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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 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 5 5 5 5 7 9 5 8 8 9 75
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/13/1, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$and$(2,5)$is a point on the curve. Find the equation of the curve.$[4]$Question 2 Code: 9709/12/M/J/19/2, Topic: Coordinate geometry Two points$A$and$B$have coordinates$(1,3)$and$(9,-1)$respectively. The perpendicular bisector of$A B$intersects the$y$-axis at the point$C$. Find the coordinates of$C$.$[5]$Question 3 Code: 9709/11/M/J/11/3, Topic: Integration$\text{(i)}$Sketch the curve$y=(x-2)^{2}$.$[1]\text{(ii)}$The region enclosed by the curve, the$x$-axis and the$y$-axis is rotated through$360^{\circ}$about the$x$-axis. Find the volume obtained, giving your answer in terms of$\pi$.$[4]$Question 4 Code: 9709/12/M/J/11/3, Topic: Quadratics The equation$x^{2}+p x+q=0$, where$p$and$q$are constants, has roots$-3$and 5.$\text{(i)}$Find the values of$p$and$q$.$[2]\text{(ii)}$Using these values of$p$and$q$, find the value of the constant$r$for which the equation$x^{2}+p x+q+r=0$has equal roots.$[3]$Question 5 Code: 9709/12/M/J/18/3, Topic: Series A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by$2 \%$of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained$8000 \mathrm{~kg}$of salt.$\text{(i)}$Find the amount of salt obtained in the 12 th week after the change.$[3]\text{(ii)}$Find the total amount of salt obtained in the first 12 weeks after the change.$[2]$Question 6 Code: 9709/12/M/J/15/4, Topic: Differentiation Variables$u, x$and$y$are such that$u=2 x(y-x)$and$x+3 y=12$. Express$u$in terms of$x$and hence find the stationary value of$u$.$[5]$Question 7 Code: 9709/11/M/J/17/4, Topic: Series$\text{(a)}$An arithmetic progression has a first term of 32 , a 5 th term of 22 and a last term of$-28$. Find the sum of all the terms in the progression.$[4]\text{(b)}$Each year a school allocates a sum of money for the library. The amount allocated each year increases by$2.5 \%$of the amount allocated the previous year. In 2005 the school allocated$\$2000$. Find the total amount allocated in the years 2005 to 2014 inclusive. $[3]$

Question 8 Code: 9709/13/M/J/18/7, Topic: Trigonometry

$\text{(a)}$ $\quad\text{(i)}$ Express $\displaystyle \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}$ in the form $a \sin ^{2} \theta+b$, where $a$ and $b$ are constants to be found. $[3]$

$\qquad \text{(ii)}$ Hence, or otherwise, and showing all necessary working, solve the equation. $[2]$

$$\frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}=\frac{1}{4}$$

$\qquad$ for $-90^{\circ} \leqslant \theta \leqslant 0^{\circ}$

$\text{(b)}$

The diagram shows the graphs of $y=\sin x$ and $y=2 \cos x$ for $-\pi \leqslant x \leqslant \pi$. The graphs intersect at the points $A$ and $B$.

$\text{(i)}$ Find the $x$-coordinate of $A$. $[2]$

$\text{(ii)}$ Find the $y$-coordinate of $B$. $[2]$

Question 9 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 10 Code: 9709/13/M/J/14/8, Topic: Quadratics

$\text{(i)}$ Express $2 x^{2}-10 x+8$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants, and use your answer to state the minimum value of $2 x^{2}-10 x+8$. $[4]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $2 x^{2}-10 x+8=k x$ has no real roots. $[4]$

Question 11 Code: 9709/12/M/J/20/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are such that

\begin{aligned} &\mathrm{f}(x)=2-3 \sin 2 x \text { for } 0 \leqslant x \leqslant \pi, \\ &\mathrm{g}(x)=-2 \mathrm{f}(x) \quad \text { for } 0 \leqslant x \leqslant \pi \end{aligned}

$\text{(a)}$ State the ranges of $\mathrm{f}$ and $\mathrm{g}$. $[3]$

The diagram below shows the graph of $y=\mathrm{f}(x)$.

$\text{(b)}$ Sketch, on this diagram, the graph of $y=\mathrm{g}(x)$. $[2]$

The function $\mathrm{h}$ is such that

$$\mathrm{h}(x)=\mathrm{g}(x+\pi) \quad \text { for }-\pi \leqslant x \leqslant 0$$

$\text{(c)}$ Describe fully a sequence of transformations that maps the curve $y=\mathrm{f}(x)$ on to $y=\mathrm{h}(x)$. $[3]$

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. $[5]$

Worked solutions: P1, P3 & P6 (S1)

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