Engineering Mat...

Consider the differential equation:

\(\frac{{{d^4}y}}{{d{x^4}}}\cos \left( {\frac{{{d^3}y}}{{d{x^3}}}} \right) = 0\)

Which of the following vector function is solenoidal?

Consider a cube defined by

x, y, z ∈ [1, 3]

If vector, \(\vec A = 2{x^2}y{\hat a_x} + 3{x^2}{y^2}{\hat a_y}\)

Let f = y^x. What is \(\frac{{{\partial ^2}f}}{{\partial x\partial y}}\) at x = 2, y = 1?

All the eigenvalues of the matrix \(\left[ {\begin{array}{*{20}{c}} 1&2&0\\ 2&1&0\\ 0&0&{ - 1} \end{array}} \right]\) are in the disc.

Let X be a random variable with distribution

Let the characteristic equation of a 3 X 3 Matrix A be \({\lambda ^3} + a{\lambda ^2} + 47\lambda \; - \;60 = 0\) if one eigenvalue of A is 4 and a is an integer value, then what is the smallest eigenvalue of A?

A function y = 5x² + 10x is defined over an open interval x = (1, 2). Atleast at one point in this interval, dy/dx is exactly ________.

Determine the volume of the solid of revolution formed when the curve y = 2 is rotated 360° about the x-axis between the limits x = 0 to x = 3.

A missile can successfully hit a target with probability 0.75. If three successful hits can destroy the target completely, how many minimum missiles must be fired so that the probability of the completely destroying the target is not less than 0.95?

The solution of the differential equation \(\frac{{{d^2}u}}{{d{x^2}}} - k\frac{{du}}{{dx}} = 0\) where k is a constant, subjected to the boundary conditions u(0) = 0 and u(L) = U, is

Consider the following system of equations in x, y, z:

x + 2y + 2z = 1

x + ay + 3z = 3

x + 11y + az = b

Consider the hemisphere x² + y² + (z - 2)² = 9, 2 ≤ z ≤ 5 and the vector field F = xi + yj + (z - 2)k The surface integral ∬ (F ⋅ n) dS, evaluated over the hemisphere with n denoting the unit outward normal vector, is

If y(x) is the solution of the differential equation \(\frac{{dy}}{{dx}} = 2\left( {1 + y} \right)\sqrt y \) satisfying y(0) = 0 and \(y\left( {\frac{x}{2}} \right) = 1\), the largest interval (to the right of the origin) on which the solution exists is

In the Laurent series expansion of \(f\left( z \right)=\frac{1}{z-1}-\frac{1}{z-2}\) valid in the region |z| > 2, then the coefficient of 1/z² is:

Let \(A = \left( {\begin{array}{*{20}{c}} 1&{ - 1}\\ 0&1 \end{array}} \right)\) then which of the following statements is/are true?

If the directional derivative of the function z = y²e^2x at (2, -1) along the unit vector \(\vec b = \alpha \hat i + \beta \hat j\) is zero, then |α + β| equals

An analytic function of a complex variable \(z = x + iy\;\left( { i= \sqrt { - 1} } \right)\) is defined as

f(z) = x²- y² + iψ (x, y),

where ψ(x,y) is a real function. The value of the imaginary part of f(Z) at z = (1 + i) is _______ (round off to 2 decimal places).

Consider a sequence of independent Bernoulli trials with probability of success in each trial being \(\frac{1}{5}\). Then which of the following statements is/are TRUE?

The solution of DE y(x² + y² + 1) dy + [2x(x² + y²) – 1] dx = 0 is

If the triple integral over the region bounded by the planes 2x + y + z = 4, x = 0, y = 0 and z = 0 is given by  \(\mathop \smallint \limits_0^2 \mathop \smallint \limits_0^{\lambda \left( x \right)} \mathop \smallint \limits_0^{\mu\left( {x,y} \right)} dx\;dy\;dz\)  then the function λ(x) – μ(x,y) is_________.

For real constants a and b, let \(M = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\ a&b \end{array}} \right]\) be an orthogonal matrix. Then which of the following statements is/are always TRUE?