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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 4 4 7 5 7 7 7 7 7 11 11 81
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/15/2, Topic: Circular measure In the diagram,$A Y B$is a semicircle with$A B$as diameter and$O A X B$is a sector of a circle with centre$O$and radius$r$. Angle$A O B=2 \theta$radians. Find an expression, in terms of$r$and$\theta$, for the area of the shaded region.$$Question 2 Code: 9709/11/M/J/16/2, Topic: Trigonometry Solve the equation$3 \sin ^{2} \theta=4 \cos \theta-1$for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$$Question 3 Code: 9709/12/M/J/16/2, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{8}{(5-2 x)^{2}}.$Given that the curve passes through$(2,7)$, find the equation of the curve.$$Question 4 Code: 9709/13/M/J/11/4, Topic: Differentiation, Integration$\text{(a)}$Differentiate$\displaystyle\frac{2 x^{3}+5}{x}$with respect to$x$.$\text{(b)}$Find$\displaystyle\int(3 x-2)^{5} \mathrm{~d} x$and hence find the value of$\displaystyle\int_{0}^{1}(3 x-2)^{5} \mathrm{~d} x$.$$Question 5 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$u$.$$Question 6 Code: 9709/11/M/J/17/4, Topic: Series$\text{(a)}$An arithmetic progression has a first term of 32 , a 5 th term of 22 and a last term of$-28$. Find the sum of all the terms in the progression.$\text{(b)}$Each year a school allocates a sum of money for the library. The amount allocated each year increases by$2.5 \%$of the amount allocated the previous year. In 2005 the school allocated$\$2000$. Find the total amount allocated in the years 2005 to 2014 inclusive. $$

Question 7 Code: 9709/12/M/J/10/6, Topic: Series

$\text{(i)}$ Find the first 3 terms in the expansion of $(1+a x)^{5}$ in ascending powers of $x$. $$

$\text{(ii)}$ Given that there is no term in $x$ in the expansion of $(1-2 x)(1+a x)^{5}$, find the value of the constant $a$. $$

$\text{(iii)}$ For this value of $a$, find the coefficient of $x^{2}$ in the expansion of $(1-2 x)(1+a x)^{5}$. $$

Question 8 Code: 9709/13/M/J/12/6, Topic: Series

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

$\text{(i)}$ Find the common difference of the progression. $$

The first term, the ninth term and the $n$th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.

$\text{(ii)}$ Find the common ratio of the geometric progression and the value of $n$. $$

Question 9 Code: 9709/12/M/J/17/6, Topic: Integration

The diagram shows the straight line $x+y=5$ intersecting the curve $\displaystyle y=\frac{4}{x}$ at the points $A(1,4)$ and $B(4,1)$. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $$

Question 10 Code: 9709/13/M/J/21/8, Topic: Functions

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

\begin{aligned}&\mathrm{f}: x \mapsto x^{2}-1 \text { for } x<0, \\&\mathrm{~g}: x \mapsto \frac{1}{2 x+1} \text { for } x<-\frac{1}{2}.\end{aligned}

$\text{(a)}$ Solve the equation $\mathrm{fg}(x)=3$. $$

$\text{(b)}$ Find an expression for $(\mathrm{fg})^{-1}(x)$. $$

Question 11 Code: 9709/12/M/J/17/10, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3 \tan \left(\frac{1}{2} x\right)-2$, for $-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)+4=0$, giving your answer correct to 1 decimal place. $$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and find the domain of $\mathrm{f}^{-1}$. $$

$\text{(iii)}$ Sketch, on the same diagram, the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$. $$

Question 12 Code: 9709/13/M/J/21/11, Topic: Differentiation, Coordinate geometry, Integration The diagram shows part of the curve with equation $\displaystyle y=x^{\frac{1}{2}}+k^{2} x^{-\frac{1}{2}}$, where $k$ is a positive constant.

$\text{(a)}$ Find the coordinates of the minimum point of the curve, giving your answer in terms of $k$. $$

The tangent at the point on the curve where $x = 4k^{2}$ intersects the $y$-axis at $P$.

$\text{(b)}$ Find the $y$-coordinate of $P$ in terms of $k$. $$

The shaded region is bounded by the curve, the $x$-axis and the lines $x = \frac{9}{4} k^{2}$ and $x = 4 k^{2}$.

$\text{(c)}$ Find the area of the shaded region in terms of $k$. $$

Worked solutions: P1, P3 & P6 (S1)

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