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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 | 5 | 6 | 7 | 6 | 4 | 7 | 9 | 9 | 10 | 7 | 9 | 84 |

Score |

Question 1 Code: 9709/12/M/J/18/1, Topic: Series

The coefficient of $x^{2}$ in the expansion of $\displaystyle \left(2+\frac{x}{2}\right)^{6}+(a+x)^{5}$ is 330 . Find the value of the constant $a$. $[5]$

Question 2 Code: 9709/13/M/J/10/2, Topic: Series

$\text{(i)}$ Find the first three terms, in descending powers of $x$, in the expansion of $\displaystyle \left(x-\frac{2}{x}\right)^{6}$. $[3]$

$\text{(ii)}$ Find the coefficient of $x^{4}$ in the expansion of $\displaystyle \left(1+x^{2}\right)\left(x-\frac{2}{x}\right)^{6}$. $[2]$

Question 3 Code: 9709/13/M/J/18/4, Topic: Coordinate geometry

A curve with equation $y=\mathrm{f}(x)$ passes through the point $A(3,1)$ and crosses the $y$-axis at $B$. It is given that $\mathrm{f}^{\prime}(x)=(3 x-1)^{-\frac{1}{3}}$. Find the $y$-coordinate of $B$. $[6]$

Question 4 Code: 9709/11/M/J/18/5, Topic: Coordinate geometry

The diagram shows a kite $O A B C$ in which $A C$ is the line of symmetry. The coordinates of $A$ and $C$ are $(0,4)$ and $(8,0)$ respectively and $O$ is the origin.

$\text{(i)}$ Find the equations of $A C$ and $O B$. $[4]$

$\text{(ii)}$ Find, by calculation, the coordinates of $B$. $[3]$

Question 5 Code: 9709/12/M/J/20/5, Topic: Functions

The function $\mathrm{f}$ is defined for $x \in \mathbb{R}$ by

$$ \text { f: } x \mapsto a-2 x $$where $a$ is a constant.

$\text{(a)}$ Express $\mathrm{ff}(x)$ and $\mathrm{f}^{-1}(x)$ in terms of $a$ and $x$. $[4]$

$\text{(b)}$ Given that $\mathrm{ff}(x)=\mathrm{f}^{-1}(x)$, find $x$ in terms of $a$. $[2]$

Question 6 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry

Points $A$ and $B$ have coordinates $(8,3)$ and $(p, q)$ respectively. The equation of the perpendicular bisector of $A B$ is $y=-2 x+4$.Find the values of $p$ and $q$. $[4]$

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

$\text{(a)}$ The first two terms of an arithmetic progression are 1 and $\cos ^{2} x$ respectively. Show that the sum of the first ten terms can be expressed in the form $a-b \sin ^{2} x$, where $a$ and $b$ are constants to be found. $[3]$

$\text{(b)}$ The first two terms of a geometric progression are 1 and $\frac{1}{3} \tan ^{2} \theta$ respectively, where $0< \theta <\frac{1}{2} \pi$.

$\text{(i)}$ Find the set of values of $\theta$ for which the progression is convergent. $[2]$

$\text{(ii)}$ Find the exact value of the sum to infinity when $\theta=\frac{1}{6} \pi$. $[2]$

Question 8 Code: 9709/12/M/J/15/8, Topic: Series

$\text{(a)}$ The first, second and last terms in an arithmetic progression are 56,53 and $-22$ respectively. Find the sum of all the terms in the progression. $[4]$

$\text{(b)}$ The first, second and third terms of a geometric progression are $2 k+6,2 k$ and $k+2$ respectively, where $k$ is a positive constant.

$\text{(i)}$ Find the value of $k$. $[3]$

$\text{(ii)}$ Find the sum to infinity of the progression. $[2]$

Question 9 Code: 9709/12/M/J/16/9, Topic: Series

A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

$\text{(i)}$ On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.

$\text{(a)}$ How many litres will be lost on the 30 th day after filling? $[2]$

$\text{(b)}$ The tank becomes empty during the $n$th day after filling. Find the value of $n$. $[3]$

$\text{(ii)}$ Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by $10 \%$ on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30 th day after filling. $[4]$

Question 10 Code: 9709/11/M/J/17/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$ for $x \in \mathbb{R}, x \neq \frac{3}{2}$.

$\text{(i)}$ Find an expression for $\mathrm{f}^{-1}(x)$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 4 x+a$ for $x \in \mathbb{R}$, where $a$ is a constant.

$\text{(ii)}$ Find the value of $a$ for which $\operatorname{gf}(-1)=3$. $[3]$

$\text{(iii)}$ Find the possible values of $a$ given that the equation $\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$ has two equal roots. $[4]$

Question 11 Code: 9709/11/M/J/19/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2-3 \cos x$ for $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ State the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2-3 \cos x$ for $0 \leqslant x \leqslant p$, where $p$ is a constant. $[2]$

$\text{(iii)}$ State the largest value of $p$ for which $\mathrm{g}$ has an inverse. $[1]$

$\text{(iv)}$ For this value of $p$, find an expression for $\mathrm{g}^{-1}(x)$. $[3]$

Question 12 Code: 9709/13/M/J/21/9, Topic: Series

$\text{(a)}$ A geometric progression is such that the second term is equal to 24% of the sum to infinity.

Find the possible values of the common ratio. $[3]$

$\text{(b)}$ An arithmetic progression $P$ has first term $a$ and common difference $d$. An arithmetic progression $Q$ has first term $2(a+1)$ and common difference $(d+1)$. It is given that

$$ \frac{\text { 5th term of } P}{12\text{th term of } Q}=\frac{1}{3} \quad \text { and } \quad \frac{\text { Sum of first } 5 \text { terms of } P}{\text { Sum of first } 5 \text { terms of } Q}=\frac{2}{3} $$Find the value of $a$ and the value of $d$. $[6]$