List of top Mathematics Questions on Complex Numbers and Quadratic Equations asked in MET

If \(\alpha\) and \(\beta\) are roots of \(4x^2 + 3x + 7 = 0\), then the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is

The equation \((\cos p - 1)x^2 + \cos p \, x + \sin p = 0\) has real roots. Then \(p\) lies in

If the roots \(x^2 + ax + 9 = 0\) are complex, then

The sum of the real solutions of equation \(2|x|^2 + 51 = |1 + 20x|\) is

The quadratic equation whose roots are \(\frac{1}{3+\sqrt{2}}\) and \(\frac{1}{3-\sqrt{2}}\), will be:

If \( 1, \omega, \omega^2, \ldots, \omega^{n-1} \) are \( n \)th roots of unity, then the value of \( (9-\omega)(9-\omega^2)\cdots(9-\omega^{n-1}) \) is

If \( a = \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} \), then the quadratic equation whose roots are \( \alpha = a + a^2 + a^4 \) and \( \beta = a^3 + a^5 + a^6 \) is

If the line \(lx + my + n = 0\) cuts the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{25} = 1\) in points whose eccentric angles differ by \(\frac{\pi}{2}\), then \(\frac{a^2 l^2 + b^2 m^2}{n^2}\) is equal to

If the tangent to ellipse \(x^2 + 2y^2 = 1\) at point \(P\left(\frac{1}{\sqrt{2}}, \frac{1}{2}\right)\) meets the auxiliary circle at the points \(R\) and \(Q\), then tangents to circle at \(Q\) and \(R\) intersect at

The number of rational values of \(m\) for which the \(y\)-coordinate of the point of intersection of the lines \(3x + 2y = 10\) and \(x = my + 2\) is an integer is

A straight line cuts intercepts from the axis of coordinates the sum of the reciprocals of which is a constant \(K\). Then it always passes through a fixed point

If \( 2x + 3b + 6c = 0 \), then at least one root of the equation \( ax^2 + bx + c = 0 \) lies in the interval

If the roots of the equation \( \frac{\alpha}{x-\alpha} + \frac{\beta}{x-\beta} = 1 \) are equal in magnitude but opposite in sign, then \( \alpha + \beta \) is equal to

If \( \alpha, \beta \) are the roots of \( x^2 - 3x + 1 = 0 \), then the equation whose roots are \( \frac{1}{\alpha - 2} \) and \( \frac{1}{\beta - 2} \) is:

If \( x \) is real, then the value of \[ \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \] does not lie between:

Let \( f(x) = x^{2} + ax + b \), where \( a, b \in \mathbb{R} \). If \( f(x)=0 \) has all its roots imaginary, then the roots of \( f(x) + f'(x) + f''(x) = 0 \) are

If \( \alpha, \beta, \gamma \) are the roots of \( x^{3} + 4x + 1 = 0 \), then the equation whose roots are \[ \frac{\alpha^{2}}{\beta+\gamma}, \quad \frac{\beta^{2}}{\gamma+\alpha}, \quad \frac{\gamma^{2}}{\alpha+\beta} \] is

If \( f(x) = 2x^{4} - 13x^{2} + ax + b \) is divisible by \( x^{2} - 3x + 2 \), then \( (a, b) \) is equal to

If \( \alpha \) and \( \beta \) are the roots of \( x^{2} - 2x + 4 = 0 \), then the value of \( \alpha^{6} + \beta^{6} \) is

The solution of the equation \( 4^{x} - 3 \cdot 2^{x+2} + 32 = 0 \) is:

The roots of $(x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0, a \in R$ are always

The number of real solutions of \( x^{2} - 3|x| + 2 = 0 \) is: