List of practice Questions

For $ x \in \mathbb{R} $, let $ f(x) = \begin{cases} x^3 \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0, & x = 0 \end{cases} $ . Then which one of the following is FALSE?

Suppose that $ f, g : \mathbb{R} \to \mathbb{R} $ are differentiable functions such that $ f $ is strictly increasing and $ g $ is strictly decreasing. Define $ p(x) = f(g(x)) $ and $ q(x) = g(f(x)) $, $ \forall x \in \mathbb{R} $. Then, for $ t > 0 $, the sign of $ \int_0^t p'(x) (q'(x)-3) \, dx $ is

Let $ a_n = \dfrac{(-1)^{n}}{\sqrt{1+n}} $ and let $ c_n = \sum_{k=0}^{n} a_{n-k} a_k $, where $ n \in \mathbb{N} \cup \{0\} $. Then which one of the following is TRUE?

Let $ a_n = n + \frac{1}{n} $, $ n \in \mathbb{N} $. Then the sum of the series $ \sum_{n=1}^{\infty} (-1)^{n+1} \dfrac{a_{n+1}}{n!} $ is

Let $ a, b, c \in \mathbb{R} $. Which of the following values of $ a, b, c $ do NOT result in the convergence of the series
\[ \sum_{n=3}^{\infty} \frac{a^n}{n^b (\log_e n)^c} ? \]

$$\text{Let } a_n = \begin{cases} 2 + \dfrac{(-1)^{\frac{n-1}{2}}}{n}, & \text{if } n \text{ is odd} \\ 1 + \dfrac{1}{2^n}, & \text{if } n \text{ is even} \end{cases}, \quad n \in \mathbb{N}.$$
Then which one of the following is TRUE?

Let $ s_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} $ for $ n \in \mathbb{N} $. Then which one of the following is TRUE for the sequence $ \{ s_n \}_{n=1}^{\infty} $?

Consider the vector space $ V $ over $ \mathbb{R} $ of polynomial functions of degree less than or equal to 3 defined on $ \mathbb{R} $. Let $ T : V \to V $ be defined by $ (T f)(x) = f(x) - x f'(x) $. Then the rank of $ T $ is

In $ \mathbb{R}^2 $, the family of trajectories orthogonal to the family of asteroids $ x^{2/3} + y^{2/3} = a^{2/3} $ is given by

Let $ f : \mathbb{R}^3 \to \mathbb{R} $ be a scalar field, $ \vec{v} : \mathbb{R}^3 \to \mathbb{R}^3 $ be a vector field and let $ \vec{a} \in \mathbb{R}^3 $ be a constant vector. If $ \vec{r} $ represents the position vector $ x\hat{i} + y\hat{j} + z\hat{k} $, then which one of the following is FALSE?

In $ \mathbb{R}^3 $, the cosine of the acute angle between the surfaces $ x^2 + y^2 + z^2 - 9 = 0 $ and $ z - x^2 - y^2 + 3 = 0 $ at the point (2, 1, 2) is

The tangent plane to the surface $ z = \sqrt{x^2 + 3y^2} $ at (1, 1, 2) is given by

A point object $O$ is placed in front of a glass rod having spherical end of radius of curvature $30\, cm$. The image would be formed at .

If the Planck?s constant $h = 6.6 \times 10^{-34} \, Js$, the de Broglie wavelength of a particle having momentum of $3.3 \times 10^{-24} \, kg \, ms^{-1}$ will be

In a binomial distribution, the mean is $4$ and variance is $3$. Then its mode is :

The ratio of the specific heats of a gas is $\frac{C_p}{C_v} = 1.66$, then the gas may be

Let $ a $ be a positive real number. If $ f $ is a continuous and even function defined on the interval $ [-a, a] $, then $ \int_{-a}^{a} \dfrac{f(x)}{1+e^x} dx $ is equal to

If $ \{ v_1, v_2, v_3 \} $ is a linearly independent set of vectors in a vector space over $ \mathbb{R} $, then which one of the following sets is also linearly independent?

Let $ a_n = \dfrac{b_{n+1}}{b_n} $, where $ b_1 = 1 $, $ b_2 = 1 $, and $ b_{n+2} = b_n + b_{n+1} $, $ n \in \mathbb{N} $. Then $ \lim_{n \to \infty} a_n $ is

Which one of the following is TRUE?

Suppose $ \alpha, \beta, \gamma \in \mathbb{R} $. Consider the following system of linear equations.
\[ x + y + z = \alpha, x + \beta y + z = \gamma, x + y + \alpha z = \beta. \] If this system has at least one solution, then which of the following statements is (are) TRUE?

Let $ m, n \in \mathbb{N}, m < n, P \in M_{n \times m}(\mathbb{R}), Q \in M_{m \times n}(\mathbb{R}) $. Which of the following is (are) NOT possible?

If $ \vec{\mathbf{F}} (x, y, z) = (2x + 3yz) \hat{i} + (3xz + 2y) \hat{j} + (3xy + 2z) \hat{k} $ for $ (x, y, z) \in \mathbb{R}^3 $, then which among the following is (are) TRUE?

Which of the following subsets of $ \mathbb{R} $ is (are) connected?

Let $ S $ be a subset of $ \mathbb{R} $ such that 2018 is an interior point of $ S $. Which of the following is (are) TRUE?