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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 | 5 | 3 | 5 | 4 | 6 | 5 | 7 | 7 | 7 | 5 | 11 | 11 | 76 |

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: 9709/11/M/J/13/1, Topic: Differentiation

It is given that $\mathrm{f}(x)=(2 x-5)^{3}+x$, for $x \in \mathbb{R}$. Show that $\mathrm{f}$ is an increasing function. $[3]$

Question 3 Code: 9709/13/M/J/10/2, Topic: Series

$\text{(i)}$ Find the first three terms, in descending powers of $x$, in the expansion of $\displaystyle \left(x-\frac{2}{x}\right)^{6}$. $[3]$

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

Question 4 Code: 9709/12/M/J/16/3, Topic: Vectors

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

$$ \overrightarrow{O A}=2 \mathbf{i}-5 \mathbf{j}-2 \mathbf{k} \quad \text { and } \quad \overrightarrow{O B}=4 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k} $$The point $C$ is such that $\overrightarrow{A B}=\overrightarrow{B C}$. Find the unit vector in the direction of $\overrightarrow{O C}$. $[4]$

Question 5 Code: 9709/12/M/J/17/3, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2} \equiv \frac{1-\sin \theta}{1+\sin \theta}$. $[3]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2}=\frac{1}{2}$, for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

Question 6 Code: 9709/13/M/J/18/3, Topic: Series

The common ratio of a geometric progression is $0.99$. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. $[5]$

Question 7 Code: 9709/11/M/J/11/7, Topic: Differentiation

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

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

$\text{(ii)}$ Find the set of values of $x$ for which the gradient of the curve is less than $\frac{1}{3}$. $[3]$

Question 8 Code: 9709/12/M/J/11/7, Topic: Coordinate geometry

The line $L_{1}$ passes through the points $A(2,5)$ and $B(10,9)$. The line $L_{2}$ is parallel to $L_{1}$ and passes through the origin. The point $C$ lies on $L_{2}$ such that $A C$ is perpendicular to $L_{2}.$ Find

$\text{(i)}$ the coordinates of $C$, $[5]$

$\text{(ii)}$ the distance $A C$. $[2]$

Question 9 Code: 9709/11/M/J/16/7, Topic: Circular measure

In the diagram, $A O B$ is a quarter circle with centre $O$ and radius $r$. The point $C$ lies on the arc $A B$ and the point $D$ lies on $O B.$ The line $C D$ is parallel to $A O$ and angle $A O C=\theta$ radians.

$\text{(i)}$ Express the perimeter of the shaded region in terms of $r, \theta$ and $\pi$. $[4]$

$\text{(ii)}$ For the case where $r=5 \mathrm{~cm}$ and $\theta=0.6$, find the area of the shaded region. $[3]$

Question 10 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 11 Code: 9709/11/M/J/15/10, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{8}{\sqrt{(} 3 x+4)}.$ The curve intersects the $y$-axis at $A(0,4).$ The normal to the curve at $A$ intersects the line $x=4$ at the point $B$.

$\text{(i)}$ Find the coordinates of $B$. $[5]$

$\text{(ii)}$ Show, with all necessary working, that the areas of the regions marked $P$ and $Q$ are equal. $[6]$

Question 12 Code: 9709/12/M/J/16/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 6 x-x^{2}-5$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find the set of values of $x$ for which $\mathrm{f}(x) \leqslant 3$. $[3]$

$\text{(ii)}$ Given that the line $y=m x+c$ is a tangent to the curve $y=\mathrm{f}(x)$, show that $4 c=m^{2}-12 m+16$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 6 x-x^{2}-5$ for $x \geqslant k$, where $k$ is a constant.

$\text{(iii)}$ Express $6 x-x^{2}-5$ in the form $a-(x-b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iv)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{g}^{-1}(x)$. $[2]$