List of practice Questions

The coefficient of \( (x - \frac{\pi}{2}) \) in the Taylor series expansion of the function

\[ f(x) = \begin{cases} \frac{4(1 - \sin(x))}{2x - \pi}, & x \neq \frac{\pi}{2} \\ 0, & x = \frac{\pi}{2} \end{cases} \]

about \( x = \frac{\pi}{2} \) is ............

Let \( \vec{F}(x, y) = -y \hat{i} + x \hat{j} \) and let \( C \) be the ellipse

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

with counterclockwise orientation. Then the value of \( \oint_C \vec{F} \cdot d\vec{r} \) (rounded to 2 decimal places) is .............

Let

\[ f(x, y) = \begin{cases} \frac{|x|}{|x| + |y|} \sqrt{x^4 + y^2}, & (x, y) \neq (0, 0), \\ 0, & (x, y) = (0, 0). \end{cases} \]

Then at \( (0, 0) \),

For \( \beta \in \mathbb{R} \), define

\[ f(x, y) = \begin{cases} x^2 |x|^\beta y & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases} \]

Then, at \( (0, 0) \), the function \( f \) is

The set of eigenvalues of which one of the following matrices is NOT equal to the set of eigenvalues of

\[ \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \]

Let \( \mathbf{F}(x, y, z) = 2y \hat{i} + x^2 \hat{j} + xy \hat{k} \) and let \( C \) be the curve of intersection of the plane

\[ x + y + z = 1 \]

and the cylinder

\[ x^2 + y^2 = 1. \]

Then the value of

\[ \left| \int_C \mathbf{F} \cdot d\mathbf{r} \right| \]

The number of critical points of the function \[ f(x, y) = (x^2 + 3y^2)^2 e^{-(x^2 + y^2)} \] is .....................

The number of elements in the set \[ \{ x \in S_3 : x^4 = e \}, \quad \text{where} \quad e \text{ is the identity element of the permutation group } S_3, \text{ is} ............... \]

The value of the integral \[ \int_{-1}^{1} \int_{-1}^{1} |x + y| \, dx \, dy \] (round off to 2 decimal places) is ................

The volume of the solid bounded by the surfaces \[ x = 1 - y^2, \quad x = y^2 - 1, \quad \text{and the planes} \quad z = 0 \quad \text{and} \quad z = 2 \] (round off to 2 decimal places) is ........

The volume of the solid of revolution of the loop of the curve \[ y^2 = x^4(x + 2) \] about the x-axis (round off to 2 decimal places) is .............

The greatest lower bound of the set \[ \left\{ \left( e^n + 2^n \right)^{\frac{1}{n}} : n \in \mathbb{N} \right\}, \] (round off to 2 decimal places) is .............

Let \[ G = \{ n \in \mathbb{N} : n \leq 55, \, \gcd(n, 55) = 1 \} \] be the group under multiplication modulo 55. Let \( x \in G \) be such that \( x^2 = 26 \) and \( x>30 \). Then \( x \) is equal to ................

Let \( \{a_n\} \) be the sequence of real numbers such that \[ a_1 = 1 \quad \text{and} \quad a_{n+1} = a_n + a_n^2 \quad \text{for all } n \geq 1. \] Then

Let \( x \) be the 100-cycle \( (1 \, 2 \, 3 \, \dots \, 100) \) and let \( y \) be the transposition \( (49 \, 50) \) in the permutation group \( S_{100} \). Then the order of \( xy \) is

Let \( W_1 \) and \( W_2 \) be subspaces of the real vector space \( \mathbb{R}^{100} \) defined by \[ W_1 = \{ (x_1, x_2, \dots, x_{100}) : x_i = 0 \text{ if } i \text{ is divisible by } 4 \}, \] \[ W_2 = \{ (x_1, x_2, \dots, x_{100}) : x_i = 0 \text{ if } i \text{ is divisible by } 5 \}. \] Then the dimension of \( W_1 \cap W_2 \) is

Let \( f : [0,1] \to \mathbb{R} \) be given by \[ f(x) = \frac{\left( \frac{1}{1+x^3} \right)^3 + \left( \frac{1}{1-x^3} \right)^3}{8(1+x)}. \] Then \( \max \{ f(x) : x \in [0,1] \} - \min \{ f(x) : x \in [0,1] \} \) is ...........

Let \( S \) and \( T \) be linear transformations from a finite dimensional vector space \( V \) to itself such that \( S(T(v)) = 0 \) for all \( v \in V \). Then

Let \( \mathbf{F} \) and \( \mathbf{G} \) be differentiable vector fields and let \( g \) be a differentiable scalar function. Then

Consider the intervals \( S = (0, 2] \) and \( T = [1, 3] \). Let \( S^\circ \) and \( T^\circ \) be the sets of interior points of \( S \) and \( T \), respectively. Then the set of interior points of \( S \setminus T \) is equal to

Let \( a_n \) be the sequence given by \[ a_n = \max \left( \sin \left( \frac{n \pi}{3} \right), \cos \left( \frac{n \pi}{3} \right) \right), \quad n \geq 1. \] Then which of the following statements is/are TRUE about the subsequences \( \{a_{6n-1}\} \) and \( \{a_{6n+1}\} \)?

Let \[ f(x) = \cos(|\pi - x|) + (x - \pi) \sin |x| \quad \text{and} \quad g(x) = x^2 \text{ for } x \in \mathbb{R}. \] If \( h(x) = f(g(x)) \), then}

Let \[ f(x) = (\sin x)^{\pi} - \pi \sin x + \pi. \] Then which of the following statements is/are TRUE?

Let \( (a_n) \) be a sequence of positive real numbers. The series \[ \sum_{n=1}^{\infty} a_n^2 \quad \text{converges if the series} \]