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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 1 (P1) | Variant(s) | P13 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 3 | 3 | 3 | 4 | 8 | 7 | 8 | 8 | 8 | 10 | 9 | 9 | 80 |

Score |

Question 1 Code: 9709/13/M/J/14/1, Topic: Series

Find the coefficient of $x$ in the expansion of $\displaystyle\left(x^{2}-\frac{2}{x}\right)^{5}$. $[3]$

Question 2 Code: 9709/13/M/J/16/1, Topic: Series

Find the coefficient of $x$ in the expansion of $\displaystyle\left(\frac{1}{x}+3 x^{2}\right)^{5}$. $[3]$

Question 3 Code: 9709/13/M/J/21/1, Topic: Integration

A curve with equation $y=\mathrm{f}(x)$ is such that $\displaystyle \mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}.$ It is given that the curve passes through the point $(2,7)$.

Find $\mathrm{f}(x)$. $[3]$

Question 4 Code: 9709/13/M/J/15/2, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=(2 x+1)^{\frac{1}{2}}$ and the point $(4,7)$ lies on the curve. Find the equation of the curve. $[4]$

Question 5 Code: 9709/13/M/J/14/7, Topic: Vectors

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

$$ \overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -1 \\ 7 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 2 \\ 4 \\ 7 \end{array}\right) $$$\text{(i)}$ Show that angle $B A C=\cos ^{-1}\left(\frac{1}{3}\right)$. $[5]$

$\text{(ii)}$ Use the result in part $\text{(i)}$ to find the exact value of the area of triangle $A B C$. $[3]$

Question 6 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry

The point $A$ has coordinates $(p, 1)$ and the point $B$ has coordinates $(9,3 p+1)$, where $p$ is a constant.

$\text{(i)}$ For the case where the distance $A B$ is 13 units, find the possible values of $p$. $[3]$

$\text{(ii)}$ For the case in which the line with equation $2 x+3 y=9$ is perpendicular to $A B$, find the value of $p$. $[4]$

Question 7 Code: 9709/13/M/J/20/7, Topic: Trigonometry

$\text{(a)}$ Show that $\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta} \equiv \frac{2}{\sin \theta \cos \theta}$. $[4]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta}=\frac{6}{\tan \theta}$ for $0^{\circ}< \theta <180^{\circ}$. $[4]$

Question 8 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 9 Code: 9709/13/M/J/16/8, Topic: Trigonometry

$\text{(i)}$ Show that $3 \sin x \tan x-\cos x+1=0$ can be written as a quadratic equation in $\cos x$ and hence solve the equation $3 \sin x \tan x-\cos x+1=0$ for $0 \leqslant x \leqslant \pi$. $[5]$

$\text{(ii)}$ Find the solutions to the equation $3 \sin 2 x \tan 2 x-\cos 2 x+1=0$ for $0 \leqslant x \leqslant \pi$. $[3]$

Question 10 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 11 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration

Points $A(2,9)$ and $B(3,0)$ lie on the curve $y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at $A$ intersects the $x$-axis at $C$. Showing all necessary working,$\text{(i)}$ find the equation of the tangent $A C$ and hence find the $x$-coordinate of $C$, $[4]$

$\text{(ii)}$ find the area of the shaded region $A B C$. $[5]$

Question 12 Code: 9709/13/M/J/13/11, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{8}{\sqrt{x}}-x$ and points $A(1,7)$ and $B(4,0)$ which lie on the curve. The tangent to the curve at $B$ intersects the line $x=1$ at the point $C$.

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

$\text{(ii)}$ Find the area of the shaded region. $[5]$