List of top Questions asked in MET

If \(iz^4 + 1 = 0\) then \(z\) can take the value

Which one of the following is independent of \(\alpha\) in the hyperbola \((0<\alpha<\pi/2)\) \(\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1\)?

If \(\cos \alpha + \cos \beta + \cos \gamma = \sin \alpha + \sin \beta + \sin \gamma = 0\) then the value of \(\cos 3\alpha + \cos 3\beta + \cos 3\gamma\) is

The circles whose equations are \(x^2 + y^2 + c^2 = 2ax\) and \(x^2 + y^2 + c^2 - 2by = 0\) will touch each other externally if

Minimum distance between the curves \(y^2 = 4x\) and \(x^2 + y^2 - 12x + 31 = 0\) is

An equilateral triangle \(SAB\) is inscribed in the parabola \(y^2 = 4ax\) having its focus at \(S\). If chord \(AB\) lies towards the left of \(S\), then side length of this triangle 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 tangents to \(x^2 + y^2 = a^2\) having inclinations \(\alpha\) and \(\beta\) intersect at \(P\). If \(\cot \alpha + \cot \beta = 0\), then the locus of \(P\) is

The values of \(\alpha\) for which the point \((\alpha - 1, \alpha + 1)\) lies in the larger segment of the circle \(x^2 + y^2 - x - y - 6 = 0\) made by the chord whose equation is \(x + y - 2 = 0\) is

Let \(\alpha_1, \alpha_2\) and \(\beta_1, \beta_2\) be the roots of \(ax^2 + bx + c = 0\) and \(px^2 + qx + r = 0\) respectively. If the system of equations \(\alpha_1 y + \alpha_2 z = 0\) and \(\beta_1 y + \beta_2 z = 0\) has a non-trivial solution, then

Coefficient of \(x\) in \(f(x) = \begin{vmatrix} x & (1 + \sin x)^3 & \cos x \\ 1 & \log(1 + x) & 2 \\ x^2 & (1 + x)^2 & 0 \end{vmatrix}\) is

For two unimodular complex numbers \(z_1\) and \(z_2\), \(\begin{bmatrix} z_1 & z_2 \\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}^{-1} \begin{bmatrix} \bar{z}_1 & -z_2 \\ \bar{z}_2 & z_1 \end{bmatrix}^{-1}\) is equal to

If \(A\) is a square matrix of order \(n\) such that \(|\operatorname{adj}(\operatorname{adj} A)| = |A|^9\), then the value of \(n\) can be

Let \(A = \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix}\) and \((A + I)^{50} - 50A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) then the value of \(a + b + c + d\) is

If \(\mathbf{a} \cdot \mathbf{i} = \mathbf{a} \cdot (\mathbf{j} + \mathbf{i}) = \mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k})\), then \(\mathbf{a}\) is equal to

If \(\theta\) be the angle between the unit vectors \(\mathbf{a}\) and \(\mathbf{b}\), then \(\cos \frac{\theta}{2}\) is equal to

The minimum value of the expression \(\sin \alpha + \sin \beta + \sin \gamma\) where \(\alpha, \beta, \gamma\) are real numbers satisfying \(\alpha + \beta + \gamma = \pi\) is

If \(\mathbf{a} \times \mathbf{b} = \mathbf{c}\), \(\mathbf{b} \times \mathbf{c} = \mathbf{a}\) and \(a, b, c\) be the moduli of the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) respectively, then

A unit vector coplanar with \(\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) and perpendicular to \(\mathbf{i} + \mathbf{j} + \mathbf{k}\) is

If the scalar projection of the vector \(x\mathbf{i} - \mathbf{j} + \mathbf{k}\) on the vector \(2\mathbf{i} - \mathbf{j} + 5\mathbf{k}\) is \(\frac{1}{\sqrt{30}}\) then value of \(x\) is equal to

If \(f(x)\) defined by \(f(x) = \begin{cases} \frac{|x^2 - x|}{x^2 - x}, & x \neq 0, 1 \\ 1, & x = 0 \\ -1, & x = 1 \end{cases}\) then \(f(x)\) is continuous for all

Function \(f(x)\) is defined as \(f(x) = \begin{cases} 3x, & x<1 \\ a-b, & x = 1 \\ 4b-a, & x>1 \end{cases}\) If \(f(x)\) is continuous at \(x = 1\), but discontinuous at \(x = 2\) then the locus of the point \((a, b)\) is a straight line excluding the point where it cuts the line

The value of \(\lim_{x \to \infty} \frac{x}{x + \frac{\sqrt[3]{x}}{x + \frac{\sqrt[3]{x}}{x + \dots}}}\) is

The value of \(\lim_{x \to 1} \frac{\sum_{k=1}^{100} x^k - 100}{x - 1}\) is