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MATHEMATICS 9709

Cambridge International AS and A Level

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 6 7 8 9 10 11 12 Total
Marks 4 4 5 4 6 6 7 7 8 8 11 12 82
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
Attempt all the 12 questions

Question 1 Code: 9709/11/M/J/10/1, Topic: Trigonometry

The acute angle $x$ radians is such that $\tan x=k$, where $k$ is a positive constant. Express, in terms of $k$,

$\text{(i)}$ $\tan (\pi-x)$, $[1]$

$\text{(ii)}$ $\tan \left(\frac{1}{2} \pi-x\right)$, $[1]$

$\text{(iii)}$ $\sin x$. $[2]$

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

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$ and $(2,5)$ is a point on the curve. Find the equation of the curve. $[4]$

Question 3 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry

Find the coordinates of the point at which the perpendicular bisector of the line joining $(2,7)$ to $(10,3)$ meets the $x$-axis. $[5]$

Question 4 Code: 9709/12/M/J/15/2, Topic: Circular measure

 

In the diagram, $A Y B$ is a semicircle with $A B$ as diameter and $O A X B$ is a sector of a circle with centre $O$ and radius $r$. Angle $A O B=2 \theta$ radians. Find an expression, in terms of $r$ and $\theta$, for the area of the shaded region. $[4]$

Question 5 Code: 9709/13/M/J/19/3, Topic: Circular measure

 

The diagram shows triangle $A B C$ which is right-angled at $A$. Angle $A B C=\frac{1}{5} \pi$ radians and $A C=8 \mathrm{~cm}$. The points $D$ and $E$ lie on $B C$ and $B A$ respectively. The sector $A D E$ is part of a circle with centre $A$ and is such that $B D C$ is the tangent to the $\operatorname{arc} D E$ at $D$.

$\text{(i)}$ Find the length of $A D$. $[3]$

$\text{(ii)}$ Find the area of the shaded region. $[3]$

Question 6 Code: 9709/12/M/J/12/5, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\tan x+\frac{1}{\tan x} \equiv \frac{1}{\sin x \cos x}$. $[2]$

$\text{(ii)}$ Solve the equation $\displaystyle\frac{2}{\sin x \cos x}=1+3 \tan x$, for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$

Question 7 Code: 9709/12/M/J/15/7, Topic: Coordinate geometry

The point $C$ lies on the perpendicular bisector of the line joining the points $A(4,6)$ and $B(10,2)$. $C$ also lies on the line parallel to $A B$ through $(3,11)$.

$\text{(i)}$ Find the equation of the perpendicular bisector of $A B$. $[4]$

$\text{(ii)}$ Calculate the coordinates of $C$. $[3]$

Question 8 Code: 9709/12/M/J/19/7, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} \mathrm{f}: x & \mapsto 3 x-2, \quad x \in \mathbb{R} \\ \mathrm{g}: x & \mapsto \frac{2 x+3}{x-1}, \quad x \in \mathbb{R}, x \neq 1 \end{aligned} $$

$\text{(i)}$ Obtain expressions for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Solve the equation $\mathrm{fg}(x)=\frac{7}{3}$. $[3]$

Question 9 Code: 9709/12/M/J/13/8, Topic: Differentiation

The volume of a solid circular cylinder of radius $r \mathrm{~cm}$ is $250 \pi \mathrm{cm}^{3}$.

$\text{(i)}$ Show that the total surface area, $S \mathrm{~cm}^{2}$, of the cylinder is given by $[2]$

$$ S=2 \pi r^{2}+\frac{500 \pi}{r} $$

$\text{(ii)}$ Given that $r$ can vary, find the stationary value of $S$. $[4]$

$\text{(iii)}$ Determine the nature of this stationary value. $[2]$

Question 10 Code: 9709/13/M/J/15/8, Topic: Differentiation

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$ for $x>-1$.

$\text{(i)}$ Find $\mathrm{f}^{\prime}(x)$. $[3]$

$\text{(ii)}$ State, with a reason, whether $\mathrm{f}$ is an increasing function, a decreasing function or neither. $[1]$

The function $\mathrm{g}$ is defined by $\displaystyle\mathrm{g}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$ for $x<-1$

$\text{(iii)}$ Find the coordinates of the stationary point on the curve $y=\mathrm{g}(x)$. $[4]$

Question 11 Code: 9709/11/M/J/18/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-2 \\ &\mathrm{~g}: x \mapsto 4+x-\frac{1}{2} x^{2} \end{aligned} $$

$\text{(i)}$ Find the points of intersection of the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{g}(x)$. $[3]$

$\text{(ii)}$ Find the set of values of $x$ for which $\mathrm{f}(x)>\mathrm{g}(x)$. $[2]$

$\text{(iii)}$ Find an expression for $\mathrm{fg}(x)$ and deduce the range of $\mathrm{fg}$. $[4]$

The function h is defined by h : $x \mapsto 4+x-\frac{1}{2} x^{2}$ for $x \geqslant k$.

$\text{(iv)}$ Find the smallest value of $k$ for which $\mathrm{h}$ has an inverse. $[2]$

Question 12 Code: 9709/12/M/J/15/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x^{2}-6 x+5$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find the set of values of $p$ for which the equation $\mathrm{f}(x)=p$ has no real roots. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 2 x^{2}-6 x+5$ for $0 \leqslant x \leqslant 4$.

$\text{(ii)}$ Express $\mathrm{g}(x)$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

$\text{(iii)}$ Find the range of $\mathrm{g}$. $[2]$

The function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto 2 x^{2}-6 x+5$ for $k \leqslant x \leqslant 4$, where $k$ is a constant.

$\text{(iv)}$ State the smallest value of $k$ for which $\mathrm{h}$ has an inverse. $[1]$

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{h}^{-1}(x)$. $[3]$

Worked solutions: P1, P3 & P6 (S1)

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