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Cambridge International AS and A Level

Name of student Date
Adm. number Year/grade 11 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 Total
Marks 5 4 5 5 6 10 9 8 9 61

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
Attempt all the 9 questions

Question 1

Find the coordinates of the point at which the perpendicular bisector of the line joining $(2,7)$ to $(10,3)$ meets the $x$-axis. $[5]$

Question 2

$\text{(i)}$ Express $4 x^{2}-12 x$ in the form $(2 x+a)^{2}+b$. $[2]$

$\text{(ii)}$ Hence, or otherwise, find the set of values of $x$ satisfying $4 x^{2}-12 x>7$. $[2]$

Question 3

$\text{(a)}$ 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 4

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 5

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=-2 x^{2}+12 x-3$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $-2 x^{2}+12 x-3$ in the form $-2(x+a)^{2}+b$, where $a$ and $b$ are constants. $[2]$

$\text{(ii)}$ State the greatest value of $\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2 x+5$ for $x \in \mathbb{R}$. $[1]$

$\text{(iii)}$ Find the values of $x$ for which $\operatorname{gf}(x)+1=0$. $[3]$

Question 6

$\text{(i)}$ Express $2 x^{2}-12 x+13$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

$\text{(ii)}$ The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2 x^{2}-12 x+13$ for $x \geqslant k$, where $k$ is a constant. It is given that $\mathrm{f}$ is a one-one function. State the smallest possible value of $k$. $[1]$

The value of $k$ is now given to be 7.

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

$\text{(iv)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$. $[5]$

Question 7

The function $\mathrm{f}: x \mapsto 5+3 \cos \left(\frac{1}{2} x\right)$ is defined for $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)=7$, giving your answer correct to 2 decimal places. $[3]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. $[2]$

$\text{(iii)}$ Explain why $\mathrm{f}$ has an inverse. $[1]$

$\text{(iv)}$ Obtain an expression for $\mathrm{f}^{-1}(x)$. $[3]$

Question 8

Three points have coordinates $A(0,7), B(8,3)$ and $C(3 k, k).$ Find the value of the constant $k$ for which

$\text{(i)}$ $C$ lies on the line that passes through $A$ and $B$, $[4]$

$\text{(ii)}$ $C$ lies on the perpendicular bisector of $A B$. $[4]$

Question 9

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 $(f g)^{-1}$. $[3]$

Worked solutions: P1, P3 & P6 (S1)

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