List of practice Questions

Which one of the following statements is true?
Let \( y \) be a twice differentiable function on \( \mathbb{R} \) satisfying \[ y''(x) = 2 + e^{-|x|}, \quad x \in \mathbb{R}, \quad y(0) = -1, \quad y'(0) = 0. \] Then
Let \( f : [0,1] \to [0,1] \) be a non-constant continuous function such that \( f \circ f = f. \) Define \[ E_f = \{x \in [0,1] : f(x) = x\}. \] Then
Consider the following statements. I. The group \((\mathbb{Q}, +)\) has no proper subgroup of finite index.
II. The group \((\mathbb{C} \setminus \{0\}, \cdot)\) has no proper subgroup of finite index.
Which one of the following statements is true?
Let \( f : \mathbb{N} \to \mathbb{N} \) be a bijective map such that \[ \sum_{n=1}^{\infty} \frac{f(n)}{n^2}<+\infty. \] The number of such bijective maps is
Define \[ S = \lim_{n \to \infty} \left(1 - \frac{1}{2^2}\right)\left(1 - \frac{1}{3^2}\right)\cdots\left(1 - \frac{1}{n^2}\right). \] Then
Let \( f : \mathbb{R} \to \mathbb{R} \) be an infinitely differentiable function such that for all \(a, b \in \mathbb{R}\) with \(a<b\), \[ \frac{f(b) - f(a)}{b - a} = f''\left(\frac{a+b}{2}\right). \] Then
Consider the family of curves \(x^2 - y^2 = ky\) with parameter \(k \in \mathbb{R}\). The equation of the orthogonal trajectory to this family passing through \((1,1)\) is given by
Which one of the following statements is true?
For every \( n \in \mathbb{N} \), let \( f_n : \mathbb{R} \to \mathbb{R} \) be a function. From the given choices, pick the statement that is the negation of \[ \text{“For every } x \in \mathbb{R} \text{ and for every real number } \varepsilon>0, \text{ there exists an integer } N>0 \text{ such that } \sum_{i=1}^p |f_{N+i}(x)|<\varepsilon \text{ for every integer } p>0.” \]
Which one of the following subsets of \(\mathbb{R}\) has a non-empty interior?
For an integer \(k \ge 0\), let \(P_k\) denote the vector space of all real polynomials in one variable of degree less than or equal to \(k\). Define a linear transformation \(T : P_2 \to P_3\) by \[ T(f(x)) = f''(x) + x f(x). \] Which one of the following polynomials is not in the range of \(T\)?
Let \(n>1\) be an integer. Consider the following two statements for an arbitrary \(n \times n\) matrix \(A\) with complex entries. I. If \(A^k = I_n\) for some integer \(k \ge 1\), then all the eigenvalues of \(A\) are \(k^{\text{th}}\) roots of unity.
II. If, for some integer \(k \ge 1\), all the eigenvalues of \(A\) are \(k^{\text{th}}\) roots of unity, then \(A^k = I_n.\)
Then
Let \( M_n(\mathbb{R}) \) be the real vector space of all \( n \times n \) matrices with real entries, \( n \ge 2 \). Let \( A \in M_n(\mathbb{R}) \). Consider the subspace \( W \) of \( M_n(\mathbb{R}) \) spanned by \(\{I_n, A, A^2, A^3, \ldots\}\). Then the dimension of \( W \) over \(\mathbb{R}\) is necessarily
Let \(y\) be the solution of \[ (1+x)y''(x) + y'(x) - \frac{1}{1+x} y(x) = 0, \quad x \in (-1,\infty), \] with initial conditions \(y(0) = 1, \ y'(0) = 0.\) Then
Consider the surface \[ S = \{(x,y,xy) \in \mathbb{R}^3 : x^2 + y^2 \le 1\}. \] Let \(\vec{F} = y\hat{i} + x\hat{j} + \hat{k}\). If \(\hat{n}\) is the continuous unit normal field to the surface \(S\) with positive \(z\)-component, then \[ \iint_S \vec{F} \cdot \hat{n} \, dS \] equals
Let \( P : \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( P(x)>0 \) for all \(x \in \mathbb{R}\). Let \(y\) be a twice differentiable function on \(\mathbb{R}\) satisfying \[ y''(x) + P(x)y'(x) - y(x) = 0 \] for all \(x \in \mathbb{R}\). Suppose that there exist two real numbers \(a,b\) (\(a<b\)) such that \(y(a) = y(b) = 0\). Then
Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(x) = f(x+1) \) for all \(x \in \mathbb{R}\). Then
Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function such that for all \(x \in \mathbb{R}\), \[ \int_0^1 f(xt) \, dt = 0. \quad \text{(*)} \] Then
Let \(p\) and \(t\) be positive real numbers. Let \(D_t\) be the closed disc of radius \(t\) centered at \((0,0)\), i.e., \[ D_t = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le t^2 \}. \] Define \[ I(p,t) = \iint_{D_t} \frac{dx\,dy}{(p^2 + x^2 + y^2)^2}. \] Then \(\lim_{t \to \infty} I(p,t)\) is finite
How many elements of the group \(\mathbb{Z}_{50}\) have order 10?
Let \( 0<\alpha<1 \) be a real number. The number of differentiable functions \( y : [0,1] \to [0,\infty) \), having continuous derivative on \([0,1]\) and satisfying \[ y'(t) = (y(t))^{\alpha}, \ t \in [0,1], \quad y(0) = 0, \] is

In a structurally controlled fluvial setting, an asymmetric flight of river terraces T1, T2, and T3 shown in the figure was sampled at location L1. The age of the sample at L1 was 30 ka (kiloyears). Assuming that the terraces were formed entirely due to deformation-related uplift, the average uplift rate in the past 30 ka in the region was ________ mm/yr. (Answer in integer.) 

In the uvarovite garnet (Ca3Cr2Si3O12), Ca is in cubic coordination, Cr is in octahedral coordination, and Si is in tetrahedral coordination. The electrostatic bond strength of the Ca2+ central ion is ________. (Round off to two decimal places.)

Refer to the schematic sketch given (not to scale). Assume average saturated density of oceanic crustal rocks = 3200 kg/m3, density of ocean water = 1000 kg/m3, and acceleration due to gravity = 10 m/s2. The overburden pressure at a point (P) located 2 km below seabed and 4 km below the ocean surface is ________ MPa. (Answer in integer.)