List of practice Questions

Let U and W are distinct 4-dimensional subspaces of a vector space V of dimension 6. Consider the following statements:
A. The dimension of U ∩ W is either 2 or 3.
B. The dimension of U + W is either 5 or 6.
C. The dimension of U ∩ W is always greater than 4.
D. The dimension of U + W is always greater than 4.
Choose the correct answer from the options given below:

The solution of the Linear Programming Problem
maximize Z = 107x + y
subject to constraints x + y ≤2
-3x + y ≥ 3
x, y ≥ 0 is

The natural domain of definition of the function f(z) = \(\frac{1}{1-|z|^2}\) is ________.

For what value(s) of k the set of vectors {(1, k, 5), (1, -3, 2), (2, -1, 1)} form a basis in R³ ?

Evaluate the integral\(\oint\limits_C\frac{dz}{(z^2+4)^2},C:|z-i|=2\)

The area bounded by the curves y = x² and y = 4 - x² is

Which one of the following is a cyclic group?

A scalar potential \(\Psi\) has the gradient defined as  \(\nabla\Psi=yz\hat{i}+xz\hat{j}+xy\hat{k}\). The value of the integral \(\int_c\nabla\Psi.d\overrightarrow{r}\) on the curve \(\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\), where curve C: x=t, y = t², z = 3t² (1 ≤ t ≤ 3) is:

The work done by the force \(\overrightarrow F = (x^2-y^2)\hat{i} + (x+y)\hat{j}\) in moving a particle along the closed path C containing the curves x + y = 0, x²+ y² = 16 and y = x in the first and fourth quadrant is

The line integral \(\overrightarrow V = x^2\hat{î}-2y\hat{j} + z^2\hat{k}\) over the straight line path from the point (-1, 2, 3) to (2, 3, 5) is

Let \(\overrightarrow V\) be a vector field and f be a scalar point function, then curl \((f\overrightarrow V)\) is equivalent to________.

Let S be a piecewise smooth surface of the sphere x² + y² + z² = 16, z> 0, bounded by a simple closed curve C. Let \(\overrightarrow V= (3x-y)\hat{i}-2yz^2\hat{j}-2y^2z\hat{k}\) be a vector field which is continuous and has continuous first order partial derivatives in a domain which contains S. Then the value of \(\int\int(\nabla\times\overrightarrow V).\hat{n}dA\), where \(\hat{n}\) the unit normal vector to S is:

The all values of z, such that
√2 sin z = coshβ + isinħβ, where β is real, are

The solution of the differential equation
{x⁴+6x²+2(x+y)} dx-xdy=0
subject to the condition y(1)=0 is

Which of the following statement is not correct?

Let A and B be 2 × 2 matrices, then which of the following is correct?

Let f (x) be defined on [0, 3] by \(f(x) =   \begin{cases}     x,\text{if x is a rational number}       \\  3-x\text{, if x is an irrational number}   \end{cases}\) Then f(x) is continuous in the interval at:

If A is symmetric real valued matrix of dimension 2022, then eigenvalues of A are

Which one of the following rings is an integral domain?

Let f: G→H be a group homomorphism from group G into group H with kernel K. If the order of G, H and K are 50, 25 and 10 respectively then the order of f(G) is

The equation (2x + y + 1) dx + (x + 2y +1) dy = 0 represents a family of:

The value of surface integral \(\iint_S(9x\hat{i}-2\hat{j}-z\hat{k}).\hat{n}dS\) over the surface S of the sphere x²+y²+z²=9 where \(\hat{n}\) is the unit outward normal to surface element dS is:

The line integral of function \(F = yz\hat{i}\), in the counter clockwise direction, along the circle x² + y² = 1 at z = 1 is

If \(\overrightarrow{r}=x_1\hat{a}_{x_1}+x_2\hat{a}_{x_2}+x_3\hat{a}_{x_3}\) and \(|\overrightarrow{r}|=r\) then \(div(r^2\nabla(In\;r))\)is

The area of surface of solid generated by the revolution of line segment y = 2x from x = 0 to x = 2 about x-axis is equal to: