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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 | 4 | 5 | 5 | 3 | 5 | 7 | 8 | 7 | 5 | 9 | 11 | 73 |

Score |

Question 1 Code: 9709/11/M/J/17/1, Topic: Series

The coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(3-2 x)^{6}$ are $a$ and $b$ respectively. Find the value of $\frac{a}{b}$. $[4]$

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

A point is moving along the curve $\displaystyle y=2 x+\frac{5}{x}$ in such a way that the $x$-coordinate is increasing at a constant rate of $0.02$ units per second. Find the rate of change of the $y$-coordinate when $x=1$. $[4]$

Question 3 Code: 9709/12/M/J/15/3, Topic: Series

$\text{(i)}$ Find the coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(2-x)^{6}$. $[3]$

$\text{(ii)}$ Find the coefficient of $x^{3}$ in the expansion of $(3 x+1)(2-x)^{6}$. $[2]$

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

The common ratio of a geometric progression is $0.99$. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. $[5]$

Question 5 Code: 9709/11/M/J/21/4, Topic: Trigonometry

The diagram shows part of the graph of $y=a \tan (x-b)+c$

Given that $0 < b < \pi$, state the values of the constants $a, b$ and $c$. $[3]$

Question 6 Code: 9709/12/M/J/16/5, Topic: Circular measure

In the diagram, triangle $A B C$ is right-angled at $C$ and $M$ is the mid-point of $B C$. It is given that angle $A B C=\frac{1}{3} \pi$ radians and angle $B A M=\theta$ radians. Denoting the lengths of $B M$ and $M C$ by $x$,

$\text{(i)}$ find $A M$ in terms of $x$, $[3]$

$\text{(ii)}$ show that $\displaystyle\theta=\frac{1}{6} \pi-\tan ^{-1}\left(\frac{1}{2 \sqrt{3}}\right)$. $[2]$

Question 7 Code: 9709/11/M/J/10/6, Topic: Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-6$ and the point $(9,2)$ lies on the curve.

$\text{(i)}$ Find the equation of the curve. $[4]$

$\text{(ii)}$ Find the $x$-coordinate of the stationary point on the curve and determine the nature of the stationary point. $[3]$

Question 8 Code: 9709/13/M/J/11/6, Topic: Series

$\text{(a)}$ A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term. $[4]$

$\text{(b)}$ An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms. $[4]$

Question 9 Code: 9709/13/M/J/13/6, Topic: Differentiation

The non-zero variables $x, y$ and $u$ are such that $u=x^{2} y$. Given that $y+3 x=9$, find the stationary value of $u$ and determine whether this is a maximum or a minimum value. $[7]$

Question 10 Code: 9709/11/M/J/21/7, Topic: Trigonometry

$\text{(a)}$ Prove the identity $\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta} \equiv 1-\tan ^{2} \theta$. $[2]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta}=2 \tan ^{4} \theta$ for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[3]$

Question 11 Code: 9709/11/M/J/11/9, Topic: Circular measure

In the diagram, $O A B$ is an isosceles triangle with $O A=O B$ and angle $A O B=2 \theta$ radians. Arc $P S T$ has centre $O$ and radius $r$, and the line $A S B$ is a tangent to the $\operatorname{arc} P S T$ at $S$.

$\text{(i)}$ Find the total area of the shaded regions in terms of $r$ and $\theta$. $[4]$

$\text{(ii)}$ In the case where $\theta=\frac{1}{3} \pi$ and $r=6$, find the total perimeter of the shaded regions, leaving your answer in terms of $\sqrt{3}$ and $\pi$. $[5]$

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

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned} $$$\text{(i)}$ Find and simplify expressions for $\mathrm{fg}(x)$ and $\operatorname{gf}(x)$. $[2]$

$\text{(ii)}$ Hence find the value of $a$ for which $\mathrm{fg}(a)=\operatorname{gf}(a)$. $[3]$

$\text{(iii)}$ Find the value of $b(b \neq a)$ for which $\mathrm{g}(b)=b$. $[2]$

$\text{(iv)}$ Find and simplify an expression for $\mathrm{f}^{-1} \mathrm{~g}(x)$. $[2]$

The function $\mathrm{h}$ is defined by

$$ \mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0 $$$\text{(v)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[2]$