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MIKE STEPHEN

Cambridge International AS and A Level

Name of student MIKE STEPHEN Date
Adm. number Year/grade 1981 Stream Mike Stephen
Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 10 10 10 10 10 10 60
Score

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Question 1 Code: 9709/31/M/J/19/9, Topic: Vectors

 

The diagram shows a set of rectangular axes $O x, O y$ and $O z$, and four points $A, B, C$ and $D$ with position vectors $\overrightarrow{O A}=3 \mathbf{i}, \overrightarrow{O B}=3 \mathbf{i}+4 \mathbf{j}, \overrightarrow{O C}=\mathbf{i}+3 \mathbf{j}$ and $\overrightarrow{O D}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}$.

$\text{(i)}$ Find the equation of the plane $B C D$, giving your answer in the form $a x+b y+c z=d$. $[6]$

$\text{(ii)}$ Calculate the acute angle between the planes $B C D$ and $O A B C$. $[4]$

Question 2 Code: 9709/32/M/J/19/9, Topic: Vectors

The points $A$ and $B$ have position vectors $\mathbf{i}+2 \mathbf{j}-\mathbf{k}$ and $3 \mathbf{i}+\mathbf{j}+\mathbf{k}$ respectively. The line $l$ has equation $\mathbf{r}=2 \mathbf{i}+\mathbf{j}+\mathbf{k}+\mu(\mathbf{i}+\mathbf{j}+2 \mathbf{k})$

$\text{(i)}$ Show that $l$ does not intersect the line passing through $A$ and $B$. $[5]$

$\text{(ii)}$ The plane $m$ is perpendicular to $A B$ and passes through the mid-point of $A B$. The plane $m$ intersects the line $l$ at the point $P$. Find the equation of $m$ and the position vector of $P$. $[5]$

Question 3 Code: 9709/33/M/J/19/9, Topic: Algebra

Let $\displaystyle\mathrm{f}(x)=\frac{2 x(5-x)}{(3+x)(1-x)^{2}}$

$\text{(i)}$ Express $\mathrm{f}(x)$ in partial fractions. $[5]$

$\text{(ii)}$ Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$ up to and including the term in $x^{3}$. $[5]$

Question 4 Code: 9709/31/O/N/19/9, Topic: Trigonometry, Integration

$\text{(i)}$ By first expanding $\cos (2 x+x)$, show that $\cos 3 x \equiv 4 \cos ^{3} x-3 \cos x$. $[4]$

$\text{(ii)}$ Hence solve the equation $\cos 3 x+3 \cos x+1=0$, for $0 \leqslant x \leqslant \pi$. $[2]$

$\text{(iii)}$ Find the exact value of $\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \cos ^{3} x \mathrm{~d} x$. $[4]$

Question 5 Code: 9709/32/O/N/19/9, Topic: Integration, Numerical solutions of equations

It is given that $\displaystyle\int_{0}^{a} x \cos \frac{1}{3} x \mathrm{~d} x=3$, where the constant $a$ is such that $0 < a < \frac{3}{2} \pi$.

$\text{(i)}$ Show that $a$ satisfies the equation

$$ a=\frac{4-3 \cos \frac{1}{3} a}{\sin \frac{1}{3} a} $$

$\text{(ii)}$ Verify by calculation that $a$ lies between $2.5$ and 3.

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

Question 6 Code: 9709/33/O/N/19/9, Topic: Differential equations

The variables $x$ and $t$ satisfy the differential equation $\displaystyle 5 \displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=(20-x)(40-x).$ It is given that $x=10$ when $t=0$.

$\text{(i)}$ Using partial fractions, solve the differential equation, obtaining an expression for $x$ in terms of $t$. $[9]$

$\text{(ii)}$ State what happens to the value of $x$ when $t$ becomes large. $[1]$

Worked solutions: P1, P3 & P6 (S1)

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