Condition for a Conic to Be a Pair of Straight Lines

Q. SECTION-A1. Find the values of m and in for which the following system of linear equations has infinitely manysolutions:3x + 4y = 12(m + n) x + 2 (m-n) y = 5m-1

Q. The second degree equation $x^2+3xy+2y^2-x-4y-6=0$ represent a pair of straight lines

1. True
2. False

Q. Prove that the equation kx + y + z = 1 , x+ ky + z = k and x + y + kz = k²​ will have unique solution when k​ ​​≠ -2. Solve the equations in this case

Q. The value of k for which the system of equations:
2x + 3y - 2z = 0; 2x - y + 3z = 0 and 7x + ky - z = 0 has non-trivial solution, is .

1. $k=-\frac{57}{5}$
2. $k=\frac{57}{5}$
3. $k=-\frac{57}{10}$
4. $k=\frac{57}{10}$

Q. How many real solutions does the system of equations x³ - 3x = y y³ - 3y = z z³ - 3z = x have?