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

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 | 4 | 4 | 5 | 6 | 5 | 7 | 7 | 8 | 8 | 9 | 12 | 9 | 84 |

Score |

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

The diagram shows the region enclosed by the curve $\displaystyle y=\frac{6}{2 x-3}$, the $x$-axis and the lines $x=2$ and $x=3$. Find, in terms of $\pi$, the volume obtained when this region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

Question 2 Code: 9709/13/M/J/16/2, Topic: Integration

The diagram shows part of the curve $\displaystyle y=\left(x^{3}+1\right)^{\frac{1}{2}}$ and the point $P(2,3)$ lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

Question 3 Code: 9709/11/M/J/16/3, Topic: Integration

The diagram shows part of the curve $\displaystyle x=\frac{12}{y^{2}}-2$. The shaded region is bounded by the curve, the $y$-axis and the lines $y=1$ and $y=2$. Showing all necessary working, find the volume, in terms of $\pi$, when this shaded region is rotated through $360^{\circ}$ about the $y$-axis. $[5]$

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

A line with equation $y=m x-6$ is a tangent to the curve with equation $y=x^{2}-4 x+3$.

Find the possible values of the constant $m$, and the corresponding coordinates of the points at which the line touches the curve. $[6]$

Question 5 Code: 9709/13/M/J/19/4, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle \mathrm{f}(x)=\frac{48}{x-1}$ for $3 \leqslant x \leqslant 7$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2 x-4$ for $a \leqslant x \leqslant b$, where $a$ and $b$ are constants.

$\text{(i)}$ Find the greatest value of $a$ and the least value of $b$ which will permit the formation of the composite function $\mathrm{gf}$. $[2]$

It is now given that the conditions for the formation of gf are satisfied.

$\text{(ii)}$ Find an expression for $\operatorname{gf}(x)$. $[1]$

$\text{(iii)}$ Find an expression for $(\mathrm{gf})^{-1}(x)$. $[2]$

Question 6 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 7 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 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: 9709/12/M/J/11/9, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=3-4 \cos ^{k} x$, for $0 \leqslant x \leqslant \pi$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=2$,

$\text{(a)}$ find the range of $\mathrm{f}$, $[2]$

$\text{(b)}$ find the exact solutions of the equation $\mathrm{f}(x)=1$. $[3]$

$\text{(ii)}$ In the case where $k=1$,

$\text{(a)}$ sketch the graph of $y=\mathrm{f}(x)$, $[2]$

$\text{(b)}$ state, with a reason, whether $\mathrm{f}$ has an inverse. $[1]$

Question 10 Code: 9709/12/M/J/17/9, Topic: Differentiation, Coordinate geometry

The equation of a curve is $y=8 \sqrt{x}-2 x$.

$\text{(i)}$ Find the coordinates of the stationary point of the curve. $[3]$

$\text{(ii)}$ Find an expression for $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ and hence, or otherwise, determine the nature of the stationary point. $[2]$

$\text{(iii)}$ Find the values of $x$ at which the line $y=6$ meets the curve. $[3]$

$\text{(iv)}$ State the set of values of $k$ for which the line $y=k$ does not meet the curve. $[1]$

Question 11 Code: 9709/12/M/J/12/10, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned} $$$\text{(i)}$ Obtain expressions, in terms of $x$, for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram, making clear the relationship between the two graphs. $[3]$

$\text{(iii)}$ Given that the equation $f g(x)=5-k x$, where $k$ is a constant, has no solutions, find the set of possible values of $k$. $[5]$

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

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=3$, solve the equation $\mathrm{ff}(x)=25$. $[2]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.

$\text{(ii)}$ Find the set of values of $k$ for which the equation $\mathrm{f}(x)=\mathrm{g}(x)$ has no real solutions. $[3]$

The function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.

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