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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 | 5 | 4 | 6 | 6 | 5 | 6 | 7 | 8 | 7 | 9 | 8 | 9 | 80 |

Score |

Question 1 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry

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 Code: 9709/13/M/J/17/3, Topic: Coordinate geometry

Find the coordinates of the points of intersection of the curve $y=x^{\frac{2}{3}}-1$ with the curve $y=x^{\frac{1}{3}}+1$. $[4]$

Question 3 Code: 9709/12/M/J/10/4, Topic: Coordinate geometry

In the diagram, $A$ is the point $(-1,3)$ and $B$ is the point $(3,1)$. The line $L_{1}$ passes through $A$ and is parallel to $O B$. The line $L_{2}$ passes through $B$ and is perpendicular to $A B$. The lines $L_{1}$ and $L_{2}$ meet at $C$. Find the coordinates of $C$. $[6]$

Question 4 Code: 9709/12/M/J/20/6, Topic: Quadratics

The equation of a curve is $y=2 x^{2}+k x+k-1$, where $k$ is a constant.

$\text{(a)}$ Given that the line $y=2 x+3$ is a tangent to the curve, find the value of $k$. $[3]$

It is now given that $k=2$.

$\text{(b)}$ Express the equation of the curve in the form $y=2(x+a)^{2}+b$, where $a$ and $b$ are constants, and hence state the coordinates of the vertex of the curve. $[3]$

Question 5 Code: 9709/11/M/J/21/6, Topic: Quadratics

The equation of a curve is $y=(2 k-3) x^{2}-k x-(k-2)$, where $k$ is a constant. The line $y=3 x-4$ is a tangent to the curve.

Find the value of $k$ $[5]$

Question 6 Code: 9709/13/M/J/21/6, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}(x)=x^{2}-2 x+5 \\ &\mathrm{~g}(x)=x^{2}+4 x+13 \end{aligned} $$$\text{(a)}$ By first expressing each of $\mathrm{f}(x)$ and $\mathrm{g}(x)$ in completed square form, express $\mathrm{g}(x)$ in the form $\mathrm{f}(x+p)+q$, where $p$ and $q$ are constants. $[4]$

$\text{(b)}$ Describe fully the transformation which transforms the graph of $y = f(x)$ to the graph of $y = g(x)$. $[2]$

Question 7 Code: 9709/12/M/J/19/7, Topic: Functions

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

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

$\text{(ii)}$ Solve the equation $\mathrm{fg}(x)=\frac{7}{3}$. $[3]$

Question 8 Code: 9709/12/M/J/16/8, Topic: Coordinate geometry

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 Code: 9709/12/M/J/18/8, Topic: Coordinate geometry

Points $A$ and $B$ have coordinates $(h, h)$ and $(4 h+6,5 h)$ respectively. The equation of the perpendicular bisector of $A B$ is $3 x+2 y=k$. Find the values of the constants $\mathrm{h}$ and $k$. $[7]$

Question 10 Code: 9709/11/M/J/12/9, Topic: Coordinate geometry

The coordinates of $A$ are $(-3,2)$ and the coordinates of $C$ are $(5,6).$ The mid-point of $A C$ is $M$ and the perpendicular bisector of $A C$ cuts the $x$-axis at $B$.

$\text{(i)}$ Find the equation of $M B$ and the coordinates of $B$. $[5]$

$\text{(ii)}$ Show that $A B$ is perpendicular to $B C$. $[2]$

$\text{(iii)}$ Given that $A B C D$ is a square, find the coordinates of $D$ and the length of $A D$. $[2]$

Question 11 Code: 9709/11/M/J/21/10, Topic: Coordinate geometry

The equation of a circle is $x^{2}+y^{2}-4 x+6 y-77=0$.

$\text{(a)}$ Find the $x$-coordinates of the points $A$ and $B$ where the circle intersects the $x$-axis. $[2]$

$\text{(b)}$ Find the point of intersection of the tangents to the circle at A and B. $[6]$

Question 12 Code: 9709/13/M/J/14/11, Topic: Coordinate geometry

The diagram shows a parallelogram $A B C D$, in which the equation of $A B$ is $y=3 x$ and the equation of $A D$ is $4 y=x+11$. The diagonals $A C$ and $B D$ meet at the point $E\left(6 \frac{1}{2}, 8 \frac{1}{2}\right)$. Find, by calculation, the coordinates of $A, B, C$ and $D$. $[8]$