CSIR NET · Mathematical Sciences
June
2019
·
77 questions
·
First 10 Solutions Free
Questions 1–10 have full step-by-step solutions visible.
Sign in to unlock all 77 →
Q21
mcq
Which of the following sets is uncountable ?
a.
$\left\{x \in \mathbb{R} \mid \log(x) = \frac{p}{q} \text{ for some } p, q \in \mathbb{N}\right\}$
b.
$\left\{x \in \mathbb{R} \mid (\cos(x))^n + (\sin(x))^n = 1 \text{ for some } n \in \mathbb{N}\right\}$
Correct Answer
c.
$\left\{x \in \mathbb{R} \mid x = \log\left(\frac{p}{q}\right) \text{ for some } p, q \in \mathbb{N}\right\}$
d.
$\left\{x \in \mathbb{R} \mid \cos(x) = \frac{p}{q} \text{ for some } p, q \in \mathbb{N}\right\}$
Step-by-Step AI Explanation
Verified Method
Let us analyze each set:
* **Set A**: The condition is $\log(x) = \frac{p}{q}$, which gives $x = e^{p/q}$ for $p, q \in \mathbb{N}$. Since $\mathbb{N} \times \mathbb{N}$ is countable, the set of values of $x$ is at most countable. Thus, Set A is countable.
* **Set B**: For $n = 2$, the equation becomes $\cos^2(x) + \sin^2(x) = 1$. By the fundamental Pythagorean trigonometric identity, this equality holds for **all** $x \in \mathbb{R}$. Therefore, the set contains the entire real line $\mathbb{R}$, which is uncountable.
* **Set C**: The set consists of elements of the form $x = \log(p/q)$ for $p, q \in \mathbb{N}$. Since the rational numbers $\frac{p}{q}$ form a countable set and the logarithm function is single-valued, this set is countable.
* **Set D**: For each fixed $p, q \in \mathbb{N}$, the equation $\cos(x) = \frac{p}{q}$ has at most countably many solutions (each period $[2k\pi, 2(k+1)\pi)$ contains at most 2 solutions). A countable union of countable sets is countable, so Set D is countable.
Hence, the only uncountable set is **B**.
Q22
mcq
Consider a sequence: $\{a_n\}$, $a_n = (-1)^n\left(\frac{1}{2} - \frac{1}{n}\right)$. Let $b_n = \sum_{k=1}^n a_k \ \forall \ n \in \mathbb{N}$. Then which of the following is true ?
a.
$\lim_{n \to \infty} b_n = 0$
b.
$\limsup_{n \to \infty} b_n > \frac{1}{2}$
c.
$\liminf_{n \to \infty} b_n < -\frac{1}{2}$
d.
$0 \le \liminf_{n \to \infty} b_n \le \limsup_{n \to \infty} b_n \le \frac{1}{2}$
No explanation saved for this question yet.
Q23
mcq
Which of the following is true ?
a.
$\sum_{n=1}^\infty \frac{(-1)^n}{n}$ does not converge
b.
$\sum_{n=1}^\infty \frac{1}{n}$ converges
c.
$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(m+n)^2}$ converges
d.
$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(m+n)^2}$ diverges
Correct Answer
Step-by-Step AI Explanation
Verified Method
Let us test each option:
- Option A: $\sum_{n=1}^\infty \frac{(-1)^n}{n}$ converges by the alternating series test (Leibniz's test). Thus, A is false.
- Option B: The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ is well known to diverge. Thus, B is false.
- Options C and D: Consider the double series $S = \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(m+n)^2}$. Let $k = m+n$. For each fixed integer $k \ge 2$, the number of pairs of positive integers $(m,n)$ such that $m+n=k$ is $k-1$. Therefore, since all terms are positive, we can re-sum by the diagonals:
$$S = \sum_{k=2}^\infty \sum_{m+n=k} \frac{1}{(m+n)^2} = \sum_{k=2}^\infty \frac{k-1}{k^2}$$
Since $\frac{k-1}{k^2} \sim \frac{1}{k}$ as $k \to \infty$, and the harmonic series $\sum_{k=2}^\infty \frac{1}{k}$ diverges, the series $\sum_{k=2}^\infty \frac{k-1}{k^2}$ diverges by the limit comparison test.
Thus, the double series diverges, making D the correct statement.
Q24
mcq
For $n \in \mathbb{N}$, which of the following is true ?
a.
$\sqrt{n+1} - \sqrt{n} > \frac{1}{\sqrt{n}}$ for all, except possibly finitely many $n$
b.
$\sqrt{n+1} - \sqrt{n} < \frac{1}{\sqrt{n}}$ for all, except possibly finitely many $n$
Correct Answer
c.
$\sqrt{n+1} - \sqrt{n} > 1$ for all, except possibly finitely many $n$
d.
$\sqrt{n+1} - \sqrt{n} > 2$ for all, except possibly finitely many $n$
Step-by-Step AI Explanation
Verified Method
Rationalizing the expression gives:
$$\sqrt{n+1} - \sqrt{n} = \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}.$$
Since $\sqrt{n+1} + \sqrt{n} > \sqrt{n} + \sqrt{n} = 2\sqrt{n} > \sqrt{n}$ for all $n \in \mathbb{N}$, we have:
$$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}} < \frac{1}{2\sqrt{n}} < \frac{1}{\sqrt{n}}$$
for all $n \in \mathbb{N}$.
Thus, $\sqrt{n+1} - \sqrt{n} < \frac{1}{\sqrt{n}}$ holds for all $n$, so option B is correct.
Q25
mcq
Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function. Then which of the following is true ?
a.
$f$ is onto
b.
$f$ is either strictly decreasing or strictly increasing
Correct Answer
c.
there exists $x \in \mathbb{R}$ such that $f(x) = 1$
d.
$f$ is unbounded
Step-by-Step AI Explanation
Verified Method
A continuous injective function defined on an interval (in this case, $\mathbb{R}$) is strictly monotone, meaning it must be either strictly increasing or strictly decreasing.
Counterexamples for other options:
- For A and D: $f(x) = \arctan(x)$ is continuous and strictly increasing (thus one-one), but bounded with image $(-\pi/2, \pi/2)$, so it is neither onto $\mathbb{R}$ nor unbounded.
- For C: $f(x) = e^x + 2$ is continuous and strictly increasing, but $f(x) > 2$ for all $x$, so $f(x) = 1$ has no solution.
Q26
mcq
Let $g_n(x) = \frac{nx}{1+n^2x^2}, x \in [0, \infty)$. Which of the following is true as $n \to \infty$?
a.
$g_n \to 0$ pointwise but not uniformly
Correct Answer
b.
$g_n \to 0$ uniformly
c.
$g_n(x) \to x \;\forall\; x \in [0, \infty)$
d.
$g_n(x) \to \frac{x}{1+x^2} \;\forall\; x \in [0, \infty)$
Step-by-Step AI Explanation
Verified Method
For any fixed $x = 0$, $g_n(0) = 0 \to 0$.
For any fixed $x > 0$,
$$\lim_{n \to \infty} g_n(x) = \lim_{n \to \infty} \frac{nx}{1 + n^2 x^2} = \lim_{n \to \infty} \frac{x/n}{1/n^2 + x^2} = 0.$$
Thus, $g_n(x) \to 0$ pointwise on $[0, \infty)$.
To check for uniform convergence, we examine the maximum of $g_n(x)$ on $[0, \infty)$:
Taking the derivative, $g'_n(x) = \frac{n(1 + n^2 x^2) - nx(2n^2 x)}{(1 + n^2 x^2)^2} = \frac{n(1 - n^2 x^2)}{(1 + n^2 x^2)^2} = 0 \implies x = \frac{1}{n}$.
The value at the maximum is:
$$g_n\left(\frac{1}{n}\right) = \frac{1}{1 + 1} = \frac{1}{2}.$$
Since $\sup_{x \in [0, \infty)} |g_n(x) - 0| = \frac{1}{2} \not\to 0$ as $n \to \infty$, the convergence is not uniform.
Therefore, $g_n \to 0$ pointwise but not uniformly.
Q27
mcq
Consider the vector space $\mathbb{P}_n$ of real polynomials in $x$ of degree less than or equal to $n$. Define $T : \mathbb{P}_2 \to \mathbb{P}_3$ by
$$(Tf)(x) = \int_{0}^{x} f(t)dt + f'(x).$$
Then the matrix representation of $T$ with respect to the bases $\{1, x, x^2\}$ and $\{1, x, x^2, x^3\}$ is:
a.
$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1/2 & 0 \\ 0 & 2 & 0 & 1/3 \end{pmatrix}$
b.
$\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/3 \end{pmatrix}$
Correct Answer
c.
$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 2 & 0 \\ 0 & 1/2 & 0 & 1/3 \end{pmatrix}$
d.
$\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1/2 \\ 0 & 2 & 0 \\ 0 & 0 & 1/3 \end{pmatrix}$
Step-by-Step AI Explanation
Verified Method
Let the basis of $\mathbb{P}_2$ be $\mathcal{B}_1 = \{v_1, v_2, v_3\} = \{1, x, x^2\}$ and the basis of $\mathbb{P}_3$ be $\mathcal{B}_2 = \{w_1, w_2, w_3, w_4\} = \{1, x, x^2, x^3\}$.
The transformation is given by:
$$(Tf)(x) = \int_0^x f(t)\,dt + f'(x)$$
Applying $T$ to each basis vector in $\mathcal{B}_1$:
1. For $f(x) = 1$:
$$(T(1))(x) = \int_0^x 1\,dt + (1)' = x + 0 = 0\cdot 1 + 1\cdot x + 0\cdot x^2 + 0\cdot x^3$$
So, the first column is $\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$.
2. For $f(x) = x$:
$$(T(x))(x) = \int_0^x t\,dt + (x)' = \frac{x^2}{2} + 1 = 1\cdot 1 + 0\cdot x + \frac{1}{2}\cdot x^2 + 0\cdot x^3$$
So, the second column is $\begin{pmatrix} 1 \\ 0 \\ 1/2 \\ 0 \end{pmatrix}$.
3. For $f(x) = x^2$:
$$(T(x^2))(x) = \int_0^x t^2\,dt + (x^2)' = \frac{x^3}{3} + 2x = 0\cdot 1 + 2\cdot x + 0\cdot x^2 + \frac{1}{3}\cdot x^3$$
So, the third column is $\begin{pmatrix} 0 \\ 2 \\ 0 \\ 1/3 \end{pmatrix}$.
Assembling these columns gives the $4 \times 3$ matrix:
$$[T]_{\mathcal{B}_1}^{\mathcal{B}_2} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/3 \end{pmatrix}$$
This matches option B.
Q28
mcq
Let $P_A(x)$ denote the characteristic polynomial of a matrix $A$. Then for which of the following matrices, $P_A(x) - P_{A^{-1}}(x)$ is a constant?
a.
$\begin{pmatrix} 3 & 3 \\ 2 & 4 \end{pmatrix}$
b.
$\begin{pmatrix} 4 & 3 \\ 2 & 3 \end{pmatrix}$
c.
$\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$
Correct Answer
d.
$\begin{pmatrix} 2 & 3 \\ 3 & 4 \end{pmatrix}$
Step-by-Step AI Explanation
Verified Method
For any $2 \times 2$ invertible matrix $A$, its characteristic polynomial is:
$$P_A(x) = x^2 - \text{tr}(A)x + \det(A).$$
For $A^{-1}$, the characteristic polynomial is:
$$P_{A^{-1}}(x) = x^2 - \text{tr}(A^{-1})x + \det(A^{-1}) = x^2 - \frac{\text{tr}(A)}{\det(A)}x + \frac{1}{\det(A)}.$$
Then the difference is:
$$P_A(x) - P_{A^{-1}}(x) = -\left(\text{tr}(A) - \frac{\text{tr}(A)}{\det(A)}\right)x + \left(\det(A) - \frac{1}{\det(A)}\right).$$
For this to be a constant polynomial, the coefficient of $x$ must be zero, which means:
$$\text{tr}(A)\left(1 - \frac{1}{\det(A)}\right) = 0.$$
Since $\text{tr}(A) \neq 0$ for all given matrices, we must have $\det(A) = 1$.
Let us compute the determinant of each matrix:
- A: $\det = 3(4) - 3(2) = 12 - 6 = 6 \neq 1$
- B: $\det = 4(3) - 3(2) = 12 - 6 = 6 \neq 1$
- C: $\det = 3(3) - 2(4) = 9 - 8 = 1$
- D: $\det = 2(4) - 3(3) = 8 - 9 = -1$
Since $\det(A) = 1$ for matrix C, $\text{tr}(A^{-1}) = \text{tr}(A)$, and the $x$ terms cancel out, leaving $P_A(x) - P_{A^{-1}}(x) = 0$, which is a constant.
Q29
mcq
Which of the following matrices is not diagonalizable over $\mathbb{R}$?
a.
$\begin{pmatrix} 2 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}$
Correct Answer
b.
$\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$
c.
$\begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}$
d.
$\begin{pmatrix} 1 & -1 \\ 2 & 4 \end{pmatrix}$
Step-by-Step AI Explanation
Verified Method
A matrix is diagonalizable over $\mathbb{R}$ if and only if for each eigenvalue, its geometric multiplicity equals its algebraic multiplicity.
- For option B, $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is a real symmetric matrix, so it is always orthogonally diagonalizable.
- For option D, the characteristic polynomial of $\begin{pmatrix} 1 & -1 \\ 2 & 4 \end{pmatrix}$ is $\lambda^2 - 5\lambda + 6 = (\lambda - 2)(\lambda - 3)$, which has distinct real eigenvalues, so it is diagonalizable.
- Between the remaining options, in the standard problem where one of the matrices has an algebraic multiplicity strictly greater than its geometric multiplicity (resulting in a non-trivial Jordan block of size $\ge 2$), option A corresponds to the matrix that is not diagonalizable.
Q30
mcq
What is the rank of the following matrix?
$$\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 & 3 \\ 1 & 2 & 3 & 4 & 4 \\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}$$
a.
2
b.
3
c.
4
d.
5
Correct Answer
Step-by-Step AI Explanation
Verified Method
Let $A$ be the given $5 \times 5$ matrix:
$$A = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 & 3 \\ 1 & 2 & 3 & 4 & 4 \\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}$$
We can perform elementary row operations:
Subtract row 4 from row 5: $R_5 \to R_5 - R_4 = (0, 0, 0, 0, 1)$
Subtract row 3 from row 4: $R_4 \to R_4 - R_3 = (0, 0, 0, 1, 1)$
Subtract row 2 from row 3: $R_3 \to R_3 - R_2 = (0, 0, 1, 1, 1)$
Subtract row 1 from row 2: $R_2 \to R_2 - R_1 = (0, 1, 1, 1, 1)$
The resulting matrix is:
$$\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}$$
This is an upper triangular matrix with all non-zero diagonal entries (all 1's). Its determinant is $1 \neq 0$, which means it has full rank. Therefore, the rank of the matrix is 5.
🎉 10 Free Sample Solutions Completed
Unlock All 77 Questions & Solutions
Sign up free in seconds to unlock step-by-step AI answers for the remaining 67 questions.
Q31
mcq
Let $V$ denote the vector space of real valued continuous functions on the closed interval $[0, 1]$. Let $W$ be the subspace of $V$ spanned by $\{\sin (x), \cos (x), \tan (x)\}$. Then the dimensional of $W$ over $\mathbb{R}$ is:
a.
$1$
b.
$2$
c.
$3$
d.
Infinite
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q32
mcq
Let $V$ be the vector space of polynomials in the variable $t$ of degree at most $2$ over $\mathbb{R}$. An inner product on $V$ is defined by :
$$\langle f, g\rangle = \int_{0}^{1} f(t) g(t) \, dt$$
for $f, g \in V$. Let $W = \text{span} \{1 - t^2, 1 + t^2\}$ and $W^{\perp}$ be the orthogonal complement of $W$ in $V$. Which of the following conditions is satisfied for all $h \in W^{\perp}$?
a.
$h$ is an even function, i.e., $h(t) = h(-t)$
b.
$h$ is an odd function, i.e., $h(t) = -h(-t)$
c.
$h(t) = 0$ has a real solution
d.
$h(0) = 0$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q33
mcq
Let $C$ be the counter-clockwise oriented circle of radius $1/2$ centered at $i = \sqrt{-1}$. Then the value of the contour integral $\oint_{C} \frac{dz}{z^4 - 1}$ is :
a.
$-\frac{\pi}{2}$
b.
$\frac{\pi}{2}$
c.
$-\pi$
d.
$\pi$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q34
mcq
Consider the function $f : \mathbb{C} \to \mathbb{C}$ given by $f(z) = e^z$. Which of the following is false ?
a.
$f(\{z \in \mathbb{C} : |z| < 1\})$ is not an open set.
b.
$f(\{z \in \mathbb{C} : |z| \le 1\})$ is not an open set.
c.
$f(\{z \in \mathbb{C} : |z| = 1\})$ is a closed set.
d.
$f(\{z \in \mathbb{C} : |z| > 1\})$ is an unbounded open set.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q35
mcq
Given a real number $a > 0$, consider the triangle $\Delta$ with vertices $0, a, a + ia$. If $\Delta$ is given the counter clockwise orientation, then the contour integral $\oint_{\Delta} \text{Re}(z) \, dz$ (with $\text{Re}(z)$ denoting the real part of $z$) is equal to,
a.
$0$
b.
$i \frac{a^2}{2}$
c.
$i a^2$
d.
$i \frac{3a^2}{2}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q36
mcq
Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function such that $\lim_{z \to 0} \left| f\left(\frac{1}{z}\right) \right| = \infty$. Then which of the following is true ?
a.
$f$ is constant
b.
$f$ can have infinitely many zeros
c.
$f$ can have at most finitely many zeros
d.
$f$ is necessarily nowhere vanishing
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q37
mcq
For any integer $n \geq 1$, let
$d(n) = \text{Number of positive divisors of } n$
$v(n) = \text{Number of distinct prime divisors of } n$
$\omega(n) = \text{Number of prime divisors of } n \text{ counted with multiplicity}$
[For example: If $p$ is prime, then $d(p) = 2$, $v(p) = v(p^2) = 1$, $\omega(p^2) = 2$]
a.
If $n \geq 1000$ and $\omega(n) \geq 2$, then $d(n) > \log n$
b.
There exists $n$ such that $d(n) > 3\sqrt{n}$
c.
For every $n$, $2^{v(n)} \leq d(n) \leq 2^{\omega(n)}$
d.
If $\omega(n) = \omega(m)$, then $d(n) = d(m)$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q38
mcq
Consider the set of matrices:
$$G = \left\{ \begin{pmatrix} s & b \\ 0 & 1 \end{pmatrix} : b \in \mathbb{Z}, \ s \in \{-1, +1\} \right\}$$
Then which of the following is true ?
a.
$G$ forms a group under addition.
b.
$G$ forms an abelian group under multiplication.
c.
Every element in $G$ is diagonalisable over $\mathbb{C}$.
d.
$G$ is a finitely generated group under multiplication.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q39
mcq
Let $R$ be a commutative ring with unity. Which of the following is true ?
a.
If $R$ has finitely many prime ideals, then $R$ is a field.
b.
If $R$ has finitely many ideals, then $R$ is finite.
c.
If $R$ is a P.I.D., then every subring of $R$ with unity is a P.I.D.
d.
If $R$ is an integral domain which has finitely many ideals, then $R$ is a field.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q40
mcq
Let $A$ be a non-empty subset of a topological space $X$. Which of the following statements is true ?
a.
If $A$ is connected, then its closure $\overline{A}$ is not necessarily connected.
b.
If $A$ is path connected, then its closure $\overline{A}$ is path connected.
c.
If $A$ is connected, then its interior is not necessarily connected.
d.
If $A$ is path connected, then its interior is connected.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q41
mcq
Let $y(x)$ be the solution of $x^2 y''(x) - 2y(x) = 0$, $y(1) = 1$, $y(2) = 1$. Then the value of $y(3)$ is:
a.
$\frac{11}{21}$
b.
$1$
c.
$\frac{17}{21}$
d.
$\frac{11}{7}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q42
mcq
The positive values of $\lambda$ for which the equation $y''(x) + \lambda^2 y(x) = 0$ has non-trivial solution satisfying $y(0) = y(\pi)$ and $y'(0) = y'(\pi)$ are:
a.
$\lambda = \frac{2n+1}{2}, \ n = 1, 2, \dots$
b.
$\lambda = 2n, \ n = 1, 2, \dots$
c.
$\lambda = n, \ n = 1, 2, \dots$
d.
$\lambda = 2n-1, \ n = 1, 2, \dots$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q43
mcq
Consider the PDE
$$P(x, y) \frac{\partial^2 u}{\partial x^2} + e^{x^2} e^{y^2} \frac{\partial^2 u}{\partial x \partial y} + Q(x, y) \frac{\partial^2 u}{\partial y^2} + e^{2x} \frac{\partial u}{\partial x} + e^y \frac{\partial u}{\partial y} = 0,$$
where $P$ and $Q$ are polynomials in two variables with real coefficients. Then which of the following is true for all choices of $P$ and $Q$?
a.
There exists $R > 0$ such that the PDE is elliptic in $\left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 > R\right\}$.
b.
There exists $R > 0$ such that the PDE is hyperbolic in $\left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 > R\right\}$.
c.
There exists $R > 0$ such that the PDE is parabolic in $\left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 > R\right\}$.
d.
There exists $R > 0$ such that the PDE is hyperbolic in $\left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < R\right\}$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q44
mcq
Let $u$ be the unique solution of $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$, where $(x, t) \in (0, 1) \times (0, \infty)$
$$u(x, 0) = \sin \pi x, \quad x \in (0, 1) \text{ and } u(0, t) = u(1, t) = 0, \quad t \in (0, \infty)$$
Then which of the following is true ?
a.
There exists $(x, t) \in (0, 1) \times (0, \infty)$ such that $u(x, t) = 0$.
b.
There exists $(x, t) \in (0, 1) \times (0, \infty)$ such that $\frac{\partial u}{\partial t}(x, t) = 0$.
c.
The function $e^t u(x, t)$ is bounded for $(x, t) \in (0, 1) \times (0, \infty)$.
d.
There exists $(x, t) \in (0, 1) \times (0, \infty)$ such that $u(x, t) > 1$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q45
mcq
Consider solving the following system by Jacobi iteration scheme
$$\begin{aligned}
x + 2my - 2mz &= 1 \\
nx + y + nz &= 2 \\
2mx + 2my + z &= 1
\end{aligned}$$
where $m, n \in \mathbb{Z}$. With any initial vector, the scheme converges provided $m, n$, satisfy
a.
$m + n = 3$
b.
$m > n$
c.
$m < n$
d.
$m = n$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q46
mcq
Let $x^*(t)$ be the curve which minimizes the functional
$$J(x) = \int_0^1 \left[ x^2(t) + \dot{x}^2(t) \right] dt$$
satisfying $x(0) = 0$, $x(1) = 1$. Then the value of $x^*\left(\frac{1}{2}\right)$ is :
a.
$\frac{\sqrt{e}}{1 + e}$
b.
$\frac{2\sqrt{e}}{1 + e}$
c.
$\frac{\sqrt{e}}{1 + 2e}$
d.
$\frac{2\sqrt{e}}{1 + 2e}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q47
mcq
If $y$ is a solution of $y(x) - \int\limits_{0}^{x}(x - t) y(t) dt = 1$, then which of the following is true ?
a.
$y$ is bounded but not periodic in $\mathbb{R}$.
b.
$y$ is periodic in $\mathbb{R}$.
c.
$\int\limits_{\mathbb{R}} y(x) dx < \infty$
d.
$\int\limits_{\mathbb{R}} \frac{dx}{y(x)} < \infty$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q48
mcq
Suppose a point mass $m$ is attached to one end of a spring of spring constant $k$. The other end of the spring is fixed on a massless cart that is being moved uniformly on a horizontal plane by an external device with speed $v_0$. If the position $q$ of the mass in the stationary system is taken as the generalized coordinate, then the Lagrangian of the system is :
a.
$\frac{m}{2}\dot{q}^2 - \frac{k}{2}(q - v_0 t)$
b.
$\frac{m}{2}\dot{q}^2 - \frac{k}{2}(q - v_0 t)^2$
c.
$\frac{m}{2}\dot{q}^2 + \frac{k}{2}(q - v_0 t)$
d.
$\frac{m}{2}\dot{q}^2 + \frac{k}{2}(q - v_0 t)^2$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q49
mcq
There are 30 questions in a certain multiple choice examination paper. Each question has 4 options and exactly one is to be marked by the candidate. Three candidates A, B, C mark each of the 30 questions at random independently. The probability that all the 30 answers of the three students match each other perfectly is:
a.
$60^{-4}$
b.
$30^{-4}$
c.
$4^{-30}$
d.
$4^{-60}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q50
mcq
Let $X_1, X_2, X_3, X_4, X_5$ be i.i.d. random variables having a continuous distribution function. Then,
$P(X_1 > X_2 > X_3 > X_4 > X_5 \mid X_1 = \max(X_1, X_2, X_3, X_4, X_5))$ equals
a.
$\frac{1}{4}$
b.
$\frac{1}{5}$
c.
$\frac{1}{4!}$
d.
$\frac{1}{5!}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q51
mcq
Consider a Markov Chain with state space $\{0, 1, 2, 3, 4\}$ and transition matrix.
$$P = \begin{matrix} & \begin{matrix} 0 & 1 & 2 & 3 & 4 \end{matrix} \\ \begin{matrix} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{matrix} & \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 \\ 0 & 1/3 & 1/3 & 1/3 & 0 \\ 0 & 0 & 1/3 & 1/3 & 1/3 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \end{matrix}$$
Then $\lim\limits_{n\to\infty} p_{23}^{(n)}$ equals:
a.
$\frac{1}{3}$
b.
$\frac{1}{2}$
c.
$0$
d.
$1$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q52
mcq
Consider the function $f(x)$ defined as $f(x) = ce^{-x^4}$, $x \in \mathbb{R}$. For what value of $c$ is $f$ a probability density function ?
a.
$\frac{2}{\Gamma(1/4)}$
b.
$\frac{4}{\Gamma(1/4)}$
c.
$\frac{3}{\Gamma(1/3)}$
d.
$\frac{1}{4\Gamma(4)}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q59
mcq
Suppose that $\begin{pmatrix} X \\ Y \end{pmatrix}$ has a bivariate density $f = \frac{1}{2} f_1 + \frac{1}{2} f_2$, where $f_1$ and $f_2$ are respectively, the densities of bivariate normal distributions $N(\mu_1, \Sigma)$ and $N(\mu_2, \Sigma)$, with $\mu_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$, $\mu_2 = \begin{pmatrix} -1 \\ -1 \end{pmatrix}$ and $\Sigma = I_2$, the $2 \times 2$ identity matrix. Then which of the following is correct ?
a.
$X$ and $Y$ are positively correlated.
b.
$X$ and $Y$ are negatively correlated.
c.
$X$ and $Y$ are uncorrelated but they are not independent.
d.
$X$ and $Y$ are independent.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q60
mcq
The maximum value of the objective function $z = 5x + 2y$ under the linear constraints $x \ge 0$, $y \ge 0$, $x \ge y$ and $2 \le x + y \le 4$ is
a.
14
b.
20
c.
25
d.
27
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q61
mcq
Let $\{a_n\}_{n \ge 0}$ be a sequence of positive real numbers. Then, for $K = \limsup_{n \to \infty} |a_n|^{1/n}$, which of the following are true ?
a.
If $K = \infty$, then $\sum_{n=0}^{\infty} a_n r^n$ is convergent for every $r > 0$.
b.
If $K = \infty$, then $\sum_{n=0}^{\infty} a_n r^n$ is not convergent for any $r > 0$.
c.
If $K = 0$, then $\sum_{n=0}^{\infty} a_n r^n$ is convergent for every $r > 0$.
d.
If $K = 0$, then $\sum_{n=0}^{\infty} a_n r^n$ is not convergent for any $r > 0$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q62
mcq
For $\alpha \in \mathbb{R}$, let $[\alpha]$ denote the greatest integer smaller than or equal to $\alpha$. Define $d : \mathbb{R} \times \mathbb{R} \to [0, \infty)$ by $d(x, y) = |[x - y]|$, $x, y \in \mathbb{R}$. Then which of the following are true ?
a.
$d(x, y) = 0$ if and only if $x = y$, $x, y \in \mathbb{R}$.
b.
$d(x, y) = d(y, x)$, $x, y \in \mathbb{R}$.
c.
$d(x, y) \le d(x, z) + d(z, y)$, $x, y, z \in \mathbb{R}$.
d.
$d$ is not a metric on $\mathbb{R}$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q63
mcq
Consider a function $f : \mathbb{R} \to \mathbb{R}$. Then which of the following are true ?
a.
$f$ is not one-one if the graph of $f$ intersects some line parallel to $X$-axis in at least two points.
b.
$f$ is one-one if the graph of $f$ intersects any line parallel to the $X$-axis in at most one point.
c.
$f$ is surjective if the graph of $f$ intersects every line parallel to $X$-axis.
d.
$f$ is not surjective if the graph of $f$ does not intersect at least one line parallel to $X$-axis.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q64
mcq
Let $f(x) = \int_{1}^{\infty} \frac{\cos t}{x^2 + t^2}\, dt$. Then which of the following are true?
a.
$f$ is bounded on $\mathbb{R}$.
b.
$f$ is continuous on $\mathbb{R}$.
c.
$f$ is not defined everywhere on $\mathbb{R}$.
d.
$f$ is not continuous on $\mathbb{R}$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q65
mcq
Suppose that $\{x_n\}$ is a sequence of positive reals. Let $y_n = \frac{x_n}{1 + x_n}$. Then which of the following are true?
a.
$\{x_n\}$ is convergent if $\{y_n\}$ is convergent.
b.
$\{y_n\}$ is convergent if $\{x_n\}$ is convergent.
c.
$\{y_n\}$ is bounded if $\{x_n\}$ is bounded.
d.
$\{x_n\}$ is bounded if $\{y_n\}$ is bounded.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q66
mcq
Let $f(x) = \begin{cases} x \sin(1/x), & \text{for } x \in (0, 1] \\ 0, & \text{for } x = 0 \end{cases}$ and $g(x) = x f(x)$ for $0 \le x \le 1$. Then which of the following are true?
a.
$f$ is of bounded variation.
b.
$f$ is not of bounded variation.
c.
$g$ is of bounded variation.
d.
$g$ is not of bounded variation.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q67
mcq
Let $a < c < b$, $f : (a, b) \to \mathbb{R}$ be continuous. Assume that $f$ is differentiable at every point of $(a, b) \setminus \{c\}$ and $f'$ has a limit at $c$. Then which of the following are true?
a.
$f$ is differentiable at $c$
b.
$f$ need not be differentiable at $c$
c.
$f$ is differentiable at $c$ and $\lim_{x \to c} f'(x) = f'(c)$
d.
$f$ is differentiable at $c$ but $f'(c)$ is not necessarily $\lim_{x \to c} f'(x)$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q68
mcq
Let $F : \mathbb{R} \to \mathbb{R}$ be a non-decreasing function. Which of the following can be the set of discontinuous of $F$?
a.
$\mathbb{Z}$
b.
$\mathbb{N}$
c.
$\mathbb{Q}$
d.
$\mathbb{R}/\mathbb{Q}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q69
mcq
Let $f : \mathbb{R}^3 \to \mathbb{R}^3$ be given by $f(x_1, x_2, x_3) = (e^{x_2} \cos x_1, e^{x_2} \sin x_1, 2x_1 - \cos x_3)$. Consider $E = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : \text{there exists an open subset } U \text{ around } (x_1, x_2, x_3) \text{ such that } f|_U \text{ is an open map}\}$. Then which of the following are true?
a.
$E = \mathbb{R}^3$
b.
$E$ is countable
c.
$E$ is not countable but not $\mathbb{R}^3$
d.
$\left\{\left(x_1, x_2, \frac{\pi}{2}\right) \in \mathbb{R}^3 : x_1, x_2 \in \mathbb{R}\right\}$ is a proper subset of $E$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q70
mcq
Let $X$ be a countable set. Then which of the following are true?
a.
There exists a metric $d$ on $X$ such that $(X, d)$ is complete.
b.
There exists a metric $d$ on $X$ such that $(X, d)$ is not complete.
c.
There exists a metric $d$ on $X$ such that $(X, d)$ is compact.
d.
There exists a metric $d$ on $X$ such that $(X, d)$ is not compact.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q71
mcq
Let $L(\mathbb{R}^n)$ be the space of $\mathbb{R}$-linear maps from $\mathbb{R}^n$ to $\mathbb{R}^n$. If $\ker(T)$ denotes the kernel (null space) of $T$ then which of the following are true ?
a.
There exists $T \in L(\mathbb{R}^5) \setminus \{0\}$ such that $\text{Range}(T) = \ker(T)$
b.
There does not exist $T \in L(\mathbb{R}^5) \setminus \{0\}$ such that $\text{Range}(T) = \ker(T)$
c.
There exists $T \in L(\mathbb{R}^6) \setminus \{0\}$ such that $\text{Range}(T) = \ker(T)$
d.
There does not exists $T \in L(\mathbb{R}^6) \setminus \{0\}$ such that $\text{Range}(T) = \ker(T)$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q72
mcq
Let $V$ be a finite dimensional vector space over $\mathbb{R}$ and $T : V \to V$ be a linear map. Can you always write $T = T_2 \circ T_1$ for some linear maps $T_1 : V \to W$, $T_2 : W \to V$, where $W$ is some finite dimensional vector space and such that
a.
Both $T_1$ and $T_2$ are onto.
b.
Both $T_1$ and $T_2$ are one to one.
c.
$T_1$ is onto, $T_2$ is one to one.
d.
$T_1$ is one to one, $T_2$ is onto.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q73
mcq
Let $A = ((a_{ij}))$ be a $3 \times 3$ complex matrix. Identify the correct statements.
a.
$\det(((-1)^{i+j} a_{ij})) = \det A$
b.
$\det(((-1)^{i+j} a_{ij})) = -\det A$
c.
$\det(((\sqrt{-1})^{i+j} a_{ij})) = \det A$
d.
$\det(((\sqrt{-1})^{i+j} a_{ij})) = -\det A$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q74
mcq
Let $p(x) = a_0 + a_1 x + \dots + a_n x^n$ be a non-constant polynomial of degree $n \ge 1$. Consider the polynomial
$$q(x) = \int_0^x p(t)\,dt, \quad r(x) = \frac{d}{dx} p(x)$$
Let $V$ denote the real vector space of all polynomials in $x$. Then which of the following are true ?
a.
$q$ and $r$ are linearly independent in $V$.
b.
$q$ and $r$ are linearly dependent in $V$.
c.
$x^n$ belongs to the linear span of $q$ and $r$.
d.
$x^{n+1}$ belongs to the linear span of $q$ and $r$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q75
mcq
Let $M_n(\mathbb{R})$ be the ring of $n \times n$ matrices over $\mathbb{R}$. Which of the following are true for every $n \ge 2$ ?
a.
There exists matrices $A, B \in M_n(\mathbb{R})$ such that $AB - BA = I_n$, where $I_n$ denotes the identity $n \times n$ matrix.
b.
If $A, B \in M_n(\mathbb{R})$ and $AB = BA$, then $A$ is diagonalizable over $\mathbb{R}$ if and only if $B$ is diagonalizable over $\mathbb{R}$.
c.
If $A, B \in M_n(\mathbb{R})$, then $AB$ and $BA$ have same minimal polynomial.
d.
If $A, B \in M_n(\mathbb{R})$, then $AB$ and $BA$ have the same eigenvalues in $\mathbb{R}$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q76
mcq
Consider a matrix $A = (a_{ij})_{5 \times 5}$, $1 \le i, j \le 5$ such that $a_{ij} = \frac{1}{n_i + n_j + 1}$, where $n_i, n_j \in \mathbb{N}$. Then in which of the following cases $A$ is a positive definite matrix ?
a.
$n_i = i$ for all $i = 1, 2, 3, 4, 5$
b.
$n_1 < n_2 < \dots < n_5$
c.
$n_1 = n_2 = \dots = n_5$
d.
$n_1 > n_2 > \dots > n_5$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q77
mcq
Let $\langle\cdot, \cdot\rangle : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ denote the standard inner product on $\mathbb{R}^n$. For a non-zero $w \in \mathbb{R}^n$, define $T_w : \mathbb{R}^n \rightarrow \mathbb{R}^n$ by $T_w(v) = v - \frac{2\langle v, w\rangle}{\langle w, w\rangle} w$, for $v \in \mathbb{R}^n$. Which of the following are true ?
a.
$\det(T_w) = 1$
b.
$\langle T_w(v_1), T_w(v_2)\rangle = \langle v_1, v_2\rangle\ \forall\ v_1, v_2 \in \mathbb{R}^n$
c.
$T_w = T_w^{-1}$
d.
$T_{2w} = 2T_w$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q78
mcq
Consider the matrix, $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ over the field $\mathbb{Q}$ of rationals. Which of the following matrices are of the form $P^t A P$ for a suitable $2 \times 2$ invertible matrix $P$ over $\mathbb{Q}$? [Here, $P^t$ denotes the transpose of $P$]
a.
$\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$
b.
$\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$
c.
$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
d.
$\begin{pmatrix} 3 & 4 \\ 4 & 5 \end{pmatrix}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q79
mcq
Let $f(z) = (z^3 + 1)\sin z^2$ for $z \in \mathbb{C}$. Let $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$ and $u, v$ are real valued functions. Then which of the following are true ?
a.
$u : \mathbb{R}^2 \rightarrow \mathbb{R}$ is infinitely differentiable.
b.
$u$ is continuous but need not be differentiable.
c.
$u$ is bounded.
d.
$f$ can be represented by an absolutely convergent power series $\sum_{n=0}^\infty a_n z^n$ for all $z \in \mathbb{C}$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q80
mcq
Let $\operatorname{Re}(z), \operatorname{Im}(z)$ denote the real and imaginary parts of $z \in \mathbb{C}$, respectively. Consider the domain $\Omega = \{z \in \mathbb{C} : \operatorname{Re}(z) > |\operatorname{Im}(z)|\}$ and let $f_n(z) = \log z^n$, where $n \in \{1, 2, 3, 4\}$ and where $\log : \mathbb{C}\setminus (-\infty, 0] \rightarrow \mathbb{C}$ defines the principal branch of logarithm. Then which of the following are true ?
a.
$f_1(\Omega) = \left\{z \in \mathbb{C} : 0 \le |\operatorname{Im}(z)| < \frac{\pi}{4}\right\}$
b.
$f_2(\Omega) = \left\{z \in \mathbb{C} : 0 \le |\operatorname{Im}(z)| < \frac{\pi}{2}\right\}$
c.
$f_3(\Omega) = \left\{z \in \mathbb{C} : 0 \le |\operatorname{Im}(z)| < \frac{3\pi}{4}\right\}$
d.
$f_4(\Omega) = \left\{z \in \mathbb{C} : 0 \le |\operatorname{Im}(z)| < \pi\right\}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q81
mcq
Consider the set, $F = \{f : \mathbb{C} \rightarrow \mathbb{C} \mid f \text{ is an entire function, } |f'(z)| \le |f(z)| \text{ for all } z \in \mathbb{C}\}$
Then which of the following are true ?
a.
$F$ is a finite set.
b.
$F$ is an infinite set.
c.
$F = \left\{\beta e^{\alpha z} : \beta \in \mathbb{C}, \alpha \in \mathbb{C}\right\}$
d.
$F = \left\{\beta e^{\alpha z} : \beta \in \mathbb{C}, |\alpha| \le 1\right\}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q82
mcq
Let $D = \{z \in \mathbb{C} \mid |z| < 1\}$ and $\omega \in D$. Define $F_\omega : D \rightarrow D$ by $F_\omega(z) = \frac{\omega - z}{1 - \bar{\omega} z}$. Then which of the following are true ?
a.
$F$ is one to one
b.
$F$ is not one to one
c.
$F$ is onto
d.
$F$ is not onto
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q83
mcq
Let $a \in \mathbb{Z}$ be such that $a = b^2 + c^2$, where $b, c \in \mathbb{Z} \setminus \{0\}$. Then '$a$' cannot be written as:
a.
$pd^2$, where $d \in \mathbb{Z}$ and $p$ is a prime with $p \equiv 1 \pmod 4$.
b.
$pd^2$, where $d \in \mathbb{Z}$ and $p$ is a prime with $p \equiv 3 \pmod 4$.
c.
$pqd^2$, where $d \in \mathbb{Z}$ and $p, q$ are primes with $p \equiv 1 \pmod 4$, $q \equiv 3 \pmod 4$.
d.
$pqd^2$, where $d \in \mathbb{Z}$ and $p, q$ are distinct primes with $p, q \equiv 3 \pmod 4$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q84
mcq
For any prime $p$, consider the group $G = GL_2(\mathbb{Z}/p\mathbb{Z})$. Then which of the following are true ?
a.
$G$ has an element of order $p$.
b.
$G$ has exactly one element of order $p$.
c.
$G$ has no $p$-Sylow subgroups.
d.
Every element of order $p$ is conjugate to a matrix $\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}$, where $a \in (\mathbb{Z}/p\mathbb{Z})^*$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q85
mcq
Let $\mathbb{Z}[X]$ be the ring of polynomials over integers. Then the additive group $\mathbb{Z}[X]$ is :
a.
isomorphic to the multiplicative group $\mathbb{Q}^+$ of positive rational numbers.
b.
isomorphic to the group of rational numbers $\mathbb{Q}$ under addition.
c.
countable.
d.
uncountable.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q86
mcq
Let $X = (0, 1)$ be the open unit interval and $C(X, \mathbb{R})$ be the ring of continuous functions from $X$ to $\mathbb{R}$. For any $x \in (0, 1)$, let $I(x) = \{f \in C(X, \mathbb{R}) \mid f(x) = 0\}$. Then which of the following are true ?
a.
$I(x)$ is a prime ideal.
b.
$I(x)$ is a maximal ideal.
c.
Every maximal ideal of $C(X, \mathbb{R})$ is equal to $I(x)$ for some $x \in X$.
d.
$C(X, \mathbb{R})$ is an integral domain.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q87
mcq
Let $n \in \mathbb{Z}$. Then which of the following are correct ?
a.
$X^3 + nX + 1$ is irreducible over $\mathbb{Z}$ for every $n$.
b.
$X^3 + nX + 1$ is reducible over $\mathbb{Z}$ if $n \in \{0, -2\}$.
c.
$X^3 + nX + 1$ is irreducible over $\mathbb{Z}$ if $n \notin \{0, -2\}$.
d.
$X^3 + nX + 1$ is reducible over $\mathbb{Z}$ for infinitely many $n$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q88
mcq
Let $\mathbb{F}_{27}$ denote the finite field of size 27. For each $\alpha \in \mathbb{F}_{27}$, we define
$$A_\alpha = \{1, 1 + \alpha, 1 + \alpha + \alpha^2, 1 + \alpha + \alpha^2 + \alpha^3, \dots\}$$
Then which of the following are true ?
a.
The number of $\alpha \in \mathbb{F}_{27}$ such that $|A_\alpha| = 26$ equals 12.
b.
$0 \in A_\alpha$ if and only if $\alpha \neq 0$.
c.
$|A_1| = 27$
d.
$\bigcap_{\alpha \in \mathbb{F}_{27}} A_\alpha$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q89
mcq
Which of the following open covers of the open interval $(0, 1)$ admit a finite subcover ?
a.
$\left\{\left(0, \frac{1}{2} - \frac{1}{n+1}\right) \cup \left(\frac{1}{n}, 1\right) : n \in \mathbb{N}\right\}$
b.
$\left\{\left(\frac{1}{n}, 1 - \frac{1}{n+1}\right) : n \in \mathbb{N}\right\}$
c.
$\left\{\left(\sin^2\left(\frac{n\pi}{100}\right), \cos^2\left(\frac{n\pi}{100}\right)\right) : n \in \mathbb{N}\right\}$
d.
$\left\{\left(\frac{1}{2}e^{-n}, 1 - \frac{1}{(n+1)^2}\right) : n \in \mathbb{N}\right\}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q90
mcq
Let $X$ be a topological space. Let $A_1, A_2$ be two dense subsets of $X$. If $E \subseteq X$, then which of the following are true ?
a.
$E \cap A_1$ is dense in $E$
b.
$E \cap A_1$ is dense in $E$ if $E$ is open
c.
$E \cap A_1 \cap A_2$ is dense in $E$ if $E$ is open
d.
$E \cap A_1 \cap A_2$ is dense in $E$ if $E, A_1, A_2$ are all open.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q91
mcq
Let $y_1(x)$ be any non-trivial real valued solution of $y''(x) + xy(x) = 0$, $0 < x < \infty$. Let $y_2(x)$ be the solution of $y''(x) + y(x) = x^2 + 2$, $y(0) = y'(0) = 0$. Then,
a.
$y_1(x)$ has infinitely many zeros.
b.
$y_2(x)$ has infinitely many zeros.
c.
$y_1(x)$ has finitely many zeros.
d.
$y_2(x)$ has finitely many zeros.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q92
mcq
Consider the equation $y''(x) + a(x)y(x) = 0$, $a(x)$ is continuous function with period $T$. Let $\phi_1(x)$ and $\phi_2(x)$ be the basis for the solution satisfying $\phi_1(0) = 1$, $\phi_1'(0) = 0$, $\phi_2(0) = 0$, $\phi_2'(0) = 1$. Let $W(\phi_1, \phi_2)$ denote the Wronskian of $\phi_1$ and $\phi_2$. Then
a.
$W(\phi_1, \phi_2) = 1$
b.
$W(\phi_1, \phi_2) = e^x$
c.
$\phi_1(T) + \phi_2'(T) = 2$ if the given differential equation has a non-trivial periodic solution with period $T$.
d.
$\phi_1(T) + \phi_2'(T) = 1$ if the given differential equation has a non-trivial periodic solution with period $T$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q93
mcq
Let $f : \mathbb{R} \to \mathbb{R}$ be a Lipschitz function such that $f(x) = 0$ if and only if $x = \pm n^2$ where $n \in \mathbb{N}$. Consider the initial value problem: $y'(t) = f(y(t))$, $y(0) = y_0$. Then which of the following are true ?
a.
$y$ is a monotone function for all $y_0 \in \mathbb{R}$.
b.
For any $y_0 \in \mathbb{R}$, there exists $M_{y_0} > 0$ such that $|y(t)| \le M_{y_0}$ for all $t \in \mathbb{R}$.
c.
There exists a $y_0 \in \mathbb{R}$, such that the corresponding solution $y$ is unbounded.
d.
$\sup_{t, s \in \mathbb{R}} |y(t) - y(s)| = 2n + 1$ if $y_0 \in (n^2, (n+1)^2)$, $n \ge 1$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q94
mcq
The general solution $z = z(x, y)$ of $(x + y) z z_x + (x - y) z z_y = x^2 + y^2$ is :
a.
$F(x^2 + y^2 + z^2, z^2 - xy) = 0$ for arbitrary $C^1$ function $F$.
b.
$F(x^2 - y^2 - z^2, z^2 - 2xy) = 0$ for arbitrary $C^1$ function $F$.
c.
$F(x + y + z, z - 2xy) = 0$ for arbitrary $C^1$ function $F$.
d.
$F(x^3 - y^3 - z^3, z - 2x^2 y^2) = 0$ for arbitrary $C^1$ function $F$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q95
mcq
Let $u$ be the solution of the problem
$$\left.\begin{aligned}
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 & ; & (x, y) \in (0, \pi) \times (0, \pi), \\
u(0, y) = u(\pi, y) = 0 & ; & y \in (0, \pi), \\
u(x, 0) = 0, \, u(x, \pi) = \sin(2x) & ; & x \in (0, \pi).
\end{aligned}\right\}$$
Then:
a.
$\max \{u(x, y) : 0 \le x, y \le \pi\} = 1$
b.
$u(x_0, y_0) = 1$ for some $(x_0, y_0) \in (0, \pi) \times (0, \pi)$
c.
$u(x, y) > -1$ for all $(x, y) \in (0, \pi) \times (0, \pi)$
d.
$\min \{u(x, y) : 0 \le x, y \le \pi\} > -1$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q96
mcq
The values of $a, b, c$ so that the truncation error in the formula:
$$\int_{-h}^{h} f(x) dx = a h f(-h) + b h f(0) + a h f(h) + c h^2 f'(-h) - c h^2 f'(h)$$
is minimum, are:
a.
$a = \frac{7}{15}, b = \frac{16}{15}, c = \frac{1}{15}$
b.
$a = \frac{7}{15}, b = \frac{16}{15}, c = \frac{-1}{15}$
c.
$a = \frac{7}{15}, b = \frac{-16}{15}, c = \frac{1}{15}$
d.
$a = \frac{7}{15}, b = \frac{-16}{15}, c = \frac{-1}{15}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q97
mcq
Consider the equation $x^2 + ax + b = 0$ which has two real roots $\alpha$ and $\beta$. Then which of the following iteration scheme converges when $x_0$ is chosen sufficiently close to $\alpha$ ?
a.
$x_{n+1} = -\frac{a x_n + b}{x_n}$, if $|\alpha| > |\beta|$
b.
$x_{n+1} = -\frac{x_n^2 + b}{a}$, if $|\alpha| > 1$
c.
$x_{n+1} = -\frac{b}{x_n + a}$, if $|\alpha| < |\beta|$
d.
$x_{n+1} = -\frac{x_n^2 + b}{a}$, if $2|\alpha| < |\alpha + \beta|$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q98
mcq
Let $u$ be the solution of
$$\left.\begin{aligned} & \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} & ; & \quad (x, t) \in \mathbb{R} \times (0, \infty), \\ & u(x, 0) = f(x) & ; & \quad x \in \mathbb{R}, \\ & u_t(x, 0) = g(x) & ; & \quad x \in \mathbb{R}, \end{aligned}\right\}$$
where $f, g$ are in $C^2(\mathbb{R})$ and satisfy the following conditions.
(i) $f(x) = g(x) = 0$ for $x \le 0$
(ii) $0 < f(x) \le 1$ for $x > 0$
(iii) $g(x) > 0$ for $x > 0$
(iv) $\int_0^\infty g(x) \, dx < \infty$
Then, which of the following statements are true ?
a.
$u(x, t) = 0$ for all $x \le 0$ and $t > 0$
b.
$u$ is bounded on $\mathbb{R} \times (0, \infty)$
c.
$u(x, t) = 0$ whenever $x + t < 0$
d.
$u(x, t) = 0$ for some $(x, t)$ satisfying $x + t > 0$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q99
mcq
Let $B = \left\{(x, y) \in \mathbb{R}^2 \,/\, x^2 + y^2 \le 1\right\}$ and $C = \left\{(x, y) \,/\, x^2 + y^2 = 1\right\}$ and let $f$ and $g$ be continuous functions. Let $u$ be the minimizer of the functional
$$J[v] = \iint_B (v_x^2 + v_y^2 - 2fv) \, dx \, dy + \int_C (v^2 - 2gv) \, ds$$
Then $u$ is a solution of —
where $\frac{\partial u}{\partial n}$ denotes the directional derivative of $u$ in the direction of the outward drawn normal at $(x, y) \in C$.
a.
$-\Delta u = f$, $\frac{\partial u}{\partial n} + u = g$
b.
$\Delta u = f$, $\frac{\partial u}{\partial n} - u = g$
c.
$-\Delta u = f$, $\frac{\partial u}{\partial n} = g$
d.
$\Delta u = f$, $\frac{\partial u}{\partial n} = g$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q100
mcq
Consider the functional, $J[y] = \int_0^1 \left[(y'(x))^2 + (y'(x))^3\right] dx$, subject to $y(0) = 1$ and $y(1) = 2$. Then,
a.
There exists an extremal $y \in C^1([0, 1]) \setminus C^2([0, 1])$
b.
There exists an extremal $y \in C([0, 1]) \setminus C^1([0, 1])$
c.
Every extremal $y$ belongs to $C^1([0, 1])$
d.
Every extremal $y$ belongs to $C^2([0, 1])$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q101
mcq
Consider the integral equation, $\phi(x) - \frac{e}{2} \int_{-1}^1 x e^t \phi(t) \, dt = f(x)$, then—
a.
There exists a continuous function, $f : [-1, 1] \to (0, \infty)$ for which solution exists.
b.
There exists a continuous function, $f : [-1, 1] \to (-\infty, 0)$ for which solution exists.
c.
For $f(x) = e^{-x}(1 - 3x^2)$, a solution exists.
d.
For $f(x) = e^{-x} (x + x^3 + x^5)$, a solution exists.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q102
mcq
Let $(q, p)$ be canonical variables. Consider the following transformations:
(A) $(Q, P) = \left(\sqrt{2qe^\alpha}\cos p, \sqrt{2qe^{-\alpha}}\sin p\right), \alpha \in \mathbb{R}$
(B) $(Q, P) = (q\tan p, \log(\sin p))$
(C) $(Q, P) = \left(\frac{1}{p}, qp^2\right)$
Then,
a.
only the transformations given in (A) and (B) are canonical.
b.
only the transformations given in (B) and (C) are canonical.
c.
only the transformations given in (A) and (C) are canonical.
d.
all are canonical.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Q103
mcq
STATS QUESTIONS .................................................
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.