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

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 | 5 | 4 | 4 | 6 | 7 | 8 | 7 | 8 | 8 | 9 | 9 | 11 | 86 |

Score |

Question 1 Code: 9709/13/M/J/11/1, Topic: Trigonometry

The coefficient of $x^{3}$ in the expansion of $(a+x)^{5}+(1-2 x)^{6}$, where $a$ is positive, is 90. Find the value of $a$. $[5]$

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

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$ and $(2,5)$ is a point on the curve. Find the equation of the curve. $[4]$

Question 3 Code: 9709/11/M/J/12/2, Topic: Series

Find the coefficient of $x^{6}$ in the expansion of $\displaystyle\left(2 x^{3}-\frac{1}{x^{2}}\right)^{7}$. $[4]$

Question 4 Code: 9709/12/M/J/12/5, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\tan x+\frac{1}{\tan x} \equiv \frac{1}{\sin x \cos x}$. $[2]$

$\text{(ii)}$ Solve the equation $\displaystyle\frac{2}{\sin x \cos x}=1+3 \tan x$, for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$

Question 5 Code: 9709/11/M/J/10/6, Topic: Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-6$ and the point $(9,2)$ lies on the curve.

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

$\text{(ii)}$ Find the $x$-coordinate of the stationary point on the curve and determine the nature of the stationary point. $[3]$

Question 6 Code: 9709/12/M/J/12/6, Topic: Circular measure

The diagram shows a metal plate made by removing a segment from a circle with centre $O$ and radius $8 \mathrm{~cm}.$ The line $A B$ is a chord of the circle and angle $A O B=2.4$ radians. Find

$\text{(i)}$ the length of $A B$, $[2]$

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

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

Question 7 Code: 9709/11/M/J/12/7, Topic: Series

$\text{(a)}$ The first two terms of an arithmetic progression are 1 and $\cos ^{2} x$ respectively. Show that the sum of the first ten terms can be expressed in the form $a-b \sin ^{2} x$, where $a$ and $b$ are constants to be found. $[3]$

$\text{(b)}$ The first two terms of a geometric progression are 1 and $\frac{1}{3} \tan ^{2} \theta$ respectively, where $0< \theta <\frac{1}{2} \pi$.

$\text{(i)}$ Find the set of values of $\theta$ for which the progression is convergent. $[2]$

$\text{(ii)}$ Find the exact value of the sum to infinity when $\theta=\frac{1}{6} \pi$. $[2]$

Question 8 Code: 9709/13/M/J/20/7, Topic: Trigonometry

$\text{(a)}$ Show that $\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta} \equiv \frac{2}{\sin \theta \cos \theta}$. $[4]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta}=\frac{6}{\tan \theta}$ for $0^{\circ}< \theta <180^{\circ}$. $[4]$

Question 9 Code: 9709/13/M/J/14/8, Topic: Quadratics

$\text{(i)}$ Express $2 x^{2}-10 x+8$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants, and use your answer to state the minimum value of $2 x^{2}-10 x+8$. $[4]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $2 x^{2}-10 x+8=k x$ has no real roots. $[4]$

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

In the diagram, $O A B$ is an isosceles triangle with $O A=O B$ and angle $A O B=2 \theta$ radians. Arc $P S T$ has centre $O$ and radius $r$, and the line $A S B$ is a tangent to the $\operatorname{arc} P S T$ at $S$.

$\text{(i)}$ Find the total area of the shaded regions in terms of $r$ and $\theta$. $[4]$

$\text{(ii)}$ In the case where $\theta=\frac{1}{3} \pi$ and $r=6$, find the total perimeter of the shaded regions, leaving your answer in terms of $\sqrt{3}$ and $\pi$. $[5]$

Question 11 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 12 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. $[2]$