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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 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 10 | 8 | 9 | 7 | 7 | 7 | 5 | 5 | 6 | 4 | 3 | 3 | 74 |

Score |

Question 1

A line has equation $y=2 x-7$ and a curve has equation $y=x^{2}-4 x+c$, where $c$ is a constant. Find the set of possible values of $c$ for which the line does not intersect the curve. $[3]$

Question 2

Find the set of values of $k$ for which the curve $y=k x^{2}-3 x$ and the line $y=x-k$ do not meet. $[3]$

Question 3

Find the set of values of $a$ for which the curve $\displaystyle y=-\frac{2}{x}$ and the straight line $y=a x+3 a$ meet at two distinct points. $[4]$

Question 4

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 5

The equation $x^{2}+p x+q=0$, where $p$ and $q$ are constants, has roots $-3$ and 5.

$\text{(i)}$ Find the values of $p$ and $q$. $[2]$

$\text{(ii)}$ Using these values of $p$ and $q$, find the value of the constant $r$ for which the equation $x^{2}+p x+q+r=0$ has equal roots. $[3]$

Question 6

A curve has equation $y=2 x^{2}-6 x+5$

$\text{(i)}$ Find the set of values of $x$ for which $y>13$. $[3]$

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

Question 7

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 8

A function $\mathrm{f}$ is such that $\displaystyle\mathrm{f}(x)=\frac{15}{2 x+3}$ for $0 \leqslant x \leqslant 6$

$\text{(i)}$ Find an expression for $\mathrm{f}^{\prime}(x)$ and use your result to explain why $\mathrm{f}$ has an inverse. $[3]$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$, and state the domain and range of $\mathrm{f}^{-1}$. $[4]$

Question 9

A piece of wire of length $24 \mathrm{~cm}$ is bent to form the perimeter of a sector of a circle of radius $r \mathrm{~cm}$.

$\text{(i)}$ Show that the area of the sector, $A \mathrm{~cm}^{2}$, is given by $A=12 r-r^{2}$. $[3]$

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

$\text{(iii)}$ Given that $r$ can vary, state the greatest value of $A$ and find the corresponding angle of the sector. $[2]$

Question 10

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 11

$\text{(i)}$ Express $2 x^{2}-10 x+8$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants, and use your answer to state the minimum value of $2 x^{2}-10 x+8$. $[4]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $2 x^{2}-10 x+8=k x$ has no real roots. $[4]$

Question 12

A line has equation $y=2 x+c$ and a curve has equation $y=8-2 x-x^{2}$.

$\text{(i)}$ For the case where the line is a tangent to the curve, find the value of the constant $c$. $[3]$

$\text{(ii)}$ For the case where $c=11$, find the $x$-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve. $[7]$