Let $f : X \to X$ such that $f(f(x)) = x$ for all $x \in X$. Then,
a.
$f$ is one-to-one and onto.
Correct Answer
b.
$f$ is one-to-one, but not onto.
c.
$f$ is onto but not one-to-one.
d.
$f$ need not be either one-to-one or onto.
Step-by-Step AI Explanation
Verified Method
Given $f(f(x)) = x$ for all $x \in X$, we have $f \circ f = I_X$, which means $f$ is its own inverse ($f^{-1} = f$). Since any function having a two-sided inverse is a bijection, $f$ must be both one-to-one (injective) and onto (surjective). Thus, option A is correct.
Q22mcq
Let $V$ be the space of twice differentiable functions on $\mathbb{R}$ satisfying $f'' - 2f' + f = 0$. Define $T : V \to \mathbb{R}^2$ by $T(f) = (f'(0), f(0))$. Then $T$ is :
a.
one-to-one and onto
Correct Answer
b.
one-to-one but not onto
c.
onto but not one-to-one
d.
neither one-to-one nor onto
Step-by-Step AI Explanation
Verified Method
The differential equation is $f'' - 2f' + f = 0$. Its characteristic equation is $r^2 - 2r + 1 = 0$, giving repeated root $r = 1$. The general solution in $V$ is $f(x) = (c_1 + c_2 x)e^x$ for constants $c_1, c_2 \in \mathbb{R}$.
Then $f(0) = c_1$, and $f'(x) = c_2 e^x + (c_1 + c_2 x)e^x$, so $f'(0) = c_1 + c_2$.
Thus, $T(f) = (c_1 + c_2, c_1)$.
- If $T(f) = (0, 0)$, then $c_1 = 0$ and $c_1 + c_2 = 0 \implies c_2 = 0$, so $f(x) = 0$. Hence, $\ker(T) = \{0\}$, which means $T$ is one-to-one.
- Given any $(u, v) \in \mathbb{R}^2$, choosing $c_1 = v$ and $c_2 = u - v$ gives $T(f) = (u, v)$, so $T$ is onto.
Therefore, $T$ is an isomorphism, which is both one-to-one and onto. Thus, option A is correct.
Q23mcq
The limit $\lim_{x \to 0} \frac{1}{x} \int_x^{2x} e^{-t^2} dt$
a.
does not exist
b.
is infinite
c.
exists and equals 1
Correct Answer
d.
exists and equals 0
Step-by-Step AI Explanation
Verified Method
Let $F(x) = \int_x^{2x} e^{-t^2} dt$. As $x \to 0$, $F(x) \to \int_0^0 e^{-t^2} dt = 0$. The limit is of the form $\frac{0}{0}$. Applying L'Hôpital's Rule and using Leibniz's rule for differentiation of an integral:
$$\lim_{x \to 0} \frac{\frac{d}{dx} \int_x^{2x} e^{-t^2} dt}{\frac{d}{dx}(x)} = \lim_{x \to 0} \frac{2 e^{-(2x)^2} - e^{-x^2}}{1} = 2 e^0 - e^0 = 2 - 1 = 1.$$
Thus, the limit exists and equals 1, making option C correct.
Q24mcq
The sum of the series: $\frac{1}{1!} + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \cdots$ equals
a.
$e$
b.
$\frac{e}{2}$
c.
$\frac{3e}{2}$
Correct Answer
d.
$1 + \frac{e}{2}$
Step-by-Step AI Explanation
Verified Method
The $n$-th term of the series is given by:
$$a_n = \frac{1 + 2 + \dots + n}{n!} = \frac{n(n+1)}{2 \cdot n!} = \frac{n+1}{2(n-1)!}$$
for $n \geq 1$. We can rewrite the numerator as $(n - 1) + 2$:
$$a_n = \frac{(n-1) + 2}{2(n-1)!} = \frac{1}{2} \frac{n-1}{(n-1)!} + \frac{1}{(n-1)!} = \frac{1}{2(n-2)!} + \frac{1}{(n-1)!}$$
(where the term $\frac{1}{2(n-2)!}$ is zero for $n=1$).
Summing from $n = 1$ to $\infty$:
$$\sum_{n=1}^\infty a_n = \frac{1}{2} \sum_{n=2}^\infty \frac{1}{(n-2)!} + \sum_{n=1}^\infty \frac{1}{(n-1)!} = \frac{1}{2} e + e = \frac{3e}{2}.$$
Hence, option C is correct.
Q25mcq
Which of the following subsets of $\mathbb{R}^n$ is compact (with respect to the usual topology of $\mathbb{R}^n$)?
By the Heine-Borel theorem, a subset of $\mathbb{R}^n$ is compact with respect to the usual topology if and only if it is closed and bounded.
- Option A: The set is an open cube, hence not closed (not compact).
- Option B: The set is a hyperplane through the origin, which is unbounded (not compact).
- Option C: The set is the non-negative orthant, which is unbounded (not compact).
- Option D: The set is $\prod_{i=1}^n [1, 2^i]$, which is a product of closed and bounded intervals in $\mathbb{R}$. Hence it is closed and bounded in $\mathbb{R}^n$, and thus compact.
Therefore, option D is correct.
Q26mcq
A polynomial of odd degree with real coefficient must have
a.
at least one real root.
Correct Answer
b.
no real root.
c.
only real roots.
d.
at least one root which is not real.
Step-by-Step AI Explanation
Verified Method
Let $P(x)$ be a polynomial of odd degree $n$ with real coefficients. As $x \to \infty$ and $x \to -\infty$, $P(x)$ approaches opposite signs (one is $+\infty$ and the other is $-\infty$) because the highest-degree term dominates. Since real polynomials are continuous on $\mathbb{R}$, by the Intermediate Value Theorem, $P(x)$ must cross the $x$-axis at least once. Hence, it has at least one real root. Alternatively, complex non-real roots of real polynomials occur in conjugate pairs, so an odd-degree polynomial must leave at least one root real. Therefore, option A is correct.
Q27mcq
Let $A, B$ be $n \times n$ matrices. Which of the following equals $\text{trace}(A^2 B^2)$?
a.
$(\text{trace}(AB))^2$
b.
$\text{trace}(AB^2 A)$
Correct Answer
c.
$\text{trace}((AB)^2)$
d.
$\text{trace}(BABA)$
Step-by-Step AI Explanation
Verified Method
Using the cyclic property of the trace, $\text{trace}(XY) = \text{trace}(YX)$ for any matrices $X$ and $Y$ of compatible dimensions. Let $X = A$ and $Y = AB^2$. Then:
$$\text{trace}(A^2 B^2) = \text{trace}(A(AB^2)) = \text{trace}((AB^2)A) = \text{trace}(AB^2 A).$$
None of the other options generally equal $\text{trace}(A^2 B^2)$. Thus, option B is correct.
Q28mcq
Let $A$ be an $m \times n$ matrix of rank $n$ with real entries. Choose the correct statement.
a.
$Ax = b$ has a solution for any $b$.
b.
$Ax = 0$ does not have a solution.
c.
If $Ax = b$ has a solution, then it is unique.
Correct Answer
d.
$y' A = 0$ for some non-zero $y$, where $y'$ denotes the transpose of the vector $y$.
Step-by-Step AI Explanation
Verified Method
The matrix $A$ is of size $m \times n$ with rank $n$. This means the columns of $A$ are linearly independent, so the null space of $A$ contains only the zero vector, i.e., $\ker(A) = \{0\}$.
- If $Ax = b$ has a solution, say $x_1$, and another solution $x_2$, then $A(x_1 - x_2) = 0$, which implies $x_1 - x_2 \in \ker(A) = \{0\}$, so $x_1 = x_2$. Thus, the solution is unique whenever it exists. Hence, statement C is correct.
- A is false because $Ax = b$ only has a solution if $b$ is in the column space of $A$, and if $m > n$, the column space is a proper subspace of $\mathbb{R}^m$.
- B is false because $Ax = 0$ always has the trivial solution $x = 0$.
- D is not necessarily true if $m = n$, because then $A$ has rank $n$, so $A^T$ is invertible and $A^T y = 0$ implies $y = 0$.
Q29mcq
The row space of a $20 \times 50$ matrix $A$ has dimension 13. What is the dimension of the space of solutions of $Ax = 0$?
a.
7
b.
13
c.
33
d.
37
Correct Answer
Step-by-Step AI Explanation
Verified Method
By definition, the dimension of the row space of a matrix $A$ is the rank of $A$, so $\text{rank}(A) = 13$. The matrix $A$ is of size $20 \times 50$, meaning it represents a linear map from $\mathbb{R}^{50}$ to $\mathbb{R}^{20}$ (i.e., the number of columns is $n = 50$). By the Rank-Nullity Theorem:
$$\text{dim}(\ker(A)) + \text{rank}(A) = n$$
$$\text{dim}(\text{space of solutions of } Ax = 0) = 50 - 13 = 37$$
Thus, option D is correct.
Q30mcq
Let $T$ be a $4 \times 4$ real matrix such that $T^4 = 0$. Let $k_i := \dim \ker T^i$ for $1 \le i \le 4$. Which of the following is NOT a possibility for the sequence $k_1 \le k_2 \le k_3 \le k_4$?
a.
$3 \le 4 \le 4 \le 4$
b.
$1 \le 3 \le 4 \le 4$
Correct Answer
c.
$2 \le 4 \le 4 \le 4$
d.
$2 \le 3 \le 4 \le 4$
Step-by-Step AI Explanation
Verified Method
Let $d_i = k_i - k_{i-1}$ with $k_0 = 0$. The sequence of differences $d_i = \dim \ker T^i - \dim \ker T^{i-1}$ must be non-increasing, i.e., $d_1 \ge d_2 \ge d_3 \ge d_4$.
For option B: the sequence is $1 \le 3 \le 4 \le 4$.
Here $k_1 = 1$, $k_2 = 3$, $k_3 = 4$, $k_4 = 4$.
The differences are:
$d_1 = 1 - 0 = 1$
$d_2 = 3 - 1 = 2$
Since $d_1 < d_2$ ($1 < 2$), this violates the property that the sequence of differences of kernel dimensions must be non-increasing (which corresponds to the dual partition of the Jordan block sizes). Therefore, $1 \le 3 \le 4 \le 4$ is NOT a possibility, making option B the correct answer.
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Given a $4 \times 4$ real matrix $A$, let $T : \mathbb{R}^4 \to \mathbb{R}^4$ be the linear transformation defined by $Tv = Av$, where we think of $\mathbb{R}^4$ as the set of real $4 \times 1$ matrices. For which choices of $A$ given below, do $\text{Image } (T)$ and $\text{Image } (T^2)$ have respective dimensions 2 and 1? [* denotes a non-zero entry]
Which of the following is a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$?
(1) $f \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 4 \\ x+y \end{pmatrix}$
(2) $g \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} xy \\ x+y \end{pmatrix}$
(3) $h \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} z-x \\ x+y \end{pmatrix}$
a.
only $f$
b.
only $g$
c.
only $h$
d.
all the transformation $f$, $g$ and $h$
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Let $f$ be a real valued harmonic function on $\mathbb{C}$, that is, $f$ satisfies the equation $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$. Define the functions:
$$g = \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y}; \quad h = \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y}$$
Then,
a.
$g$ and $h$ are both holomorphic functions.
b.
$g$ is holomorphic, but $h$ need not be holomorphic.
c.
$h$ is holomorphic, but $g$ need not be holomorphic.
d.
both $g$ and $h$ are identically equal to the zero function.
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Let $D$ be the set of tuples $(w_1, \dots, w_{10})$, where $w_i \in \{1, 2, 3\}$, $1 \le i \le 10$ and $w_i + w_{i+1}$ is an even number for each $i$ with $1 \le i \le 9$. Then the number of elements in $D$ is :
a.
$2^{11} + 1$
b.
$2^{10} + 1$
c.
$3^{10} + 1$
d.
$3^{11} + 1$
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Let, for each $n \ge 1$, $C_n$ be the open disc in $\mathbb{R}^2$, with centre at the point $(n, 0)$ and radius equal to $n$. Then, $C = \bigcup_{n \ge 1} C_n$ is:
a.
$\{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } |y| < x\}$
b.
$\{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } |y| < 2x\}$
c.
$\{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } |y| < 3x\}$
d.
$\{(x, y) \in \mathbb{R}^2 : x > 0\}$
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Let $a, b \in \mathbb{R}$ be such that $a^2 + b^2 \ne 0$. Then the Cauchy problem
$$a \frac{\partial u}{\partial x} + b \frac{\partial u}{\partial y} = 1; \quad x, y \in \mathbb{R}$$
$$u(x, y) = x \text{ on } ax + by = 1$$
a.
has more than one solution if either $a$ or $b$ is zero.
b.
has no solution.
c.
has a unique solution.
d.
has infinitely many solutions.
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Let $f : \mathbb{R} \to \mathbb{R}$ be a polynomial of the form $f(x) = a_0 + a_1 x + a_2 x^2$ with $a_0, a_1, a_2 \in \mathbb{R}$ and $a_2 \ne 0$, if
$$E_1 = \int_{-1}^1 f(x) dx - [f(-1) + f(1)], \quad E_2 = \int_{-1}^1 f(x) dx - \frac{1}{2}(f(-1) + 2f(0) + f(1))$$
and $|x|$ is the absolute value of $x \in \mathbb{R}$, then —
a.
$|E_1| < |E_2|$
b.
$|E_1| = 2|E_2|$
c.
$|E_1| = 4|E_2|$
d.
$|E_1| = 8|E_2|$
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Let $y(x)$ be a continuous solution of the initial value problem
$$y' + 2y = f(x), \quad y(0) = 0, \quad \text{where } f(x) = \begin{cases} 1 & ; & 0 \le x \le 1 \\ 0 & ; & x > 1 \end{cases}$$
Then $y\left(\frac{3}{2}\right)$ is equal to :
a.
$\frac{\sinh (1)}{e^3}$
b.
$\frac{\cosh (1)}{e^3}$
c.
$\frac{\sinh (1)}{e^2}$
d.
$\frac{\cosh (1)}{e^2}$
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Consider two weightless, inextensible rods $AB$ and $BC$, suspended at $A$ and joined by a flexible joint at $B$. Then the degrees of freedom of the system is :
a.
3
b.
4
c.
5
d.
6
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Consider the initial value problem
$$\frac{\partial u}{\partial x} + 2\frac{\partial u}{\partial y} = 0, \quad u(0, y) = 4e^{-2y}$$
Then the value of $u(1, 1)$ is :
a.
$4e^{-2}$
b.
$4e^2$
c.
$2e^4$
d.
$4e^4$
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Let $X, Y$ be independent random variables and let $Z = \frac{X - Y}{2} + 3$. If $X$ has characteristic function $\psi$, then $Z$ has characteristic function $\theta$ where
Assume that $X \sim \text{Binomial }(n, p)$ for some $n \ge 1$ and $0 < p < 1$ and $Y \sim \text{Poisson }(\lambda)$ for some $\lambda > 0$. Suppose $E[X] = E[Y]$. Then
a.
$Var(X) = Var(Y)$
b.
$Var(X) < Var(Y)$
c.
$Var(Y) < Var(X)$
d.
$Var(X)$ may be larger or smaller than $Var(Y)$ depending on the values of $n$, $p$ and $\lambda$.
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Let $X_1, X_2, \dots, X_7$ be a random sample from $N(\mu, \sigma^2)$, where $\mu$ and $\sigma^2$ are unknown. Consider the problem of testing $H_0 : \mu = 2$ against $H_1 : \mu > 2$. Suppose the observed values of $x_1, x_2, \dots, x_7$ are $1.2, 1.3, 1.7, 1.8, 2.1, 2.3, 2.7$. If we use the Uniformly Most Powerful test, which of the following is true ?
a.
$H_0$ is accepted both at $5\%$ and $1\%$ levels of significance.
b.
$H_0$ is rejected both at $5\%$ and $1\%$ levels of significance.
c.
$H_0$ is rejected at $5\%$ level of significance, but accepted at $1\%$ level of significance.
d.
$H_0$ is rejected at $1\%$ level of significance, but accepted at $5\%$ level of significance.
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Suppose $X_i | \theta_i \sim N(\theta_i, \sigma^2)$, $i = 1, 2$ are independently distributed. Under the prior distribution, $\theta_1$ and $\theta_2$ are i.i.d. $N(\mu, \tau^2)$, where $\sigma^2$, $\mu$ and $\tau^2$ are known. Then which of the following is true about the marginal distribution of $X_1$ and $X_2$ ?
a.
$X_1$ and $X_2$ are i.i.d. $N(\mu, \tau^2 + \sigma^2)$.
b.
$X_1$ and $X_2$ are not normally distributed.
c.
$X_1$ and $X_2$ are $N(\mu, \tau^2 + \sigma^2)$ but they are not independent.
d.
$X_1$ and $X_2$ are normally distributed but are not identically distributed.
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Consider the model $Y_i = i\beta + \varepsilon_i, i = 1, 2, 3$ where $\varepsilon_1, \varepsilon_2, \varepsilon_3$ are independent with mean $0$ and variance $\sigma^2, 2\sigma^2, 3\sigma^2$ respectively. Which of the following is the best linear unbiased estimate of $\beta$ ?
Let $Y = (Y_1, ..., Y_n)'$ have the multivariate normal distribution $N_n(0, I)$. Which of the following is the covariance matrix of the conditional distribution of $Y$ given $\sum_{i=1}^n Y_i$ ?
[$1$ denotes the $n \times 1$ vector with all elements $1$]
a.
$I$
b.
$I + \frac{11'}{n}$
c.
$I - \frac{11'}{n}$
d.
$\frac{11'}{n}$
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Suppose there are $k$ strata of $N = kM$ units each with size $M$. Draw a sample of size $n_i$ with replacement from the $i$-th stratum and denote by $\overline{y}_i$ the sample mean of the study variable selected in the $i$-th stratum, $i = 1, 2, ..., k$. Define $\overline{y}_s = \frac{1}{k}\sum_{i=1}^k \overline{y}_i$ and $\overline{y}_w = \frac{\sum_{i=1}^k n_i \overline{y}_i}{n}$. Which of the following is necessarily true ?
a.
$\overline{y}_s$ is unbiased but $\overline{y}_w$ is not unbiased for the population mean.
b.
$\overline{y}_s$ is not unbiased but $\overline{y}_w$ is unbiased for the population mean.
c.
Both $\overline{y}_s$ and $\overline{y}_w$ are unbiased for the population mean.
d.
Neither $\overline{y}_s$ nor $\overline{y}_w$ is unbiased for the population mean.
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Consider a Balanced Incomplete Block Design (BIBD) with parameters $(b, k, v, r, \lambda)$. Which of the following cannot possibly by the parameters of a BIBD ?
a.
$(b-1, k-\lambda, b-k, k, \lambda)$
b.
$(b, v-k, v, b-r, b-2r+\lambda)$
c.
$\left(\frac{v(v-1)}{2}, 2, v, v-1, 1\right)$
d.
$(k, b, r, v, \lambda-1)$
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Let $X_1, X_2, ..., X_n$ be independent and identically distributed random variables having an exponential distribution with mean $1/\lambda$. Let $S_n = X_1 + X_2 + \dots + X_n$ and $N = \inf \{n \ge 1: S_n > 1\}$. Then $Var(N)$ equals
a.
$1$
b.
$\lambda$
c.
$\lambda^2$
d.
$\infty$
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Which of the following sets in $\mathbb{R}^2$ have positive Lebesgue measure ?
$$\left[\text{For two sets } A, B \subseteq \mathbb{R}^2, A + B = \{a + b \mid a \in A, b \in B\}\right]$$
a.
$S = \{(x, y) \mid x^2 + y^2 = 1\}$
b.
$S = \{(x, y) \mid x^2 + y^2 < 1\}$
c.
$S = \{(x, y) \mid x = y\} + \{(x, y) \mid x = -y\}$
d.
$S = \{(x, y) \mid x = y\} + \{(x, y) \mid x = y\}$
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Let $f$ be a bounded function on $\mathbb{R}$ and $a \in \mathbb{R}$. For $\delta > 0$, let $\omega(a, \delta) = \sup |f(x) - f(a)|$, $x \in [a - \delta, a + \delta]$. Then,
a.
$\omega(a, \delta_1) \le \omega(a, \delta_2)$ if $\delta_1 \le \delta_2$
b.
$\lim_{\delta \to 0^+} \omega(a, \delta) = 0$ for all $a \in \mathbb{R}$
c.
$\lim_{\delta \to 0^+} \omega(a, \delta)$ need not exist
d.
$\lim_{\delta \to 0^+} \omega(a, \delta) = 0$ if and only if $f$ is continuous at $a$
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Let $\{a_0, a_1, a_2, \dots\}$ be a sequence of real numbers. For any $k \ge 1$, let $s_n = \sum_{k=0}^n a_{2k}$. Which of the following statements are correct?
a.
If $\lim_{n \to \infty} s_n$ exists, then $\sum_{m=0}^\infty a_m$ exists.
b.
If $\lim_{n \to \infty} s_n$ exists, then $\sum_{m=0}^\infty a_m$ need not exist.
c.
If $\sum_{m=0}^\infty a_m$ exists, then $\lim_{n \to \infty} s_n$ exists.
d.
If $\sum_{m=0}^\infty a_m$ exists, then $\lim_{n \to \infty} s_n$ need not exist.
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Let $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be the function $F(x, y) = \langle Ax, y \rangle$, where $\langle , \rangle$ is the standard inner product of $\mathbb{R}^n$ and $A$ is a $n \times n$ real matrix. Here, $D$ denotes the total derivative. Which of the following statements are correct?
a.
$(DF(x, y))(u, v) = \langle Au, y \rangle + \langle Ax, v \rangle$
b.
$(DF(x, y))(0, 0) = 0$
c.
$DF(x, y)$ may not exist for some $(x, y) \in \mathbb{R}^n \times \mathbb{R}^n$
d.
$DF(x, y)$ does not exist at $(x, y) = (0, 0)$
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Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function such that $\int_{\mathbb{R}^n} |f(x)| dx < \infty$. Let $A$ be a real $n \times n$ invertible matrix and for $x, y \in \mathbb{R}^n$, let $\langle x, y \rangle$ denote the standard inner product in $\mathbb{R}^n$. Then $\int_{\mathbb{R}^n} f(Ax) e^{i \langle y, x \rangle} dx =$
a.
$\int_{\mathbb{R}^n} f(x) e^{i \langle (A^{-1})^T y, x \rangle} \frac{dx}{|\det A|}$
b.
$\int_{\mathbb{R}^n} f(x) e^{i \langle A^T y, x \rangle} \frac{dx}{|\det A|}$
c.
$\int_{\mathbb{R}^n} f(x) e^{i \langle (A^T)^{-1} y, x \rangle} dx$
d.
$\int_{\mathbb{R}^n} f(x) e^{i \langle A^{-1} y, x \rangle} \frac{dx}{|\det A|}$
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An $n \times n$ complex matrix $A$ satisfies $A^k = I_n$, the $n \times n$ identity matrix, where $k$ is a positive integer $>1$. Suppose $1$ is not an eigenvalue of $A$. Then which of the following statements are necessarily true?
a.
$A$ is diagonalizable
b.
$A + A^2 + \dots + A^{k-1} = 0$, the $n \times n$ zero matrix.
Let $S$ be the set of $3 \times 3$ real matrices $A$ with $A^T A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$. Then the set $S$ contains
a.
a nilpotent matrix.
b.
a matrix of rank one.
c.
a matrix of rank two.
d.
a non-zero skew-symmetric matrix.
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Let $S : \mathbb{R}^n \to \mathbb{R}^n$ be given by $S(v) = \alpha v$ for a fixed $\alpha \in \mathbb{R}$, $\alpha \ne 0$. Let $T : \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation such that $B = \{v_1, \dots, v_n\}$ is a set of linearly independent eigenvectors of $T$. Then,
a.
The matrix of $T$ with respect to $B$ is diagonal.
b.
The matrix of $(T - S)$ with respect to $B$ is diagonal.
c.
The matrix of $T$ with respect to $B$ is not necessarily diagonal, but is upper triangular.
d.
The matrix of $T$ with respect to $B$ is diagonal but the matrix of $(T - S)$ with respect to $B$ is not diagonal.
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Let $A = \begin{pmatrix} a & b & c \\ 0 & a & d \\ 0 & 0 & a \end{pmatrix}$ be a $3 \times 3$ matrix where $a, b, c, d$ are integers. Then, we must have:
a.
If $a \neq 0$, there is a polynomial $p \in \mathbb{Q}[x]$ such that $p(A)$ is the inverse of $A$.
b.
For each polynomial $q \in \mathbb{Z}[x]$, the matrix $q(A) = \begin{pmatrix} q(a) & q(b) & q(c) \\ 0 & q(a) & q(d) \\ 0 & 0 & q(a) \end{pmatrix}$
c.
If $A^n = 0$ for some positive integer $n$, then $A^3 = 0$.
d.
$A$ commutes with every matrix of the form \begin{pmatrix} a' & 0 & c' \\ 0 & a' & 0 \\ 0 & 0 & a' \end{pmatrix}.
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Let $u$ be a real $n \times 1$ vector satisfying $u'u = 1$, where $u'$ is the transpose of $u$. Define $A = I - 2uu'$, where $I$ is the $n$-th order identity matrix. Which of the following statements are true ?
a.
$A$ is singular
b.
$A^2 = A$
c.
$\text{Trace}\,(A) = n - 2$
d.
$A^2 = I$
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Let $p$ be a polynomial in 1-complex variable. Suppose all zeros of $p$ are in the upper half plane $H = \{z \in \mathbb{C} \mid \text{Im}\,(z) > 0\}$. Then,
a.
$\text{Im}\,\frac{p'(z)}{p(z)} > 0$ for $z \in \mathbb{R}$
b.
$\text{Re}\,i\frac{p'(z)}{p(z)} < 0$ for $z \in \mathbb{R}$
c.
$\text{Im}\,\frac{p'(z)}{p(z)} > 0$ for $z \in \mathbb{C}$, with $\text{Im}\,z < 0$
d.
$\text{Im}\,\frac{p'(z)}{p(z)} > 0$ for $z \in \mathbb{C}$, with $\text{Im}\,z > 0$
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Let $\sigma : \{1, 2, 3, 4, 5\} \rightarrow \{1, 2, 3, 4, 5\}$ be a permutation (one-to-one and onto function) such that $\sigma^{-1}(j) \le \sigma(j)$ $\forall\ j, 1 \le j \le 5$. Then which of the following are true ?
a.
$\sigma \circ \sigma(j) = j$ for all $j, 1 \le j \le 5$
b.
$\sigma^{-1}(j) = \sigma(j)$ for all $j, 1 \le j \le 5$
c.
The set $\{k : \sigma(k) \ne k\}$ has an even number of elements.
d.
The set $\{k : \sigma(k) = k\}$ has an odd number of elements.
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Consider the set $\mathbb{Z}$ of integers, with the topology $\tau$ in which a subset is closed if and only if it is empty, or $\mathbb{Z}$, or finite. Which of the following statements are true ?
a.
$\tau$ is the subspace topology induced from the usual topology on $\mathbb{R}$.
b.
$\mathbb{Z}$ is compact in the topology $\tau$.
c.
$\mathbb{Z}$ is Hausdorff in the topology $\tau$.
d.
Every infinite subset of $\mathbb{Z}$ is dense in the topology $\tau$.
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For the initial value problem
$$\frac{dy}{dx} = y^2 + \cos^2 x, \ x > 0$$
$$y(0) = 0$$
The largest interval of existence of the solution predicted by Picard's theorem is :
a.
$[0, 1]$
b.
$[0, 1/2]$
c.
$[0, 1/3]$
d.
$[0, 1/4]$
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Let $P$ be a continuous function on $\mathbb{R}$ and $W$ the Wronskian of two linearly independent solutions $y_1$ and $y_2$ of the ODE.
$$\frac{d^2 y}{dx^2} + (1 + x^2)\frac{dy}{dx} + P(x) y = 0, \ x \in \mathbb{R}$$
Let $W(1) = a, W(2) = b$ and $W(3) = c$, then
a.
$a < 0$ and $b > 0$
b.
$a < b < c$ or $a > b > c$
c.
$\frac{a}{|a|} = \frac{b}{|b|} = \frac{c}{|c|}$
d.
$0 < a < b$ and $b > c > 0$
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A particle of mass $m$ is contrained to move on the surface of a cylinder $x^2 + y^2 = a^2$ under the influence of a force directed towards the origin and proportional to the distance of the particle from the origin. Then,
a.
the angular momentum about z-axis is constant.
b.
the angular momentum about z-axis is not constant.
c.
the motion is simple harmonic in z-direction.
d.
the motion is not simple harmonic in z-direction.
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The function $G(x, \zeta) = \begin{cases} a + b\log \zeta, & 0 < x \le \zeta \\ c + d\log x, & \zeta \le x \le 1 \end{cases}$ is a Green's function for $x y'' + y' = 0$, subject in $y$ being bounded as $x \to 0$ and $y(1) = y'(1)$, if
a.
$a = 1, b = 1, c = 1, d = 1$
b.
$a = 1, b = 0, c = 1, d = 0$
c.
$a = 0, b = 1, c = 0, d = 1$
d.
$a = 0, b = 0, c = 0, d = 0$
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