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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Probability & Statistics 1 (S1) | Variant(s) | P61, P62, P63 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 3 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 12 | 9 | 81 |

Score |

Question 1 Code: 9709/13/M/J/17/1, Topic: Series

The coefficients of $x$ and $x^{2}$ in the expansion of $(2+a x)^{7}$ are equal. Find the value of the non-zero constant $a$. $[3]$

Question 2 Code: 9709/11/M/J/18/2, Topic: Differentiation

A point is moving along the curve $\displaystyle y=2 x+\frac{5}{x}$ in such a way that the $x$-coordinate is increasing at a constant rate of $0.02$ units per second. Find the rate of change of the $y$-coordinate when $x=1$. $[4]$

Question 3 Code: 9709/11/M/J/16/3, Topic: Integration

The diagram shows part of the curve $\displaystyle x=\frac{12}{y^{2}}-2$. The shaded region is bounded by the curve, the $y$-axis and the lines $y=1$ and $y=2$. Showing all necessary working, find the volume, in terms of $\pi$, when this shaded region is rotated through $360^{\circ}$ about the $y$-axis. $[5]$

Question 4 Code: 9709/12/M/J/12/4, Topic: Coordinate geometry

The point $A$ has coordinates $(-1,-5)$ and the point $B$ has coordinates $(7,1)$. The perpendicular bisector of $A B$ meets the $x$-axis at $C$ and the $y$-axis at $D$. Calculate the length of $C D$. $[6]$

Question 5 Code: 9709/13/M/J/18/4, Topic: Coordinate geometry

A curve with equation $y=\mathrm{f}(x)$ passes through the point $A(3,1)$ and crosses the $y$-axis at $B$. It is given that $\mathrm{f}^{\prime}(x)=(3 x-1)^{-\frac{1}{3}}$. Find the $y$-coordinate of $B$. $[6]$

Question 6 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 7 Code: 9709/12/M/J/20/5, Topic: Functions

The function $\mathrm{f}$ is defined for $x \in \mathbb{R}$ by

$$ \text { f: } x \mapsto a-2 x $$where $a$ is a constant.

$\text{(a)}$ Express $\mathrm{ff}(x)$ and $\mathrm{f}^{-1}(x)$ in terms of $a$ and $x$. $[4]$

$\text{(b)}$ Given that $\mathrm{ff}(x)=\mathrm{f}^{-1}(x)$, find $x$ in terms of $a$. $[2]$

Question 8 Code: 9709/13/M/J/10/7, Topic: Circular measure

The diagram shows a metal plate $A B C D E F$ which has been made by removing the two shaded regions from a circle of radius 10 cm and centre $O.$ The parallel edges $A B$ and $E D$ are both of length 12 cm.

$\text{(i)}$ Show that angle $D O E$ is 1.287 radians, correct to 4 significant figures. $[2]$

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

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

Question 9 Code: 9709/12/M/J/17/8, Topic: Vectors

Relative to an origin $O$, the position vectors of three points $A, B$ and $C$ are given by $\overrightarrow{O A}=3 \mathbf{i}+p \mathbf{j}-2 p \mathbf{k}, \quad \overrightarrow{O B}=6 \mathbf{i}+(p+4) \mathbf{j}+3 \mathbf{k} \quad$ and $\quad \overrightarrow{O C}=(p-1) \mathbf{i}+2 \mathbf{j}+q \mathbf{k}$ where $p$ and $q$ are constants.

$\text{(i)}$ In the case where $p=2$, use a scalar product to find angle $A O B$. $[4]$

$\text{(ii)}$ In the case where $\overrightarrow{A B}$ is parallel to $\overrightarrow{O C}$, find the values of $p$ and $q$. $[4]$

Question 10 Code: 9709/12/M/J/11/9, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=3-4 \cos ^{k} x$, for $0 \leqslant x \leqslant \pi$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=2$,

$\text{(a)}$ find the range of $\mathrm{f}$, $[2]$

$\text{(b)}$ find the exact solutions of the equation $\mathrm{f}(x)=1$. $[3]$

$\text{(ii)}$ In the case where $k=1$,

$\text{(a)}$ sketch the graph of $y=\mathrm{f}(x)$, $[2]$

$\text{(b)}$ state, with a reason, whether $\mathrm{f}$ has an inverse. $[1]$

Question 11 Code: 9709/13/M/J/11/10, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned} $$$\text{(i)}$ Evaluate $\mathrm{fg(2)}$. $[2]$

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

$\text{(iii)}$ Obtain an expression for $\mathrm{g}^{\prime}(x)$ and use your answer to explain why $\mathrm{g}$ has an inverse. $[3]$

$\text{(iv)}$ Express each of $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$ in terms of $x$. $[4]$

Question 12 Code: 9709/13/M/J/16/10, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=2 x+3$ for $x \geqslant 0$. The function $\mathrm{g}$ is such that $\mathrm{g}(x)=a x^{2}+b$ for $x \leqslant q$, where $a, b$ and $q$ are constants. The function fg is such that fg $(x)=6 x^{2}-21$ for $x \leqslant q$

$\text{(i)}$ Find the values of $a$ and $b$. $[3]$

$\text{(ii)}$ Find the greatest possible value of $q$. $[2]$

It is now given that $q=-3$.

$\text{(iii)}$ Find the range of $\mathrm{fg}$. $[1]$

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