List of practice Questions

Let \( X \) be a random variable with the probability mass function

\[P(X = n) = \begin{cases} \dfrac{1}{10}, & n = 1,2,\dots,10, \\ 0, & \text{otherwise}.\end{cases}\]

Then \( E(\max\{X, 5\}) \) equals ............

Let \( X \) be a sample observation from \( U(\theta, \theta^2) \) distribution, where \( \theta \in \Theta = \{2, 3\} \) is the unknown parameter. For testing \[ H_0 : \theta = 2 \,\, \text{against} \,\, H_1 : \theta = 3, \] let \( \alpha \) and \( \beta \) be the size and power, respectively, of the test that rejects \( H_0 \) if and only if \( X \geq 3.5 \). Then \( \alpha + \beta \) equals ...............

A fair die is rolled four times independently. For \( i = 1, 2, 3, 4 \), define \[ Y_i = \begin{cases} 1, & \text{if 6 appears in the \( i \)-th throw}, \\ 0, & \text{otherwise}. \end{cases} \] Then \( P(\max\{Y_1, Y_2, Y_3, Y_4\} = 1) \) equals ............

Let \( \mathbb{C}^* = \mathbb{C} \setminus \{0\} \) denote the group of non-zero complex numbers under multiplication. Suppose \[ Y_n = \{ z \in \mathbb{C} \mid z^n = 1 \}, n \in \mathbb{N}. \] Which of the following is (are) subgroup(s) of \( \mathbb{C}^* \)?

Let \( P \) and \( Q \) be two non-empty disjoint subsets of \( \mathbb{R} \). Which of the following is (are) FALSE?

Suppose \( f, g, h \) are permutations of the set \( \{ \alpha, \beta, \gamma, \delta \} \), where
f interchanges \( \alpha \) and \( \beta \) but fixes \( \gamma \) and \( \delta \),
g interchanges \( \beta \) and \( \gamma \) but fixes \( \alpha \) and \( \delta \),
h interchanges \( \gamma \) and \( \delta \) but fixes \( \alpha \) and \( \beta \).
Which of the following permutations interchange(s) \( \alpha \) and \( \delta \) but fix(es) \( \beta \) and \( \gamma \)?

The solution(s) of the differential equation \( \frac{dy}{dx} = (\sin 2x) y^{1/3}\) satisfying \(y(0) = 0\) is(are)

Let \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R} \) be defined by \( f(x) = x + \dfrac{1}{x^3} \). On which of the following interval(s) is \( f \) one-to-one?

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 \( (0, 0) \)?