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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 | 4 | 6 | 7 | 6 | 8 | 7 | 7 | 8 | 8 | 9 | 79 |

Score |

Question 1 Code: 9709/12/M/J/12/1, Topic: Integration

The diagram shows the region enclosed by the curve $\displaystyle y=\frac{6}{2 x-3}$, the $x$-axis and the lines $x=2$ and $x=3$. Find, in terms of $\pi$, the volume obtained when this region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

Question 2 Code: 9709/11/M/J/18/1, Topic: Series

$\text{(i)}$ Find the first three terms in the expansion, in ascending powers of $x$, of $(1-2 x)^{5}$. $[2]$

$\text{(ii)}$ Given that the coefficient of $x^{2}$ in the expansion of $\left(1+a x+2 x^{2}\right)(1-2 x)^{5}$ is 12 , find the value of the constant $a$. $[3]$

Question 3 Code: 9709/12/M/J/21/1, Topic: Quadratics

$\text{(a)}$ Express $16 x^{2}-24 x+10$ in the form $(4 x+a)^{2}+b$. $[2]$

$\text{(b)}$ It is given that the equation $16 x^{2}-24 x+10=k$, where $k$ is a constant, has exactly one root.

Find the value of this root. $[2]$

Question 4 Code: 9709/11/M/J/18/3, Topic: Integration, Coordinate geometry

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{12}{(2 x+1)^{2}}$. The point $(1,1)$ lies on the curve. Find the coordinates of the point at which the curve intersects the $x$-axis. $[6]$

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

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

$$ \overrightarrow{O A}=\left(\begin{array}{r} 3 \\ 0 \\ -4 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -3 \\ 2 \end{array}\right) $$$\text{(i)}$ Find the cosine of angle $A O B$. $[3]$

The position vector of $C$ is given by $\overrightarrow{O C}=\left(\begin{array}{c}k \\ -2 k \\ 2 k-3\end{array}\right)$

$\text{(ii)}$ Given that $A B$ and $O C$ have the same length, find the possible values of $k$. $[4]$

Question 6 Code: 9709/13/M/J/17/4, Topic: Vectors

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

$$ \overrightarrow{O A}=\left(\begin{array}{l} 5 \\ 1 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 5 \\ 4 \\ -3 \end{array}\right) $$The point $P$ lies on $A B$ and is such that $\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}$.

$\text{(i)}$ Find the position vector of $P$. $[3]$

$\text{(ii)}$ Find the distance $O P$. $[1]$

$\text{(iii)}$ Determine whether $O P$ is perpendicular to $A B$. Justify your answer. $[2]$

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

The diagram shows part of the curve $\displaystyle y=2-\frac{18}{2 x+3}$, which crosses the $x$-axis at $A$ and the $y$-axis at $B$. The normal to the curve at $A$ crosses the $y$-axis at $C$.

$\text{(i)}$ Show that the equation of the line $A C$ is $9 x+4 y=27$. $[6]$

$\text{(ii)}$ Find the length of $B C$. $[2]$

Question 8 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 9 Code: 9709/12/M/J/19/7, Topic: Functions

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

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

$\text{(ii)}$ Solve the equation $\mathrm{fg}(x)=\frac{7}{3}$. $[3]$

Question 10 Code: 9709/12/M/J/14/8, Topic: Integration, Differentiation

The equation of a curve is such that $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 x-1$. Given that the curve has a minimum point at $(3,-10)$, find the coordinates of the maximum point. $[8]$

Question 11 Code: 9709/13/M/J/15/8, Topic: Differentiation

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$ for $x>-1$.

$\text{(i)}$ Find $\mathrm{f}^{\prime}(x)$. $[3]$

$\text{(ii)}$ State, with a reason, whether $\mathrm{f}$ is an increasing function, a decreasing function or neither. $[1]$

The function $\mathrm{g}$ is defined by $\displaystyle\mathrm{g}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$ for $x<-1$

$\text{(iii)}$ Find the coordinates of the stationary point on the curve $y=\mathrm{g}(x)$. $[4]$

Question 12 Code: 9709/13/M/J/20/8, Topic: Series

The first term of a progression is $\sin ^{2} \theta$, where $0 < \theta < \frac{1}{2} \pi$. The second term of the progression is $\sin ^{2} \theta \cos ^{2} \theta$.

$\text{(a)}$ Given that the progression is geometric, find the sum to infinity. It is now given instead that the progression is arithmetic. $[3]$

$\text{(b)} \quad \text{(i)}$ Find the common difference of the progression in terms of $\sin \theta$. $[3]$

$ \quad \quad \text{(ii)}$ Find the sum of the first 16 terms when $\theta=\frac{1}{3} \pi$. $[3]$