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

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 | 5 | 4 | 6 | 6 | 6 | 5 | 6 | 8 | 7 | 11 | 11 | 11 | 86 |

Score |

Question 1 Code: 9709/12/M/J/11/2, Topic: Series

$\text{(i)}$ Find the terms in $x^{2}$ and $x^{3}$ in the expansion of $\displaystyle\left(1-\frac{3}{2} x\right)^{6}$. $[3]$

$\text{(ii)}$ Given that there is no term in $x^{3}$ in the expansion of $\displaystyle (k+2 x)\left(1-\frac{3}{2} x\right)^{6}$, find the value of the constant $k$. $[2]$

Question 2 Code: 9709/13/M/J/20/2, Topic: Integration

The equation of a curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-3 x^{-\frac{1}{2}}.$ It is given that the point $(4,7)$ lies on the curve. Find the equation of the curve. $[4]$

Question 3 Code: 9709/13/M/J/14/4, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$. $[3]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$ for $0 \leqslant x \leqslant 2 \pi$. $[3]$

Question 4 Code: 9709/12/M/J/18/4, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=a+b \cos x$ for $0 \leqslant x \leqslant 2 \pi$. It is given that $\mathrm{f}\left(\frac{1}{3} \pi\right)=5$ and $\mathrm{f}(\pi)=11$.

$\text{(i)}$ Find the values of the constants $a$ and $b$. $[3]$

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

Question 5 Code: 9709/13/M/J/21/4, Topic: Trigonometry

$\text{(a)}$ Show that the equation $[2]$

$$ \frac{\tan x+\sin x}{\tan x-\sin x}=k $$where $k$ is a constant, may be expressed as

$$ \frac{1+\cos x}{1-\cos x}=k $$$\text{(b)}$ Hence express $\cos x$ in terms of $k$. $[2]$

$\text{(c)}$ Hence solve the equation $\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$ for $-\pi < x < \pi$. $[2]$

Question 6 Code: 9709/11/M/J/14/5, Topic: Series

An arithmetic progression has first term $a$ and common difference $d$. It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.

$\text{(i)}$ Find $d$ in terms of $a$. $[3]$

$\text{(ii)}$ Find the 100 th term in terms of $a$. $[2]$

Question 7 Code: 9709/12/M/J/15/6, Topic: Trigonometry

A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, $h \mathrm{~m}$, of a passenger above the ground is given by the formula $h=60(1-\cos k t)$. In this formula, $k$ is a constant, $t$ is the time in minutes that has elapsed since the passenger started the ride at ground level and $k t$ is measured in radians.

$\text{(i)}$ Find the greatest height of the passenger above the ground. $[1]$

One complete revolution of the wheel takes 30 minutes.

$\text{(ii)}$ Show that $k=\frac{1}{15} \pi$. $[2]$

$\text{(iii)}$ Find the time for which the passenger is above a height of $90 \mathrm{~m}$. $[3]$

Question 8 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 9 Code: 9709/12/M/J/20/7, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $2 r$, and angle $A O B=\frac{1}{6} \pi$ radians. The point $C$ is the midpoint of $O A$.

$\text{(a)}$ Show that the exact length of $B C$ is $r \sqrt{5-2 \sqrt{3}}$. $[2]$

$\text{(b)}$ Find the exact perimeter of the shaded region. $[2]$

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

Question 10 Code: 9709/11/M/J/21/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:

$$ \begin{aligned} &\mathrm{f}(x)=(x-2)^{2}-4 \text { for } x \geqslant 2, \\ &\mathrm{~g}(x)=a x+2 \text { for } x \in \mathbb{R}, \end{aligned} $$where $a$ is a constant.

$\text{(a)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(b)}$ Find $\mathrm{f}^{-1}(x)$. $[2]$

$\text{(c)}$ Given that $a=-\frac{5}{3}$, solve the equation $\mathrm{f}(x)=\mathrm{g}(x)$. $[3]$

$\text{(d)}$ Given instead that $\operatorname{ggf}^{-1}(12)=62$, find the possible values of $a$. $[5]$

Question 11 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 12 Code: 9709/12/M/J/16/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 6 x-x^{2}-5$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find the set of values of $x$ for which $\mathrm{f}(x) \leqslant 3$. $[3]$

$\text{(ii)}$ Given that the line $y=m x+c$ is a tangent to the curve $y=\mathrm{f}(x)$, show that $4 c=m^{2}-12 m+16$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 6 x-x^{2}-5$ for $x \geqslant k$, where $k$ is a constant.

$\text{(iii)}$ Express $6 x-x^{2}-5$ in the form $a-(x-b)^{2}$, where $a$ and $b$ are constants. $[2]$

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

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{g}^{-1}(x)$. $[2]$