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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 | 4 | 6 | 7 | 8 | 7 | 6 | 8 | 8 | 11 | 8 | 10 | 88 |

Score |

Question 1 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry

Find the coordinates of the point at which the perpendicular bisector of the line joining $(2,7)$ to $(10,3)$ meets the $x$-axis. $[5]$

Question 2 Code: 9709/11/M/J/21/1, Topic: Integration

The equation of a curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{x^{4}}+32 x^{3}$. It is given that the curve passes through the point $\left(\frac{1}{2}, 4\right)$.

Find the equation of the curve. $[4]$

Question 3 Code: 9709/11/M/J/13/4, Topic: Series

The third term of a geometric progression is $-108$ and the sixth term is 32. Find

$\text{(i)}$ the common ratio, $[3]$

$\text{(ii)}$ the first term, $[1]$

$\text{(iii)}$ the sum to infinity. $[2]$

Question 4 Code: 9709/11/M/J/15/5, Topic: Quadratics

A piece of wire of length $24 \mathrm{~cm}$ is bent to form the perimeter of a sector of a circle of radius $r \mathrm{~cm}$.

$\text{(i)}$ Show that the area of the sector, $A \mathrm{~cm}^{2}$, is given by $A=12 r-r^{2}$. $[3]$

$\text{(ii)}$ Express $A$ in the form $a-(r-b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iii)}$ Given that $r$ can vary, state the greatest value of $A$ and find the corresponding angle of the sector. $[2]$

Question 5 Code: 9709/11/M/J/17/6, Topic: Differentiation

The horizontal base of a solid prism is an equilateral triangle of side $x \mathrm{~cm}$. The sides of the prism are vertical. The height of the prism is $h \mathrm{~cm}$ and the volume of the prism is $2000 \mathrm{~cm}^{3}$.

$\text{(i)}$ Express $\mathrm{h}$ in terms of $x$ and show that the total surface area of the prism, $A \mathrm{~cm}^{2}$, is given by $[3]$

$$ A=\frac{\sqrt{3}}{2} x^{2}+\frac{24000}{\sqrt{3}} x^{-1} $$$\text{(ii)}$ Given that $x$ can vary, find the value of $x$ for which $A$ has a stationary value. $[3]$

$\text{(iii)}$ Determine, showing all necessary working, the nature of this stationary value. $[2]$

Question 6 Code: 9709/12/M/J/11/7, Topic: Coordinate geometry

The line $L_{1}$ passes through the points $A(2,5)$ and $B(10,9)$. The line $L_{2}$ is parallel to $L_{1}$ and passes through the origin. The point $C$ lies on $L_{2}$ such that $A C$ is perpendicular to $L_{2}.$ Find

$\text{(i)}$ the coordinates of $C$, $[5]$

$\text{(ii)}$ the distance $A C$. $[2]$

Question 7 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 8 Code: 9709/12/M/J/11/8, Topic: Vectors

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

$$ \overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 3 \\ 5 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 4 \\ 2 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 10 \\ 0 \\ 6 \end{array}\right) $$$\text{(i)}$ Find angle $A B C$. $[6]$

The point $D$ is such that $A B C D$ is a parallelogram.

$\text{(ii)}$ Find the position vector of $D$. $[2]$

Question 9 Code: 9709/12/M/J/17/8, Topic: Vectors

Relative to an origin $O$, the position vectors of three points $A, B$ and $C$ are given by $\overrightarrow{O A}=3 \mathbf{i}+p \mathbf{j}-2 p \mathbf{k}, \quad \overrightarrow{O B}=6 \mathbf{i}+(p+4) \mathbf{j}+3 \mathbf{k} \quad$ and $\quad \overrightarrow{O C}=(p-1) \mathbf{i}+2 \mathbf{j}+q \mathbf{k}$ where $p$ and $q$ are constants.

$\text{(i)}$ In the case where $p=2$, use a scalar product to find angle $A O B$. $[4]$

$\text{(ii)}$ In the case where $\overrightarrow{A B}$ is parallel to $\overrightarrow{O C}$, find the values of $p$ and $q$. $[4]$

Question 10 Code: 9709/13/M/J/11/9, Topic: Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{\sqrt{x}}-1$ and $P(9,5)$ is a point on the curve.

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

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

$\text{(iii)}$ Find an expression for $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ and determine the nature of the stationary point. $[2]$

$\text{(iv)}$ The normal to the curve at $P$ makes an angle of $\tan ^{-1} k$ with the positive $x$-axis. Find the value of $k$. $[2]$

Question 11 Code: 9709/13/M/J/13/9, Topic: Series

$\text{(a)}$ In an arithmetic progression, the sum, $S_{n}$, of the first $n$ terms is given by $S_{n}=2 n^{2}+8 n$. Find the first term and the common difference of the progression. $[3]$

$\text{(b)}$ The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the $1 \mathrm{st}$ term, the 9 th term and the $n$th term respectively of an arithmetic progression. Find the value of $n$. $[5]$

Question 12 Code: 9709/13/M/J/12/11, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=8-(x-2)^{2}$, for $x \in \mathbb{R}$.

$\text{(i)}$ Find the coordinates and the nature of the stationary point on the curve $y=\mathrm{f}(x)$. $[3]$

The function $\mathrm{g}$ is such that $\mathrm{g}(x)=8-(x-2)^{2}$, for $k \leqslant x \leqslant 4$, where $k$ is a constant.

$\text{(ii)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

For this value of $k$,

$\text{(iii)}$ find an expression for $\mathrm{g}^{-1}(x)$, $[3]$

$\text{(iv)}$ sketch, on the same diagram, the graphs of $y=\mathrm{g}(x)$ and $y=\mathrm{g}^{-1}(x)$. $[3]$