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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 | 7 | 6 | 8 | 9 | 8 | 9 | 9 | 11 | 67 |

Score |

Question 1 Code: 9709/12/O/N/11/4, Topic: Quadratics

The equation of a curve is $y^{2}+2 x=13$ and the equation of a line is $2 y+x=k$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=8$, find the coordinates of the points of intersection of the line and the curve. $[4]$

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

Question 2 Code: 9709/13/M/J/17/6, Topic: Quadratics

The line $3 y+x=25$ is a normal to the curve $y=x^{2}-5 x+k$. Find the value of the constant $k$. $[6]$

Question 3 Code: 9709/12/O/N/17/6, Topic: Functions

$\text{(a)}$ The function $\mathrm{f}$, defined by $\mathrm{f}: x \mapsto a+b \sin x$ for $x \in \mathbb{R}$, is such that $\mathrm{f}\left(\frac{1}{6} \pi\right)=4$ and $\mathrm{f}\left(\frac{1}{2} \pi\right)=3$.

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

$\text{(ii)}$ Evaluate $\mathrm{ff}(0)$. $[2]$

$\text{(b)}$ The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto c+d \sin x$ for $x \in \mathbb{R}$. The range of $\mathrm{g}$ is given by $-4 \leqslant \mathrm{~g}(x) \leqslant 10$. Find the values of the constants $c$ and $d$. $[3]$

Question 4 Code: 9709/11/M/J/13/7, Topic: Quadratics, Differentiation, Coordinate geometry

A curve has equation $y=x^{2}-4 x+4$ and a line has equation $y=m x$, where $m$ is a constant.

$\text{(i)}$ For the case where $m=1$, the curve and the line intersect at the points $A$ and $B$. Find the coordinates of the mid-point of $A B$. $[4]$

$\text{(ii)}$ Find the non-zero value of $m$ for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve. $[5]$

Question 5 Code: 9709/11/M/J/17/7, Topic: Integration, Quadratics, Differentiation

A curve for which $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=7-x^{2}-6 x$ passes through the point $(3,-10)$.

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

$\text{(ii)}$ Express $7-x^{2}-6 x$ in the form $a-(x+b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iii)}$ Find the set of values of $x$ for which the gradient of the curve is positive. $[3]$

Question 6 Code: 9709/13/M/J/17/9, Topic: Functions

$\text{(i)}$ Express $9 x^{2}-6 x+6$ in the form $(a x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=9 x^{2}-6 x+6$ for $x \geqslant p$, where $p$ is a constant.

$\text{(ii)}$ State the smallest value of $p$ for which $\mathrm{f}$ is a one-one function. $[1]$

$\text{(iii)}$ For this value of $p$, obtain an expression for $\mathrm{f}^{-1}(x)$, and state the domain of $\mathrm{f}^{-1}$. $[4]$

$\text{(iv)}$ State the set of values of $q$ for which the equation $\mathrm{f}(x)=q$ has no solution. $[1]$

Question 7 Code: 9709/11/M/J/14/10, Topic: Functions

The diagram shows the function $\mathrm{f}$ defined for $-1 \leqslant x \leqslant 4$, where

$$ \mathrm{f}(x)= \begin{cases}3 x-2 & \text { for }-1 \leqslant x \leqslant 1 \\ \frac{4}{5-x} & \text { for } 1 < x \leqslant 4\end{cases} $$$\text{(i)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(ii)}$ Copy the diagram and on your copy sketch the graph of $y=\mathrm{f}^{-1}(x)$. $[2]$

$\text{(iii)}$ Obtain expressions to define the function $\mathrm{f}^{-1}$, giving also the set of values for which each expression is valid. $[6]$

Question 8 Code: 9709/11/M/J/16/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 4 \sin x-1$ for $-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.

$\text{(i)}$ State the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ Find the coordinates of the points at which the curve $y=\mathrm{f}(x)$ intersects the coordinate axes. $[3]$

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

$\text{(iv)}$ Obtain an expression for $\mathrm{f}^{-1}(x)$, stating both the domain and range of $\mathrm{f}^{-1}$. $[4]$