List of top Mathematics Questions

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:

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

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

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

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. \( BA = (AB)^T \)

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:

Which one of the following mathematical structure forms a group?

If \( A = \begin{pmatrix} 2 & 4 & 1 \\ 0 & 2 & -1 \\ 0 & 0 & 1 \end{pmatrix} \) satisfies \( A^3 + \mu A^2 + \lambda A - 4I_3 = 0 \), then the respective values of \( \lambda \) and \( \mu \) are:

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. \( \{(x,y,z) : x-y=1\} \)}

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

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

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

The value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) is:

Let f be a continuous function on \(\mathbb{R}\) and \( F(x) = \int_{x-2^{x+2} f(t) dt \), then F'(x) is}

The function \( f(z) = |z|^2 \) is differentiable, at

If C is the positively oriented circle represented by \( |z|=2 \), then \( \int_C \frac{e^{2z}}{z-4} dz \) is:

If C is a triangle with vertices (0,0), (1,0) and (1,1) which are oriented counter clockwise, then \( \oint_C 2xydx + (x^2+2x)dy \) is equal to:

For the function \( f(x, y) = x^3 + y^3 - 3x - 12y + 12 \), which of the following are correct:
A. minima at (1,2)
B. maxima at (-1,-2)
C. neither a maxima nor a minima at (1,-2) and (-1,2)
D. the saddle points are (-1,2) and (1,-2)

Consider the following: Let f(z) be a complex valued function defined on a subset \( S \subset \mathbb{C} \) of complex numbers. Then which of the following are correct?
A. The order of a zero of a polynomial equals to the order of its first non-vanishing derivative at that zero of the polynomial
B. Zeros of non-zero analytic function are isolated
C. Zeros of f(z) are obtained by equating the numerator to zero if there is no common factor in the numerator and the denominator of f(z)
D. Limit points of zeros of an analytic function is an isolated essential singularity

Which of the following statement is true:

Let [x] be the greatest integer function, where x is a real number, then \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} ([x] + [y] + [z]) \, dx \, dy \, dz = \)

Let V(F) be a finite dimensional vector space and T: V \(\to\) V be a linear transformation. Let R(T) denote the range of T and N(T) denote the null space of T. If rank(T) = rank(T\textsuperscript{2}), then which of the following are correct?
A. N(T) = R(T)
B. N(T) = N(T\textsuperscript{2})
C. N(T) \(\cap\) R(T) = \{0\
D. R(T) = R(T\textsuperscript{2})}

If \( S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right) ... \left(1-\frac{1}{n^2}\right) \), then S is equal to:

Which of the following subsets form subgroups of the group <ℤ, +>?

(A). H₁ = {0}
(B). H₂ = {n+1 : n ∈ ℤ}
(C). H₃ = {2n : n ∈ ℤ}
(D). H₄ = {2n+1 : n ∈ ℤ}

Choose the correct answer from the options given below: