List of practice Questions

Let \( f(x, y) = |xy| + x \) for all \( (x, y) \in \mathbb{R}^2 \). Determine the existence of the partial derivative of \( f \) with respect to \( x \) exists:

Let \( f_i: \mathbb{R} \to \mathbb{R} \) for \( i = 1, 2 \) be defined as follows:
\[f_1(x) = \begin{cases} \sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\0, & \text{if } x = 0 \end{cases}\]
and
\[f_2(x) = \begin{cases} x\left(\sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right)\right), & \text{if } x \neq 0 \\0, & \text{if } x = 0 \end{cases}\]
Then, determine the continuity of these functions at \( x = 0 \):

For \( n \in \mathbb{N} \), let \[a_n = \sqrt{n} \sin^2\left(\frac{1}{n}\right) \cos n,\] and \[b_n = \sqrt{n} \sin\left(\frac{1}{n^2}\right) \cos n.\] Then

Let \( x_1, x_2, x_3, x_4 \) be the observed values from a random sample drawn from a \( N(\mu, \sigma^2) \) distribution, where \( \mu \in \mathbb{R} \) and \( \sigma \in (0, \infty) \) are unknown parameters. Let \( \bar{x} \) and \( s = \sqrt{\frac{1}{3} \sum_{i=1}^{4} (x_i - \bar{x})^2} \) be the observed be the observed sample mean sample standard deviation,repectively. For testing the hypotheses \( H_0: \mu = 0 \) against \( H_1: \mu \neq 0 \), the likelihood ratio test of size \( \alpha = 0.05 \) rejects \( H_0 \) if and only if \[\frac{|\bar{x}|}{s} > k.\] Then the value of \( k \) is given by:

Let \( X_1, X_2, \ldots, X_{20} \) be a random sample from the \( N(5, 2) \) distribution, and let \( Y_i = X_{2i} - X_{2i-1} \) for \( i = 1, 2, \ldots, 10 \). Then \( W = \frac{1}{4} \sum_{i=1}^{10} Y_i^2 \) has the distribution:

For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:

Let \( X \) be a random variable having the Poisson distribution with mean \( 1 \). Let \( g: \mathbb{N} \cup \{0\} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} 1 & \text{if } x \in \{0, 2\} \\ 0 & \text{if } x \notin \{0, 2\} \end{cases}\]
Then \( E(g(X)) \) is equal to:

Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?

Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?

Let \( A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \) be a real matrix, where \( ad = 1 \) and \( c \ne 0 \). If \( A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \), then \( (\alpha, \beta, \gamma, \delta) \) is equal to

Let \( f(x,y) = x^2 + 3y^2 - \frac{2}{3} xy \) at \( (x,y) \in \mathbb{R}^2 \). Then

Let \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then

For a positive integer \( n \), let \( U(n) = \{ r \in \mathbb{Z}_n : \gcd(r, n) = 1 \} \) be the group under multiplication modulo \( n \). Then, which one of the following statements is TRUE?

Let \( G \) be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE?

Consider the following statements. P: If a system of linear equations \( Ax = b \) has a unique solution, where \( A \) is an \( m \times n \) matrix and \( b \) is an \( m \times 1 \) matrix, then \( m = n \). Q: For a subspace \( W \) of a nonzero vector space \( V \), whenever \( u \in V \setminus W \) and \( v \in V \setminus W \), then \( u + v \in V \setminus W \). Which one of the following holds?

Let \( g: \mathbb{R} \to \mathbb{R} \) be a continuous function. Which one of the following is the solution of the differential equation \[ \frac{d^2 y}{dx^2} + y = g(x) \quad \text{for} \quad x \in \mathbb{R}, \] satisfying the conditions \( y(0) = 0 \), \( y'(0) = 1 \)?

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6?

Consider the group \( G = \{ A \in M_2(\mathbb{R}) : AA^T = I_2 \ \) with respect to matrix multiplication. Let \[ Z(G) = \{ A \in G : AB = BA, \, for all \, B \in G \}. \] Then, the cardinality of \( Z(G) \) is:}

Let \( V \) be a nonzero subspace of the complex vector space \( M_7(\mathbb{C}) \) such that every nonzero matrix in \( V \) is invertible. Then, the dimension of \( V \) over \( \mathbb{C} \) is:

For a twice continuously differentiable function \(g: \mathbb{R} \to \mathbb{R}\), define \[ u_g(x, y) = \frac{1}{y} \int_{-y}^{y} g(x + t) \, dt \quad \text{for} \quad (x, y) \in \mathbb{R}^2, \, y>0. \] Which one of the following holds for all such \(g\)?

Let \( y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} = 1 + y \sec x \quad \text{for} \quad x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right), \] that satisfies \( y(0) = 0 \). Then, the value of \( y\left( \frac{\pi}{6} \right) \) equals:

Let \( F \) be the family of curves given by \[ x^2 + 2hxy + y^2 = 1, \quad -1<h<1. \] Then, the differential equation for the family of orthogonal trajectories to \( F \) is:

Let \( G \) be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE?

Which one of the following is TRUE for the symmetric group \( S_{13} \)?

Let \( y_c : \mathbb{R} \to (0, \infty) \) be the solution of the Bernoulli’s equation \[ \frac{dy}{dx} - y + y^3 = 0, \quad y(0) = c>0. \] Then, for every \( c>0 \), which one of the following is true?