Family of Straight Lines

Q.

The points (0, 0), (a, 0) and $(\frac{a}{2},\frac{a\sqrt{3}}{2})$ are vertices of

1. Isosceles triangle

2. Equilateral triangle

3. Scalene triangle

4. None of these

Q. The equation of the straight line passing through the point of intersection of $x+2y=5$ and $3x+7y=17$ and perpendicular to the straight line $3x+4y=10$ is

1. $4x-3y+1=0$
2. $4x-3y+12=0$
3. $4x-3y-2=0$
4. $4x-3y+2=0$

Q.

Column 1 has the condition on circle and column 2 has the equation of circles.which one of them are matched correct?

Column 1 column 2

1) Centre of origin and radius = 10 P) $(x-8)(x+8)+(y-6)(y+6)=0$

2) Extremeties of diameter are (8, 6) and (-8, -6) Q) $(x-6)(x+6)+(y-8)(y+8)=0$

3) Centre is (1, -2) and radius is 4 R) $x^2+y^2-2x+4y-11=0$

Q. If $A(-3,2)$ and $B(1,-4)$. Find the equation of locus of point P, such that $3PA=2PB$

Q.

Find the equation of lines passing through intersection of lines x+y+4 = 0 and 3x-y-8 = 0 and equally inclined to axis..

1. x-y-6 = 0

2. x+y+4 = 0

3. x+y+6 = 0

4. 3x-y-8 = 0

Q. Solve the equation $\sqrt{3}{x}^2-\sqrt{2}x+3\sqrt{3}=0$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one entry of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots \text{up to}\infty }}},& & \\ & \text{then}x\text{is equal to}& \text{(P)}& 4\\ \text{(B)}& \text{If}a\text{and}x\text{are positive integers such}& & \\ & \text{that}x<a\text{and}\sqrt{a-x},\sqrt{x},\sqrt{a+x}& \text{(Q)}& 5\\ & \text{are in A.P., then least possible value of}a\text{is}& & \\ \text{(C)}& \text{If}3a+2b+4c=0,a,b,c\in R\text{and the}& & \\ & \text{line}ax+by+c=0\text{always passes}& & \\ & \text{through a fixed point}(p,q)\text{, then the}& & \\ & \text{value of}2p+q\text{is}& \text{(R)}& 2\\ \text{(D)}& \text{If}k(\sin {18}^{\circ }+\cos {36}^{\circ }{)}^2=5,\text{then the}& & \\ & \text{value of}k\text{is}& \text{(S)}& 3\\ & & \text{(T)}& 6\end{array}$

Which of the following is the only CORRECT combination?

1. $C\to R;D\to S$
2. $C\to S;D\to T$
3. $C\to T;D\to R$
4. $C\to R;D\to P$

Q. Solve the equation $x^2+x+\frac{1}{\sqrt{2}}=0$

Q. If $G=\{7,8\}$ and $H=\{5,4,2\}$, find $G\times H$ and $H\times G$.

Q. Let a and b be non-zero and real numbers. Then, the equation
$(a{x}^2+b{y}^2+c)(x^2-5xy+6y^2)=0$ represents

1. Four straight lines, when c=0 and a, b are of the same sign
2. Two straight lines and a circle, when a=b and c is of sign opposite to that of a
3. Two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
4. A circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Q. A diagonal of rhombus $ABCD$ is member of both the families of lines
$(x+y-1)+{\lambda }_1(2x+3y-2)=0$ and
$(x-y+2)+{\lambda }_2(2x-3y+5)=0$
where ${\lambda }_1$ and ${\lambda }_2\in R$ and one of the vertex of rhombus is $(3,2)$. If area of the rhombus is $12\sqrt{5}$ square units, then the length of the longer diagonal of the rhombus is

Q. The $x$-intercept of angle bisector of angle between the lines $2x+y-2=0$ and $2x+4y+7=0$ which contains the fixed point on the family of lines $(2\cos \alpha +3\sin \alpha )x+(3\cos \alpha -5\sin \alpha )y=5\cos \alpha -2\sin \alpha$ for different values of $\alpha$, is equal to

1. $\frac{11}{2}$
2. $-\frac{11}{2}$
3. $-\frac{1}{2}$
4. $\frac{1}{2}$

Q. Solve the equation $x^2+\frac{x}{\sqrt{2}}+1=0$

Q. Locus of the image of the point (2, 3) in the line $(2x-3y+4)+k(x-2y+3)=0,kϵR,$ is a

1. Straight line parallel to x-axis
2. Straight line parallel to y-axis
3. Circle of radius $\sqrt{2}$
4. Circle of adius 3