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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 | 4 | 7 | 7 | 7 | 5 | 10 | 6 | 8 | 10 | 12 | 11 | 91 |

Score |

Question 1 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 2 Code: 9709/12/M/J/16/2, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{8}{(5-2 x)^{2}}.$ Given that the curve passes through $(2,7)$, find the equation of the curve. $[4]$

Question 3 Code: 9709/11/M/J/18/5, Topic: Coordinate geometry

The diagram shows a kite $O A B C$ in which $A C$ is the line of symmetry. The coordinates of $A$ and $C$ are $(0,4)$ and $(8,0)$ respectively and $O$ is the origin.

$\text{(i)}$ Find the equations of $A C$ and $O B$. $[4]$

$\text{(ii)}$ Find, by calculation, the coordinates of $B$. $[3]$

Question 4 Code: 9709/12/M/J/14/6, Topic: Series

The $1 \mathrm{st}, 2 \mathrm{nd}$ and $3 \mathrm{rd}$ terms of a geometric progression are the $1 \mathrm{st}, 9$ th and $21 \mathrm{st}$ terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is $r$, where $r \neq 1$. Find

$\text{(i)}$ the value of $r$, $[4]$

$\text{(ii)}$ the 4 th term of each progression. $[3]$

Question 5 Code: 9709/13/M/J/17/7, Topic: Circular measure

The diagram shows two circles with centres $A$ and $B$ having radii $8 \mathrm{~cm}$ and $10 \mathrm{~cm}$ respectively. The two circles intersect at $C$ and $D$ where $C A D$ is a straight line and $A B$ is perpendicular to $C D$.

$\text{(i)}$ Find angle $A B C$ in radians. $[1]$

$\text{(ii)}$ Find the area of the shaded region. $[6]$

Question 6 Code: 9709/11/M/J/21/7, Topic: Trigonometry

$\text{(a)}$ Prove the identity $\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta} \equiv 1-\tan ^{2} \theta$. $[2]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta}=2 \tan ^{4} \theta$ for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[3]$

Question 7 Code: 9709/11/M/J/10/8, Topic: Coordinate geometry

The diagram shows a triangle $A B C$ in which $A$ is $(3,-2)$ and $B$ is $(15,22)$. The gradients of $A B, A C$ and $B C$ are $2 m,-2 m$ and $m$ respectively, where $m$ is a positive constant.

$\text{(i)}$ Find the gradient of $A B$ and deduce the value of $m$. $[2]$

$\text{(ii)}$ Find the coordinates of $C$. $[4]$

The perpendicular bisector of $A B$ meets $B C$ at $D$.

$\text{(iii)}$ Find the coordinates of $D$. $[4]$

Question 8 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 9 Code: 9709/13/M/J/15/9, Topic: Series

$\text{(a)}$ The first term of an arithmetic progression is $-2222$ and the common difference is 17. Find the value of the first positive term. $[3]$

$\text{(b)}$ The first term of a geometric progression is $\sqrt{3}$ and the second term is $2 \cos \theta$, where $0 < \theta < \pi$. Find the set of values of $\theta$ for which the progression is convergent. $[5]$

Question 10 Code: 9709/13/M/J/21/10, Topic: Coordinate geometry

Points $A(-2,3), B(3,0)$ and $C(6,5)$ lie on the circumference of a circle with centre $D$.

$\text{(a)}$ Show that angle $A B C=90^{\circ}$. $[2]$

$\text{(b)}$ Hence state the coordinates of $D$. $[1]$

$\text{(c)}$ Find an equation of the circle. $[2]$

The point $E$ lies on the circumference of the circle such that $BE$ is a diameter.

$\text{(d)}$ Find an equation of the tangent to the circle at $E$. $[5]$

Question 11 Code: 9709/12/M/J/19/11, Topic: Differentiation, Integration

The diagram shows part of the curve $\displaystyle y=\sqrt{(} 4 x+1)+\frac{9}{\sqrt{(4 x+1)}}$ and the minimum point $M$.

$\text{(i)}$ Find expressions for $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\int y \mathrm{~d} x$. $[6]$

$\text{(ii)}$ Find the coordinates of $M$. $[3]$

The shaded region is bounded by the curve, the $y$-axis and the line through $M$ parallel to the $x$-axis.

$\text{(iii)}$ Find, showing all necessary working, the area of the shaded region. $[3]$

Question 12 Code: 9709/12/M/J/20/11, Topic: Coordinate geometry

The equation of a circle with centre $C$ is $x^{2}+y^{2}-8 x+4 y-5=0$.

$\text{(a)}$ Find the radius of the circle and the coordinates of $C$. $[3]$

The point $P(1,2)$ lies on the circle.

$\text{(b)}$ Show that the equation of the tangent to the circle at $P$ is $4 y=3 x+5$. The point $Q$ also lies on the circle and $P Q$ is parallel to the $x$-axis. $[3]$

$\text{(c)}$ Write down the coordinates of $Q$. $[2]$

The tangents to the circle at $P$ and $Q$ meet at $T$.

$\text{(d)}$ Find the coordinates of $T$. $[3]$