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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 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 11 | 11 | 15 | 96 |

Score |

Question 1 Code: 9709/13/M/J/11/3, Topic: Coordinate geometry

The line $\displaystyle\frac{x}{a}+\frac{y}{b}=1$, where $a$ and $b$ are positive constants, meets the $x$-axis at $P$ and the $y$-axis at $Q$. Given that $P Q=\sqrt{(} 45)$ and that the gradient of the line $P Q$ is $-\frac{1}{2}$, find the values of $a$ and $b$. $[5]$

Question 2 Code: 9709/13/M/J/12/4, Topic: Trigonometry

$\text{(i)}$ Solve the equation $\sin 2 x+3 \cos 2 x=0$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$. $[5]$

$\text{(ii)}$ How many solutions has the equation $\sin 2 x+3 \cos 2 x=0$ for $0^{\circ} \leqslant x \leqslant 1080^{\circ}$? $[1]$

Question 3 Code: 9709/13/M/J/21/4, Topic: Trigonometry

$\text{(a)}$ Show that the equation $[2]$

$$ \frac{\tan x+\sin x}{\tan x-\sin x}=k $$where $k$ is a constant, may be expressed as

$$ \frac{1+\cos x}{1-\cos x}=k $$$\text{(b)}$ Hence express $\cos x$ in terms of $k$. $[2]$

$\text{(c)}$ Hence solve the equation $\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$ for $-\pi < x < \pi$. $[2]$

Question 4 Code: 9709/13/M/J/10/5, Topic: Differentiation, Integration

The equation of a curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{\sqrt{(} 3 x-2)}.$ Given that the curve passes through the point $P(2,11)$, find

$\text{(i)}$ the equation of the normal to the curve at $P$, $[3]$

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

Question 5 Code: 9709/12/M/J/13/5, Topic: Trigonometry

It is given that $a=\sin \theta-3 \cos \theta$ and $b=3 \sin \theta+\cos \theta$, where $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.

$\text{(i)}$ Show that $a^{2}+b^{2}$ has a constant value for all values of $\theta$. $[3]$

$\text{(ii)}$ Find the values of $\theta$ for which $2 a=b$. $[4]$

Question 6 Code: 9709/12/M/J/17/6, Topic: Integration

The diagram shows the straight line $x+y=5$ intersecting the curve $\displaystyle y=\frac{4}{x}$ at the points $A(1,4)$ and $B(4,1)$. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[7]$

Question 7 Code: 9709/12/M/J/15/7, Topic: Coordinate geometry

The point $C$ lies on the perpendicular bisector of the line joining the points $A(4,6)$ and $B(10,2)$. $C$ also lies on the line parallel to $A B$ through $(3,11)$.

$\text{(i)}$ Find the equation of the perpendicular bisector of $A B$. $[4]$

$\text{(ii)}$ Calculate the coordinates of $C$. $[3]$

Question 8 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry

The point $A$ has coordinates $(p, 1)$ and the point $B$ has coordinates $(9,3 p+1)$, where $p$ is a constant.

$\text{(i)}$ For the case where the distance $A B$ is 13 units, find the possible values of $p$. $[3]$

$\text{(ii)}$ For the case in which the line with equation $2 x+3 y=9$ is perpendicular to $A B$, find the value of $p$. $[4]$

Question 9 Code: 9709/12/M/J/20/7, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $2 r$, and angle $A O B=\frac{1}{6} \pi$ radians. The point $C$ is the midpoint of $O A$.

$\text{(a)}$ Show that the exact length of $B C$ is $r \sqrt{5-2 \sqrt{3}}$. $[2]$

$\text{(b)}$ Find the exact perimeter of the shaded region. $[2]$

$\text{(c)}$ Find the exact area of the shaded region. $[3]$

Question 10 Code: 9709/13/M/J/10/9, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=x+\frac{4}{x}$ which has a minimum point at $M .$ The line $y=5$ intersects the curve at the points $A$ and $B$.

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

$\text{(ii)}$ Find the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[6]$

Question 11 Code: 9709/11/M/J/13/9, Topic: Differentiation, Integration

A curve has equation $y=\mathrm{f}(x)$ and is such that $\mathrm{f}^{\prime}(x)=3 x^{\frac{1}{2}}+3 x^{-\frac{1}{2}}-10$.

$\text{(i)}$ By using the substitution $u=x^{\frac{1}{2}}$, or otherwise, find the values of $x$ for which the curve $y=\mathrm{f}(x)$ has stationary points. $[4]$

$\text{(ii)}$ Find $\mathrm{f}^{\prime \prime}(x)$ and hence, or otherwise, determine the nature of each stationary point. $[3]$

$\text{(iii)}$ It is given that the curve $y=\mathrm{f}(x)$ passes through the point $(4,-7)$. Find $\mathrm{f}(x)$. $[4]$

Question 12 Code: 9709/12/M/J/14/10, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x-3, \quad x \in \mathbb{R}, \\ &\mathrm{g}: x \mapsto x^{2}+4 x, \quad x \in \mathbb{R}. \end{aligned} $$$\text{(i)}$ Solve the equation $\mathrm{ff}(x)=11$. $[2]$

$\text{(ii)}$ Find the range of $\mathrm{g}$. $[2]$

$\text{(iii)}$ Find the set of values of $x$ for which $\mathrm{g}(x)>12$. $[3]$

$\text{(iv)}$ Find the value of the constant $p$ for which the equation $\operatorname{gf}(x)=p$ has two equal roots. $[3]$

Function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto x^{2}+4 x$ for $x \geqslant k$, and it is given that $\mathrm{h}$ has an inverse.

$\text{(v)}$ State the smallest possible value of $k$. $[1]$

$\text{(vi)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[4]$