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

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

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. $[4]$

Question 2 Code: 9709/11/M/J/16/2, Topic: Trigonometry

Solve the equation $3 \sin ^{2} \theta=4 \cos \theta-1$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

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. $[4]$

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 $x$. $[3]$

$\text{(b)}$ Find $\displaystyle\int(3 x-2)^{5} \mathrm{~d} x$ and hence find the value of $\displaystyle\int_{0}^{1}(3 x-2)^{5} \mathrm{~d} x$. $[4]$

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 $u$. $[5]$

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. $[4]$

$\text{(b)}$ 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 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 $x$. $[2]$

$\text{(ii)}$ 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 $a$. $[2]$

$\text{(iii)}$ For this value of $a$, find the coefficient of $x^{2}$ in the expansion of $(1-2 x)(1+a x)^{5}$. $[3]$

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. $[2]$

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 $n$. $[5]$

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. $[7]$

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 $\mathrm{fg}(x)=3$. $[4]$

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

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. $[3]$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and find the domain of $\mathrm{f}^{-1}$. $[5]$

$\text{(iii)}$ Sketch, on the same diagram, the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$. $[3]$

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 $k$. $[4]$

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 $k$. $[4]$

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 $k$. $[3]$