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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 | 6 | 6 | 6 | 4 | 8 | 7 | 7 | 11 | 10 | 9 | 9 | 87 |

Score |

Question 1 Code: 9709/13/M/J/21/2, Topic: Differentiation

The function $\mathrm{f}$ is defined by $\displaystyle \mathrm{f}(x)=\frac{1}{3}(2 x-1)^{\frac{3}{2}}-2 x$ for $\frac{1}{2}< x< a$. It is given that $\mathrm{f}$ is a decreasing function.

Find the maximum possible value of the constant $a$. $[4]$

Question 2 Code: 9709/11/M/J/17/3, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta} \equiv \frac{2}{\sin \theta}$. $[3]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{3}{\cos \theta}$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

Question 3 Code: 9709/11/M/J/10/4, Topic: Integration

The diagram shows the curve $y=6 x-x^{2}$ and the line $y=5$. Find the area of the shaded region. $[6]$

Question 4 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 5 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry

Points $A$ and $B$ have coordinates $(8,3)$ and $(p, q)$ respectively. The equation of the perpendicular bisector of $A B$ is $y=-2 x+4$.Find the values of $p$ and $q$. $[4]$

Question 6 Code: 9709/13/M/J/10/7, Topic: Circular measure

The diagram shows a metal plate $A B C D E F$ which has been made by removing the two shaded regions from a circle of radius 10 cm and centre $O.$ The parallel edges $A B$ and $E D$ are both of length 12 cm.

$\text{(i)}$ Show that angle $D O E$ is 1.287 radians, correct to 4 significant figures. $[2]$

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

$\text{(iii)}$ Find the area of the metal plate. $[3]$

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

A television quiz show takes place every day. On day 1 the prize money is $\$ 1000$. If this is not won the prize money is increased for day 2. The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money.

Model 1: Increase the prize money by $\$ 1000$ each day.

Model 2: Increase the prize money by $10 \%$ each day.

On each day that the prize money is not won the television company makes a donation to charity. The amount donated is $5 \%$ of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity

$\text{(i)}$ if Model 1 is used, $[4]$

$\text{(ii)}$ if Model 2 is used. $[3]$

Question 9 Code: 9709/13/M/J/10/9, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=x+\frac{4}{x}$ which has a minimum point at $M .$ The line $y=5$ intersects the curve at the points $A$ and $B$.

$\text{(i)}$ Find the coordinates of $A, B$ and $M$. $[5]$

$\text{(ii)}$ Find the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[6]$

Question 10 Code: 9709/11/M/J/17/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$ for $x \in \mathbb{R}, x \neq \frac{3}{2}$.

$\text{(i)}$ Find an expression for $\mathrm{f}^{-1}(x)$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 4 x+a$ for $x \in \mathbb{R}$, where $a$ is a constant.

$\text{(ii)}$ Find the value of $a$ for which $\operatorname{gf}(-1)=3$. $[3]$

$\text{(iii)}$ Find the possible values of $a$ given that the equation $\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$ has two equal roots. $[4]$

Question 11 Code: 9709/12/M/J/10/10, Topic: Differentiation

The equation of a curve is $y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.

$\text{(i)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $[3]$

$\text{(ii)}$ Find the equation of the tangent to the curve at the point where the curve intersects the $y$-axis. $[3]$

$\text{(iii)}$ Find the set of values of $x$ for which $\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$ is an increasing function of $x$. $[3]$

Question 12 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration

Points $A(2,9)$ and $B(3,0)$ lie on the curve $y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at $A$ intersects the $x$-axis at $C$. Showing all necessary working,$\text{(i)}$ find the equation of the tangent $A C$ and hence find the $x$-coordinate of $C$, $[4]$

$\text{(ii)}$ find the area of the shaded region $A B C$. $[5]$