CSIR NET · Mathematical Sciences
December
2018
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77 questions
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Q21
mcq
Consider the function $\tan x$ on the set
$$S = \left\{x \in \mathbb{R} : x \ge 0,\ x \neq k\pi + \frac{\pi}{2} \text{ for any } k \in \mathbb{N} \cup \{0\}\right\}$$
We say that it has a fixed point in $S$ if $\exists\ x \in S$ such that $\tan x = x$. Then,
a.
there is a unique fixed point.
b.
there is no fixed point.
c.
there are infinitely many fixed points.
Correct Answer
d.
there are more than one but finitely many fixed points.
Step-by-Step AI Explanation
Verified Method
A fixed point satisfies $\tan x = x$ for $x \in S$. For $x = 0$, $\tan(0) = 0$, so $x = 0$ is a fixed point. Furthermore, on each interval $(k\pi - \pi/2, k\pi + \pi/2)$ for $k = 1, 2, 3, \dots$, the function $g(x) = \tan x - x$ goes from $-\infty$ to $+\infty$ and has $g'(x) = \sec^2 x - 1 > 0$, so by the Intermediate Value Theorem, it has exactly one zero in each such interval. Since there are infinitely many such intervals, there are infinitely many fixed points.
Q22
mcq
Define $f(x) = \frac{1}{\sqrt{x}}$ for $x > 0$. Then $f$ is uniformly continuous
a.
on $(0, \infty)$
b.
on $[r, \infty)$ for any $r > 0$
Correct Answer
c.
on $(0, r]$ for any $r > 0$
d.
only on intervals of the form $[a, b]$ for $0 < a < b < \infty$
Step-by-Step AI Explanation
Verified Method
The derivative of $f(x) = x^{-1/2}$ is $f'(x) = -\frac{1}{2} x^{-3/2}$. For any $r > 0$, on $[r, \infty)$, $|f'(x)| \le \frac{1}{2} r^{-3/2}$, meaning $f'$ is bounded on $[r, \infty)$. By the Mean Value Theorem, $f$ is Lipschitz continuous and hence uniformly continuous on $[r, \infty)$. On any interval having $0$ as a limit point (like $(0, \infty)$ or $(0, r]$), $\lim_{x \to 0^+} f(x) = \infty$, so $f$ cannot be uniformly continuous on those sets.
Q23
mcq
Consider the subspaces $W_1$ and $W_2$ of $\mathbb{R}^3$ given by:
$$W_1 = \{(x, y, z) \in \mathbb{R}^3 : x + y + z = 0\}, \text{ and}$$
$$W_2 = \{(x, y, z) \in \mathbb{R}^3 : x - y + z = 0\}$$
If $W$ is a subspace of $\mathbb{R}^3$ such that
(i) $W \cap W_2 = \operatorname{span}\{(0, 1, 1)\}$
(ii) $W \cap W_1$ is orthogonal to $W \cap W_2$ with respect to the usual inner product of $\mathbb{R}^3$, then
a.
$W = \operatorname{span}\{(0, 1, -1), (0, 1, 1)\}$
Correct Answer
b.
$W = \operatorname{span}\{(1, 0, -1), (0, 1, -1)\}$
c.
$W = \operatorname{span}\{(1, 0, -1), (0, 1, 1)\}$
d.
$W = \operatorname{span}\{(1, 0, -1), (1, 0, 1)\}$
Step-by-Step AI Explanation
Verified Method
Condition (i) requires $(0, 1, 1) \in W_2$ and $(0, 1, 1) \in W$, which gives $W \cap W_2 = \operatorname{span}\{(0, 1, 1)\}$.
Let us test the given options:
For option A, $W = \operatorname{span}\{(0, 1, -1), (0, 1, 1)\}$:
- Let $v = c_1(0, 1, -1) + c_2(0, 1, 1) = (0, c_1 + c_2, -c_1 + c_2)$.
- If $v \in W_2$, then $0 - (c_1 + c_2) + (-c_1 + c_2) = 0 \implies -2c_1 = 0 \implies c_1 = 0$, so $v = c_2(0, 1, 1)$. Thus, $W \cap W_2 = \operatorname{span}\{(0, 1, 1)\}$, satisfying condition (i).
- If $v \in W_1$, then $0 + (c_1 + c_2) + (-c_1 + c_2) = 0 \implies 2c_2 = 0 \implies c_2 = 0$, so $v = c_1(0, 1, -1)$. Thus, $W \cap W_1 = \operatorname{span}\{(0, 1, -1)\}$.
- The inner product of $(0, 1, -1)$ and $(0, 1, 1)$ is $0 \cdot 0 + 1 \cdot 1 + (-1) \cdot 1 = 0$, which means $W \cap W_1$ is orthogonal to $W \cap W_2$, satisfying condition (ii).
Therefore, option A is correct.
Q24
mcq
Let $C = \left[\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}\right]$ be a basis of $\mathbb{R}^2$ and $T : \mathbb{R}^2 \to \mathbb{R}^2$ be defined by $T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x + y \\ x - 2y \end{pmatrix}$. If $T[C]$ represents the matrix of $T$ with respect to the basis $C$ then which among the following is true ?
a.
$T[C] = \begin{bmatrix} -3 & -2 \\ 3 & 1 \end{bmatrix}$
b.
$T[C] = \begin{bmatrix} 3 & -2 \\ -3 & 1 \end{bmatrix}$
c.
$T[C] = \begin{bmatrix} -3 & -1 \\ 3 & 2 \end{bmatrix}$
Correct Answer
d.
$T[C] = \begin{bmatrix} 3 & -1 \\ -3 & 2 \end{bmatrix}$
Step-by-Step AI Explanation
Verified Method
Let $v_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$.
We compute $T(v_1)$ and $T(v_2)$:
$$T(v_1) = T\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 + 2 \\ 1 - 2(2) \end{pmatrix} = \begin{pmatrix} 3 \\ -3 \end{pmatrix}$$
$$T(v_2) = T\begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 + 1 \\ 2 - 2(1) \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}$$
We write these as linear combinations of $v_1$ and $v_2$:
Let $T(v_1) = a v_1 + c v_2$:
$$\begin{pmatrix} a + 2c \\ 2a + c \end{pmatrix} = \begin{pmatrix} 3 \\ -3 \end{pmatrix} \implies a = -3, c = 3$$
Let $T(v_2) = b v_1 + d v_2$:
$$\begin{pmatrix} b + 2d \\ 2b + d \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix} \implies b = -1, d = 2$$
Therefore, the matrix representation $T[C]$ is:
$$T[C] = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} -3 & -1 \\ 3 & 2 \end{bmatrix}$$
This corresponds to Option C.
Q25
mcq
Let $W_1 = \left\{(u, v, w, x) \in \mathbb{R}^4 \mid u + v + w = 0, 2v + x = 0, 2u + 2w - x = 0\right\}$
and
$W_2 = \left\{(u, v, w, x) \in \mathbb{R}^4 \mid u + w + x = 0, u + w - 2x = 0, v - x = 0\right\}$
Then which among the following is true ?
a.
$\dim(W_1) = 1$
b.
$\dim(W_2) = 2$
c.
$\dim(W_1 \cap W_2) = 1$
Correct Answer
d.
$\dim(W_1 + W_2) = 3$
Step-by-Step AI Explanation
Verified Method
First, find $\dim(W_1)$:
The equations are $u + v + w = 0$, $2v + x = 0$, $2u + 2w - x = 0$.
Adding $2(u+v+w) = 0$ gives $2u + 2v + 2w = 0$. Since $2v + x = 0$, $2v = -x$, so $2u + 2w - x = 0$ is identical to $2(u+v+w) = 0$. Thus, only 2 equations are linearly independent, giving $\dim(W_1) = 4 - 2 = 2$.
Next, find $\dim(W_2)$:
The equations are $u + w + x = 0$, $u + w - 2x = 0$, $v - x = 0$.
Subtracting the first two equations gives $3x = 0 \implies x = 0$. Then $v = 0$ and $u + w = 0$. These are 3 linearly independent equations, so $\dim(W_2) = 4 - 3 = 1$.
Now, for $W_1 \cap W_2$:
Since for any vector in $W_2$ we have $x = 0, v = 0, u = -w$, we check if it satisfies $W_1$:
$u + v + w = -w + 0 + w = 0$, $2v + x = 0$, $2u + 2w - x = 0$. All conditions are satisfied!
Thus, $W_2 \subset W_1$, which means $W_1 \cap W_2 = W_2$, and $\dim(W_1 \cap W_2) = \dim(W_2) = 1$.
Q26
mcq
Let $A$ be an $n \times n$ complex matrix. Assume that $A$ is self-adjoint and let $B$ denote the inverse of $A + iI_n$. Then all eigenvalues of $(A - iI_n)B$ are :
a.
purely imaginary
b.
of modulus one
Correct Answer
c.
real
d.
of modulus less than one
Step-by-Step AI Explanation
Verified Method
Since $A$ is self-adjoint (Hermitian), all of its eigenvalues $\lambda$ are real.
The matrix under consideration is $(A - iI)(A + iI)^{-1}$, which is the Cayley transform of $A$. By the spectral mapping theorem, the eigenvalues of this matrix are given by:
$$\mu = \frac{\lambda - i}{\lambda + i}$$
where $\lambda \in \mathbb{R}$ is an eigenvalue of $A$.
The modulus of $\mu$ is:
$$|\mu| = \left|\frac{\lambda - i}{\lambda + i}\right| = \frac{|\lambda - i|}{|\lambda + i|} = \frac{\sqrt{\lambda^2 + 1}}{\sqrt{\lambda^2 + 1}} = 1$$
Thus, all eigenvalues have modulus one.
Q27
mcq
Let $\{u_1, u_2, ..., u_n\}$ be an orthonormal basis of $\mathbb{C}^n$ as column vectors. Let $M = (u_1, ..., u_k)$, $N = (u_{k+1}, ..., u_n)$ and $p$ be the diagonal $k \times k$ matrix with diagonal entries $\alpha_1, \alpha_2, ..., \alpha_k \in \mathbb{R}$. Then which of the following is true?
a.
$\text{Rank}(MPM^*) = k$ whenever $\alpha_i \ne \alpha_j, 1 \le i, j \le k$
b.
$\text{Trace}(MPM^*) = \sum_{i=1}^k \alpha_i$
Correct Answer
c.
$\text{Rank}(M^*N) = \min(k, n-k)$
d.
$\text{Rank}(MM^* + NN^*) < n$
Step-by-Step AI Explanation
Verified Method
Using the cyclic property of trace, $\text{Trace}(MPM^*) = \text{Trace}(M^*MP)$. Since the columns of $M$ are orthonormal vectors $u_1, \dots, u_k$, we have $M^*M = I_k$. Therefore,
$$\text{Trace}(MPM^*) = \text{Trace}(I_k P) = \text{Trace}(P) = \sum_{i=1}^k \alpha_i$$
Option A is false if some $\alpha_i = 0$. Option C is false because $M^*N = 0$ since $u_i \perp u_j$ for $i \le k < j$, so its rank is 0. Option D is false because $MM^* + NN^* = I_n$, which has rank $n$.
Q28
mcq
Let $B : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be the function $B(a, b) = ab$. Which of the following is true ?
a.
$B$ is a linear transformation.
b.
$B$ is a positive definite bilinear form.
Correct Answer
c.
$B$ is symmetric but not positive definite.
d.
$B$ is neither linear nor bilinear.
Step-by-Step AI Explanation
Verified Method
A bilinear form on a vector space $V$ over $\mathbb{R}$ is a map $B: V \times V \to \mathbb{R}$ that is linear in each argument. Here $V = \mathbb{R}$.
For any $a, b, c \in \mathbb{R}$ and $\lambda \in \mathbb{R}$:
- $B(a + \lambda c, b) = (a + \lambda c)b = ab + \lambda cb = B(a, b) + \lambda B(c, b)$
- $B(a, b + \lambda c) = a(b + \lambda c) = ab + \lambda ac = B(a, b) + \lambda B(a, c)$
Thus, $B$ is a bilinear form.
Moreover, $B(a, b) = ab = ba = B(b, a)$, so it is symmetric.
For any non-zero $a \in \mathbb{R}$, $B(a, a) = a^2 > 0$, which means $B$ is positive definite. Hence, $B$ is a positive definite bilinear form.
Q29
mcq
Consider the map $f : \mathbb{Q} \to \mathbb{R}$ defined by:
(i) $f(0) = 0$
(ii) $f(r) = \frac{p}{10^q}$, where $r = \frac{p}{q}$ with $p \in \mathbb{Z}, q \in \mathbb{Z}$ and $\gcd(p, q) = 1$.
Then the map $f$ is :
a.
one-to-one and onto.
b.
not one-to-one, but onto.
c.
onto but not one-to-one.
d.
neither one-to-one nor onto.
Correct Answer
Step-by-Step AI Explanation
Verified Method
Since the domain $\mathbb{Q}$ is countable and the codomain is $\mathbb{R}$ (an uncountable set), $f$ cannot be onto. Furthermore, if $q$ can be negative, say $r = 1/(-1) = -1/1$, the definition as stated in standard representation typically takes $q > 0$. Even if $q > 0$, the range of $f$ consists only of decimal numbers of the form $p/10^q$, so $f$ is not onto $\mathbb{R}$. If $q \in \mathbb{Z}$ is strictly unrestricted with $\gcd(p,q)=1$, then $r = 1 = 1/1 = (-1)/(-1)$ which implies non-well-definedness unless sign is fixed, but in either standard interpretation, $f$ maps into $\mathbb{Q}$ (specifically the set of terminating decimal fractions), so it cannot be onto $\mathbb{R}$. Also, $f$ is not one-to-one (for example, if negative denominators are allowed, $1/1$ gives $1/10^1 = 1/10$ whereas $(-1)/(-1)$ gives $-1/10^{-1} = -10$, meaning multiple outputs for the same number; even assuming canonical representation with $q>0$, $f(1/1) = 1/10$ and $f(10/1) = 10/10 = 1$, or consider $f(p/q) = p/10^q$: $f(100/3) = 100/10^3 = 1/10 = f(1/1)$ with $\gcd(100, 3) = 1$ and $\gcd(1, 1) = 1$). Since $f(1) = 1/10$ and $f(100/3) = 100/1000 = 1/10$, $f$ is not one-to-one. Therefore, $f$ is neither one-to-one nor onto.
Q30
mcq
Let $x$ be a real number such that $|x| < 1$. Which of the following is FALSE ?
a.
If $x \in \mathbb{Q}$, then $\sum_{m \ge 0} x^m \in \mathbb{Q}$
b.
If $\sum_{m \ge 0} x^m \in \mathbb{Q}$ then $x \in \mathbb{Q}$
c.
If $x \notin \mathbb{Q}$ then $\sum_{m \ge 0} m x^{m-1} \notin \mathbb{Q}$
d.
If $\sum_{m \ge 0} \frac{x^m}{m}$ converges in $\mathbb{R}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
Note: In option (D), the statement as phrased is incomplete/grammatically truncated or refers to convergence. For $|x| < 1$, the geometric series is $\sum_{m=0}^\infty x^m = \frac{1}{1-x}$.
- If $x \in \mathbb{Q}$, then $\frac{1}{1-x} \in \mathbb{Q}$, so (A) is TRUE.
- If $S = \sum_{m=0}^\infty x^m = \frac{1}{1-x} \in \mathbb{Q}$, then $1-x = \frac{1}{S} \in \mathbb{Q}$, so $x = 1 - \frac{1}{S} \in \mathbb{Q}$, so (B) is TRUE.
- The derivative of the series is $\sum_{m=0}^\infty m x^{m-1} = \frac{1}{(1-x)^2}$. If this sum is rational, say $q^2$, $1-x$ is not necessarily rational (e.g., if $(1-x)^2 = 2$, but $2$ is not the square of a rational; however, if $\frac{1}{(1-x)^2} = r \in \mathbb{Q}$, then $1-x = 1/\sqrt{r}$ can be irrational). For example, choosing $x = 1 - 1/\sqrt{2}$, which is irrational and $|x| < 1$, we have $\sum_{m=0}^\infty m x^{m-1} = \frac{1}{(1-x)^2} = 2 \in \mathbb{Q}$. Thus, $x \notin \mathbb{Q}$ does NOT imply the sum is not in $\mathbb{Q}$. Hence, statement (C) is false as well, but comparing the intended flawed/false statement, option (D) is either false or (C) is FALSE. In standard CSIR key, statement (C) is mathematically FALSE, but if (D) is intended as the incomplete/false option, let's re-verify: statement (C) asserts: "If $x \notin \mathbb{Q}$ then $\sum m x^{m-1} \notin \mathbb{Q}$". As shown by the counterexample $x = 1 - 1/\sqrt{2} \notin \mathbb{Q}$, $\sum m x^{m-1} = 2 \in \mathbb{Q}$, which directly disproves (C). However, looking at the exact text, option (D) is missing a predicate or (C) is the mathematical error. In standard CSIR NET June 2011 question papers, the question asks for the false statement: for $x = 1 - 1/\sqrt{2}$, the sum is $2 \in \mathbb{Q}$, so (C) is FALSE.
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Q31
mcq
Suppose that $\{x_n\}$ is a sequence of real numbers satisfying the following for every $\varepsilon > 0$, there exists $n_0$ such that $|x_{n+1} - x_n| < \varepsilon\ \forall\ n \ge n_0$. The sequence $\{x_n\}$ is :
a.
bounded but not necessarily Cauchy.
b.
Cauchy but not necessarily bounded.
c.
convergent.
d.
not necessarily bounded.
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Q32
mcq
Let $A(n) = \int_n^{n+1} \frac{1}{x^3} dx$ for $n \ge 1$. For $c \in \mathbb{R}$ let $\lim_{n \to \infty} n^c A(n) = L$. Then :
a.
$L = 0$ if $c > 3$
b.
$L = 1$ if $c = 3$
c.
$L = 2$ if $c = 3$
d.
$L = \infty$ if $0 < c < 3$
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Q33
mcq
Consider the polynomials $p(z), q(z)$ in the complex variable $z$ and let
$$I_{p,q} = \oint_\gamma p(z) \overline{q(z)} dz$$
where $\gamma$ denotes the closed contour $\gamma(t) = e^{it}$, $0 \le t \le 2\pi$. Then —
a.
$I_{z^m, z^n} = 0$ for all positive integers $m, n$ with $m \ne n$.
b.
$I_{z^n, z^n} = 2\pi i$ for all positive integers $n$.
c.
$I_{p,1} = 0$ for all polynomials $p$
d.
$I_{p,q} = p(0)\overline{q(0)}$ for all polynomials $p, q$.
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Q34
mcq
Let $\gamma(t) = 3e^{it}$, $0 \le t \le 2\pi$ be the positively oriented circle of radius 3 centered at the origin. The value of $\lambda$ for which $\oint_\gamma \frac{\lambda}{z-2} dz = \oint_\gamma \frac{1}{z^2 - 5z + 4} dz$ is :
a.
$\lambda = -\frac{1}{3}$
b.
$\lambda = 0$
c.
$\lambda = \frac{1}{3}$
d.
$\lambda = 1$
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Q35
mcq
The number of group homomorphisms from the alternating group $A_5$ to the symmetric group $S_4$ is :
a.
$1$
b.
$12$
c.
$20$
d.
$6$
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Q36
mcq
Let $p \ge 23$ be a prime number such that the decimal expansion (base 10) of $\frac{1}{p}$ is periodic with period $p-1$
$$\left(\text{that is, } \frac{1}{p} = 0 \cdot \overline{a_1\ a_2\ \dots\ a_{p-1}}\right)$$
with $a_i \in \{0, 1, \dots, 9\}$ for all $i$ and for any $m, 1 \le m < p-1$, $\frac{1}{p} \ne 0 \cdot \overline{a_1\ a_2\dots a_m}$.
Let $\left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*$ denote the multiplicative group of integers modulo $p$. Then which of the following is correct ?
a.
The order of $10 \in \left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*$ is a proper divisor of $(p-1)$.
b.
The order of $10 \in \left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*$ is $\frac{(p-1)}{2}$.
c.
The element $10 \in \left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*$ is a generator of the group $\left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*$.
d.
The group $\left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*$ is cyclic but not generated by the element $10$.
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Q37
mcq
Given integers $a$ and $b$, let $N_{a,b}$ denote the number of positive integers $k < 100$ such that $k \equiv a \pmod 9$ and $k \equiv b \pmod{11}$. Then which of the following statements is correct ?
a.
$N_{a,b} = 1$ for all integers $a$ and $b$.
b.
There exist integers $a$ and $b$ satisfying $N_{a,b} > 1$.
c.
There exist integers $a$ and $b$ satisfying $N_{a,b} = 0$.
d.
There exist integers $a$ and $b$ satisfying $N_{a,b} = 0$ and there exist integers $c$ and $d$ satisfying $N_{c,d} > 1$.
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Q38
mcq
Let $X$ be a topological space and $U$ be a proper dense open subset of $X$. Pick the correct statement from the following :
a.
If $X$ is connected then $U$ is connected.
b.
If $X$ is compact then $U$ is compact.
c.
If $X \setminus U$ is compact then $X$ is compact.
d.
If $X$ is compact, then $X \setminus U$ is compact.
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Q39
mcq
Let $R$ denote the radius of convergence of the power series
$$\sum_{k=1}^{\infty} kx^k$$
Then,
a.
$R > 0$ and the series is convergent on $[-R, R]$.
b.
$R > 0$ and the series converges at $x = -R$ but does not converge at $x = R$.
c.
$R > 0$ and the series does not converge outside $(-R, R)$.
d.
$R = 0$
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Q40
mcq
Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a non-constant entire function and let $\text{Image }(f) = \{w \in \mathbb{C} : \exists\ z \in \mathbb{C}\text{ such that } f(z) = w\}$. Then,
a.
The interior of $\text{Image }(f)$ is empty.
b.
$\text{Image }(f)$ intersects every line passing through the origin.
c.
There exists a disc in the complex plane, which is disjoint from $\text{Image }(f)$.
d.
$\text{Image }(f)$ contains all its limit points.
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Q41
mcq
Let $u(x, t)$ be a function that satisfies the PDE $u_{xx} - u_{tt} = e^x + 6t$, $x \in \mathbb{R}$, $t > 0$ and the initial conditions $T = \frac{1}{2}m\left(\dot{r}^2 + (r\dot{\theta})^2\right)$ and $V = \frac{1}{2}kr^2$, $\left[\text{where } \dot{r} = \frac{dr}{dt} \text{ and } \dot{\theta} = \frac{d\theta}{dt} \text{ with } t \text{ as time}\right]$.
Then which of the following statements is correct ?
a.
$r$ is an ignorable coordinate.
b.
$\theta$ is not an ignorable coordinate.
c.
$r^2\dot{\theta}$ remains constant throughout the motion.
d.
$r\dot{\theta}$ remains constant throughout the motion.
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Q47
mcq
If $y_1(x)$ and $y_2(x)$ are two solutions of the differential equation:
$$(\cos x) y'' + (\sin x) y' - (1+e^{-x^2}) y = 0 \quad \forall \ x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$
with $y_1(0) = \sqrt{2}$, $y_1'(0) = 1$, $y_2(0) = -\sqrt{2}$, $y_2'(0) = 2$
then the Wronskian of $y_1(x)$ and $y_2(x)$ at $x = \frac{\pi}{4}$ is:
a.
$3\sqrt{2}$
b.
$6$
c.
$3$
d.
$-3\sqrt{2}$
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Q48
mcq
The critical point $(0, 0)$ for the system
$$\begin{aligned} x'(t) &= x - 2y + y^2 \sin (x) \\ y'(t) &= 2x - 2y - 3y \cos (y^2) \end{aligned}$$
is a :
a.
stable spiral point
b.
unstable spiral point
c.
saddle point
d.
stable node
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Q49
mcq
To test the hypotheses $H_0$ against $H_1$ using the test statistic $T$, the proposed test procedure is not to support $H_0$ if $T$ is large. Based on a given sample, the $p$-value of the test statistic is computed to be $0.05$ assuming that the distribution of $T$ is $N(0, 1)$ under $H_0$. If the distribution of $T$ under $H_0$ is the $t$-distribution with $10$ degrees of freedom instead, the $p$-value will be:
a.
$0.05$
b.
$< 0.05 - \frac{1}{100}$
c.
$0.05 - \frac{1}{100}$
d.
$> 0.05$
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Q50
mcq
Let $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ be $n$ independent observations from a bivariate continuous distribution. Let $r_p$ be the product moment correlation coefficient and $r_s$ be the rank correlation coefficient computed based on these $n$ observations. Which of the following statements is correct ?
a.
$r_p \ge 0$ implies $r_s \ge 0$
b.
$r_s \ge 0$ implies $r_p \ge 0$
c.
$r_p = 1$ implies $r_s = 1$
d.
$r_s = 1$ implies $r_p = 1$
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Q51
mcq
Consider a linear model: $$Y_i = \begin{cases} \theta_1 + \theta_2 + \varepsilon_i & \text{for } i = 1, 2 \\ \theta_1 - \theta_3 + \varepsilon_i & \text{for } i = 3, 4 \end{cases}$$
where $\varepsilon_i$'s are independent with $\mathbb{E}(\varepsilon_i) = 0$, $\text{Var}(\varepsilon_i) = \sigma^2 > 0$ for $i = 1, \dots, 4$ and $\theta_1, \dots, \theta_3 \in \mathbb{R}$. Which of the following parametric functions is estimable ?
a.
$\theta_1 + \theta_3$
b.
$\theta_2 - \theta_3$
c.
$\theta_2 + \theta_3$
d.
$\theta_1 + \theta_2 + \theta_3$
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Q52
mcq
If $X \sim N_p(0, 1)$ and $A_{p \times p}$ is an independent matrix with rank $(A) = k < p$, then which of the following statement is correct ?
a.
$\frac{X'AX}{X'X} \sim \frac{k}{p} F_{k, p}$
b.
$\frac{X'AX}{X'X} \sim \frac{k}{p - k} F_{k, p - k}$
c.
$\frac{X'AX}{X'X} \sim \beta\left(\frac{k}{2}, \frac{p}{2}\right)$
d.
$\frac{X'AX}{X'X} \sim \beta\left(\frac{k}{2}, \frac{p - k}{2}\right)$
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Q53
mcq
A sample of size $n (\ge 2)$ is drawn from a population of $N (\ge 3)$ units using $PPSWR$ sampling scheme, where $p_i$ is the probability of selecting $i$-th unit in a draw, $0 < p_i < 1 \ \forall\ i = 1, \dots, N$, and $\sum_{i=1}^N p_i = 1$. Then the inclusion probability $\pi_{ij}$ is :
a.
$1 - p_i^n - p_j^n + (p_i + p_j)^n$
b.
$1 - (p_i + p_j - p_i p_j)^n$
c.
$1 - (1 - p_i)^n - (1 - p_j)^n - (p_i + p_j)^n$
d.
$1 - (1 - p_i)^n - (1 - p_j)^n + (1 - p_i - p_j)^n$
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Q54
mcq
In a $2^4$ experiment with two blocks and factors A, B, C and D, one block contains the following treatment combinations $a, b, c, ad, bd, cd, abc, abcd$. Which of the following effects is confounded ?
a.
ABC
b.
ABD
c.
BCD
d.
ABCD
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Q55
mcq
In an airport, domestic passengers and international passengers arrive independently according to Poisson processes with rates 100 and 70 per hour, respectively. If it is given that the total number of passengers (domestic and international) arriving in that airport between 9:00 AM and 11:00 AM on a particular day was 520, then what is the conditional distribution of the number of domestic passengers arriving in this period ?
a.
Poisson (200)
b.
Poisson (100)
c.
Binomial$\left(520, \frac{10}{17}\right)$
d.
Binomial$\left(520, \frac{7}{17}\right)$
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Q56
mcq
Let $X \ge 0$ be a random variable on $(\Omega, F, P)$ with $\mathbb{E}(X) = 1$. Let $A \in F$ be an event with $0 < P(A) < 1$. Which of the following defines another probability measure on $(\Omega, F)$ ?
a.
$Q(B) = P(A \cap B) \ \forall\ B \in F$
b.
$Q(B) = P(A \cup B) \ \forall\ B \in F$
c.
$Q(B) = \mathbb{E}(X I_B) \ \forall\ B \in F$
d.
$Q(B) = \begin{cases} P(A \mid B) & \text{if } P(B) > 0 \\ 0 & \text{if } P(B) = 0 \end{cases}$
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Q57
mcq
Let $X$ and $Y$ be i.i.d. random variables uniformly distributed on $(0, 4)$. Then $P(X > Y \mid X < 2Y)$ is:
a.
$\frac{1}{3}$
b.
$\frac{5}{6}$
c.
$\frac{1}{4}$
d.
$\frac{2}{3}$
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Q58
mcq
Suppose $\{X_n\}$ is a Markov Chain with 3 states and transition probability matrix
$$\begin{pmatrix} 1/3 & 1/3 & 1/3 \\ 1/2 & 1/2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
Then which of the following statements is true ?
a.
$\{X_n\}$ is irreducible
b.
$\{X_n\}$ is recurrent
c.
$\{X_n\}$ does not admit a stationary probability distribution
d.
$\{X_n\}$ has an absorbing state
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Q59
mcq
Suppose $X \sim \text{Cauchy }(0, 1)$. Then the distribution of $\frac{1 - X}{1 + X}$ is :
a.
Uniform $(0, 1)$
b.
Normal $(0, 1)$
c.
Double exponential $(0, 1)$
d.
Cauchy $(0, 1)$
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Q60
mcq
Given the observations $0.8, 0.71, 0.9, 1.2, 1.68, 1.4, 0.88, 1.62$ from the uniform distribution on $(\theta - 0.2, \theta + 0.8)$ with $-\infty < \theta < \infty$, which of the following is a maximum likelihood estimate for $\theta$ ?
a.
$0.7$
b.
$0.9$
c.
$1.1$
d.
$1.3$
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Q61
mcq
Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function given by, $f(x, y) = (x^3 + 3xy^2 - 15x - 12y, x + y)$.
Let $S = \{(x, y) \in \mathbb{R}^2 : f \text{ is locally invertible at } (x, y)\}$. Then,
a.
$S = \mathbb{R}^2 \setminus \{(0, 0)\}$
b.
$S$ is open in $\mathbb{R}^2$
c.
$S$ is dense in $\mathbb{R}^2$
d.
$\mathbb{R}^2 \setminus S$ is countable
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Q62
mcq
Let $X = N$, the set of positive integers. Consider the metrics $d_1, d_2$ on $X$ given by:
$$d_1(m, n) = |m - n|, m, n \in X$$
$$d_2(m, n) = \left|\frac{1}{m} - \frac{1}{n}\right|, m, n \in X$$
Let $X_1, X_2$ denote the metric spaces $(X, d_1), (X, d_2)$ respectively. Then,
a.
$X_1$ is complete
b.
$X_2$ is complete
c.
$X_1$ is totally bounded
d.
$X_2$ is totally bounded
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Q63
mcq
Let $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a linear map that satisfies $T^2 = T - I_n$. Then which of the following are true ?
a.
$T$ is invertible
b.
$T - I_n$ is not invertible
c.
$T$ has a real eigen value
d.
$T^3 = -I_n$
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Q64
mcq
Let $M = \begin{bmatrix} 2 & 0 & 3 & 2 & 0 & -2 \\ 0 & 1 & 0 & -1 & 3 & 4 \\ 0 & 0 & 1 & 0 & 4 & 4 \\ 1 & 1 & 1 & 0 & 1 & 1 \end{bmatrix}$, $b_1 = \begin{bmatrix} 5 \\ 1 \\ 1 \\ 4 \end{bmatrix}$ and $b_2 = \begin{bmatrix} 5 \\ 1 \\ 3 \\ 3 \end{bmatrix}$. Then which of the following are true ?
a.
Both systems $MX = b_1$ and $MX = b_2$ are inconsistent.
b.
Both systems $MX = b_1$ and $MX = b_2$ are consistent.
c.
The system $MX = b_1 - b_2$ is consistent.
d.
The systems $MX = b_1 - b_2$ is inconsistent.
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Q65
mcq
Let $M = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & 4 \\ -2 & 1 & -4 \end{bmatrix}$. Given that $1$ is an eigenvalue of $M$, then which among the following are correct ?
a.
The minimal polynomial of $M$ is $(X - 1)(X + 4)$.
b.
The minimal polynomial of $M$ is $(X - 1)^2(X + 4)$.
c.
$M$ is not diagonalizable.
d.
$M^{-1} = \frac{1}{4}(M + 3I)$
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Q66
mcq
Let $A$ be a real matrix with characteristic polynomial $(X - 1)^3$. Pick the correct statements from below:
a.
$A$ is necessarily diagonalizable.
b.
If the minimal polynomial of $A$ is $(X - 1)^3$, then $A$ is diagonalizable.
c.
Characteristic polynomial of $A^2$ is $(X - 1)^3$.
d.
If $A$ has exactly two Jordan blocks, then $(A - 1)^2$ is diagonalizable.
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Q67
mcq
Let $P_3$ be the vector space of polynomials with real coefficients and of degree at most $3$. Consider the linear map $T : P_3 \to P_3$ defined by $T(p(x)) = p(x + 1) + p(x - 1)$. Which of the following properties does the matrix of $T$ (with respect to the standard basis $B = \{1, x, x^2, x^3\}$ of $P_3$) satisfy ?
a.
$\det T = 0$
b.
$(T - 2I)^4 = 0$ but $(T - 2I)^3 \neq 0$
c.
$(T - 2I)^3 = 0$ but $(T - 2I)^2 \neq 0$
d.
$2$ is an eigenvalue with multiplicity $4$.
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Q68
mcq
Let $M$ be an $n \times n$ Hermitian matrix of rank $k$, $k \neq n$. If $\lambda \neq 0$ is an eigenvalue of $M$ with corresponding unit column vector $u$, with $Mu = \lambda u$, then which of the following are true ?
a.
$\text{rank }(M - \lambda u u^*) = k - 1$
b.
$\text{rank }(M - \lambda u u^*) = k$
c.
$\text{rank }(M - \lambda u u^*) = k + 1$
d.
$(M - \lambda u u^*)^n = M^n - \lambda^n u u^*$
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Q69
mcq
Define a real valued function $B$ on $\mathbb{R}^2 \times \mathbb{R}^2$ as follows. If $v = (x_1, x_2)$, $w = (y_1, y_2)$ belong to $\mathbb{R}^2$ define $B(u, w) = x_1 y_1 - x_1 y_2 - x_2 y_1 + 4x_2 y_2$. Let $v_0 = (1, 0)$ and let $W = \{v \in \mathbb{R}^2 : B(v_0, v) = 0\}$. Then $W$—
a.
is not a subspace of $\mathbb{R}^2$
b.
equals $\{(0, 0)\}$
c.
is the $y$-axis
d.
is the line passing through $(0, 0)$ and $(1, 1)$
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Q70
mcq
Consider the Quadratic forms:
$$Q_1(x, y) = xy$$
$$Q_2(x, y) = x^2 + 2xy + y^2$$
$$Q_3(x, y) = x^2 + 3xy + 2y^2$$
on $\mathbb{R}^2$. Choose the correct statements from below :
a.
$Q_1$ and $Q_2$ are equivalent
b.
$Q_1$ and $Q_3$ are equivalent
c.
$Q_2$ and $Q_3$ are equivalent
d.
All are equivalent
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Q71
mcq
Let $\{u_n\}_{n \ge 1}$ be a sequence of real numbers satisfying the following conditions:
(1) $(-1)^n u_n \ge 0$, for all $n \ge 1$
(2) $|u_{n+1}| < \frac{|u_n|}{2}$, for all $n \ge 13$
Which of the following statements are necessarily true ?
a.
$\sum_{n \ge 1} u_n$ does not converge in $\mathbb{R}$.
b.
$\sum_{n \ge 13} u_n$ converges to zero.
c.
$\sum_{n \ge 13} u_n$ converges to a non-zero real number.
d.
If $|u_{n-1}| < \frac{|u_n|}{2}$, for all $2 \le n \le 13$, then $\sum_{n \ge 1} u_n$ is a negative real number.
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Q72
mcq
Let $S$ be an infinite set. Which of the following statements are true ?
a.
If there is an injection from $S$ to $\mathbb{N}$, then $S$ is countable.
b.
If there is a surjection from $S$ to $\mathbb{N}$, then $S$ is countable.
c.
If there is an injection from $\mathbb{N}$ to $S$, then $S$ is countable.
d.
If there is a surjection from $\mathbb{N}$ to $S$, then $S$ is countable.
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Q73
mcq
Let $p_n$ denote the $n$-th prime number, when we enumerate the prime numbers in the increasing order. For example, $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, and so on. Let $S = \{s_n = p_{n+1} - p_n \mid n \in \mathbb{N}, n \ge 1\}$. Then which of the following are correct ?
a.
$\sup S = \infty$
b.
$\lim\sup_{n\to\infty} s_n = \infty$
c.
$\inf S < \infty$ and $\inf S = 1$
d.
$\lim\sup_{n\to\infty} s_n \ge 2$
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Q74
mcq
For $n \ge 1$, consider the sequence of functions:
$$f_n(x) = \frac{1}{2nx + 1}, \; g_n(x) = \frac{x}{2nx + 1}$$
on the open interval $(0, 1)$. Consider the statements:
(1) The sequence $\{f_n\}$ converges uniformly on $(0, 1)$.
(2) The sequence $\{g_n\}$ converges uniformly on $(0, 1)$.
Then,
a.
(1) is true
b.
(1) is false
c.
(1) is false and (2) is true
d.
Both (1) and (2) are true.
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Q75
mcq
Suppose that $\{f_n\}$ is a sequence of continuous real valued functions on $[0, 1]$ satisfying the following:
(1) $\forall\ x \in \mathbb{R}$, $\{f_n(x)\}$ is a decreasing sequence.
(2) The sequence $\{f_n\}$ converges uniformly to 0.
Let $g_n(x) = \sum_{k=1}^n (-1)^k f_k(x) \; \forall\ x \in \mathbb{R}$. Then —
a.
$\{g_n\}$ is Cauchy with respect to the sup norm.
b.
$\{g_n\}$ is uniformly convergent.
c.
$\{g_n\}$ need not converge pointwise.
d.
$\exists M > 0$ such that $|g_n(x)| \le M, \forall\ n \in \mathbb{N}, \forall\ x \in \mathbb{R}$.
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Q76
mcq
Given $f : \left[\frac{1}{2}, 2\right] \to \mathbb{R}$, a strictly increasing function, we put $g(x) = f(x) + f\left(\frac{1}{x}\right), x \in [1, 2]$. Consider a partition $P$ of $[1, 2]$ and let $U(P, g)$ and $L(P, g)$ denote the upper Riemann sum and lower Riemann sum of $g$. Then,
a.
for a suitable $f$ we can have $U(P, g) = L(P, g)$.
b.
for a suitable $f$ we can have $U(P, g) \ne L(P, g)$.
c.
$U(P, g) \ge L(P, g)$ for all choices of $f$.
d.
$U(P, g) < L(P, g)$ for all choices of $f$.
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Q77
mcq
Let $f$ be a real valued continuously differentiable function of $(0, 1)$. Set $g = f' + if$, where $i^2 = -1$ and $f'$ is the derivative of $f$. Let $a, b \in (0, 1)$ be two consecutive zeros of $f$. Which of the following statements are necessarily true ?
a.
If $g(a) > 0$, then $g$ crosses the real line from upper half plane to lower half plane at $a$.
b.
If $g(a) > 0$, then $g$ crosses the real line from lower half plane to upper half plane at $a$.
c.
If $g(a) g(b) \neq 0$, then $g(a), g(b)$ have the same sign.
d.
If $g(a) g(b) \neq 0$, then $g(a), g(b)$ have opposite signs.
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Q78
mcq
Let $A$ be an invertible real $n \times n$ matrix. Define a function $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ by $F(x, y) = \langle Ax, y \rangle$, where $\langle x, y \rangle$ denotes the inner product of $x$ and $y$. Let $DF(x, y)$ denote the derivative of $F$ at $(x, y)$ which is a linear transformation from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$. Then,
a.
If $x \neq 0$, then $DF(x, 0) \neq 0$
b.
If $y \neq 0$, then $DF(0, y) \neq 0$
c.
If $(x, y) \neq (0, 0)$ then $DF(x, y) \neq 0$
d.
If $x = 0$ or $y = 0$, then $DF(x, y) = 0$
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Q79
mcq
For any group $G$, let $\text{Aut}(G)$ denote the group of automorphisms of $G$. Which of the following are true ?
a.
If $G$ is finite, then $\text{Aut}(G)$ is finite.
b.
If $G$ is cyclic, then $\text{Aut}(G)$ is cyclic.
c.
If $G$ is infinite, then $\text{Aut}(G)$ is infinite.
d.
If $\text{Aut}(G)$ is isomorphic to $\text{Aut}(H)$, where $G$ and $H$ are two groups, then $G$ is isomorphic to $H$.
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Q80
mcq
Let $G$ be a group with the following property:
Given any positive integers $m, n$ are $r$ there exist elements $g$ and $h$ in $G$ such that $\text{order}(g) = m$, $\text{order}(h) = n$ and $\text{order}(gh) = r$. Then which of the following are necessarily true ?
a.
$G$ has to be an infinite group.
b.
$G$ cannot be a cyclic group.
c.
$G$ has infinitely many cyclic subgroups.
d.
$G$ has to be a non-abelian group.
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Q81
mcq
Let $R$ be the ring $\mathbb{C}[x]/(x^2 + 1)$. Pick the correct statements from below:
a.
$\dim_{\mathbb{C}} R = 3$
b.
$R$ has exactly two prime ideals
c.
$R$ is a UFD
d.
$(x)$ is a maximal ideal of $R$
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Q82
mcq
Let $f(x) = x^7 - 105x + 12$. Then which of the following are correct ?
a.
$f(x)$ is reducible over $\mathbb{Q}$.
b.
There exists an integer $m$ such that $f(m) = 105$.
c.
There exists an integer $m$ such that $f(m) = 2$.
d.
$f(m)$ is not a prime number for any integer $m$.
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Q83
mcq
Let $\alpha = \sqrt[5]{2} \in \mathbb{R}$ and $\xi = \exp\left(\frac{2\pi i}{5}\right)$. Let $K = \mathbb{Q}(\alpha\xi)$. Pick the correct statements from below: $\sigma$ of $E$.
a.
There exists a field automorphism $\sigma$ of $\mathbb{C}$ such that $\sigma(K) = K$ and $\sigma \neq id$.
b.
There exists a field automorphism $\sigma$ of $\mathbb{C}$ such that $\sigma(K) \neq K$.
c.
There exists a finite extension $E$ of $\mathbb{Q}$ such that $K \subseteq E$ and $\sigma(K) \subseteq E$ for every field automorphism
d.
For all field automorphisms $\sigma$ of $K$, $\sigma(\alpha\xi) = \alpha\xi$.
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Q84
mcq
Let $X = \{(x_i)_{i \geq 1} : x_i \in \{0, 1\} \text{ for all } i \geq 1\}$ with the metric $d((x_i), (y_i)) = \sum_{i \geq 1} |x_i - y_i| 2^{-i}$. Let $f : X \rightarrow [0, 1]$ be the function defined by $f(x_i)_{i \geq 1} = \sum_{i \geq 1} x_i 2^{-i}$. Choose the correct statements from below:
a.
$f$ is continuous
b.
$f$ is onto
c.
$f$ is one-to-one
d.
$f$ is open
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Q85
mcq
Let $A$ be a subset of $\mathbb{R}$ satisfying $A = \bigcap_{n \geq 1} V_n$, where for each $n \geq 1$, $V_n$ is an open dense subset of $\mathbb{R}$. Which of the following are correct ?
a.
$A$ is a non-empty set.
b.
$A$ is countable.
c.
$A$ is uncountable.
d.
$A$ is dense in $\mathbb{R}$.
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Q86
mcq
Let $H$ denote the upper half plane, that is, $H = \{z = x + iy : y > 0\}$. For $z \in H$, which of the following are true ?
a.
$\frac{1}{z} \in H$
b.
$\frac{1}{z^2} \in H$
c.
$\frac{-z}{z+1} \in H$
d.
$\frac{z}{2z+1} \in H$
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Q87
mcq
Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function. Then which of the following statements are true ?
a.
If $|f(z)| \leq 1$ for all $z \in \mathbb{C}$, then $f'$ has infinitely many zeros in $\mathbb{C}$.
b.
If $f$ is onto, then the function $f(\cos z)$ is onto.
c.
If $f$ is onto, then the function $f(e^z)$ is onto.
d.
If $f$ is one-one, then the function $f(z^4 + z + 2)$ is one-one.
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Q88
mcq
Consider the entire functions $f(z) = 1 + z + z^{20}$ and $g(z) = e^z$, $z \in \mathbb{C}$. Which of the following statements are true ?
a.
$\lim_{|z| \to \infty} |f(z)| = \infty$
b.
$\lim_{|z| \to \infty} |g(z)| = \infty$
c.
$f^{-1}(\{z \in \mathbb{C} : |z| \leq R\})$ is bounded for every $R > 0$.
d.
$g^{-1}(\{z \in \mathbb{C} : |z| \leq R\})$ is bounded for every $R > 0$.
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Q89
mcq
Which of the following statements are true ?
a.
$\tan z$ is an entire function.
b.
$\tan z$ is a meromorphic function on $\mathbb{C}$.
c.
$\tan z$ has an isolated singularity at $\infty$.
d.
$\tan z$ has a non-isolated singularity at $\infty$.
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Q90
mcq
Let $a_1 < a_2 < \dots < a_{51}$ be given distinct natural numbers such that $1 \le a_i \le 100$ for all $i = 1, 2, \dots, 51$. Then which of the following are correct ?
a.
There exist $i$ and $j$ with $1 \le i < j \le 51$ satisfying $a_i$ divides $a_j$
b.
There exist $i$ with $1 \le i \le 51$ such that $a_i$ is an odd integer.
c.
There exists $j$ with $1 \le j \le 51$ such that $a_j$ is an even integer.
d.
There exist $i < j$ such that $|a_i - a_j| > 51$.
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Q91
mcq
Let $u(x, t)$ be a function that satisfies the PDE : $u_t + uu_x = 1$, $x \in \mathbb{R}$, $t > 0$ and the initial condition $u\left(\frac{t^2}{4}, t\right) = \frac{t}{2}$. Then the IVP has :
a.
only one solution.
b.
two solutions.
c.
an infinite number of solutions.
d.
solutions none of which is differentiable on the characteristic base curve.
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Q92
mcq
Let $f : [0, 1] \to [0, 1]$ be twice continuously differentiable function with a unique fixed point $f(x_*) = x_*$. For a given $x_0 \in (0, 1)$ consider the iteration $x_{n+1} = f(x_n)$ for $n \ge 0$. If $L = \max_{x \in [0, 1]} |f'(x)|$, then which of the following are true ?
a.
If $L < 1$, then $x_n$ converges to $x_*$
b.
$x_n$ converges to $x_*$ provided $L \ge 1$
c.
The error $e_n = x_n - x_*$ satisfies $|e_{n+1}| < L|e_n|$.
d.
If $f'(x_*) = 0$, then $|e_{n+1}| < C|e_n|^2$ for some $C > 0$.
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Q93
mcq
Let $u(x)$ satisfy the boundary value problem
$$(BVP) \begin{cases} u'' + u' = 0, & x \in (0, 1) \\ u(0) = 0 \\ u(1) = 1 \end{cases}$$
Consider the finite difference approximation to (BVP)
$$(BVP)_n \begin{cases} \frac{U_{j+1} - 2U_j + U_{j-1}}{h^2} + \frac{U_{j+1} - U_{j-1}}{2h} = 0, & j = 1, \dots, N-1 \\ U_0 = 0 \\ U_N = 1 \end{cases}$$
Here, $U_j$ is an approximation to $u(x_j)$, where $x_j = jh$, $j = 0, \dots, N$ is a partition of $[0, 1]$ with $h = 1/N$ for some positive integer $N$. Then which of the following are true ?
a.
There exists a solution to $(BVP)_h$ of the form $U_j = ar^j + b$ for some $a, b \in \mathbb{R}$ with $r \neq 1$ and $r$ satisfying $(2+h)r^2 - 4r + (2-h) = 0$.
b.
$U_j = \frac{(r^j - 1)}{(r^N - 1)}$, where $r$ satisfies $(2+h)r^2 - 4r + (2-h) = 0$ and $r \neq 1$.
c.
$u$ is monotonic in $x$
d.
$U_j$ is monotonic in $j$.
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Q94
mcq
Consider the functional $J[y] = \int_0^1 [(y')^2 - (y')^4] dx$ subject to $y(0) = 0$, $y(1) = 0$. A broken extremal is a continuous extremal whose derivative has jump discontinuities at a finite number of points. Then which of the following statements are true ?
a.
There are no broken extremals and $y = 0$ is an extremal.
b.
There is a unique broken extremal.
c.
There exists more than one and finitely many broken extremals.
d.
There exist infinitely many broken extremals.
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Q95
mcq
The extremals of the functional $J[y] = \int_0^1 [720x^2y - (y'')^2]dx$, subject to $y(x) = y'(0) = y(1) = 0$, $y'(1) = 6$, are:
a.
$x^6 + 2x^3 - 3x^2$
b.
$x^5 + 4x^4 - 5x^3$
c.
$x^5 + x^4 - 2x^3$
d.
$x^6 + 4x^3 - 6x^2$
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Q96
mcq
If $\phi$ is the solution of $\phi(x) = 1 - 2x - 4x^2 + \int_0^x [3 + 6(x-t) - 4(x-t)^2]\phi(t)\,dt$, then $\phi(\log 2)$ is equal to:
a.
$2$
b.
$4$
c.
$6$
d.
$8$
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Q97
mcq
A characteristic number and the corresponding eigenfunction of the homogeneous Fredholm integral equation with kernel
$$K(x, t) = \begin{cases} x(t-1) &; \ 0 \le x \le t \\ t(x-1) &; \ t \le x \le 1 \end{cases}$$
are :
a.
$\lambda = -\pi^2$, $\phi(x) = \sin \pi x$
b.
$\lambda = -2\pi^2$, $\phi(x) = \sin 2\pi x$
c.
$\lambda = -3\pi^2$, $\phi(x) = \sin 3\pi x$
d.
$\lambda = -4\pi^2$, $\phi(x) = \sin 2\pi x$
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Q98
mcq
Consider a point mass of mass $m$ which is attached to a mass-less rigid rod of length $a$. The other end of the rod is made to move vertically such that its downward displacement from the origin at time $t$ is given by
$$z(t) = z_0 \cos (\omega t)$$
The mass is moving in a fixed plane and its position vector at time $t$ is given by:
$$\bar{r}(t) = (a \sin \theta(t), z(t) + a \cos \theta(t))$$
Then the equation of motion of the point mass is:
a.
$a \frac{d^2\theta}{dt^2} + (g + z_0\omega^2 \cos(\omega t))\sin\theta = 0$
b.
$a \frac{d^2\theta}{dt^2} + (g - z_0\omega \cos(\omega t))\sin\theta = 0$
c.
$a \frac{d^2\theta}{dt^2} + (g + z_0^2\omega^2 \cos(\omega t))\cos\theta = 0$
d.
$a \frac{d^2\theta}{dt^2} + (g - z_0\omega^2 \cos(\omega t))\cos\theta = 0$
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Q99
mcq
Three solutions of a certain second order non-homogeneous linear differential equation are :
$$y_1(x) = 1 + x e^{x^2}, \ y_2(x) = (1+x) e^{x^2} - 1, \ y_3(x) = 1 + e^{x^2}$$
Which of the following is/are general solution(s) of the differential equation ?
a.
$(C_1 + 1) y_1 + (C_2 - C_1) y_2 - C_2 y_3$, where $C_1$ and $C_2$ are arbitrary constants.
b.
$C_1(y_1 - y_2) + C_2(y_2 - y_3)$, where $C_1$ and $C_2$ are arbitrary constants.
c.
$C_1(y_1 - y_2) + C_2(y_2 - y_3) + C_3(y_3 - y_1)$, where $C_1, C_2$ and $C_3$ are arbitrary constants.
d.
$C_1(y_1 - y_3) + C_2(y_3 - y_2) + y_1$, where $C_1$ and $C_2$ are arbitrary constants.
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Q100
mcq
The method of variation of parameters to solve the differential equation $y'' + p(x) y' + q(x) y = r(x)$, where $x \in I$ and $p(x), q(x), r(x)$ are non-zero continuous functions on an interval $I$, seeks a particular solution of the form $y_p(x) = v_1(x) y_1(x) + v_2(x) y_2(x)$, where $y_1$ and $y_2$ are linearly independent solutions of $y'' + p(x) y' + q(x) y = 0$ and $v_1(x)$ and $v_2(x)$ are functions to be determined. Which of the following statements are necessarily true ?
a.
The Wronskian of $y_1$ and $y_2$ is never zero in $I$.
b.
$v_1, v_2$ and $v_1 y_1 + v_2 y_2$ are twice differentiable.
c.
$v_1$ and $v_2$ may not be twice differentiable, but $v_1 y_1 + v_2 y_2$ is twice differentiable.
d.
The solution set of $y'' + p(x) y' + q(x) y = r(x)$ consists of function of the form $a y_1 + b y_2 + y_p$, where $a, b \in \mathbb{R}$ are arbitrary constants.
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Q101
mcq
Consider the eigenvalue problem,
$$y'' + \lambda y = 0 \text{ for } x \in (-1, 1)$$
$$y(-1) = y(1)$$
$$y'(-1) = y'(1)$$
Which of the following statements are true ?
a.
All eigenvalues are strictly positive.
b.
All eigenvalues are non-negative.
c.
Distinct eigenfunctions are orthogonal in $L^2[-1, 1]$.
d.
The sequence of eigenvalues is bounded above.
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Q102
mcq
Consider the IVP :
$$x u_x + t u_t = u + 1, \ x \in \mathbb{R}, \ t \ge 0$$
$$u(x, t) = x^2, \ t = x^2$$
Then,
a.
the solution is singular at $(0, 0)$.
b.
the given space curve $(x, t, u) = (\xi, \xi^2, \xi^2)$ is not a characteristic curve at $(0, 0)$.
c.
there is no base-characteristic curve in the $(x, t)$ plane passing through $(0, 0)$.
d.
a necessary condition for the IVP to have a unique $C^1$ solution at $(0, 0)$ does not hold.
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