List of top Mathematics Questions asked in MET

The function $y = a(1-\cos x)$ is maximum when x is equal to

The volume of the parallelepiped whose sides are given by $\overrightarrow{OA} = 2\hat{i} - 3\hat{j}$, $\overrightarrow{OB} = \hat{i} + \hat{j} - \hat{k}$, $\overrightarrow{OC} = 3\hat{i} - \hat{k}$ is

Is it possible for wheel $W_n$ ($n \ge 3$) to be bipartite?

The GCD of 364 and 462 is

The system $x + y - 4z = 2$, $3x + y + 5z = 7$, $2x + 3y + z = 5$ has

If \[ t_n = \sum_{r=0}^n \frac{1}{\left({}^nC_r\right)^k} \quad \text{and} \quad S_n = \sum_{r=0}^n \frac{r}{\left({}^nC_r\right)^k}, \] where \( k \in \mathbb{Z}^+ \), then \[ \cos^{-1}\left( \frac{S_n}{n t_n} \right) \] is:

Sum of the last 30 coefficients in the expansion of \[ (1 + x)^{59}, \] when expanded in ascending powers of \( x \), is:

If \[ 2\tan^{-1}(\cos x) = \tan^{-1}(2 \csc x), \] then the value of \( x \) is:

The coefficient of the term independent of \( x \) in the expansion of \[ \left( \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x - 1}{x - x^{1/2}} \right)^{10} \] is:

\[ \int \frac{(2x^{12} + 5x^9)}{(1 + x^3 + x^5)^3} \, dx \] equals:

\[ \begin{vmatrix} \sin\theta & \cos\theta & \sin 2\theta \\ \sin\left(\theta + \frac{2\pi}{3}\right) & \cos\left(\theta + \frac{2\pi}{3}\right) & \sin\left(2\theta + \frac{4\pi}{3}\right) \\ \sin\left(\theta - \frac{2\pi}{3}\right) & \cos\left(\theta - \frac{2\pi}{3}\right) & \sin\left(2\theta - \frac{4\pi}{3}\right) \end{vmatrix} \] equals:

If the function \[ f(x) = \begin{cases} (\cos x)^{1/x^2}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \] is continuous at \( x = 0 \), then the value of \( k \) is:

If \[ f(x) = \frac{e^{1/x} - 1}{e^{1/x} + 1}, \quad x \neq 0 \quad \text{and} \quad f(0) = 0, \] then \(f(x)\) is:

The domain of definition of the function \[ f(x) = \frac{1}{| |x| - 1 | - 5} \] is:

The period of the function \[ f(x) = \begin{vmatrix} \sin x & \cos x & \sin x \cos x \\ \sin x & \cos x & \sin x \\ \cos x & \sin x & \sin x \end{vmatrix} \] is:

If \( \mathbf{b} \) and \( \mathbf{c} \) are any two non-collinear unit vectors and \( \mathbf{a} \) is any vector, then \[ (\mathbf{a} \cdot \mathbf{b})\mathbf{b} + (\mathbf{a} \cdot \mathbf{c})\mathbf{c} + \frac{(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}))}{|\mathbf{b} \times \mathbf{c}|^2} (\mathbf{b} \times \mathbf{c}) \] is equal to:

The degree of the differential equation \( x = \frac{dy}{dx} + \frac{1}{2!} \left(\frac{dy}{dx}\right)^2 + \frac{1}{3!} \left(\frac{dy}{dx}\right)^3 + \cdots \) is:

Let \(A\) and \(B\) be two events such that \(P(A \cup B) = \frac{1}{6}\), \(P(A \cap B) = \frac{1}{4}\) and \(P(\overline{A}) = \frac{1}{4}\), where \(\overline{A}\) stands for the complement of event \(A\). Then, the events \(A\) and \(B\) are:

If \( f(x) = \int_0^1 e^{|t - x|} \, dt \) for \( 0 \leq x \leq 1 \), then the maximum value of \( f(x) \) is:

A line passing through origin and is perpendicular to two given lines \(2x + y + 6 = 0\) and \(4x + 2y - 9 = 0\). The ratio in which the origin divides this line, is

The intercepts on the straight line \(y = mx\) by the lines \(y = 2\) and \(y = 6\) is less than 5, then \(m\) belongs to

The number of positive integers satisfying the inequality \(^{n+1}C_{n-1} + ^{n}C_{n-1} \leq 50\) is

The locus of the mid-points of the chords of the circle \(x^2 + y^2 = 16\) which are tangent to the hyperbola \(9x^2 - 16y^2 = 144\) is

The vector equation of the plane through the point \(\hat{i} + \hat{j} - 2\hat{k}\) and perpendicular to the line of intersection of the planes \(\mathbf{r} \cdot (\hat{i} - \hat{j} + 3\hat{k}) = 1\) and \(\mathbf{r} \cdot (4\hat{i} - 2\hat{j} + 2\hat{k}) = 2\) is

If \(P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\), \(A = \begin{bmatrix} 1 & 1 0 & 1 \end{bmatrix}\), and \(Q = PAP'\), then \(P'Q^{2005}P\) is