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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 | Total |
---|---|---|---|---|---|---|---|---|---|

Marks | 6 | 5 | 5 | 5 | 9 | 9 | 9 | 12 | 60 |

Score |

Question 1 Code: 9709/12/O/N/17/2, Topic: Functions

A function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 4-5 x$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and find the point of intersection of the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$. $[3]$

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

Question 2 Code: 9709/12/M/J/18/2, Topic: Quadratics

The equation of a curve is $y=x^{2}-6 x+k$, where $k$ is a constant.

$\text{(i)}$ Find the set of values of $k$ for which the whole of the curve lies above the $x$-axis. $[2]$

$\text{(ii)}$ Find the value of $k$ for which the line $y+2 x=7$ is a tangent to the curve. $[3]$

Question 3 Code: 9709/11/O/N/10/3, Topic: Quadratics

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+3 \\ &\mathrm{~g}: x \mapsto x^{2}-2 x \end{aligned} $$Express $\operatorname{gf}(x)$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[5]$

Question 4 Code: 9709/12/M/J/11/6, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{x+3}{2 x-1}, x \in \mathbb{R}, x \neq \frac{1}{2}$.

$\text{(i)}$ Show that $\mathrm{ff}(x)=x$. $[3]$

$\text{(ii)}$ Hence, or otherwise, obtain an expression for $\mathrm{f}^{-1}(x)$. $[2]$

Question 5 Code: 9709/13/O/N/13/10, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto x^{2}+4 x$ for $x \geqslant c$, where $c$ is a constant. It is given that $\mathrm{f}$ is a one-one function.

$\text{(i)}$ State the range of $\mathrm{f}$ in terms of $c$ and find the smallest possible value of $c$. $[3]$

The function $\mathrm{g}$ is defined by $g: x \mapsto a x+b$ for $x \geqslant 0$, where $a$ and $b$ are positive constants. It is given that, when $c=0, \operatorname{gf}(1)=11$ and $f g(1)=21$.

$\text{(ii)}$ Write down two equations in $a$ and $b$ and solve them to find the values of $a$ and $b$. $[6]$

Question 6 Code: 9709/13/M/J/18/10, Topic: Functions

The one-one function $\mathrm{f}$ is defined by $\mathrm{f}(x)=(x-2)^{2}+2$ for $x \geqslant c$, where $c$ is a constant.

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

In parts $\text{(ii)}$ and $\text{(iii)}$ the value of $c$ is 4.

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

$\text{(iii)}$ Solve the equation $\mathrm{ff}(x)=51$, giving your answer in the form $a+\sqrt{b}$. $[5]$

Question 7 Code: 9709/12/O/N/18/10, Topic: Quadratics, Coordinate geometry

The equation of a curve is $\displaystyle y=2 x+\frac{12}{x}$ and the equation of a line is $y+x=k$, where $k$ is a constant.

$\text{(i)}$ Find the set of values of $k$ for which the line does not meet the curve. $[3]$

In the case where $k=15$, the curve intersects the line at points $A$ and $B$.

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

$\text{(iii)}$ Find the equation of the perpendicular bisector of the line joining $A$ and $B$. $[3]$

Question 8 Code: 9709/12/M/J/15/11, Topic: Functions

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

$\text{(i)}$ Find the set of values of $p$ for which the equation $\mathrm{f}(x)=p$ has no real roots. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 2 x^{2}-6 x+5$ for $0 \leqslant x \leqslant 4$.

$\text{(ii)}$ Express $\mathrm{g}(x)$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

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

The function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto 2 x^{2}-6 x+5$ for $k \leqslant x \leqslant 4$, where $k$ is a constant.

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

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{h}^{-1}(x)$. $[3]$