List of practice Questions

The minimum size of glass tubing that can be used to measure water level, if the capillary rise in the tube is not to exceed 0.25 cm. (Take surface tension of water in contact air as 0.0075 kg(f)/m).
For the flow through a pipe, if Reynolds's number is greater than 4000, it indicates:
Dynamic viscosity of a fluid is 2.2 poise and specific gravity is 0.7. Then kinematic viscosity in SI units is:
In order to determine the workability of a concrete mixture, the recommended tests are: (A) Flow test
(B) Compaction Factor test
(C) Flexural strength test
(D) Slump test
Choose the correct answer from the options given below:
The value of \( v_3 \) for which the vector \( \vec{v} = e^y \sin x \hat{i} + e^y \cos x \hat{j} + v_3 \hat{k} \) is solenoidal, is:
The value of the integral \( \iint_R (x+y) \,dy\,dx \) in the region R bounded by \( x=0, x=2, y=x, y=x+2 \), is
For any Linear Programming Problem (LPP), choose the correct statement:
A. There exists only finite number of basic feasible solutions to LPP
B. Any convex combination of k - different optimum solution to a LPP is again an optimum solution to the problem
C. If a LPP has feasible solution, then it also has a basic feasible solution
D. A basic solution to AX = b is degenerate if one or more of the basic variables vanish
Which of the following are correct:
A. Every infinite bounded set of real number has a limit point
B. The set \( S = \{x : 0<x \le 1, x \in \mathbb{R}\} \) is a closed set
C. The set of whole real numbers is open as well closed set
D. The set \( S = \{1, -1, \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3}, ... \} \) is neither open set nor closed set

Match List-I with List-II and choose the correct option:

\[ \begin{array}{|l|l|} \hline \textbf{LIST-I (Function)} & \textbf{LIST-II (Expansion)} \\ \hline A. \log(1-x) & I. 1 + \frac{1}{3} + \frac{1}{6} + \frac{3}{40} + \frac{15}{336} + \dots \\ \hline B. \sin^{-1} x & II. 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \\ \hline C. \log 2 & III. x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} + \dots, -1 < x \le 1 \\ \hline D. \frac{\pi}{2} & IV. -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots, -1 \le x < 1 \\ \hline \end{array} \]

If U and W are distinct 4-dimensional subspaces of a vector space V of dimension 6, then the possible dimensions of \( U \cap W \) is:
Which of the following forms a linear transformation:
Let \( f(x) = |x| + |x-1| + |x+1| \) be a function defined on \( \mathbb{R} \), then \( f(x) \) is:
Let \(f(x)\) be a real valued function defined for all \(x \in \mathbb{R}\), such that \(|f(x)-f(y)| \le (x-y)^2\), for all \(x,y \in \mathbb{R}\), then
The value of the integral \( \int_0^\infty \int_0^x x e^{-x^2/y} dy \, dx \) is:
If, \( u=y^3-3x^2y \) be a harmonic function then its corresponding analytic function \( f(z) \), where \( z=x+iy \), is:
A complete solution of \( y'' + a_1 y' + a_2 y = 0 \) is \( y = b_1 e^{-x} + b_2 e^{-3x} \), where \( a_1, a_2, b_1 \) and \( b_2 \) are constants, then the respective values of \( a_1 \) and \( a_2 \) are:
If 'a' is an imaginary cube root of unity, then \( (1-a+a^2)^5 + (1+a-a^2)^5 \) is equal to:
The solution of the differential equation \( xdy - ydx = (x^2+y^2)dx \), is
The value of \( \iiint_E \frac{dx\,dy\,dz}{x^2+y^2+z^2} \), where E: \( x^2+y^2+z^2 \le a^2 \), is
The given series \( 1 - \frac{1}{2^p} + \frac{1}{3^p} - \frac{1}{4^p} + ... (p>0) \) is conditionally convergent, if 'p' lies in the interval:
Which of the following set of vectors forms the basis for \( \mathbb{R}^3 \)?
If p is a prime number and O(G) denotes the order of a group G and p divides O(G), then group G has an element of order p. Then, this is a statement of
Which of the following statements are true for group of permutations?
A. Every permutation of a finite set can be written as a cycle or a product of disjoint cycles
B. The order of a permutation of a finite set written in a disjoint cycle form is the least common multiple of the lengths of the cycles
C. If \(A_n\) is a group of even permutation of n-symbol (\(n>1\)), then the order of \(A_n\) is n!
D. The pair of disjoint cycles commute
A ring (R, +, .), where all elements are idempotent is always:
If \( v = \sin^{-1} \left( \frac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}} \right) \), then \( x\frac{\partial v}{\partial x} + y\frac{\partial v}{\partial y} \) is equal to :