List of top Engineering Mathematics Questions

Let \( f(x, y, z) = 4x^2 + 7xy + 3xz^2 \). The direction in which the function \( f(x, y, z) \) increases most rapidly at point \( P = (1, 0, 2) \) is
Let the probability density function of a random variable \( x \) be given as \[ f(x) = ae^{-2|x|} \] The value of ‘a’ is ________.
Consider a matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -2 \\ 0 & 1 & 1 \end{bmatrix} \). The matrix \( A \) satisfies the equation \( 6A^{-1} = A^2 + cA + dI \), where \( c \) and \( d \) are scalars and \( I \) is the identity matrix. Then \( (c + d) \) is equal to
Let \( f(x) = \int_0^x e^t (t - 1)(t - 2) \, dt \). Then \( f(x) \) decreases in the interval
e\(^A\) denotes the exponential of a square matrix A. Suppose \( \lambda \) is an eigenvalue and \( \mathbf{v} \) is the corresponding eigen-vector of matrix A. Consider the following two statements: Statement 1: e\(^\lambda\) is an eigenvalue of e\(^A\). Statement 2: \( \mathbf{v} \) is an eigen-vector of e\(^A\). Which one of the following options is correct?
Consider a 3 x 3 matrix A whose \( (i, j) \)-th element, \( a_{i,j} = (i - j)^3 \). Then the matrix A will be
The vector field \( \vec{v} = x^3 \hat{i} + y^3 \hat{j} + z^3 \hat{k} \) is a vector field where \( \hat{i}, \hat{j}, \hat{k} \) are the base vectors of a Cartesian coordinate system.
Using the Gauss divergence theorem, the value of the outward flux of the vector field over the surface of a sphere of unit radius centered at the origin is _________ (rounded off to one decimal place).
The largest eigenvalue of the given matrix is: \[ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{bmatrix} \]
The real function \( y = \sin^2(|x|) \) is
Consider the differential equation \(\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0\). The boundary conditions are \(y = 0\) and \(\frac{dy}{dx} = 1\) at \(x = 0\). Then the value of \(y\) at \(x = \frac{1}{2}\) is
Consider the partial differential equation \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0\) where \(x, y\) are real. If \(f(x, y) = a(x)b(y)\), where \(a(x)\) and \(b(y)\) are real functions, which one of the following statements can be true?
The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius \(R\) is
The arc length of the parametric curve: \( x=\cos\theta,\; y=\sin\theta,\; z=\theta \) from \( \theta=0 \) to \( \theta=2\pi \) is equal to _________ (round off to one decimal place).
If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s):
Let \(\hat{i}, \hat{j}, \hat{k}\) be the unit vectors in the x, y, and z directions, respectively. If the vector \(\hat{i} + \hat{j}\) is rotated about positive \(\hat{k}\) by \(135^\circ\), one gets
The equation of the straight line representing the tangent to the curve \(y = x^2\) at the point (1, 1) is
The signal \( x(t) = (t - 1)^2 u(t - 1) \), where \( u(t) \) is the unit-step function, has the Laplace transform \( X(s) \). The value of \( X(1) \) is_________

Truth table of a logic gate is given below: 
\[ \begin{array}{|c|c|c|} \hline \text{Input A} & \text{Input B} & \text{Output Y} \\ \hline 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & X \\ \hline \end{array} \] 

The value of X in the above table is .............. 
 

Let \( A = \begin{pmatrix} \sin \theta & \cos \theta \\ \cos \theta & \sin \theta \end{pmatrix} \) and \( A + A^T - 2I = 0 \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix. The value of \( \theta \) (in degrees) is ............