Engineering Mat...

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

$A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 4& 2\\ 2& 6& 5\end{array}\right]$

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 $\underset{-\pi }{\overset{\pi }{\int }}\left|x\right|\cos nxdx,n\ge 1$ is (are)

Consider a matrix A = uv^T where  $u={(}_2^1),v={(}_1^1)$.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  $\overrightarrow{u}=\frac{1}{3}\left(-y^3̂i+x^3̂j+z^3̂k\right)$and n̂ as the unit normal vector to the surface of the hemisphere (x² + y² + z² = 1; z ≥ 0), the value of integral $\int \left(\mathrm{\nabla }\times u\right)\bullet \hat{n}dS$ evaluated on the curved surface of the hemisphere S is

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

$\left[\begin{array}{ccc}4& 5& x\\ 5& 6& y\\ 6& k& z\end{array}\right]$

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 $\left[\begin{array}{ccc}1& 3& 2\\ 2& 2& -3\\ 4& 4& -6\\ 2& 5& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 2\\ 1\end{array}\right]$ The value of x₃ (round off to the nearest integer), is ______.

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

$\underset{\left|z\right|=1}{\stackrel{}{\oint }}\frac{e^{kz}}{z}dz$ where k is a real constant.

Then the value of integral