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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 | 5 | 5 | 6 | 5 | 7 | 6 | 6 | 8 | 7 | 6 | 10 | 11 | 82 |

Score |

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

The diagram shows a circle $C$ with centre $O$ and radius $3 \mathrm{~cm}$. The radii $O P$ and $O Q$ are extended to $S$ and $R$ respectively so that $O R S$ is a sector of a circle with centre $O$. Given that $P S=6 \mathrm{~cm}$ and that the area of the shaded region is equal to the area of circle $C$,

$\text{(i)}$ show that angle $P O Q=\frac{1}{4} \pi$ radians, $[3]$

$\text{(ii)}$ find the perimeter of the shaded region. $[2]$

Question 2 Code: 9709/11/M/J/13/3, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $8 \mathrm{~cm}$. Angle $B O A$ is $\alpha$ radians. $O A C$ is a semicircle with diameter $O A$. The area of the semicircle $O A C$ is twice the area of the sector $O A B$.

$\text{(i)}$ Find $\alpha$ in terms of $\pi$. $[3]$

$\text{(ii)}$ Find the perimeter of the complete figure in terms of $\pi$. $[2]$

Question 3 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 4 Code: 9709/12/M/J/14/4, Topic: Circular measure

The diagram shows a sector of a circle with radius $r \mathrm{~cm}$ and centre $O$. The chord $A B$ divides the sector into a triangle $A O B$ and a segment $A X B$. Angle $A O B$ is $\theta$ radians.

$\text{(i)}$ In the case where the areas of the triangle $A O B$ and the segment $A X B$ are equal, find the value of the constant $p$ for which $\theta=p \sin \theta$. $[2]$

$\text{(ii)}$ In the case where $r=8$ and $\theta=2.4$, find the perimeter of the segment $A X B$. $[3]$

Question 5 Code: 9709/12/M/J/17/4, Topic: Circular measure

The diagram shows a circle with radius $r \mathrm{~cm}$ and centre $O$. Points $A$ and $B$ lie on the circle and $A B C D$ is a rectangle. Angle $A O B=2 \theta$ radians and $A D=r \mathrm{~cm}$.

$\text{(i)}$ Express the perimeter of the shaded region in terms of $r$ and $\theta$. $[3]$

$\text{(ii)}$ In the case where $r=5$ and $\theta=\frac{1}{6} \pi$, find the area of the shaded region. $[4]$

Question 6 Code: 9709/13/M/J/20/5, Topic: Circular measure

The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre $O$ and radius $5 \mathrm{~cm}$. The thickness of the cord and the size of the pin $P$ can be neglected. The pin is situated 13 cm vertically below $O$. Points $A$ and $B$ are on the circumference of the circle such that $A P$ and $B P$ are tangents to the circle. The cord passes over the major arc $A B$ of the circle and under the pin such that the cord is taut.

Calculate the length of the cord. $[6]$

Question 7 Code: 9709/12/M/J/16/6, Topic: Circular measure

The diagram shows a circle with radius $r \mathrm{~cm}$ and centre $O$. The line $P T$ is the tangent to the circle at $P$ and angle $P O T=\alpha$ radians. The line $O T$ meets the circle at $Q.$

$\text{(i)}$ Express the perimeter of the shaded region $P Q T$ in terms of $r$ and $\alpha$. $[3]$

$\text{(ii)}$ In the case where $\alpha=\frac{1}{3} \pi$ and $r=10$, find the area of the shaded region correct to 2 significant figures. $[3]$

Question 8 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 9 Code: 9709/12/M/J/20/7, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $2 r$, and angle $A O B=\frac{1}{6} \pi$ radians. The point $C$ is the midpoint of $O A$.

$\text{(a)}$ Show that the exact length of $B C$ is $r \sqrt{5-2 \sqrt{3}}$. $[2]$

$\text{(b)}$ Find the exact perimeter of the shaded region. $[2]$

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

Question 10 Code: 9709/11/M/J/20/8, Topic: Circular measure

In the diagram, $A B C$ is a semicircle with diameter $A C$, centre $O$ and radius $6 \mathrm{~cm}$. The length of the $\operatorname{arc} A B$ is $15 \mathrm{~cm}$. The point $X$ lies on $A C$ and $B X$ is perpendicular to $A X$.

Find the perimeter of the shaded region $B X C$. $[6]$

Question 11 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 12 Code: 9709/12/M/J/21/12, Topic: Circular measure

The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are $A, B, C, D$, $E$ and $F$. Points $P$ and $Q$ are situated where straight sections of the rope meet the pipe with centre $A.$

$\text{(a)}$ Show that angle $P A Q=\frac{1}{3} \pi$ radians. $[2]$

$\text{(b)}$ Find the length of the rope. $[4]$

$\text{(c)}$ Find the area of the hexagon $A B C D E F$, giving your answer in terms of $\sqrt{3}$. $[2]$

$\text{(d)}$ Find the area of the complete region enclosed by the rope. $[3]$