List of top Questions asked in MET- 2016

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

If a and b be two perpendicular unit vectors such that \(\mathbf{x} = \mathbf{b} - (\mathbf{a} \times \mathbf{x})\), then \(|\mathbf{x}|\) is equal to

If \(aN = \{an : n \in N\}\) and \(bN \cap cN = dN\), where \(a,b,c \in N\) and \(b,c\) are coprime, then

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

The mean deviation from the mean for the set of observations \(-1, 0, 4\) is

If \(f(x)\) is a non-negative continuous function for all \(x \ge 1\) such that \(f'(x) \le p f(x)\), where \(p > 0\) and \(f(1) = 0\), then \([f(\sqrt{e}) + f(\sqrt{\pi})]\) is equal to

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:

If \(X\) follows a binomial distribution with parameters \(n = 8\) and \(p = \frac{1}{2}\), then \(P(|x - 4| \le 2)\) is equal to

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

If \(\theta = \sin^{-1}x + \cos^{-1}x - \tan^{-1}x\), \(1 \le x < \infty\), then the smallest interval in which \(\theta\) lies is

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

If \(\cos^{-1}p + \cos^{-1}q + \cos^{-1}r = 3\pi\), then \(p^2 + q^2 + r^2 + 2pqr\) is equal to

The points on the curve \(xy^2 = 1\) which are nearest to the origin, are

The statement \(p \rightarrow (q \rightarrow p)\) is equivalent to

Total number of polynomials of the form \(x^3 + ax^2 + bx + c\) that are divisible by \(x^2 + 1\), where \(a,b,c \in \{1, 2, 3, \dots, 10\}\) is equal to

Total number of regions in which 'n' coplanar lines can divide the plane, it is known that no two lines are parallel and no three of them are concurrent, is equal to

The value of \(\lim_{n \to \infty} \left[\sqrt[3]{n^2 - n^3} + n\right]\) 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:

If \(\begin{bmatrix} \alpha & \beta \gamma & -\alpha \end{bmatrix}\) is to be the square root of the two-rowed unit matrix, then \(\alpha, \beta\) and \(\gamma\) should satisfy the relation

The differential equation of the family of curves \(y = Ae^{3x} + Be^{5x}\), where \(A\) and \(B\) are arbitrary constants, is

The derivative of \(\sin^{-1}(2x\sqrt{1-x^2})\) with respect to \(\sin^{-1}(3x - 4x^3)\) 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:

Area bounded by the curves \(y = x^2\) and \(y = 2 - x^2\) is

\(\lim_{n \to \infty} \left( \frac{1^2}{1 - n^3} + \frac{2^2}{1 - n^3} + \dots + \frac{n^2}{1 - n^3} \right)\) is equal to