List of practice Questions

For $ x > -\dfrac{1}{2} $, let $ f_1(x) = \dfrac{2x}{1+2x} $, $ f_2(x) = \log_e(1 + 2x) $ and $ f_3(x) = 2x $. Then which one of the following is TRUE?

Let $ f : \mathbb{R} \to \mathbb{R} $ be a function and let $ J $ be a bounded open interval in $ \mathbb{R} $. Define
\[ W(f, J) = \sup \{ f(x) | x \in J \} - \inf \{ f(x) | x \in J \} \] Which one of the following is FALSE?

Consider the group $ \mathbb{Z}^2 = \{ (a, b) | a, b \in \mathbb{Z} \} $ under component-wise addition. Then which of the following is a subgroup of $ \mathbb{Z}^2 $?

Let $ I $ denote the $ 4 \times 4 $ identity matrix. If the roots of the characteristic polynomial of a $ 4 \times 4 $ matrix $ M $ are $ \pm \sqrt{\dfrac{1 \pm \sqrt{5}}{2}} $, then $ M^8 $ is

Let $ G $ be a group satisfying the property that $ f: G \to \mathbb{Z}_{221} $ is a homomorphism implies $ f(g) = 0 $, $ \forall g \in G $. Then a possible group $ G $ is

A particular integral of the differential equation
\[ y'' + 3y' + 2y = e^{e^x} \] is

An integrating factor of the differential equation
\[ \left( y + \frac{1}{3} y^3 + \frac{1}{2} x^2 \right) \, dx + \frac{1}{4} (x + xy^2) \, dy = 0 \] is

Let $ y(x) $ be the solution of the differential equation $ \frac{dy}{dx} + y = f(x) $, for $ x \geq 0, y(0) = 0 $, where
\[ f(x) = \begin{cases} 2, & 0 \leq x < 1 \\ 0, & x \geq 1 \end{cases} \] Then $ y(x) $ is

Let $ U, V $ and $ W $ be finite dimensional real vector spaces, $ T: U \to V $, $ S: V \to W $ and $ P: W \to U $ be linear transformations. If range $ (ST) = $ nullspace $ (P) $, nullspace $ (ST) = $ range $ (P) $ and rank $ (T) = $ rank $ (S) $, then which one of the following is TRUE?

If $ \hat{F}(x, y) = (3x - 8y) \hat{i} + (4y - 6xy) \hat{j} $ for $ (x, y) \in \mathbb{R}^2 $, then $ \oint_C \vec{F} \cdot d \vec{r} $, where $ C $ is the boundary of the triangular region bounded by the lines $ x = 0 $, $ y = 0 $, and $ x + y = 1 $ oriented in the anti-clockwise direction, is

Consider the region $ D $ in the yz-plane bounded by the line $ y = \frac{1}{2} $ and the curve $ y^2 + z^2 = 1 $, where $ y \geq 0 $. If the region $ D $ is revolved about the $ z $-axis in $ \mathbb{R}^3 $, then the volume of the resulting solid is

Let $ a, b \in \mathbb{R} $ and let $ f: \mathbb{R} \to \mathbb{R} $ be a thrice differentiable function. If $ z = e^u f(v) $, where $ u = ax + by $ and $ v = ax - by $, then which one of the following is TRUE?

Let $ f(x, y) = \begin{cases} \dfrac{xy}{(x^2 + y^2)^{\alpha}}, & (x, y) \neq (0,0) \\ 0, & (x, y) = (0,0) \end{cases} $ Then which one of the following is TRUE for $ f $ at the point $ (0, 0) $?

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