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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 3 (P3) | Variant(s) | P31, P32, P33 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 5 | 5 | 6 | 6 | 8 | 9 | 9 | 9 | 9 | 10 | 12 | 11 | 99 |

Score |

Question 1 Code: 9709/32/M/J/20/2, Topic: Logarithmic and exponential functions

The variables $x$ and $y$ satisfy the equation $y^{2}=A \mathrm{e}^{k x}$, where $A$ and $k$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(1.5,1.2)$ and $(5.24,2.7)$ as shown in the diagram.

Find the values of $A$ and $k$ correct to 2 decimal places. $[5]$

Question 2 Code: 9709/33/M/J/13/3, Topic: Trigonometry

Solve the equation $\tan 2 x=5 \cot x$, for $0^{\circ}< x <180^{\circ}$. $[5]$

Question 3 Code: 9709/31/M/J/10/4, Topic: Trigonometry, Integration

$\text{(i)}$ Using the expansions of $\cos (3 x-x)$ and $\cos (3 x+x)$, prove that $[4]$

$$ \frac{1}{2}(\cos 2 x-\cos 4 x) \equiv \sin 3 x \sin x $$$\text{(ii)}$ Hence show that $[4]$

$$ \displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \sin 3 x \sin x \mathrm{~d} x=\frac{1}{8} \sqrt{3} $$Question 4 Code: 9709/32/M/J/16/4, Topic: Differentiation

The curve with equation $\displaystyle y=\frac{(\ln x)^{2}}{x}$ has two stationary points. Find the exact values of the coordinates of these points. $[6]$

Question 5 Code: 9709/31/M/J/17/5, Topic: Numerical solutions of equations

The diagram shows a semicircle with centre $O$, radius $r$ and diameter $A B$. The point $P$ on its circumference is such that the area of the minor segment on $A P$ is equal to half the area of the minor segment on $B P$. The angle $A O P$ is $x$ radians.

$\text{(i)}$ Show that $x$ satisfies the equation $x=\frac{1}{3}(\pi+\sin x)$. $[3]$

$\text{(ii)}$ Verify by calculation that $x$ lies between 1 and $1.5.$ $[2]$

$\text{(iii)}$ Use an iterative formula based on the equation in part $\text{(i)}$ to determine $x$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$

Question 6 Code: 9709/32/M/J/11/6, Topic: Differential equations

A certain curve is such that its gradient at a point $(x, y)$ is proportional to $x y$. At the point $(1,2)$ the gradient is $4.$

$\text{(i)}$ By setting up and solving a differential equation, show that the equation of the curve is $y=2 \mathrm{e}^{x^{2}-1}$. $[7]$

$\text{(ii)}$ State the gradient of the curve at the point $(-1,2)$ and sketch the curve. $[2]$

Question 7 Code: 9709/31/M/J/13/6, Topic: Vectors

The points $P$ and $Q$ have position vectors, relative to the origin $O$, given by

$$ \overrightarrow{O P}=7 \mathbf{i}+7 \mathbf{j}-5 \mathbf{k} \quad \text { and } \quad \overrightarrow{O Q}=-5 \mathbf{i}+\mathbf{j}+\mathbf{k} $$The mid-point of $P Q$ is the point $A$. The plane $\Pi$ is perpendicular to the line $P Q$ and passes through $A$.

$\text{(i)}$ Find the equation of $\Pi$, giving your answer in the form $a x+b y+c z=d$. $[4]$

$\text{(ii)}$ The straight line through $P$ parallel to the $x$-axis meets $\Pi$ at the point $B$. Find the distance $A B$, correct to 3 significant figures. $[5]$

Question 8 Code: 9709/33/M/J/12/7, Topic: Integration, Numerical solutions of equations

The diagram shows part of the curve $y=\cos (\sqrt{x})$ for $x \geqslant 0$, where $x$ is in radians. The shaded region between the curve, the axes and the line $x=p^{2}$, where $p>0$, is denoted by $R$. The area of $R$ is equal to 1.

$\text{(i)}$ Use the substitution $x=u^{2}$ to find $\displaystyle\int_{0}^{p^{2}} \cos (\sqrt{x}) \mathrm{d} x$. Hence show that $\displaystyle \sin p=\frac{3-2 \cos p}{2 p}$. $[6]$

$\text{(ii)}$ Use the iterative formula $\displaystyle p_{n+1}=\sin ^{-1}\left(\frac{3-2 \cos p_{n}}{2 p_{n}}\right)$, with initial value $p_{1}=1$, to find the value of $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $[3]$

Question 9 Code: 9709/33/M/J/21/7, Topic: Differential equations

For the curve shown in the diagram, the normal to the curve at the point $P$ with coordinates $(x, y)$ meets the $x$-axis at $N.$ The point $M$ is the foot of the perpendicular from $P$ to the $x$-axis.

The curve is such that for all values of $x$ in the interval $0 \leqslant x < \frac{1}{2} \pi$, the area of triangle $P M N$ is equal to $\tan x$.

$\text{(a)} \, \, \text{(i)}$ Show that $\displaystyle\frac{M N}{y}=\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $[1]$

$\text{(ii)}$ Hence show that $x$ and $y$ satisfy the differential equation $\displaystyle\frac{1}{2} y^{2} \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\tan x$. $[2]$

$\text{(b)}$ Given that $y=1$ when $x=0$, solve this differential equation to find the equation of the curve, expressing $y$ in terms of $x$. $[6]$

Question 10 Code: 9709/31/M/J/13/9, Topic: Trigonometry, Integration

$\text{(i)}$ Express $4 \cos \theta+3 \sin \theta$ in the form $R \cos (\theta-\alpha)$, where $R>0$ and $0 < \alpha < \frac{1}{2} \pi$. Give the value of $\alpha$ correct to 4 decimal places. $[3]$

$\text{(ii)}$ Hence

$\text{(a)}$ solve the equation $4 \cos \theta+3 \sin \theta=2$ for $0< \theta <2 \pi$, $[4]$

$\text{(b)}$ find $\displaystyle\int \frac{50}{(4 \cos \theta+3 \sin \theta)^{2}} \mathrm{~d} \theta$. $[3]$

Question 11 Code: 9709/31/M/J/10/10, Topic: Vectors

The lines $l$ and $m$ have vector equations

$$ \mathbf{r}=\mathbf{i}+\mathbf{j}+\mathbf{k}+s(\mathbf{i}-\mathbf{j}+2 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=4 \mathbf{i}+6 \mathbf{j}+\mathbf{k}+t(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k}) $$respectively.

$\text{(i)}$ Show that $l$ and $m$ intersect. $[4]$

$\text{(ii)}$ Calculate the acute angle between the lines. $[3]$

$\text{(iii)}$ Find the equation of the plane containing $l$ and $m$, giving your answer in the form $a x+b y+c z=d$. $[5]$

Question 12 Code: 9709/32/M/J/17/10, Topic: Numerical solutions of equations, Integration

The diagram shows the curve $y=x^{2} \cos 2 x$ for $0 \leqslant x \leqslant \frac{1}{4} \pi .$ The curve has a maximum point at $M$ where $x=p$.

$\text{(i)}$ Show that $p$ satisfies the equation $\displaystyle p=\frac{1}{2} \tan ^{-1}\left(\frac{1}{p}\right)$. $[3]$

$\text{(ii)}$ Use the iterative formula $\displaystyle p_{n+1}=\frac{1}{2} \tan ^{-1}\left(\frac{1}{p_{n}}\right)$ to determine the value of $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $[3]$

$\text{(iii)}$ Find, showing all necessary working, the exact area of the region bounded by the curve and the $x$-axis. $[5]$