Engineering Mat...

The value of t so that \(\left[ {\begin{array}{*{20}{c}} 4\\ { - 1} \end{array}} \right]\) is an eigen vector of \(\left[ {\begin{array}{*{20}{c}} 3&4\\ 2&t \end{array}} \right]\) is ______

\(A = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 1&4&2\\ 2&6&5 \end{array}} \right]\)

For the above-given matrix, which of the following statement is/are correct.

The value of the contour integral in the complex plane

\(\oint \frac{{{z^3} - 2z + 3}}{{z - 2}}dz\)

An ordinary six-faced die is thrown four times. Which of the given results is/are correct?

(Ace represents the face side 1)

Equation (α xy³ + y cos x) dx + (x²y² + β sin x) dy = 0 is exact if

The temperature in an auditorium is given by T = x² + y² – 2z². A mosquito located at (2, 2, 1) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. The direction, in which it must fly is

The value(s) of the integral \(\mathop \smallint \limits_{ - \pi }^\pi \left| x \right|\cos nxdx,\;n \ge 1\) is (are)

Consider a matrix A = uv^T where  \(u=(^1_2) ,\ v=(^1_1)\).Note that v^T denotes the transpose of v. The largest eigenvalue of A is ________.

If the function defined by f(x) = 2x² + 3x – m log x is monotonic decreasing function on the open interval (0, 1), then the maximum possible value of the parameter m is

If u = x log xy where \({x^3} + {y^3} + 3xy = 1,\) find du/dx

Which of the following vector identities is / are always true?

If y = f(x) satisfies the boundary value problem y’’ + 9y = 0, y (0) = 0, y(π/2) = √2, then y(π/4) is ______  

In the Laurent expansion of \(f\left( z \right) = \frac{1}{{\left( {z - 1} \right)\left( {z - 2} \right)}}\) valid in the region 1 < |z| < 2, the co-efficient of \(\frac{1}{{{z^2}}}\) is

Given a vector  \(\vec u = \frac{1}{3}\left( { - {y^3}̂ i + {x^3}̂ j + {z^3}̂ k} \right)\)and n̂ as the unit normal vector to the surface of the hemisphere (x² + y² + z² = 1; z ≥ 0), the value of integral \(\smallint \left( {\;\nabla \times u} \right) \bullet \hat n\;dS\) evaluated on the curved surface of the hemisphere S is

Let \(A = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 1}\\ { - 1}&2&0\\ 0&0&{ - 2} \end{array}} \right]\) and B = A³ – A² – 4A + 5I, where I is the 3 × 3 identity matrix. The determinant of B is _______ (up to 1 decimal place).

\(\left[ {\begin{array}{*{20}{c}} 4&5&x\\ 5&6&y\\ 6&k&z \end{array}} \right]\)

For the given matrix, if x, y, z are in AP with a common difference d and the rank of the matrix is 2, then which of the following results is/are always correct?

The boundary value problem y” + λy = 0, y’(0) = y’(π) = 0 will have non zero solutions if and only if the values of λ are

In Taylor's series expansion of exp (x) + sin (x) about the point x = r, the coefficient of (x – π)² is

Consider the system of equations \(\left[ {\begin{array}{*{20}{c}}1&3&2\\2&2&{ - 3}\\4&4&{ - 6}\\2&5&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\1\\2\\1\end{array}} \right]\) The value of x₃ (round off to the nearest integer), is ______.

If the following integral is evaluated using Cauchy’s Integral formula

\(\underset{\left| z \right|=1}{\overset{{}}{\mathop \oint }}\,\frac{{{e}^{kz}}}{z}dz\) where k is a real constant.

Then the value of integral