List of practice Questions

Let D be the region bounded by a closed cylinder \( x^2+y^2=16 \), \( z=0 \) and \( z=4 \), then the value of \( \iiint_D (\nabla \cdot \vec{v}) dV \), where \( \vec{v} = 3x^2\hat{i} + 6y^2\hat{j} + z\hat{k} \), is:

The value of the double integral \( \iint_R e^{x^2} dxdy \), where R is a region given by \( 2y \le x \le 2 \) and \( 0 \le y \le 1 \), is:

Let A be a \( 2 \times 2 \) matrix with \( \det(A) = 4 \) and \( \text{trace}(A) = 5 \). Then the value of \( \text{trace}(A^2) \) is:

If \( \vec{F} \) be the force and C is a non-closed arc, then \( \int_C \vec{F} \cdot d\vec{r} \) represents:

In Green's theorem, \( \oint_C (x^2y dx + x^2 dy) = \iint_R f(x,y) dx dy \), where C is the boundary described counter clockwise of the triangle with vertices (0, 0), (1, 0), (1, 1) and R is the region bounded by a simple closed curve C in the x-y plane, then \( f(x,y) \) is equal to:

The value of \( \int_0^{1+i} (x^2 - iy) dz \), along the path \( y = x^2 \) is:

The system of linear equations \( x+y+z=6 \), \( x+2y+5z=10 \), \( 2x+3y+\lambda z=\mu \) has a unique solution, if

For the given linear programming problem,
Minimum Z = 6x + 10y
subject to the constraints
\( x \ge 6 \); \( y \ge 2 \); \( 2x+y \ge 10 \); \( x,y \ge 0 \),
the redundant constraints are:

The solution of the differential equation \( (x^2 - 4xy - 2y^2)dx + (y^2 - 4xy - 2x^2)dy = 0 \), is

If \( x \in \mathbb{R} \) and a particular integral (P.I.) of \( (D^2 - 2D + 4)y = e^x \sin x \) is \( \frac{1}{2} e^x f(x) \), then \( f(x) \) is:

The value of \( \int_{0}^{\pi} \frac{x \sin x}{1+\cos^2 x} dx \) is:

The orthogonal trajectory of the cardioid \( r = a(1 - \cos\theta) \), where 'a' is an arbitrary constant is:

If the vectors \( \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \begin{pmatrix} p \\ 0 \\ 1 \end{pmatrix} \) are linearly dependent, then the value of p is:

The number of maximum basic feasible solution of the system of equations AX = b, where A is m \( \times \) n matrix, b is n \( \times \) 1 column matrix and rank of A is \(\rho(A) = m\), is:

The solution of the differential equation \( (xy^3 + y)dx + (2x^2y^2 + 2x + 2y^4)dy = 0 \) is:

If G is a cyclic group of order 12, then the order of Aut(G) is:

Which of the following function is discontinuous at every point of \(\mathbb{R}\)?

Let A and B be two symmetric matrices of same order, then which of the following statement are correct:
A. AB is symmetric
B. A+B is symmetric
C. \( A^T B = AB^T \)
D. \( BA = (AB)^T \)

Consider the function \( f(x, y) = x^2 + xy^2 + y^4 \), then which of the following statement is correct:

If \( f(z) = (x^2-y^2-2xy) + i(x^2-y^2+2xy) \) and \( f'(z)=cz \), where c is a complex constant, then \( |c| \) is equals to:

The solution of the differential equation \( \frac{xdy - ydx}{xdx + ydy} = \sqrt{x^2+y^2} \) is:

The integral domain of which cardinality is not possible:
A. 5
B. 6
C. 7
D. 10

Let \(m, n \in \mathbb{N}\) such that \(m<n\) and \(P_{m \times n}(\mathbb{R})\) and \(Q_{n \times m}(\mathbb{R})\) are matrices over real numbers and let \(\rho(V)\) denotes the rank of the matrix V. Then, which of the following are NOT possible.
A. \( \rho(PQ) = n \)
B. \( \rho(QP) = m \)
C. \( \rho(PQ) = m \)
D. \( \rho(QP) = \lfloor(m+n)/2\rfloor \), where \(\lfloor \rfloor\) is the greatest integer function

Which of the following are subspaces of vector space \(\mathbb{R}^3\):
A. \( \{(x,y,z) : x+y=0\ \)
B. \( \{(x,y,z) : x-y=0\} \)
C. \( \{(x,y,z) : x+y=1\} \)
D. \( \{(x,y,z) : x-y=1\} \)}

Maximize \( Z = 2x+3y \), subject to the constraints:
\( x+y \le 2 \)
\( 2x+y \le 3 \)
\( x,y \ge 0 \)