Home » 9709 » Paper 33 May June 2019 Pure Math III – 9709/33/M/J/19

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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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