$\require{\cancel}$ $\require{\stix[upint]}$

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 | 4 | 3 | 3 | 4 | 7 | 8 | 7 | 9 | 10 | 10 | 10 | 10 | 85 |

Score |

Question 1 Code: 9709/33/M/J/11/1, Topic: Logarithmic and exponential functions

Use logarithms to solve the equation $5^{2 x-1}=2\left(3^{x}\right)$, giving your answer correct to 3 significant figures. $[4]$

Question 2 Code: 9709/31/M/J/13/1, Topic: Algebra

Find the quotient and remainder when $2 x^{2}$ is divided by $x+2$. $[3]$

Question 3 Code: 9709/33/M/J/17/1, Topic: Trigonometry

Prove the identity $\displaystyle\frac{\cot x-\tan x}{\cot x+\tan x} \equiv \cos 2 x$. $[3]$

Question 4 Code: 9709/33/M/J/17/3, Topic: Logarithmic and exponential functions

Using the substitution $u=\mathrm{e}^{x}$, solve the equation $4 \mathrm{e}^{-x}=3 \mathrm{e}^{x}+4$. Give your answer correct to 3 significant figures. $[4]$

Question 5 Code: 9709/31/M/J/14/5, Topic: Complex numbers

The complex number $z$ is defined by $\displaystyle z=\frac{9 \sqrt{3}+91}{\sqrt{3}-\mathrm{i}}$. Find, showing all your working,

$\text{(i)}$ an expression for $z$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$, where $r>0$ and $-\pi < \theta \leqslant \pi$, $[5]$

$\text{(ii)}$ the two square roots of $z$, giving your answers in the form $r \mathrm{e}^{\mathrm{i} \theta}$, where $r>0$ and $-\pi < \theta \leqslant \pi$. $[3]$

Question 6 Code: 9709/32/M/J/15/5, Topic: Numerical solutions of equations

The diagram shows a circle with centre $O$ and radius $r$. The tangents to the circle at the points $A$ and $B$ meet at $T$, and the angle $A O B$ is $2 x$ radians. The shaded region is bounded by the tangents $A T$ and $B T$, and by the minor $\operatorname{arc} A B$. The perimeter of the shaded region is equal to the circumference of the circle.

$\text{(i)}$ Show that $x$ satisfies the equation $[3]$

$$ \tan x=\pi-x $$$\text{(ii)}$ This equation has one root in the interval $0 < x < \frac{1}{2} \pi$. Verify by calculation that this root lies between 1 and $1.3$. $[2]$

$\text{(iii)}$ Use the iterative formula

$$ \displaystyle x_{n+1}=\tan ^{-1}\left(\pi-x_{n}\right) $$to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $[3]$

Question 7 Code: 9709/33/M/J/20/6, Topic: Numerical solutions of equations

$\text{(a)}$ By sketching a suitable pair of graphs, show that the equation $x^{5}=2+x$ has exactly one real root. $[2]$

$\text{(b)}$ Show that if a sequence of values given by the iterative formula

$$ \displaystyle x_{n+1}=\frac{4 x_{n}^{5}+2}{5 x_{n}^{4}-1} $$converges, then it converges to the root of the equation in part $\text{(a)}$. $[2]$

$\text{(c)}$ Use the iterative formula with initial value $x_{1}=1.5$ to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$

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: 9709/31/M/J/13/8, Topic: Integration

$\text{(a)}$ Show that $\displaystyle\int_{2}^{4} 4 x \ln x \mathrm{~d} x=56 \ln 2-12$. $[5]$

$\text{(b)}$ Use the substitution $u=\sin 4 x$ to find the exact value of $\displaystyle\int_{0}^{\frac{1}{24} \pi} \cos ^{3} 4 x \mathrm{~d} x$. $[5]$

Question 10 Code: 9709/33/M/J/10/9, Topic: Algebra

$\text{(i)}$ Express $\displaystyle\frac{4+5 x-x^{2}}{(1-2 x)(2+x)^{2}}$ in partial fractions. $[5]$

$\text{(ii)}$ Hence obtain the expansion of $\displaystyle\frac{4+5 x-x^{2}}{(1-2 x)(2+x)^{2}}$ in ascending powers of $x$, up to and including the term in $x^{2}$. $[5]$

Question 11 Code: 9709/33/M/J/13/9, Topic: Differentiation, Integration

The diagram shows the curve $y=\sin ^{2} 2 x \cos x$ for $0 \leqslant x \leqslant \frac{1}{2} \pi$, and its maximum point $M$.

$\text{(i)}$ Find the $x$-coordinate of $M$. $[6]$

$\text{(ii)}$ Using the substitution $u=\sin x$, find by integration the area of the shaded region bounded by the curve and the $x$-axis. $[4]$

Question 12 Code: 9709/32/M/J/18/10, Topic: Vectors

Two lines $l$ and $m$ have equations $\mathbf{r}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}+s(2 \mathbf{i}+3 \mathbf{j}-\mathbf{k})$ and $\mathbf{r}=\mathbf{i}+3 \mathbf{j}+4 \mathbf{k}+t(\mathbf{i}+2 \mathbf{j}+\mathbf{k})$ respectively.

$\text{(i)}$ Show that the lines are skew. $[4]$

A plane $p$ is parallel to the lines $l$ and $m$.

$\text{(ii)}$ Find a vector that is normal to $p$. $[3]$

$\text{(iii)}$ Given that $p$ is equidistant from the lines $l$ and $m$, find the equation of $p$. Give your answer in the form $a x+b y+c z=d$. $[3]$