14. Locus of the image of the point $(2,3)$ in the line $(x-2 y+3)+\lambda(2 x-3 y+4)=0$ is

(a) $x^{2}+y^{2}-3 x-4 y-4=0$

(b) $2 x^{2}+2 y^{2}+2 x+4 y-7=0$

(c) $x^{2}+y^{2}-2 x-4 y+4=0$

(d) None

Show Answer

Solution: $\left.\begin{array}{l}x-2 y+3=0 \\ 2 x-3 y+4=0\end{array}\right\} P(1,2)$

Let image be $\mathrm{B}(\mathrm{h}, \mathrm{k})$ then $\mathrm{AP}=\mathrm{BP}$

$\Rightarrow(\mathrm{h}-1)^{2}+(\mathrm{k}-2)^{2}=1^{2}+1^{2}$

$\mathrm{h}^{2}+\mathrm{k}^{2}-2 \mathrm{~h}-4 \mathrm{k}+5=2$

$\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-4 \mathrm{y}+3=0$

Answer : d

15. If the lines $a x+y+1=0, x+b y+1=0$ and $x+y+c=0 (a, b, c$ being distinct and different from 1$)$ are concurrent, then $\left(\dfrac{1}{1-a}\right)+\dfrac{1}{1-b}+\dfrac{1}{1-c}=$

(a) 0

(b) 1

(c) $\dfrac{1}{a+b+c}$

(d) None

16. The set of values of ’ $b$ ’ for which the origin and the point $(1,1)$ lie on the same side of the st. line $\mathrm{a}^{2} \mathrm{x}+\mathrm{aby}+1=0 \quad \forall \mathrm{a} \in \mathrm{R}, \mathrm{b}>\mathrm{o}$ are

(a) $\mathrm{b} \in[2,4)$

(b) $\mathrm{b} \in(0,2)$

(c) $\mathrm{b} \in[0,2]$

(d) None of these

17. Let $P(-1,0), Q(0,0)$ and $R(3,3 \sqrt{3})$ be three points. Then the equation of the bisector of the angle $\mathrm{PQR}$ is

(a) $\dfrac{\sqrt{3}}{2} \mathrm{x}+\mathrm{y}=0$

(b) $x+\sqrt{3} y=0$

(c) $\sqrt{3} \mathrm{x}+\mathrm{y}=0$

(d) $x+\dfrac{\sqrt{3}}{2} y=0$

Show Answer

Solution:

$\dfrac{\sqrt{3}-m}{1+\sqrt{3 m}}=\dfrac{m}{1+0}$

$\begin{aligned} \sqrt{3}-m=m+\sqrt{3} m^{2} \Rightarrow & \sqrt{3} m^{2}+2 m-\sqrt{3}=0 \\ & \sqrt{3} m^{2}+3 m-m-\sqrt{3}=0 \\ & (\sqrt{3} m-1)(3+\sqrt{3})=0 \\ & m=\dfrac{1}{\sqrt{3}},-\sqrt{3} \end{aligned}$

$y=-\sqrt{3} \mathrm{x}$

$\sqrt{3} x+y=0$

18. $\mathrm{OPQR}$ is a square and $\mathrm{M}, \mathrm{N}$ are the mid points of the sides $\mathrm{PQ}$ and $\mathrm{QR}$ respectively. If the ratio of the areas of the square and the triangle OMN is $\lambda: 6$, then $\dfrac{\lambda}{4}$ is equal to

(a) 2

(b) 4

(c) 12

(d) 16

Show Answer

Solution: ar. of square $=a^{2}$

ar of $\Delta \mathrm{OMN}=\dfrac{1}{2}\left|\begin{array}{lll}0 & 0 & 1 \\ \mathrm{a} & \dfrac{\mathrm{a}}{2} & 1 \\ \dfrac{\mathrm{a}}{2} & \mathrm{a} & 1\end{array}\right|=\dfrac{1}{2} \cdot \dfrac{3 \mathrm{a}^{2}}{4}=\dfrac{3 \mathrm{a}^{2}}{8}$

$\dfrac{\mathrm{a}^{2}}{\dfrac{3 \mathrm{a}^{2}}{8}}=\dfrac{\lambda}{6} \Rightarrow \lambda=16$

Answer : b

19. A pair of perpendicular straight lines drawn through the origin form an isoceles triangle with line $2 x+3 y=6$, then area of the triangle so formed is

(a) $\dfrac{36}{13}$

(b) $\dfrac{12}{17}$

(c) $\dfrac{13}{5}$

(d) $\dfrac{17}{13}$

Show Answer

Solution: $\mathrm{OM}=\left|\dfrac{-6}{\sqrt{4+9}}\right|=\dfrac{6}{\sqrt{13}}, \mathrm{PQ}=2 \times \dfrac{6}{\sqrt{13}}$, area of $\triangle \mathrm{OPQ}=\dfrac{1}{2} \times \dfrac{2 \times 6}{\sqrt{13}} \times \dfrac{6}{\sqrt{13}}=\dfrac{36}{13}$

20. Match the column

Exercise

1. If $t _{1}$ and $t _{2}$ are roots of the equation $t^{2}+\lambda t+1=0$, where $\lambda$ is an arbitrary constant. Then the line joining the points $\left(a t _{1}{ }^{2}, 2 a t _{1}\right)$ & $\left(a t^{2}, 2 a t _{2}\right)$ always passes through a fixed point

(a) $(a, 0)$

(b) $(-a, 0)$

(c) $(0, a)$

(d) $(0,-\mathrm{a})$

2. The equation $\mathrm{x}^{3}+\mathrm{y}^{3}=0$ represents

(a) three real straight lines

(b) three points

(c) combined equationof ast. line & a circle

(d) None of these.

3. The three lines whose combined equation is $y^{3}-4 x^{2} y=0$ form a triangle which is

(a) isosceles

(b) equilateral

(c) right angled

(d) None of these

$\mathrm{A}(1,3)$ and $\mathrm{C}\left(-\dfrac{2}{5},-\dfrac{2}{5}\right)$ are the vertices of a triangle $\mathrm{ABC}$ and the equation of the angle bisector of $\angle \mathrm{ABC}$ is $\mathrm{x}+\mathrm{y}=2$

Comprehension Type

4. Equation of side $\mathrm{BC}$ is

(a) $7 x+3 y=4$

(b) $7 x+3 y+4=0$

(c) $7 x-3 y+4=0$

(d) $7 x-3 y=4$

5. Coordinates of vertex $B$ are

(a) $\left(\dfrac{3}{10}, \dfrac{17}{10}\right)$

(b) $\left(\dfrac{17}{10}, \dfrac{3}{10}\right)$

(c) $\left(-\dfrac{5}{2}, \dfrac{9}{2}\right)$

(d) $(1,1)$

6. Equation of side $\mathrm{AB}$ is

(a) $3 x+7 y=24$

(b) $3 x+7 y+24=0$

(c) $13 x+7 y+8=0$

(d) $13 x-7 y+8=0$

7. Assertion and reasoning Type

Lines $\mathrm{L} _{1} \mathrm{~L} _{2}$ given by $\mathrm{y}-\mathrm{x}=0$ and $2 \mathrm{x}+\mathrm{y}=0$ intersect the line $\mathrm{L} _{3}$ given by $\mathrm{y}+2=0$ at $\mathrm{P}$ and $\mathrm{Q}$, respectively. The bisector of the acute angle between $\mathrm{L} _{1}$ and $\mathrm{L} _{2}$ intersects $\mathrm{L} _{3}$ at $\mathrm{R}$.

Statement 1 : The ratio PR: RQ equals $2 \sqrt{2}: \sqrt{5}$.

Statement 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

a Statement 1 is true, Statement 2 is True; Statement 2 is a correct explanation for Statement1.

b Statement 1 is True, Statement 2 is True; Statement 2 is NOT a correct explanation for Statement 1 .

c Statement 1 is True, Statement 2 is False.

d Statement 1 is False, Statement 2 is True.

8. Matrix-match

This question contains statements given in two columns which have to be matched. Statements a, b, c, d in column I have to be matched with statements p,q, r, s in column II. If the correct match is a $\rightarrow \mathrm{p}, \mathrm{a} \rightarrow \mathrm{s}, \mathrm{b} \rightarrow \mathrm{q} \mathrm{b} \rightarrow \mathrm{r}, \mathrm{c} \rightarrow \mathrm{p}, \mathrm{c} \rightarrow \mathrm{q}$ and $\mathrm{d} \rightarrow \mathrm{s}$, then the correctly dubbled $4 \times 4$ matrix should be as follows :

Consider the lines given by

$\begin{aligned} & \mathrm{L} _{1}: \mathrm{x}+3 \mathrm{y}-5=0 \\ & \mathrm{~L} _{2}: 3 \mathrm{x}-\mathrm{ky}-1=0 \\ & \mathrm{~L} _{3}: 5 \mathrm{x}+2 \mathrm{y}-12=0 \end{aligned}$