(a, 0)$ and $(b, 0)$, where $a \lt b$. It is given that $b$ is an integer.

(i) Find the value of $b$.

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(ii) Hence show that $ \ a \ $ satisfies the equation $a = -\frac{1}{3}(2+a^2+a^3).$

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(iii) Use an iterative formula based on the equation in part (ii) to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

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7. The curve $y = \sin (x + \frac{1}{3} \pi) \cos x$ has two stationary points in the interval $0 \leq x \leq \pi.$

(i) Find $\frac{\textrm{d}y}{\textrm{d}x}.$

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(ii) By considering the formula for $ \cos(A + B) $, show that, at the stationary points on the curve, $ \cos (2x + \frac{1}{3} \pi) = 0.$

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(iii) Hence find the exact $x$-coordinates of the stationary points.

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8. The complex number $u$ is defined by

$$ u = \frac{4i}{1 \ – \ (\sqrt{3})i}.$$

(i) Express $u$ in the form $x + iy$, where $x$ and $y$ are real and exact.

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(ii) Find the exact modulus and argument of $u$.

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(iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $|z| \lt 2$ and $|z − u| \lt |z|$.

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9. Let $ \displaystyle f(x) = \frac{2x (5 \ – \ x)}{(3+x)(1 \ – \ x)^{2}}. $

(i) Express $f(x)$ in partial fractions.

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(b) Hence obtain the expansion of $f(x)$ in ascending powers of $x$ up to and including the term in $x^3$.

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10. The line $l$ has equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu (-2\mathbf{i} − \mathbf{j} − 2\mathbf{k}).$

(i) The point $P$ has position vector $ 4\mathbf{i} + 2\mathbf{j} \ – \ 3\mathbf{k}$. Find the length of the perpendicular from $P$ to $l$.

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(ii) It is given that $l$ lies in the plane with equation $ax + by + 2z = 13$, where $a$ and $b$ are constants. Find the values of $a$ and $b$.

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