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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 | 3 | 6 | 7 | 6 | 6 | 7 | 7 | 7 | 10 | 8 | 11 | 11 | 89 |

Score |

Question 1 Code: 9709/12/M/J/16/1, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 10-3 x, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{10}{3-2 x}, \quad x \in \mathbb{R}, x \neq \frac{3}{2} \end{aligned} $$Solve the equation $\mathrm{ff}(x)=\operatorname{gf}(2)$. $[3]$

Question 2 Code: 9709/12/M/J/10/5, Topic: Vectors

Relative to an origin $O$, the position vectors of the points $A$ and $B$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{r} -2 \\ 3 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{c} 4 \\ 1 \\ p \end{array}\right). $$$\text{(i)}$ Find the value of $p$ for which $\overrightarrow{O A}$ is perpendicular to $\overrightarrow{O B}$. $[2]$

$\text{(ii)}$ Find the values of $p$ for which the magnitude of $\overrightarrow{A B}$ is 7. $[4]$

Question 3 Code: 9709/13/M/J/11/5, Topic: Vectors

In the diagram, $O A B C D E F G$ is a rectangular block in which $O A=O D=6 \mathrm{~cm}$ and $A B=12 \mathrm{~cm}$. The unit vectors i, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $\overrightarrow{O A}, \overrightarrow{O C}$ and $\overrightarrow{O D}$ respectively. The point $P$ is the mid-point of $D G, Q$ is the centre of the square face $C B F G$ and $R$ lies on $A B$ such that $A R=4 \mathrm{~cm}$.

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

$\text{(ii)}$ Use a scalar product to find angle $R Q P$. $[4]$

Question 4 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry

The diagram shows the curve $y=7 \sqrt{x}$ and the line $y=6 x+k$, where $k$ is a constant. The curve and the line intersect at the points $A$ and $B$.

$\text{(i)}$ For the case where $k=2$, find the $x$-coordinates of $A$ and $B$. $[4]$

$\text{(ii)}$ Find the value of $k$ for which $y=6 x+k$ is a tangent to the curve $y=7 \sqrt{x}$. $[2]$

Question 5 Code: 9709/12/M/J/15/6, Topic: Trigonometry

A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, $h \mathrm{~m}$, of a passenger above the ground is given by the formula $h=60(1-\cos k t)$. In this formula, $k$ is a constant, $t$ is the time in minutes that has elapsed since the passenger started the ride at ground level and $k t$ is measured in radians.

$\text{(i)}$ Find the greatest height of the passenger above the ground. $[1]$

One complete revolution of the wheel takes 30 minutes.

$\text{(ii)}$ Show that $k=\frac{1}{15} \pi$. $[2]$

$\text{(iii)}$ Find the time for which the passenger is above a height of $90 \mathrm{~m}$. $[3]$

Question 6 Code: 9709/12/M/J/19/6, Topic: Trigonometry

The equation of a curve is $y=3 \cos 2 x$ and the equation of a line is $\displaystyle 2 y+\frac{3 x}{\pi}=5$.

$\text{(i)}$ State the smallest and largest values of $y$ for both the curve and the line for $0 \leqslant x \leqslant 2 \pi$. $[3]$

$\text{(ii)}$ Sketch, on the same diagram, the graphs of $y=3 \cos 2 x$ and $\displaystyle 2 y+\frac{3 x}{\pi}=5$ for $0 \leqslant x \leqslant 2 \pi$. $[3]$

$\text{(iii)}$ State the number of solutions of the equation $\displaystyle 6 \cos 2 x=5-\frac{3 x}{\pi}$ for $0 \leqslant x \leqslant 2 \pi$. $[1]$

Question 7 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 8 Code: 9709/13/M/J/12/8, Topic: Circular measure

In the diagram, $A B$ is an arc of a circle with centre $O$ and radius $r$. The line $X B$ is a tangent to the circle at $B$ and $A$ is the mid-point of $O X$.

$\text{(i)}$ Show that angle $A O B=\frac{1}{3} \pi$ radians. $[2]$

Express each of the following in terms of $r, \pi$ and $\sqrt{3}$ :

$\text{(ii)}$ the perimeter of the shaded region, $[3]$

$\text{(iii)}$ the area of the shaded region. $[2]$

Question 9 Code: 9709/13/M/J/19/9, Topic: Trigonometry

The function $\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$ is defined for $0 \leqslant x \leqslant \pi$, where $p$ and $q$ are positive constants. The diagram shows the graph of $y=\mathrm{f}(x)$.

$\text{(i)}$ In terms of $p$ and $q$, state the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ State the number of solutions of the following equations.

$\quad\text{(a)}$ $\mathrm{f}(x)=p+q$ $[1]$

$\quad\text{(b)}$ $\mathrm{f}(x)=q$ $[1]$

$\quad\text{(c)}$ $\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q$ $[1]$

$\text{(iii)}$ For the case where $p=3$ and $q=2$, solve the equation $\mathrm{f}(x)=4$, showing all necessary working. $[5]$

Question 10 Code: 9709/11/M/J/21/10, Topic: Coordinate geometry

The equation of a circle is $x^{2}+y^{2}-4 x+6 y-77=0$.

$\text{(a)}$ Find the $x$-coordinates of the points $A$ and $B$ where the circle intersects the $x$-axis. $[2]$

$\text{(b)}$ Find the point of intersection of the tangents to the circle at A and B. $[6]$

Question 11 Code: 9709/13/M/J/18/11, Topic: Differentiation, Integration

The diagram shows part of the curve $y=(x+1)^{2}+(x+1)^{-1}$ and the line $x=1$. The point $A$ is the minimum point on the curve.

$\text{(i)}$ Show that the $x$-coordinate of $A$ satisfies the equation $2(x+1)^{3}=1$ and find the exact value of $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ at $A$. $[5]$

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

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. $[3]$