61.

If a is an imaginary cube root of unity, then the value of the determinant

$\left|\begin{array}{ccc}1 + \mathrm{w}& {\mathrm{w}}^2& - \mathrm{w}\\ 1 + {\mathrm{w}}^2& \mathrm{w}& - {\mathrm{w}}^2\\ \mathrm{w} + {\mathrm{w}}^2& \mathrm{w}& - {\mathrm{w}}^2\end{array}\right|$ is

• - 2w

• - 3w²

• - 1

• 0

$\mathrm{Let} \mathrm{y} = {\mathrm{e}}^{{\mathrm{x}}^2} \mathrm{and} \mathrm{y} = {\mathrm{e}}^{{\mathrm{x}}^2}\sin \left(\mathrm{x}\right)$be two given curves. Then, angle between the tangents to the curves at any point of their intersection is

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

The value of $2{\cot }^{-1}\left(\frac{1}{2}\right) - {\cot }^{-1}\left(\frac{4}{3}\right)$ is

• $- \frac{\mathrm{\pi }}{8}$

• $\frac{3\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

The integrating factor of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} + \left(3{\mathrm{x}}^2{\tan }^{-1}\left(\mathrm{y}\right) - {\mathrm{x}}^3\right)\left(1 + {\mathrm{y}}^2\right) = 0$ is

• ${\mathrm{e}}^{{\mathrm{x}}^2}$

• ${\mathrm{e}}^{{\mathrm{x}}^3}$

• ${\mathrm{e}}^{3{\mathrm{x}}^2}$

• ${\mathrm{e}}^{3{\mathrm{x}}^3}$

If $\mathrm{y} = {\mathrm{e}}^{- \mathrm{x}}\cos \left(2\mathrm{x}\right)$, then which of the following differential equation is satisfied?

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2\frac{d\mathrm{y}}{d\mathrm{x}} + 5\mathrm{y} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 5\frac{d\mathrm{y}}{d\mathrm{x}} + 2\mathrm{y} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - 5\frac{d\mathrm{y}}{d\mathrm{x}} - 2\mathrm{y} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2\frac{d\mathrm{y}}{d\mathrm{x}} - 5\mathrm{y} = 0$

For a matrix $\mathrm{A} = \left(\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right)$, if U₁, U₂, and U₃. are $3 \times 1$ column matrices satisfying ${\mathrm{AU}}_1 = \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right), {\mathrm{AU}}_2 = \left(\begin{array}{c}2\\ 3\\ 0\end{array}\right), {\mathrm{AU}}_3 = \left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and U is $3 \times 3$  matrix whose columns are U₁, U₂, and U₃.  Then, sum of the elements of U^{- 1}

• 6

• 0

• 1

• 2/3

Let f be any continuously differentiable function on [a, b] and twice differentiable on (a, b) such that f(a) = f"(a) = 0 and f(b) = 0. Then,

• f''(a) = 0

• f'(x) = 0 for some $\mathrm{x} \in \left(\mathrm{a}, \mathrm{b}\right)$

• $\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) \ne 0 \mathrm{for} \mathrm{some} \mathrm{x} \in \left(\mathrm{a}, \mathrm{b}\right)$

• $\mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right) = 0 \mathrm{for} \mathrm{some} \mathrm{x} \in \left(\mathrm{a}, \mathrm{b}\right)$

A relation p on the set of real number R is defined as {x$\mathrm{\rho }$y: xy > 0}. Then, which of the following is/are true?

• $\mathrm{\rho }$ is reflexive and symmetric

• $\mathrm{\rho }$ is symmetric but not reflexive

• $\mathrm{\rho }$ is symmetric and transitive

• $\mathrm{\rho }$ is an equivalence relation

Let f : $\mathrm{R} \to \mathrm{R}$  be such that f(2x - 1) = f(x) for all $\mathrm{x} \in \mathrm{R}$. If f is continuous at x = 1 and f(1) = 1, then

• f(2) = 1

• f(2) = 2

• f is continuous only at x = 1

• f is continuous at all points

The value of $\underset{\mathrm{x}\to 2}{\lim }{\int }_2^{\mathrm{x}}\frac{3{\mathrm{t}}^2}{\left(\mathrm{x} - 2\right)}d\mathrm{t}$ is

• 10

• 12

• 18

• 16