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

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	4	4	5	5	6	7	8	8	6	11	8	12	84
Score

Question 1 Code: 9709/13/M/J/16/2, Topic: Integration

 

The diagram shows part of the curve $\displaystyle y=\left(x^{3}+1\right)^{\frac{1}{2}}$ and the point $P(2,3)$ lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

Question 2 Code: 9709/13/M/J/20/2, Topic: Integration

The equation of a curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-3 x^{-\frac{1}{2}}.$ It is given that the point $(4,7)$ lies on the curve. Find the equation of the curve. $[4]$

Question 3 Code: 9709/12/M/J/21/2, Topic: Coordinate geometry

$\text{(a)}$ The graph of $y=\mathrm{f}(x)$ is transformed to the graph of $y=2 \mathrm{f}(x-1)$.

Describe fully the two single transformations which have been combined to give the resulting transformation. $[3]$

$\text{(b)}$ The curve $y=\sin 2 x-5 x$ is reflected in the $y$-axis and then stretched by scale factor $\frac{1}{3}$ in the $x$-direction.

Write down the equation of the transformed curve. $[2]$

Question 4 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$ The point $P(2,9)$ lies on the curve.

$\text{(i)}$ A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of $0.05$ units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. $[2]$

$\text{(ii)}$ Find the equation of the curve. $[3]$

Question 5 Code: 9709/12/M/J/10/5, Topic: Vectors

Relative to an origin $O$, the position vectors of the points $A$ and $B$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{r} -2 \\ 3 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{c} 4 \\ 1 \\ p \end{array}\right). $$

$\text{(i)}$ Find the value of $p$ for which $\overrightarrow{O A}$ is perpendicular to $\overrightarrow{O B}$. $[2]$

$\text{(ii)}$ Find the values of $p$ for which the magnitude of $\overrightarrow{A B}$ is 7. $[4]$

Question 6 Code: 9709/12/M/J/13/5, Topic: Trigonometry

It is given that $a=\sin \theta-3 \cos \theta$ and $b=3 \sin \theta+\cos \theta$, where $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.

$\text{(i)}$ Show that $a^{2}+b^{2}$ has a constant value for all values of $\theta$. $[3]$

$\text{(ii)}$ Find the values of $\theta$ for which $2 a=b$. $[4]$

Question 7 Code: 9709/11/M/J/15/7, Topic: Series

$\text{(a)}$ The third and fourth terms of a geometric progression are $\frac{1}{3}$ and $\frac{2}{9}$ respectively. Find the sum to infinity of the progression. $[4]$

$\text{(b)}$ A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector. $[4]$

Question 8 Code: 9709/12/M/J/17/7, Topic: Series

$\text{(a)}$ The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20000. $[4]$

$\text{(b)}$ A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression. $[4]$

Question 9 Code: 9709/11/M/J/14/8, Topic: Vectors

8 Relative to an origin $O$, the position vectors of points $A$ and $B$ are given by $$ \overrightarrow{O A}=\left(\begin{array}{c} 3 p \\ 4 \\ p^{2} \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{c} -p \\ -1 \\ p^{2} \end{array}\right) $$

$\text{(i)}$ Find the values of $p$ for which angle $A O B$ is $90^{\circ}$. $[3]$

$\text{(ii)}$ For the case where $p=3$, find the unit vector in the direction of $\overrightarrow{B A}$. $[3]$

Question 10 Code: 9709/11/M/J/10/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x^{2}-12 x+7$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $\mathrm{f}(x)$ in the form $a(x-b)^{2}-c$. $[3]$

$\text{(ii)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(iii)}$ Find the set of values of $x$ for which $f(x)<21$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 2 x+k$ for $x \in \mathbb{R}$.

$\text{(iv)}$ Find the value of the constant $k$ for which the equation $\operatorname{gf}(x)=0$ has two equal roots. $[4]$

Question 11 Code: 9709/12/M/J/19/9, Topic: Coordinate geometry

The curve $C_{1}$ has equation $y=x^{2}-4 x+7$. The curve $C_{2}$ has equation $y^{2}=4 x+k$, where $k$ is a constant. The tangent to $C_{1}$ at the point where $x=3$ is also the tangent to $C_{2}$ at the point $P$. Find the value of $k$ and the coordinates of $P$. $[8]$

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

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned} $$

$\text{(i)}$ Obtain expressions, in terms of $x$, for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

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

$\text{(iii)}$ Given that the equation $f g(x)=5-k x$, where $k$ is a constant, has no solutions, find the set of possible values of $k$. $[5]$