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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 | 5 | 5 | 7 | 5 | 7 | 10 | 8 | 11 | 8 | 12 | 8 | 90 |

Score |

Question 1 Code: 9709/11/M/J/20/1, Topic: Series

The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms is 91.

Find the first term and the common difference of the progression. $[4]$

Question 2 Code: 9709/12/M/J/13/3, Topic: Quadratics

The straight line $y=m x+14$ is a tangent to the curve $\displaystyle y=\frac{12}{x}+2$ at the point $P$. Find the value of the constant $m$ and the coordinates of $P$. $[5]$

Question 3 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$ The point $P(2,9)$ lies on the curve.

$\text{(i)}$ A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of $0.05$ units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. $[2]$

$\text{(ii)}$ Find the equation of the curve. $[3]$

Question 4 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 5 Code: 9709/12/M/J/15/5, Topic: Trigonometry

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

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{\tan \theta}{6}$, for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[4]$

Question 6 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 7 Code: 9709/11/M/J/10/8, Topic: Coordinate geometry

The diagram shows a triangle $A B C$ in which $A$ is $(3,-2)$ and $B$ is $(15,22)$. The gradients of $A B, A C$ and $B C$ are $2 m,-2 m$ and $m$ respectively, where $m$ is a positive constant.

$\text{(i)}$ Find the gradient of $A B$ and deduce the value of $m$. $[2]$

$\text{(ii)}$ Find the coordinates of $C$. $[4]$

The perpendicular bisector of $A B$ meets $B C$ at $D$.

$\text{(iii)}$ Find the coordinates of $D$. $[4]$

Question 8 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 9 Code: 9709/11/M/J/13/9, Topic: Differentiation, Integration

A curve has equation $y=\mathrm{f}(x)$ and is such that $\mathrm{f}^{\prime}(x)=3 x^{\frac{1}{2}}+3 x^{-\frac{1}{2}}-10$.

$\text{(i)}$ By using the substitution $u=x^{\frac{1}{2}}$, or otherwise, find the values of $x$ for which the curve $y=\mathrm{f}(x)$ has stationary points. $[4]$

$\text{(ii)}$ Find $\mathrm{f}^{\prime \prime}(x)$ and hence, or otherwise, determine the nature of each stationary point. $[3]$

$\text{(iii)}$ It is given that the curve $y=\mathrm{f}(x)$ passes through the point $(4,-7)$. Find $\mathrm{f}(x)$. $[4]$

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

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned} $$$\text{(i)}$ Obtain expressions, in terms of $x$, for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram, making clear the relationship between the two graphs. $[3]$

$\text{(iii)}$ Given that the equation $f g(x)=5-k x$, where $k$ is a constant, has no solutions, find the set of possible values of $k$. $[5]$

Question 12 Code: 9709/13/M/J/14/10, Topic: Integration

The diagram shows the curve $y=-x^{2}+12 x-20$ and the line $y=2 x+1$. Find, showing all necessary working, the area of the shaded region. $[8]$