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Name of student | Date | ||||

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

Subject | Pure Mathematics 1 (P1) | Variant(s) | P11, P12, P13 | ||

Start time | Duration | Stop time |

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

Marks | 4 | 5 | 3 | 3 | 5 | 6 | 8 | 7 | 8 | 9 | 11 | 11 | 80 |

Score |

Question 1 Code: 9709/11/M/J/10/1, Topic: Trigonometry

The acute angle $x$ radians is such that $\tan x=k$, where $k$ is a positive constant. Express, in terms of $k$,

$\text{(i)}$ $\tan (\pi-x)$, $[1]$

$\text{(ii)}$ $\tan \left(\frac{1}{2} \pi-x\right)$, $[1]$

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

Question 2 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry

Find the coordinates of the point at which the perpendicular bisector of the line joining $(2,7)$ to $(10,3)$ meets the $x$-axis. $[5]$

Question 3 Code: 9709/13/M/J/14/1, Topic: Series

Find the coefficient of $x$ in the expansion of $\displaystyle\left(x^{2}-\frac{2}{x}\right)^{5}$. $[3]$

Question 4 Code: 9709/11/M/J/16/1, Topic: Series

Find the term independent of $x$ in the expansion of $\displaystyle\left(x-\frac{3}{2 x}\right)^{6}$. $[3]$

Question 5 Code: 9709/13/M/J/11/2, Topic: Coordinate geometry

Find the set of values of $m$ for which the line $y=m x+4$ intersects the curve $y=3 x^{2}-4 x+7$ at two distinct points. $[5]$

Question 6 Code: 9709/11/M/J/20/6, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-a \\ &\mathrm{~g}: x \mapsto 3 x+b \end{aligned} $$where $a$ and $b$ are constants.

$\text{(a)}$ Given that $\operatorname{gg}(2)=10$ and $\mathrm{f}^{-1}(2)=14$, find the values of $a$ and $b$. $[4]$

$\text{(b)}$ Using these values of $a$ and $b$, find an expression for $\operatorname{gf}(x)$ in the form $c x+d$, where $c$ and $d$ are constants. $[2]$

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: 9709/13/M/J/12/8, Topic: Circular measure

In the diagram, $A B$ is an arc of a circle with centre $O$ and radius $r$. The line $X B$ is a tangent to the circle at $B$ and $A$ is the mid-point of $O X$.

$\text{(i)}$ Show that angle $A O B=\frac{1}{3} \pi$ radians. $[2]$

Express each of the following in terms of $r, \pi$ and $\sqrt{3}$ :

$\text{(ii)}$ the perimeter of the shaded region, $[3]$

$\text{(iii)}$ the area of the shaded region. $[2]$

Question 9 Code: 9709/13/M/J/13/8, Topic: Vectors

The diagram shows a parallelogram $O A B C$ in which

$$ \overrightarrow{O A}=\left(\begin{array}{r} 3 \\ 3 \\ -4 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{l} 5 \\ 0 \\ 2 \end{array}\right) $$$\text{(i)}$ Use a scalar product to find angle $B O C$. $[6]$

$\text{(ii)}$ Find a vector which has magnitude 35 and is parallel to the vector $\overrightarrow{O C}$. $[2]$

Question 10 Code: 9709/13/M/J/17/9, Topic: Functions

$\text{(i)}$ Express $9 x^{2}-6 x+6$ in the form $(a x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=9 x^{2}-6 x+6$ for $x \geqslant p$, where $p$ is a constant.

$\text{(ii)}$ State the smallest value of $p$ for which $\mathrm{f}$ is a one-one function. $[1]$

$\text{(iii)}$ For this value of $p$, obtain an expression for $\mathrm{f}^{-1}(x)$, and state the domain of $\mathrm{f}^{-1}$. $[4]$

$\text{(iv)}$ State the set of values of $q$ for which the equation $\mathrm{f}(x)=q$ has no solution. $[1]$

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

$\text{(a)}$ A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is $5 \mathrm{~cm}$, find the perimeter of the smallest sector. $[6]$

$\text{(b)}$ The first, second and third terms of a geometric progression are $2 k+3, k+6$ and $k$, respectively. Given that all the terms of the geometric progression are positive, calculate

$\text{(i)}$ the value of the constant $k$, $[3]$

$\text{(ii)}$ the sum to infinity of the progression. $[2]$

Question 12 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 $y=\mathrm{f}^{-1}(x)$. $[3]$