List of practice Questions

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}, \, x, y, z \in \mathbb{R}. \) Which of the following is FALSE?

Let \( S \) be the part of the cone \( z^2 = x^2 + y^2 \) between the planes \( z = 0 \) and \( z = 1. \) Then the value of the surface integral \( \iint_S (x^2 + y^2) \, dS \) is

The value of the triple integral \( \iiint_V (x^2y + 1) \, dx\,dy\,dz \), where \( V \) is the region given by \( x^2 + y^2 \le 1, \, 0 \le z \le 2, \) is

A solution of the differential equation \( 2x^2\dfrac{d^2y}{dx^2} + 3x\dfrac{dy}{dx} - y = 0, \, x > 0 \) that passes through the point (1, 1) is

Consider the differential equation \( L[y] = (y - y^2)dx + xdy = 0. \) The function \( f(x, y) \) is said to be an integrating factor of the equation if \( f(x, y)L[y] = 0 \) becomes exact. If \( f(x, y) = \dfrac{1}{x^2 y^2}, \) then

Let \( M \) be a real \( 6 \times 6 \) matrix. Let 2 and -1 be two eigenvalues of \( M. \) If \( M^5 = aI + bM \), where \( a, b \in \mathbb{R}, \) then

The area bounded by the curves \( x^2 + y^2 = 2x \) and \( x^2 + y^2 = 4x \), and the straight lines \( y = x \) and \( y = 0 \) is

Let \( S \) be the surface of the portion of the sphere with centre at the origin and radius 4, above the \( xy \)-plane. Let \( \vec{F} = y\hat{i} - x\hat{j} + yxz^3\hat{k}. \) If \( \hat{n} \) is the unit outward normal to \( S \), then
\[ \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \] equals

Let \( f(x, y) = \begin{cases} x^2 \sin \dfrac{1}{x} + y^2 \sin \dfrac{1}{y}, & xy \ne 0 \\ x^2 \sin \dfrac{1}{x}, & x \ne 0, y = 0 \\ \\ y^2 \sin \dfrac{1}{y}, & y \ne 0, x = 0 \\ 0, & x = y = 0 \end{cases} \).

Which of the following is true at \( (0, 0)? \)

Let \( a \in \mathbb{R} \). If \( f(x) = \begin{cases} (x + a)^2, & x \leq 0 \\ (x + a)^3, & x > 0 \end{cases} \), then

Define \( s_1 = \alpha > 0 \) and \( s_{n+1} = \sqrt{\dfrac{1 + s_n^2}{1 + \alpha}}, \, n \geq 1. \) Which of the following is true?

Consider the following group under matrix multiplication:

\[ H = \left\{ \begin{bmatrix} 1 & p & q \\ 0 & 1 & r \\ 0 & 0 & 1 \end{bmatrix} : p, q, r \in \mathbb{R} \right\}. \]

Then the center of the group is isomorphic to

The radius of convergence of the power series \( \displaystyle \sum_{n=1}^{\infty} \left( \dfrac{n+2}{n} \right)^{n^2} x^n \) is

Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation given by \( T(x, y) = (-x, y). \) Then

If \( u = x^3 \) and \( v = y^2 \) transform the differential equation \( 3x^5 dx - y(y^2 - x^3) dy = 0 \) to \( \dfrac{dv}{du} = \dfrac{\alpha u}{2(u - v)} \), then \( \alpha \) is

If the directional derivative of the function \( z = y^2 e^{2x} \) at \( (2, -1) \) along the unit vector \( \vec{b} = \alpha \hat{i} + \beta \hat{j} \) is zero, then \( |\alpha + \beta| \) equals

Let \( s_n = 1 + \dfrac{(-1)^n}{n}, \, n \in \mathbb{N}. \) Then the sequence \(\{s_n\}\) is

If \( x^2 + xy^2 = c \), where \( c \in \mathbb{R} \), is the general solution of the exact differential equation \[ M(x, y)\, dx + 2xy\, dy = 0, \] then \( M(1,1) \) is ............

Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................

Suppose that \( G \) is a group of order 57 which is not cyclic. If \( G \) contains a unique subgroup \( H \) of order 19, then for any \( g \notin H \), the order of \( g \) is ................

Consider the real vector space \( P_{2020} = \left\{ \sum_{i=0}^n a_i x^i : a_i \in \mathbb{R}, \, 0 \le n \le 2020 \right\}. \) Let \( W \) be the subspace given by \[ W = \left\{ \sum_{i=0}^n a_i x^i \in P_{2020} : a_i = 0 \text{ for all odd } i \right\}. \] Then the dimension of \( W \) is ................

Let \( \phi : S_3 \to S_1 \) be a non-trivial non-injective group homomorphism. Then the number of elements in the kernel of \( \phi \) is .............

The sum of the series \[ \frac{1}{2(2^2 - 1)} + \frac{1}{3(3^2 - 1)} + \frac{1}{4(4^2 - 1)} + \cdots \] is ...........

Let \( C \) be the boundary of the square with vertices \( (0,0), (1,0), (1,1), (0,1) \) oriented counterclockwise. Then the value of the line integral \[ \oint_C x^2y^2 dx + (x^2 - y^2) dy \] is ............ (rounded off to two decimal places).

Let \( x_n = n^{1/n} \) and \( y_n = e^{1 - x_n}, \, n \in \mathbb{N}. \) Then the value of \( \lim_{n \to \infty} y_n \) is ..............