M L Aggarwal Solutions for Chapter: Quadratic Equations, Exercise 7: Higher Order Thinking Skills (HOTS)

Find the value(s) of $k$ for which the following simultaneous linear equations have a unique solution

$\left(k-1\right)x-3y=4, 3x-\left(4k+1\right)y=5$

If $k$ is a natural number and the roots of the equation $x^2+11x+6k=0$ are rational numbers, then find the smallest value of $k$

Find the values of $k$ for which the equation $x^2+5kx+16=0$ has no real roots.

Find the value of $k$ for which the equation $5x^2-kx+1=0$ has real roots

If the equation $x^2+2cx+ab=0$ has two distinct real roots, then prove that the equation $x^2-2\left(a+b\right)x+a^2+b^2+2c^2=0$ has no real roots.