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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 | 5 | 6 | 4 | 6 | 3 | 6 | 6 | 6 | 7 | 12 | 9 | 75 |

Score |

Question 1 Code: 9709/12/M/J/21/2, Topic: Coordinate geometry

$\text{(a)}$ The graph of $y=\mathrm{f}(x)$ is transformed to the graph of $y=2 \mathrm{f}(x-1)$.

Describe fully the two single transformations which have been combined to give the resulting transformation. $[3]$

$\text{(b)}$ The curve $y=\sin 2 x-5 x$ is reflected in the $y$-axis and then stretched by scale factor $\frac{1}{3}$ in the $x$-direction.

Write down the equation of the transformed curve. $[2]$

Question 2 Code: 9709/12/M/J/14/3, Topic: Trigonometry

The reflex angle $\theta$ is such that $\cos \theta=k$, where $0 < k < 1$.

$\text{(i)}$ Find an expression, in terms of $k$, for

$\text{(a)}$ $\sin \theta$, $[2]$

$\text{(b)}$ $\tan \theta$. $[1]$

$\text{(ii)}$ Explain why $\sin 2 \theta$ is negative for $0 < k < 1$. $[2]$

Question 3 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 4 Code: 9709/13/M/J/20/3, Topic: Coordinate geometry

In each of parts $\text{(a)}, \text{(b)}$ and $\text{(c)}$, the graph shown with solid lines has equation $y=\mathrm{f}(x)$. The graph shown with broken lines is a transformation of $y=\mathrm{f}(x)$.

$\text{(a)}$

State, in terms of $\mathrm{f}$, the equation of the graph shown with broken lines. $[1]$

$\text{(b)}$

State, in terms of f, the equation of the graph shown with broken lines. $[1]$

$\text{(c)}$

State, in terms of $\mathrm{f}$, the equation of the graph shown with broken lines. $[2]$

Question 5 Code: 9709/11/M/J/10/4, Topic: Integration

The diagram shows the curve $y=6 x-x^{2}$ and the line $y=5$. Find the area of the shaded region. $[6]$

Question 6 Code: 9709/11/M/J/21/4, Topic: Trigonometry

The diagram shows part of the graph of $y=a \tan (x-b)+c$

Given that $0 < b < \pi$, state the values of the constants $a, b$ and $c$. $[3]$

Question 7 Code: 9709/12/M/J/11/5, Topic: Trigonometry

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

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\cos \theta}{\tan \theta(1-\sin \theta)}=4$, for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

Question 8 Code: 9709/12/M/J/21/5, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2 x^{2}+3$ for $x \geqslant 0$.

$\text{(a)}$ Find and simplify an expression for $\mathrm{ff}(x)$. $[2]$

$\text{(b)}$ Solve the equation $\mathrm{ff}(x)=34 x^{2}+19$. $[4]$

Question 9 Code: 9709/11/M/J/20/8, Topic: Circular measure

In the diagram, $A B C$ is a semicircle with diameter $A C$, centre $O$ and radius $6 \mathrm{~cm}$. The length of the $\operatorname{arc} A B$ is $15 \mathrm{~cm}$. The point $X$ lies on $A C$ and $B X$ is perpendicular to $A X$.

Find the perimeter of the shaded region $B X C$. $[6]$

Question 10 Code: 9709/13/M/J/21/8, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:

$$ \begin{aligned}&\mathrm{f}: x \mapsto x^{2}-1 \text { for } x<0, \\&\mathrm{~g}: x \mapsto \frac{1}{2 x+1} \text { for } x<-\frac{1}{2}.\end{aligned} $$$\text{(a)}$ Solve the equation $\mathrm{fg}(x)=3$. $[4]$

$\text{(b)}$ Find an expression for $(\mathrm{fg})^{-1}(x)$. $[3]$

Question 11 Code: 9709/12/M/J/16/10, Topic: Differentiation, Integration, Coordinate geometry

The diagram shows the part of the curve $\displaystyle y=\frac{8}{x}+2 x$ for $x>0$, and the minimum point $M$

$\text{(i)}$ Find expressions for $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}, \displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ and $\displaystyle\int y^{2} \mathrm{~d} x$. $[5]$

$\text{(ii)}$ Find the coordinates of $M$ and determine the coordinates and nature of the stationary point on the part of the curve for which $x<0$. $[5]$

$\text{(iii)}$ Find the volume obtained when the region bounded by the curve, the $x$-axis and the lines $x=1$ and $x=2$ is rotated through $360^{\circ}$ about the $x$-axis. $[2]$

Question 12 Code: 9709/13/M/J/16/10, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=2 x+3$ for $x \geqslant 0$. The function $\mathrm{g}$ is such that $\mathrm{g}(x)=a x^{2}+b$ for $x \leqslant q$, where $a, b$ and $q$ are constants. The function fg is such that fg $(x)=6 x^{2}-21$ for $x \leqslant q$

$\text{(i)}$ Find the values of $a$ and $b$. $[3]$

$\text{(ii)}$ Find the greatest possible value of $q$. $[2]$

It is now given that $q=-3$.

$\text{(iii)}$ Find the range of $\mathrm{fg}$. $[1]$

$\text{(iv)}$ Find an expression for $\mathrm{(f g)^{-1}}(x)$ and state the domain of $(f g)^{-1}$. $[3]$