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

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	4	4	5	6	8	8	7	7	9	9	12	11	90
Score

Question 1 Code: 9709/13/O/N/17/1, Topic: Series

An arithmetic progression has first term $-12$ and common difference $6.$ The sum of the first $n$ terms exceeds $3000.$ Calculate the least possible value of $n$. $[4]$

Question 2 Code: 9709/11/M/J/20/1, Topic: Series

The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms is 91.

Find the first term and the common difference of the progression. $[4]$

Question 3 Code: 9709/11/M/J/21/2, Topic: Series

The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410.

Find the 60th term of the progression. $[5]$

Question 4 Code: 9709/11/M/J/19/5, Topic: Quadratics

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=-2 x^{2}+12 x-3$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $-2 x^{2}+12 x-3$ in the form $-2(x+a)^{2}+b$, where $a$ and $b$ are constants. $[2]$

$\text{(ii)}$ State the greatest value of $\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2 x+5$ for $x \in \mathbb{R}$. $[1]$

$\text{(iii)}$ Find the values of $x$ for which $\operatorname{gf}(x)+1=0$. $[3]$

Question 5 Code: 9709/11/O/N/18/6, Topic: Differentiation, Coordinate geometry

A curve has a stationary point at $\left(3,9 \frac{1}{2}\right)$ and has an equation for which $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=a x^{2}+a^{2} x$, where $a$ is a non-zero constant.

$\text{(i)}$ Find the value of $a$. $[2]$

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

$\text{(iii)}$ Determine, showing all necessary working, the nature of the stationary point. $[2]$

Question 6 Code: 9709/11/O/N/18/7, Topic: Integration

 

The diagram shows part of the curve with equation $y=k\left(x^{3}-7 x^{2}+12 x\right)$ for some constant $k$. The curve intersects the line $y=x$ at the origin $O$ and at the point $A(2,2)$.

$\text{(i)}$ Find the value of $k$. $[1]$

$\text{(ii)}$ Verify that the curve meets the line $y=x$ again when $x=5$. $[2]$

$\text{(iii)}$ Find, showing all necessary working, the area of the shaded region. $[5]$

Question 7 Code: 9709/13/O/N/19/7, Topic: Trigonometry

$\text{(i)}$ Show that the equation $3 \cos ^{4} \theta+4 \sin ^{2} \theta-3=0$ can be expressed as $3 x^{2}-4 x+1=0$, where $x=\cos ^{2} \theta$. $[2]$

$\text{(ii)}$ Hence solve the equation $3 \cos ^{4} \theta+4 \sin ^{2} \theta-3=0$ for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[5]$

Question 8 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 9 Code: 9709/13/M/J/18/10, Topic: Functions

The one-one function $\mathrm{f}$ is defined by $\mathrm{f}(x)=(x-2)^{2}+2$ for $x \geqslant c$, where $c$ is a constant.

$\text{(i)}$ State the smallest possible value of $c$. $[1]$

In parts $\text{(ii)}$ and $\text{(iii)}$ the value of $c$ is 4.

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$. $[3]$

$\text{(iii)}$ Solve the equation $\mathrm{ff}(x)=51$, giving your answer in the form $a+\sqrt{b}$. $[5]$

Question 10 Code: 9709/12/O/N/18/10, Topic: Quadratics, Coordinate geometry

The equation of a curve is $\displaystyle y=2 x+\frac{12}{x}$ and the equation of a line is $y+x=k$, where $k$ is a constant.

$\text{(i)}$ Find the set of values of $k$ for which the line does not meet the curve. $[3]$

In the case where $k=15$, the curve intersects the line at points $A$ and $B$.

$\text{(ii)}$ Find the coordinates of $A$ and $B$. $[3]$

$\text{(iii)}$ Find the equation of the perpendicular bisector of the line joining $A$ and $B$. $[3]$

Question 11 Code: 9709/12/O/N/18/11, Topic: Differentiation, Integration, Coordinate geometry

 

The diagram shows part of the curve $y=3 \sqrt{(} 4 x+1)-2 x$. The curve crosses the $y$-axis at $A$ and the stationary point on the curve is $M$.

$\text{(i)}$ Obtain expressions for $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\int y \mathrm{~d} x$. $[5]$

$\text{(ii)}$ Find the coordinates of $M$. $[3]$

$\text{(iii)}$ Find, showing all necessary working, the area of the shaded region. $[4]$

Question 12 Code: 9709/13/O/N/19/11, Topic: Coordinate geometry, Integration

 

The diagram shows part of the curve $y=(x-1)^{-2}+2$, and the lines $x=1$ and $x=3$. The point $A$ on the curve has coordinates $(2,3)$. The normal to the curve at $A$ crosses the line $x=1$ at $B$.

$\text{(i)}$ Show that the normal $A B$ has equation $\displaystyle y=\frac{1}{2} x+2$. $[3]$

$\text{(ii)}$ Find, showing all necessary working, the volume of revolution obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[8]$