Condition of Concurrency of 3 Straight Lines

Q. $\mathrm{If}\mathrm{the}\mathrm{lines}{a}_{1}x+b_{1}y+1=0,a_{2}x+b_{2}y+1=0\mathrm{and}{a}_{3}x+b_{3}y+1=0\mathrm{are}\mathrm{concurrent},\mathrm{then}\mathrm{the}\mathrm{points}(a_1,b_1),(a_2,b_2)\mathrm{and}(a_3,b_3)\mathrm{will}\mathrm{be}\mathrm{collinear}.$

1. True
2. False

Q.

Three lines given by $L_1:a_{1}x+b_{1}y+c_1=0,L_2:a_{2}x+b_{2}y+c_2=0\text{and}{L}_3:a_{3}x+b_{3}y+c_3=0$ are always concurrent given that

$\left|\begin{array}{cccc}{a}_1& b_1& c_1& \\ a_2& b_2& c_2& \\ a_3& b_3& c_3& \end{array}\right|=0$

1. False

2. True

Q. The value of 'a' for which the lines 2ax - 2y + 3z = 0; x + ay + 2z = 0 and 2x + az = 0 are concurrent, is .

1. -2
2. 2
3. -1
4. 1

Q. From $25$ points on a plane, $8$ are on a straight line, $7$ are on another straight line and $10$ are on a third straight line.
Then the number of triangles that can be drawn by connecting some three points from these $25$ is

1. ${}^{25}{C}_3$
2. ${}^{25}{C}_3-({}^8{C}_3+{}^7{C}_3+{}^{10}{C}_3)$
3. ${}^{25}{C}_3+({}^8{C}_3+{}^7{C}_3+{}^{10}{C}_3)$
4. ${}^8{C}_3$+${}^7{C}_3+{}^{10}{C}_3$

Q. The system of linear equations
$x+\lambda y-z=0;\lambda x-y-z=1;x+y-\lambda z=0$ has a non - trivial solution for

1. infinitely many values of $\lambda$
2. exactly one value of $\lambda$
3. exactly two values of $\lambda$
4. exactly three values of $\lambda$

Q. The number of value of k, for which the system the system of equation (k + 1)x + 8y = 4k $\Rightarrow$ kx + (k + 3)y = 3k - 1 has no solution, is

1. infinite
2. 1
3. 2
4. 3

Q. Let $\lambda and\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations
$\lambda x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0,x+\left(cos\alpha \right)y+\left(sin\alpha \right)z=0and-x+\left(sin\alpha \right)y-\left(cos\alpha \right)z=0$
has a non - trivial solution.
For $\lambda =1$, the values of $\alpha$ are ___.
($n$ belongs to integers)

1. $\alpha =n\pi$
2. $\alpha =n\frac{\pi }{2}$
3. $n\pi +\frac{\pi }{4}$
4. $n\frac{\pi }{2}+\frac{\pi }{4}$

Q. $\mathrm{If}\mathrm{the}\mathrm{lines}{a}_{1}x+b_{1}y+1=0,a_{2}x+b_{2}y+1=0\mathrm{and}{a}_{3}x+b_{3}y+1=0\mathrm{are}\mathrm{concurrent},\mathrm{then}\mathrm{the}\mathrm{points}(a_1,b_1),(a_2,b_2)\mathrm{and}(a_3,b_3)\mathrm{will}\mathrm{be}\mathrm{collinear}.$

1. True
2. False

Q. The value(s) of a for which the lines 2x + y - 1 = 0; ax + 3y - 3 = 0 and 3x + 2y - 2 = 0 are concurrent, is/are:

1. 1
2. 2
3. 3
4. 4
5. 5

Q. The number of value of k, for which the system the system of equation (k + 1)x + 8y = 4k $\Rightarrow$ kx + (k + 3)y = 3k - 1 has no solution, is

1. infinite
2. 1
3. 2
4. 3

Q. The system of linear equations
$x+\lambda y-z=0;\lambda x-y-z=1;x+y-\lambda z=0$ has a non - trivial solution for

1. infinitely many values of $\lambda$
2. exactly one value of $\lambda$
3. exactly two values of $\lambda$
4. exactly three values of $\lambda$