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

Qtn No.	1	2	3	4	5	6	7	8	Total
Marks	4	4	5	7	5	7	8	9	49
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: 9709/13/O/N/11/2, Topic: Series

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

$\text{(i)}$ an arithmetic progression, $[2]$

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

Question 3 Code: 9709/12/O/N/20/3, Topic: Quadratics

The equation of a curve is $y=2 x^{2}+m(2 x+1)$, where $m$ is a constant, and the equation of a line is $y=6 x+4$.

Show that, for all values of $m$, the line intersects the curve at two distinct points. $[5]$

Question 4 Code: 9709/12/O/N/11/4, Topic: Quadratics

The equation of a curve is $y^{2}+2 x=13$ and the equation of a line is $2 y+x=k$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=8$, find the coordinates of the points of intersection of the line and the curve. $[4]$

$\text{(ii)}$ Find the value of $k$ for which the line is a tangent to the curve. $[3]$

Question 5 Code: 9709/12/O/N/20/4, Topic: Series

The sum, $S_{n}$, of the first $n$ terms of an arithmetic progression is given by

$$ S_{n}=n^{2}+4 n $$

The $k$ th term in the progression is greater than 200.

Find the smallest possible value of $k$. $[5]$

Question 6 Code: 9709/12/O/N/17/5, Topic: Trigonometry

$\text{(i)}$ Show that the equation $\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$ can be expressed as $[3]$

$$ 2 \cos ^{2} 2 x+3 \cos 2 x+1=0 $$

$\text{(ii)}$ Hence solve the equation $\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$

Question 7 Code: 9709/12/O/N/16/6, Topic: Circular measure

 

The diagram shows a metal plate $A B C D$ made from two parts. The part $B C D$ is a semicircle. The part $D A B$ is a segment of a circle with centre $O$ and radius $10 \mathrm{~cm}$. Angle $B O D$ is $1.2$ radians.

$\text{(i)}$ Show that the radius of the semicircle is $5.646 \mathrm{~cm}$, correct to 3 decimal places. $[2]$

$\text{(ii)}$ Find the perimeter of the metal plate. $[3]$

$\text{(iii)}$ Find the area of the metal plate. $[3]$

Question 8 Code: 9709/12/O/N/15/8, Topic: Functions

The function $\mathrm{f}$ is defined, for $x \in \mathbb{R}$, by $\mathrm{f}: x \mapsto x^{2}+a x+b$, where $a$ and $b$ are constants.

$\text{(i)}$ In the case where $a=6$ and $b=-8$, find the range of $\mathrm{f}$. $[3]$

$\text{(ii)}$ In the case where $a=5$, the roots of the equation $\mathrm{f}(x)=0$ are $k$ and $-2 k$, where $k$ is a constant. Find the values of $b$ and $k$. $[3]$

$\text{(iii)}$ Show that if the equation $\mathrm{f}(x+a)=a$ has no real roots, then $a^{2}<4(b-a)$. $[3]$