List of practice Questions

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:
\( x+y \le 2 \)
\( 2x+y \le 3 \)
\( x,y \ge 0 \)
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:
The value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) 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:
Let f be a continuous function on \(\mathbb{R}\) and \( F(x) = \int_{x-2^{x+2} f(t) dt \), then F'(x) 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)
Which of the following statement is true:
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
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:
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})}
(A) Comment upon the mode of pollination in Vallisneria and Zostera. (B)

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Which of the following offences may be considered a 'victimless crime'?
What are the five ordeals, treated as divine means of proof of the guilt or innocence of the accused in the traditional Indian governance system?