List of practice Questions

Let \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( S \) be the sphere given by \( (x - 2)^2 + (y - 2)^2 + (z - 2)^2 = 4. \) If \( \hat{n} \) is the unit outward normal to \( S \), then \( \dfrac{1}{\pi} \iint_S \vec{F} \cdot \hat{n} \, dS \) is .............
Let \( f(x, y) = e^x \sin y, \, x = t^3 + 1, \, y = t^4 + t. \) Then \( \dfrac{df}{dt} \) at \( t = 0 \) is ............. (rounded off to two decimal places).
Consider the differential equation \[ \frac{dy}{dx} + 10y = f(x), x > 0, \] where \( f(x) \) is a continuous function such that \( \lim_{x \to \infty} f(x) = 1. \) Then the value of \( \lim_{x \to \infty} y(x) \) is .................
If \( \displaystyle \int_0^1 \int_{2y}^2 e^{x^2} \, dx \, dy = k(e^4 - 1), \) then \( k \) equals ...............
Let \( V \) be a non-zero vector space over a field \( F \). Let \( S \subset V \) be a non-empty set. Consider the following properties of \( S \):
(I) For any vector space \( W \) over \( F \), any map \( f : S \to W \) extends to a linear map from \( V \) to \( W \).
(II) For any vector space \( W \) over \( F \) and any two linear maps \( f, g : V \to W \) satisfying \( f(s) = g(s) \) for all \( s \in S \), we have \( f(v) = g(v) \) for all \( v \in V \).
(III) \( S \) is linearly independent.
(IV) The span of \( S \) is \( V \).
Which of the following statement(s) is/are true?
Let \( a = \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \cdots + \frac{n-1}{n^2} \right) \) and \( b = \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n} \right). \) Which of the following is/are true?
Let \( S \) be that part of the surface of the paraboloid \( z = 16 - x^2 - y^2 \) which is above the plane \( z = 0 \) and \( D \) be its projection on the xy-plane. Then the area of \( S \) equals
Let \( G \) be a group with identity \( e \). Let \( H \) be an abelian non-trivial proper subgroup of \( G \) with the property that \( H \cap gHg^{-1} = \{ e \} \) for all \( g \notin H \). If \( K = \{ g \in G : gh = hg \text{ for all } h \in H \}, \) then
Let \( F = \{\omega \in \mathbb{C} : \omega^{2020} = 1\}. \) Consider the groups \\ \[ G = \left\{ \begin{pmatrix} \omega & z \\ 0 & 1 \end{pmatrix} : \omega \in F, z \in \mathbb{C} \right\} \text{and} H = \left\{ \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} : z \in \mathbb{C} \right\} \] under matrix multiplication. Then the number of cosets of \( H \) in \( G \) is \\
Let \( a, b, c \in \mathbb{R} \) such that \( a < b < c. \) Which of the following is/are true for any continuous function \( f : \mathbb{R} \to \mathbb{R} \) satisfying \( f(a) = b, f(b) = c \) and \( f(c) = a? \)
Let \( a, b \in \mathbb{R} \) and \( a < b. \) Which of the following statement(s) is/are true?
Let \( M \) be a \( 4 \times 3 \) real matrix and let \( \{e_1, e_2, e_3\} \) be the standard basis of \( \mathbb{R}^3. \) Which of the following is true?
Let \( f(x, y, z) = x^3 + y^3 + z^3 - 3xyz. \) A point at which the gradient of \( f \) is equal to zero is
Let \( M \) be an \( n \times n \) ( \( n \ge 2 \) ) non-zero real matrix with \( M^2 = 0 \) and let \( \alpha \in \mathbb{R} \setminus \{0\}. \) Then
Let \(\{a_n\}\) be a sequence of positive real numbers. Suppose that \(l = \lim_{n \to \infty} \dfrac{a_{n+1}}{a_n}.\) Which of the following is true?
Suppose that \( S \) is the sum of a convergent series \( \displaystyle \sum_{n=1}^{\infty} a_n. \) Define \( t_n = a_n + a_{n+1} + a_{n+2}. \) Then the series \( \displaystyle \sum_{n=1}^{\infty} t_n \)
Let \( f(x) = 2x^3 - 9x^2 + 7. \) Which of the following is true?
Which of the following is FALSE?
Let \( g: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. If \( f(x, y) = g(y) + x g'(y) \), then
If the equation of the tangent plane to the surface \( z = 16 - x^2 - y^2 \) at the point \( P(1, 3, 6) \) is \( ax + by + cz + d = 0 \), then the value of \( |d| \) is

Consider the schematic isobaric T-X phase diagram in the binary forsterite (Fo)-fayalite (Fa) chemical system. If there is equilibrium crystallization of melt (L), the wt.% of olivine crystallized from a melt of composition "a" at a temperature T2 is ......... 

From the data shown in the table, the weighted mean size (in micrometer, correct to two decimal places) of the sediment population is ............ 
\[ \begin{array}{|c|c|} \hline \text{Grain Size (micrometer)} & \text{Dry Sediment Weight (in gram)} \\ \hline 4 & 50 \\ 20 & 75 \\ 40 & 125 \\ 60 & 50 \\ \hline \end{array} \]

A set of sedimentary rocks A, B, and C are affected by a fault F-F. The amount of vertical throw (in m) along the fault is ............ 

A coal seam occurs in a stratigraphic sequence as shown in the figure. If a vertical borehole is drilled at location B, the coal seam will be intersected at a depth (in m) of ........ 

In an undeformed and normal stratigraphic succession, a dolerite dyke was emplaced before deposition of sandstone B. The difference between the maximum ages (in Myr) of deposition of sandstone A and sandstone B is ...............