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

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 | 6 | 4 | 6 | 8 | 9 | 11 | 9 | 9 | 11 | 10 | 12 | 11 | 106 |

Score |

Question 1 Code: 9709/12/M/J/10/4, Topic: Coordinate geometry

In the diagram, $A$ is the point $(-1,3)$ and $B$ is the point $(3,1)$. The line $L_{1}$ passes through $A$ and is parallel to $O B$. The line $L_{2}$ passes through $B$ and is perpendicular to $A B$. The lines $L_{1}$ and $L_{2}$ meet at $C$. Find the coordinates of $C$. $[6]$

Question 2 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 3 Code: 9709/11/M/J/14/7, Topic: Coordinate geometry

The coordinates of points $A$ and $B$ are $(a, 2)$ and $(3, b)$ respectively, where $a$ and $b$ are constants. The distance $A B$ is $\sqrt{(} 125)$ units and the gradient of the line $A B$ is 2. Find the possible values of $a$ and of $b$. $[6]$

Question 4 Code: 9709/11/M/J/18/7, Topic: Vectors

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

$$ \overrightarrow{O A}=\left(\begin{array}{r} 1 \\ -3 \\ 2 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} -1 \\ 3 \\ 5 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 3 \\ 1 \\ -2 \end{array}\right) $$$\text{(i)}$ Find $\overrightarrow{A C}$. $[1]$

$\text{(ii)}$ The point $M$ is the mid-point of $A C$. Find the unit vector in the direction of $\overrightarrow{O M}$. $[3]$

$\text{(iii)}$ Evaluate $\overrightarrow{A B} \cdot \overrightarrow{A C}$ and hence find angle $B A C$. $[4]$

Question 5 Code: 9709/11/M/J/17/8, Topic: Circular measure

In the diagram, $O A X B$ is a sector of a circle with centre $O$ and radius $10 \mathrm{~cm}.$ The length of the chord $A B$ is $12 \mathrm{~cm}$. The line $O X$ passes through $M$, the mid-point of $A B$, and $O X$ is perpendicular to $A B$. The shaded region is bounded by the chord $A B$ and by the arc of a circle with centre $X$ and radius $X A$.

$\text{(i)}$ Show that angle $A X B$ is $2.498$ radians, correct to 3 decimal places. $[3]$

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

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

Question 6 Code: 9709/11/M/J/10/10, Topic: Vectors

The diagram shows the parallelogram $O A B C$. Given that $\overrightarrow{O A}=\mathbf{i}+3 \mathbf{j}+3 \mathbf{k}$ and $\overrightarrow{O C}=3 \mathbf{i}-\mathbf{j}+\mathbf{k}$, find

$\text{(i)}$ the unit vector in the direction of $\overrightarrow{O B}$, $[3]$

$\text{(ii)}$ the acute angle between the diagonals of the parallelogram, $[5]$

$\text{(iii)}$ the perimeter of the parallelogram, correct to 1 decimal place. $[3]$

Question 7 Code: 9709/13/M/J/13/10, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=3$, solve the equation $\mathrm{ff}(x)=25$. $[2]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.

$\text{(ii)}$ Find the set of values of $k$ for which the equation $\mathrm{f}(x)=\mathrm{g}(x)$ has no real solutions. $[3]$

The function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.

$\text{(iii)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[4]$

Question 8 Code: 9709/12/M/J/20/10, Topic: Differentiation

The equation of a curve is $y=54 x-(2 x-7)^{3}$.

$\text{(a)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[4]$

$\text{(b)}$ Find the coordinates of each of the stationary points on the curve. $[3]$

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

Question 9 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 10 Code: 9709/13/M/J/15/11, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $r$. The point $C$ on $O B$ is such that angle $A C O$ is a right angle. Angle $A O B$ is $\alpha$ radians and is such that $A C$ divides the sector into two regions of equal area.

$\text{(i)}$ Show that $\sin \alpha \cos \alpha=\frac{1}{2} \alpha$. $[4]$

It is given that the solution of the equation in part $\text{(i)}$ is $\alpha=0.9477$, correct to 4 decimal places.

$\text{(ii)}$ Find the ratio

perimeter of region $O A C$ : perimeter of region $A C B$,

giving your answer in the form $k: 1$, where $k$ is given correct to 1 decimal place. $[5]$

$\text{(iii)}$ Find angle $A O B$ in degrees. $[1]$

Question 11 Code: 9709/11/M/J/20/11, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{8}{x+2}$ and the line $2 y+x=8$, intersecting at points $A$ and $B$. The point $C$ lies on the curve and the tangent to the curve at $C$ is parallel to $A B$.

$\text{(a)}$ Find, by calculation, the coordinates of $A, B$ and $C$. $[6]$

$\text{(b)}$ Find the volume generated when the shaded region, bounded by the curve and the line, is rotated through $360^{\circ}$ about the $x$-axis. $[6]$

Question 12 Code: 9709/11/M/J/14/12, Topic: Integration, Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{\frac{1}{2}}-x^{-\frac{1}{2}}$. The curve passes through the point $\left(4, \frac{2}{3}\right)$.

$\text{(i)}$ Find the equation of the curve. $[4]$

$\text{(ii)}$ Find $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[2]$

$\text{(iii)}$ Find the coordinates of the stationary point and determine its nature. $[5]$