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

Name of student | Date | ||||

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

Subject | Pure Mathematics 1 (P1) | Variant(s) | P11 | ||

Start time | Duration | Stop time |

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

Marks | 5 | 5 | 5 | 6 | 7 | 7 | 6 | 10 | 11 | 11 | 14 | 87 |

Score |

Question 1 Code: 9709/11/M/J/20/2, Topic: Series

The coefficient of $\displaystyle \frac{1}{x}$ in the expansion of $\displaystyle \left(k x+\frac{1}{x}\right)^{5}+\left(1-\frac{2}{x}\right)^{8}$ is 74.

Find the value of the positive constant $k$. $[5]$

Question 2 Code: 9709/11/M/J/16/3, Topic: Integration

The diagram shows part of the curve $\displaystyle x=\frac{12}{y^{2}}-2$. The shaded region is bounded by the curve, the $y$-axis and the lines $y=1$ and $y=2$. Showing all necessary working, find the volume, in terms of $\pi$, when this shaded region is rotated through $360^{\circ}$ about the $y$-axis. $[5]$

Question 3 Code: 9709/11/M/J/20/3, Topic: Series

Each year the selling price of a diamond necklace increases by $5 \%$ of the price the year before. The selling price of the necklace in the year 2000 was $\$ 36000$.

$\text{(a)}$ Write down an expression for the selling price of the necklace $n$ years later and hence find the selling price in 2008. $[3]$

$\text{(b)}$ The company that makes the necklace only sells one each year. Find the total amount of money obtained in the ten-year period starting in the year 2000. $[2]$

Question 4 Code: 9709/11/M/J/19/5, Topic: Quadratics

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=-2 x^{2}+12 x-3$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $-2 x^{2}+12 x-3$ in the form $-2(x+a)^{2}+b$, where $a$ and $b$ are constants. $[2]$

$\text{(ii)}$ State the greatest value of $\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2 x+5$ for $x \in \mathbb{R}$. $[1]$

$\text{(iii)}$ Find the values of $x$ for which $\operatorname{gf}(x)+1=0$. $[3]$

Question 5 Code: 9709/11/M/J/16/7, Topic: Circular measure

In the diagram, $A O B$ is a quarter circle with centre $O$ and radius $r$. The point $C$ lies on the arc $A B$ and the point $D$ lies on $O B.$ The line $C D$ is parallel to $A O$ and angle $A O C=\theta$ radians.

$\text{(i)}$ Express the perimeter of the shaded region in terms of $r, \theta$ and $\pi$. $[4]$

$\text{(ii)}$ For the case where $r=5 \mathrm{~cm}$ and $\theta=0.6$, find the area of the shaded region. $[3]$

Question 6 Code: 9709/11/M/J/19/7, Topic: Vectors

The diagram shows a three-dimensional shape in which the base $O A B C$ and the upper surface $D E F G$ are identical horizontal squares. The parallelograms $O A E D$ and $C B F G$ both lie in vertical planes. The point $M$ is the mid-point of $A F$.

Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $O A$ and $O C$ respectively and the unit vector $\mathbf{k}$ is vertically upwards. The position vectors of $A$ and $D$ are given by $\overrightarrow{O A}=8 \mathbf{i}$ and $\overrightarrow{O D}=3 \mathbf{i}+10 \mathbf{k}$.

$\text{(i)}$ Express each of the vectors $\overrightarrow{A M}$ and $\overrightarrow{G M}$ in terms of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$. $[3]$

$\text{(ii)}$ Use a scalar product to find angle $G M A$ correct to the nearest degree. $[4]$

Question 7 Code: 9709/11/M/J/14/8, Topic: Vectors

8 Relative to an origin $O$, the position vectors of points $A$ and $B$ are given by $$ \overrightarrow{O A}=\left(\begin{array}{c} 3 p \\ 4 \\ p^{2} \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{c} -p \\ -1 \\ p^{2} \end{array}\right) $$$\text{(i)}$ Find the values of $p$ for which angle $A O B$ is $90^{\circ}$. $[3]$

$\text{(ii)}$ For the case where $p=3$, find the unit vector in the direction of $\overrightarrow{B A}$. $[3]$

Question 8 Code: 9709/11/M/J/21/8, Topic: Circular measure

The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre $C$. The boundary of the plate consists of two arcs $P S$ and $Q R$ of the original circle and two semicircles with $P Q$ and $R S$ as diameters. The radius of the circle with centre $C$ is $4 \mathrm{~cm}$, and $P Q=R S=4 \mathrm{~cm}$ also.

$\text{(a)}$ Show that angle $P C S=\frac{2}{3} \pi$ radians. $[2]$

$\text{(b)}$ Find the exact perimeter of the plate. $[3]$

$\text{(c)}$ Show that the area of the plate is $\left(\frac{20}{3}\pi + 8\sqrt{3}\right)$ cm$^{2}$. $[5]$

Question 9 Code: 9709/11/M/J/21/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:

$$ \begin{aligned} &\mathrm{f}(x)=(x-2)^{2}-4 \text { for } x \geqslant 2, \\ &\mathrm{~g}(x)=a x+2 \text { for } x \in \mathbb{R}, \end{aligned} $$where $a$ is a constant.

$\text{(a)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(b)}$ Find $\mathrm{f}^{-1}(x)$. $[2]$

$\text{(c)}$ Given that $a=-\frac{5}{3}$, solve the equation $\mathrm{f}(x)=\mathrm{g}(x)$. $[3]$

$\text{(d)}$ Given instead that $\operatorname{ggf}^{-1}(12)=62$, find the possible values of $a$. $[5]$

Question 10 Code: 9709/11/M/J/11/11, Topic: Functions, Coordinate geometry

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned} $$$\text{(i)}$ Find and simplify expressions for $\mathrm{fg}(x)$ and $\operatorname{gf}(x)$. $[2]$

$\text{(ii)}$ Hence find the value of $a$ for which $\mathrm{fg}(a)=\operatorname{gf}(a)$. $[3]$

$\text{(iii)}$ Find the value of $b(b \neq a)$ for which $\mathrm{g}(b)=b$. $[2]$

$\text{(iv)}$ Find and simplify an expression for $\mathrm{f}^{-1} \mathrm{~g}(x)$. $[2]$

The function $\mathrm{h}$ is defined by

$$ \mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0 $$$\text{(v)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[2]$

Question 11 Code: 9709/11/M/J/21/11, Topic: Differentiation, Integration

The equation of a curve is $y=2 \sqrt{3 x+4}-x$

$\text{(a)}$ Find the equation of the normal to the curve at the point $(4,4)$, giving your answer in the form $y=m x+c$ $[5]$

$\text{(b)}$ Find the coordinates of the stationary point. $[3]$

$\text{(c)}$ Determine the nature of the stationary point. $[2]$

$\text{(d)}$ Find the exact area of the region bounded by the curve, the $x$-axis and the lines $x = 0$ and $x = 4$. $[4]$