Condition for Concurrency of Three Lines

Q.

Let $W$ denotes the words in the English dictionary Define the relation $R$ by $R=\{(x,y)\in W\times W\}$ the ward $x$ and $y$ have at least one letter in common. Then $R$ is

1. Not reflexive, symmetric, and transitive

2. Reflexive, symmetric, and not transitive

3. Reflexive, symmetric, and transitive

4. Reflexive, not symmetric, and transitive

Q.

Consider the lines given by
$L_1:x+3y-5=0$
$L_2:3x-ky-1=0$
$L_3:5x+2y-12=0$

Match the Statements/Exp[ressions in Column I with the Statements/Expressions in Column II.
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ L_1,L_2,L_3\text{are concurrent, if}& \text{k = -9}\\ \text{One of}{L}_1,L_2,L_3\text{is parallel to at least one of the other two, if}& \text{k}=\frac{6}{5}\\ L_1,L_2,L_3\text{form a triangle, if}& k=\frac{5}{6}\\ L_1,L_2,L_3\text{do not form a triangle, if}& \text{k = 5}\end{array}$

1. (a) $\to$ (s), (b) $\to$ (p, q), (c) $\to$ (r), (d) $\to$ (p, q, s)

2. (a) $\to$ (p), (b) $\to$ (p, q, s), (c) $\to$ (r), (d) $\to$ (p, q)

3. (a) $\to$ (s), (b) $\to$ (p, q, s), (c) $\to$ (s, r), (d) $\to$ (p, q, s)

4. none of these

Q. If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in

1. H.P.
2. G.P.
3. A.P.
4. A.G.P

Q.

If 4a+5b+6c=0 then the set of lines ax+by+c=0 are concurrent at the point

1. $(\frac{2}{3},\frac{5}{6})$

2. $(\frac{1}{3},\frac{1}{2})$

3. $(\frac{1}{2},\frac{4}{3})$

4. $(\frac{1}{3},\frac{7}{3})$

Q. Three lines $px+qy+r=0,qx+ry+p=0$ and $rx+py+q=0$ are concurrent, if

1. $p+q+r=0$
2. $p^2+q^2+r^2=pr+rq$
3. $p^3+q^3+r^3=3pqr$
4. None of these

Q. The value of $a$ for which the lines
$2x+y-1=0ax+3y-3=03x+2y-2=0$
are concurrent:

Q. Let $ABC$ and $PQR$ be any two triangles in the same plane. Assume that the perpendiculars from the points $A,B,C$ to the sides $QR,RP,PQ$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $P,Q,R$ to $BC,CA,AB$ respectively are also concurrent.