$\require{\cancel}$ $\require{\stix[upint]}$
Name of student | Date | ||||
Adm. number | Year/grade | Stream | |||
Subject | Pure Mathematics 1 (P1) | Variant(s) | P11, P12, P13 | ||
Start time | Duration | Stop time |
Qtn No. | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|
Marks | 4 | 6 | 6 | 5 | 8 | 29 |
Score |
Question 1 Code: 9709/13/M/J/16/2, Topic: Integration
The diagram shows part of the curve $\displaystyle y=\left(x^{3}+1\right)^{\frac{1}{2}}$ and the point $P(2,3)$ lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$
Question 2 Code: 9709/12/M/J/17/3, Topic: Trigonometry
$\text{(i)}$ Prove the identity $\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2} \equiv \frac{1-\sin \theta}{1+\sin \theta}$. $[3]$
$\text{(ii)}$ Hence solve the equation $\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2}=\frac{1}{2}$, for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$
Question 3 Code: 9709/13/M/J/14/4, Topic: Trigonometry
$\text{(i)}$ Prove the identity $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$. $[3]$
$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$ for $0 \leqslant x \leqslant 2 \pi$. $[3]$
Question 4 Code: 9709/12/M/J/19/4, Topic: Trigonometry
Angle $x$ is such that $\sin x=a+b$ and $\cos x=a-b$, where $a$ and $b$ are constants.
$\text{(i)}$ Show that $a^{2}+b^{2}$ has a constant value for all values of $x$. $[3]$
$\text{(ii)}$ In the case where $\tan x=2$, express $a$ in terms of $b$. $[2]$
Question 5 Code: 9709/12/M/J/19/8, Topic: Vectors
The position vectors of points $A$ and $B$, relative to an origin $O$, are given by
$$ \overrightarrow{O A}=\left(\begin{array}{r} 6 \\ -2 \\ -6 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 3 \\ k \\ -3 \end{array}\right) $$where $k$ is a constant.
$\text{(i)}$ Find the value of $k$ for which angle $A O B$ is $90^{\circ}$. $[2]$
$\text{(ii)}$ Find the values of $k$ for which the lengths of $O A$ and $O B$ are equal. The point $C$ is such that $\overrightarrow{A C}=2 \overrightarrow{C B}$. $[2]$
$\text{(iii)}$ In the case where $k=4$, find the unit vector in the direction of $\overrightarrow{O C}$. $[4]$