List of practice Questions

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function with the property that for every \( y \in \mathbb{R} \), the value of the expression \[ \sup_{x \in \mathbb{R}} [xy - f(x)] \] is finite. Define \( g(y) = \sup_{x \in \mathbb{R}} [xy - f(x)] \) for \(y \in \mathbb{R}\). Then

Consider the equation \[ x^{2021} + x^{2020} + \cdots + x + 1 = 0. \] Then

Let \( f : [0,1] \to [0, \infty) \) be a continuous function such that \[ (f(t))^2<1 + 2 \int_0^t f(s)\,ds, \quad \text{for all } t \in [0,1]. \] Then

Let \( D \subseteq \mathbb{R}^2 \) be defined by \[ D = \mathbb{R}^2 \setminus \{(x,0) : x \in \mathbb{R}\}. \] Consider the function \( f : D \to \mathbb{R} \) defined by \[ f(x,y) = x \sin\!\left(\frac{1}{y}\right). \] Then

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?