Home » 9709 » Paper 32 Feb Mar 2020 Pure Math III – 9709/32/F/M/20

1.     (a)     Sketch the graph of $ y \ = \ |x \ – \ 2|. $ $$\tag*{[1]} $$
        (b)     Solve the inequality $ |x \ – \ 2| \ < \ 3x \ - \ 4. $ $$\tag*{[3]} $$ 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.
$$\tag*{[4]} $$

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

4.       Find $$ \ \displaystyle \int_{\frac{1}{6} π}^{\frac{1}{3} π} x \sec^{2} ⁡x \ \textrm{d}x. $$
Give your answer in a simplified exact form.
$$ \tag*{[7]} $$

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5.     (a)     Show that $$ \frac{\cos ⁡3x}{\sin ⁡x} + \frac{\sin ⁡3x}{\cos ⁡x} \ = \ 2 \cot ⁡2x $$
$$ \tag*{[4]} $$

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

for $ {\small \ 0 < x < \pi }.$ $$ \tag*{[3]} $$ Check here for my solution

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

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

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

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

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

9709/32/F/M/20 – Paper 32 Feb Mar 2020 No 8

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

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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. $$\tag*{[5]}$$
    (b)     Hence obtain the expansion of $ {\small f(x) }$ in ascending powers of $ x $, up to and including the term in $x^{2}$. $$\tag*{[5]}$$

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

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

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