CSIR NET · Mathematical Sciences
June
2016
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82 questions
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Q1
mcq
$\lim_{n \to \infty} \left(1 - \frac{1}{n^2}\right)^n$ equal
a.
$1$
Correct Answer
b.
$e^{-1/2}$
c.
$e^{-2}$
d.
$e^{-1}$
Step-by-Step AI Explanation
Verified Method
We can evaluate the limit by factoring the base:
$$\left(1 - \frac{1}{n^2}\right)^n = \left(1 - \frac{1}{n}\right)^n \left(1 + \frac{1}{n}\right)^n$$
Taking the limit as $n \to \infty$:
$$\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = e^{-1} \quad \text{and} \quad \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e^1$$
Therefore,
$$\lim_{n \to \infty} \left(1 - \frac{1}{n^2}\right)^n = e^{-1} \cdot e^1 = e^0 = 1$$
Hence, the correct option is A.
Q2
mcq
Consider the interval $(-1, 1)$ and a sequence $\{\alpha_n\}_{n=1}^\infty$ of elements in it. Then,
a.
Every limit point of $\{\alpha_n\}$ is in $(-1, 1)$
b.
Every limit point of $\{\alpha_n\}$ is in $[-1, 1]$
Correct Answer
c.
The limit points of $\{\alpha_n\}$ can only be in $\{-1, 0\text{C}, 1\}$
d.
The limit points of $\{\alpha_n\}$ cannot be in $\{-1, 0, 1\}$
Step-by-Step AI Explanation
Verified Method
Since each term $\alpha_n \in (-1, 1)$, we have $-1 < \alpha_n < 1$ for all $n \in \mathbb{N}$. If $L$ is any limit point (subsequential limit) of the sequence $\{\alpha_n\}$, then there exists a subsequence $\{\alpha_{n_k}\}$ such that $\lim_{k \to \infty} \alpha_{n_k} = L$. By the properties of limits under weak inequalities, since $-1 < \alpha_{n_k} < 1$, it follows that $-1 \le L \le 1$. Therefore, every limit point must lie in the closure of $(-1, 1)$, which is the closed interval $[-1, 1]$. Hence, option B is correct.
Option A is incorrect because a sequence like $\alpha_n = 1 - \frac{1}{n}$ has limit point $1 \notin (-1, 1)$.
Options C and D are incorrect because the set of limit points can contain any point in $[-1, 1]$, for instance $\alpha_n = 0.5$ has $0.5$ as a limit point, which is not restricted to $\{-1, 0, 1\}$, nor are $-1, 0, 1$ forbidden.
Q3
mcq
Let $F : \mathbb{R} \to \mathbb{R}$ be a monotone function. Then
a.
$F$ has no discontinuities.
b.
$F$ has only finitely many discontinuities.
c.
$F$ can have at most countably many discontinuities.
Correct Answer
d.
$F$ can have uncountably many discontinuities.
Step-by-Step AI Explanation
Verified Method
A well-known theorem in real analysis states that any discontinuity of a monotone function $F: \mathbb{R} \to \mathbb{R}$ is a jump discontinuity. For each point of discontinuity $x$, the open interval $(F(x^-), F(x^+))$ (or $(F(x^+), F(x^-))$ if decreasing) is non-empty, and these intervals are mutually disjoint for distinct points of discontinuity. Since each disjoint non-empty interval contains a rational number, and the set of rational numbers $\mathbb{Q}$ is countable, the collection of discontinuities must be at most countable. Thus, C is correct.
Q4
mcq
Consider the function
$$f(x, y) = \frac{x^2}{y^2}, (x, y) \in [1/2, 3/2] \times [1/2, 3/2]$$
The derivative of the function at $(1, 1)$ along the direction $(1, 1)$ is:
a.
$0$
Correct Answer
b.
$1$
c.
$2$
d.
$-2$
Step-by-Step AI Explanation
Verified Method
We are given $f(x, y) = \frac{x^2}{y^2}$. Let us compute the gradient of $f$ at $(1, 1)$:
$$\frac{\partial f}{\partial x} = \frac{2x}{y^2} \implies \frac{\partial f}{\partial x}(1, 1) = 2$$
$$\frac{\partial f}{\partial y} = -\frac{2x^2}{y^3} \implies \frac{\partial f}{\partial y}(1, 1) = -2$$
Thus, $\nabla f(1, 1) = (2, -2)$.
The directional derivative along the direction vector $v = (1, 1)$ (or unit vector $u = \frac{1}{\sqrt{2}}(1, 1)$) is:
$$\nabla f(1, 1) \cdot v = 2(1) + (-2)(1) = 0$$
Even if normalized, $\nabla f(1, 1) \cdot u = \frac{2 - 2}{\sqrt{2}} = 0$.
Hence, the derivative is $0$, making A the correct option.
Q5
mcq
Consider the improper Riemann integral $\int_0^x y^{-1/2} dy$. This integral is:
a.
continuous in $[0, \infty)$
Correct Answer
b.
continuous only in $(0, \infty)$
c.
discontinuous in $(0, \infty)$
d.
discontinuous only in $\left(\frac{1}{2}, \infty\right)$
Step-by-Step AI Explanation
Verified Method
For any $x \ge 0$, we have:
$$F(x) = \int_0^x y^{-1/2} dy = \lim_{\epsilon \to 0^+} \left[ 2y^{1/2} \right]_\epsilon^x = 2\sqrt{x}$$
The function $F(x) = 2\sqrt{x}$ is well-defined and continuous on $[0, \infty)$ (including at $x = 0$ as $\lim_{x \to 0^+} 2\sqrt{x} = 0 = F(0)$).
Therefore, the integral is continuous in $[0, \infty)$, so A is the correct option.
Q6
mcq
Which one of the following statements is true for the sequence of functions
$$f_n(x) = \frac{1}{n^2 + x^2}, n = 1, 2, \dots, x \in [1/2, 1]?$$
a.
The sequence is monotonic and has $0$ as the limit for all $x \in [1/2, 1]$ as $n \to \infty$.
Correct Answer
b.
The sequence is not monotonic but has $f(x) = \frac{1}{x^2}$ as the limit as $n \to \infty$.
c.
The sequence is monotonic and has $f(x) = \frac{1}{x^2}$ as the limit as $n \to \infty$.
d.
The sequence is not monotonic but has $0$ as the limit
Step-by-Step AI Explanation
Verified Method
For any fixed $x \in [1/2, 1]$, as $n$ increases, $n^2 + x^2$ increases strictly with $n$. Therefore, the sequence
$$f_n(x) = \frac{1}{n^2 + x^2}$$
is strictly decreasing (and hence monotonic) with respect to $n$ for each $x \in [1/2, 1]$.
Furthermore, for any $x \in [1/2, 1]$:
$$\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{1}{n^2 + x^2} = 0.$$
Hence, the sequence is monotonic and converges to $0$ pointwise (in fact, uniformly) on $[1/2, 1]$. Thus, option A is correct.
Q7
mcq
Given a $n \times n$ matrix $B$ define $e^B$ by
$$e^B = \sum_{j=0}^\infty \frac{B^j}{j!}$$
Let $p$ be the characteristic polynomial of $B$. Then the matrix $e^{p(B)}$ is
a.
$I_{n \times n}$
Correct Answer
b.
$0_{n \times n}$
c.
$e I_{n \times n}$
d.
$\pi I_{n \times n}$
Step-by-Step AI Explanation
Verified Method
By the Cayley-Hamilton theorem, any square matrix $B$ satisfies its own characteristic equation, meaning $p(B) = 0_{n \times n}$.
Using the definition of the matrix exponential:
$$e^{p(B)} = e^{0_{n \times n}} = \sum_{j=0}^\infty \frac{(0_{n \times n})^j}{j!} = I_{n \times n} + 0 + 0 + \dots = I_{n \times n}$$
Thus, $e^{p(B)} = I_{n \times n}$, and option A is correct.
Q8
mcq
Let $A$ be a $n \times n$ real symmetric non-singular matrix. Suppose there exists $x \in \mathbb{R}^n$ such that $x'Ax < 0$
Then we can conclude that
a.
$\det(A) < 0$
b.
$B = -A$ is positive definite
c.
$\exists y \in \mathbb{R}^n : y'A^{-1}y < 0$
Correct Answer
d.
$\forall y \in \mathbb{R}^n : y'A^{-1}y < 0$
Step-by-Step AI Explanation
Verified Method
Since $A$ is a real symmetric matrix, it has real eigenvalues. The condition that there exists $x \in \mathbb{R}^n$ with $x'Ax < 0$ implies that $A$ has at least one negative eigenvalue, say $\lambda < 0$, with an associated eigenvector $v \neq 0$ such that $Av = \lambda v$.
Then, $A^{-1}v = \frac{1}{\lambda} v$.
Letting $y = v$, we have:
$$y' A^{-1} y = v' \left(\frac{1}{\lambda} v\right) = \frac{1}{\lambda} \|v\|^2 < 0$$
since $\lambda < 0$ and $\|v\|^2 > 0$.
Alternatively, if we let $y = Ax$, then $y'A^{-1}y = (x'A)A^{-1}(Ax) = x'Ax < 0$. Since $A$ is non-singular and $x \neq 0$, $y = Ax \neq 0$. Thus, there exists $y \in \mathbb{R}^n$ such that $y'A^{-1}y < 0$.
- Option A is false in general (e.g., $A = -I_{2 \times 2}$ has $\det(A) = 1 > 0$).
- Option B is false because $A$ could be indefinite (e.g., $A = \operatorname{diag}(1, -1)$).
- Option D is false for the same reason ($A$ could be indefinite).
Therefore, option C is correct.
Q9
mcq
Let $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$. Let $f : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(v, w) = w^T A v$.
Pick the correct statement from below:
a.
There exists an eigenvector $v$ of $A$ such that $Av$ is perpendicular to $v$
b.
The set $\{v \in \mathbb{R}^2 \mid f(v, v) = 0\}$ is a nonzero subspace of $\mathbb{R}^2$
c.
If $v, w \in \mathbb{R}^2$ are nonzero vectors such that $f(v, v) = 0 = f(w, w)$, then $v$ is a scalar multiple of $w$.
d.
For every $v \in \mathbb{R}^2$, there exists a nonzero $w \in \mathbb{R}^2$ such that $f(v, w) = 0$
Correct Answer
Step-by-Step AI Explanation
Verified Method
Let $v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \in \mathbb{R}^2$. Then $f(v, w) = w^T A v = \langle w, Av \rangle$. Since $Av \in \mathbb{R}^2$, the orthogonal complement $(Av)^\perp$ in $\mathbb{R}^2$ has dimension at least $1$ (it is either 1-dimensional if $Av \neq 0$, or the whole space $\mathbb{R}^2$ if $Av = 0$). Thus, there always exists a non-zero vector $w \in \mathbb{R}^2$ orthogonal to $Av$, meaning $w^T A v = 0$, so (D) is correct.
- (A) is false: The eigenvalues of $A$ are $1$ and $-1$, with eigenvectors along the coordinate axes; for any eigenvector, $Av = \lambda v$ with $\lambda = \pm 1 \neq 0$, so $\langle Av, v \rangle = \pm \|v\|^2 \neq 0$.
- (B) is false: $f(v, v) = v_1^2 - v_2^2 = 0 \implies v_1 = \pm v_2$, which is the union of two lines (not closed under addition, hence not a subspace).
- (C) is false: For $v = (1, 1)^T$ and $w = (1, -1)^T$, both satisfy $f(v, v) = f(w, w) = 0$, but they are linearly independent.
Q10
mcq
Let $A$ be a $n \times m$ matrix and $b$ be a $n \times 1$ vector (with real entries). Suppose the equation $Ax = b$, $x \in \mathbb{R}^m$ admits a unique solution. Then we can conclude that
a.
$m \ge n$
b.
$n \ge m$
Correct Answer
c.
$n = m$
d.
$n > m$
Step-by-Step AI Explanation
Verified Method
The linear system $Ax = b$ has a unique solution if and only if the null space of $A$ is trivial, i.e., $\operatorname{nullity}(A) = 0$. By the rank-nullity theorem, $\operatorname{rank}(A) = m$. Since the rank cannot exceed the number of rows or columns, $\operatorname{rank}(A) \le \min(n, m)$. Hence, $m \le n$, which means $n \ge m$.
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Q11
mcq
Let $V$ be the vector space of all real polynomials of degree $\le 10$. Let $T p(x) = p'(x)$ for $p \in V$ be a linear transformation from $V$ to $V$. Consider the basis $\{1, x, x^2, \dots, x^{10}\}$ of $V$. Let $A$ be the matrix of $T$ with respect to this basis. Then
a.
$\text{Trace } A = 1$
b.
$\det A = 0$
c.
there is no $m \in \mathbb{N}$ such that $A^m = 0$
d.
$A$ has a nonzero eigenvalue
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Q12
mcq
Let $x = (x_1, x_2, x_3), y = (y_1, y_2, y_3) ext{ be linearly independent}$.
Let $\delta_1 = x_2 y_3 - y_2 x_3, \delta_2 = x_1 y_3 - y_1 x_3, \delta_3 = x_1 y_2 - y_1 x_2$. If $V$ is the span of $x, y$, then
a.
$V = \{(u, v, w) : \delta_1 u - \delta_2 v + \delta_3 w = 0\}$
b.
$V = \{(u, v, w) : -\delta_1 u + \delta_2 v + \delta_3 w = 0\}$
c.
$V = \{(u, v, w) : \delta_1 u + \delta_2 v - \delta_3 w = 0\}$
d.
$V = \{(u, v, w) : \delta_1 u + \delta_2 v + \delta_3 w = 0\}$
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Q13
mcq
Let $p(x)$ be a polynomial of degree $d \geq 2$. The radius of convergence of the power series $\sum_{n=0}^{\infty} P(n) z^n$ is:
a.
$0$
b.
$1$
c.
$\infty$
d.
dependent on $d$
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Q14
mcq
Let $P(z), Q(z)$ be two complex non-constant polynomials of degree $m, n$ respectively. The number of roots of $P(z) = P(z) Q(z)$ counted with multiplicity is equal to:
a.
$\min \{m, n\}$
b.
$\max \{m, n\}$
c.
$m + n$
d.
$m - n$
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Q15
mcq
The residue of the function $f(z) = e^{e^{-1/z}}$ at $z = 0$ is
a.
$1 + e^{-1}$
b.
$e^{-1}$
c.
$-e^{-1}$
d.
$1 - e^{-1}$
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Q16
mcq
Let $D$ be the open unit disc in $\mathbb{C}$ and $H(D)$ be the collection of all holomorphic functions on it. Let
$$S = \left\{ f \in H(D) : f\left(\frac{1}{2}\right) = \frac{1}{2}, f\left(\frac{1}{4}\right) = \frac{1}{4}, \dots, f\left(\frac{1}{2n}\right) = \frac{1}{2n}, \dots \right\}$$
and
$$T = \left\{ f \in H(D) : f\left(\frac{1}{2}\right) = f\left(\frac{1}{3}\right) = \frac{1}{2}, f\left(\frac{1}{4}\right) = f\left(\frac{1}{5}\right) = \frac{1}{4}, \dots, f\left(\frac{1}{2n}\right) = f\left(\frac{1}{2n+1}\right) = \frac{1}{2n}, \dots \right\}$$
Then
a.
Both $S, T$ are singleton sets
b.
$S$ is a singleton set but $T = \phi$
c.
$T$ is a singleton set but $S = \phi$
d.
Both $S, T$ are empty
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Q17
mcq
Which of the following statements is FALSE? There exists an integer such that:
a.
$x \equiv 23 \bmod 1000$ and $x \equiv 45 \bmod 6789$
b.
$x \equiv 23 \bmod 1000$ and $x \equiv 54 \bmod 6789$
c.
$x \equiv 32 \bmod 1000$ and $x \equiv 54 \bmod 9876$
d.
$x \equiv 32 \bmod 1000$ and $x \equiv 44 \bmod 9876$
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Q18
mcq
Let $p$ be a prime number. How many distinct sub-rings (with unity) of cardinality $p$ does the field $F_{p^2}$ have ?
a.
$0$
b.
$1$
c.
$p$
d.
$p^2$
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Q19
mcq
Let $G = (\mathbb{Z} / 25\mathbb{Z})^*$ be the group of units (i.e. the elements that have a multiplicative inverse) in the ring $(\mathbb{Z} / 25\mathbb{Z})$. Which of the following is a generator of $G$?
a.
$3$
b.
$4$
c.
$5$
d.
$6$
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Q20
mcq
Let $p \ge 5$ be a prime. Then
a.
$\mathbb{F}_p \times \mathbb{F}_p$ has at least five subgroups of order $p$
b.
Every subgroup of $\mathbb{F}_p \times \mathbb{F}_p$ is of the form $\mathrm{H}_1 \times \mathrm{H}_2$ where $\mathrm{H}_1, \mathrm{H}_2$ are subgroups of $\mathbb{F}_p$
c.
Every subgroup of $\mathbb{F}_p \times \mathbb{F}_p$ is an ideal of the ring $\mathbb{F}_p \times \mathbb{F}_p$
d.
The ring $\mathbb{F}_p \times \mathbb{F}_p$ is a field
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Q21
mcq
Let $y_1$ and $y_2$ be two solutions of the problem
$$\left.\begin{aligned} y''(t) + ay'(t) + by(t) = 0, \, t \in \mathbb{R} \\ y(0) = 0 \end{aligned}\right\}$$
where $a$ and $b$ are real constants. Let $w$ be the Wronskian of $y_1$ and $y_2$. Then
a.
$w(t) = 0, \forall t \in \mathbb{R}$
b.
$w(t) = c, \forall t \in \mathbb{R}$ for some positive constant $c$
c.
$w$ is a nonconstant positive function
d.
There exists $t_1, t_2 \in \mathbb{R}$ such that $w(t_1) < 0 < w(t_2)$
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Q22
mcq
Let $A = \begin{bmatrix} -2 & 1 & 0 \\ 0 & -2 & 1 \\ 0 & 0 & -2 \end{bmatrix}$, $x(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix}$ and $|x(t)| = \left(x_1^2(t) + x_2^2(t) + x_3^2(t)\right)^{1/2}$. Then any solution of the first order system of the ordinary differential equation
$$\left.\begin{aligned} x'(t) &= Ax(t) \\ x(0) &= x_0 \end{aligned}\right\} \text{ satisfies}$$
a.
$\lim_{t \to \infty} |x(t)| = 0$
b.
$\lim_{t \to \infty} |x(t)| = \infty$
c.
$\lim_{t \to \infty} |x(t)| = 2$
d.
$\lim_{t \to \infty} |x(t)| = 12$
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Q23
mcq
Let $a, b, c, d$ be four differentiable functions defined on $\mathbb{R}^2$. Then the partial differential equation
$$\left(a(x, y)\frac{\partial}{\partial x} + b(x, y)\frac{\partial}{\partial y}\right)\left(c(x, y)\frac{\partial}{\partial x} + d(x, y)\frac{\partial}{\partial y}\right)u = 0 \text{ is}$$
a.
always hyperbolic
b.
always parabolic
c.
never parabolic
d.
never elliptic
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Q24
mcq
For the Cauchy problem
$$\begin{aligned} u_t - u u_x &= 0, \, x \in \mathbb{R}, t > 0 \\ u(x, 0) &= x, \, x \in \mathbb{R}, \end{aligned}$$
which of the following statements is true ?
a.
The solution exists for all $t > 0$.
b.
The solution $u$ exists for $t < \frac{1}{2}$ and breaks down at $t = \frac{1}{2}$.
c.
The solution $u$ exists for $t < 1$ and breaks down at $t = 1$.
d.
The solution $u$ exists for $t < 2$ and breaks down at $t = 2$.
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Q25
mcq
Let $f(x) = x^2 + 2x + 1$ and the derivative of $f$ at $x = 1$ is approximated by using the central-difference formula
$$f'(1) \approx \frac{f(1+h) - f(1-h)}{2h} \text{ with } h = \frac{1}{2}.$$
Then the absolute value of the error in the approximation of $f'(1)$ is equal to
a.
1
b.
1/2
c.
0
d.
1/12
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Q26
mcq
The curve of fixed length $l$, that joins the points $(0, 0)$ and $(1, 0)$, lies above the x-axis, and encloses the maximum area between itself and the x-axis, is a segment of
a.
a straight line
b.
a parabola
c.
an ellipse
d.
a circle.
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Q27
mcq
Consider the integral equation
$$y(x) = x^3 + \int_0^x \sin(x-t)y(t)\,dt, \quad x \in [0, \pi]$$
Then the value of $y(1)$ is
a.
19/20
b.
1
c.
17/20
d.
21/20
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Q28
mcq
Consider the equations of motion for a system
$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0, \quad i = 1, 2, 3, \dots, n$$
where
$$L = T - V \left[\begin{array}{l}\text{with } T(t, \dot{q}_i, q_i) \text{ as kinetic energy}\\\text{and } V(t, q_i) \text{ as potential energy}\end{array}\right],$$
$q_i$ the generalized coordinates, and $\dot{q}_i$ the generalized velocities. Then the equations of motion in the form as above are
a.
necessarily restricted to a conservative system but there is no unique choice of L
b.
not necessarily restricted to a conservative system and there is a unique choice of L
c.
necessarily restricted to a conservative system and there is a unique choice of L
d.
not necessarily restricted to a conservative system and there is no unique choice of L
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Q29
mcq
Hundred (100) tickets are marked $1, 2, \dots, 100$ and are arranged at random. Four tickets are picked from these tickets and are given to four persons A, B, C and D. What is the probability that A gets the ticket with the largest value (among A, B, C, D) and D gets the ticket with the smallest value (among A, B, C, D)?
a.
$\frac{1}{4}$
b.
$\frac{1}{6}$
c.
$\frac{1}{2}$
d.
$\frac{1}{12}$
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Q30
mcq
Let X and Y be independent and identically distributed random variables such that $P(X = 0) = P(X = 1) = \frac{1}{2}$.
Let $Z = X + Y$ and $W = |X - Y|$. Then which statement is not correct ?
a.
X and W are independent
b.
Y and W are independent
c.
Z and W are uncorrelated
d.
Z and W are independent.
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Q31
mcq
Let $\{X_t\}$ and $\{Y_t\}$ be two independent pure birth processes with birth rates $\lambda_1$ and $\lambda_2$ respectively.
Let $Z_t = X_t + Y_t$. Then
a.
$\{Z_t\}$ is not a pure birth process.
b.
$\{Z_t\}$ is a pure birth process with birth rate $\lambda_1 + \lambda_2$.
c.
$\{Z_t\}$ is a pure birth process with birth rate $\min(\lambda_1, \lambda_2)$.
d.
$\{Z_t\}$ is a pure birth process with birth rate $\lambda_1 \lambda_2$.
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Q32
mcq
Let $X_1 \sim N(0,1)$ and let
$$X_2 = \begin{cases} -X_1, & -2 \le X_1 \le 2 \\ X_1, & \text{otherwise} \end{cases}$$
Then identify the correct statement.
a.
$\text{Corr}(X_1, X_2) = 1$
b.
$X_2$ does not have $N(0, 1)$ distribution.
c.
$(X_1, X_2)$ has a bivariate normal distribution.
d.
$(X_1, X_2)$ does not have a bivariate normal distribution.
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Q33
mcq
Let $X_1, \dots, X_n$ be a random sample from $N(\theta, 1)$, where $\theta \in \{1, 2\}$. Then which of the following statements about the maximum likelihood estimator (MLE) of $\theta$ is correct?
a.
MLE of $\theta$ does not exist.
b.
MLE of $\theta$ is $\overline{X}$.
c.
MLE of $\theta$ exists but it is not $\overline{X}$.
d.
MLE of $\theta$ is an unbiased estimator of $\theta$.
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Q34
mcq
Let $X_1, \dots, X_n$ denote a random sample from a $N(\mu, \sigma^2)$ distribution. Let $\mu \in \mathbb{R}$ be known and $\sigma^2 (>0)$ be unknown. Let $\chi_{n, \alpha/2}^2$ be an upper $(\alpha/2)^{\text{th}}$ percentile point of a $\chi_n^2$ distribution. Then a $100(1-\alpha)\%$ confidence interval $\sigma^2$ for is given by
a.
$\left(\frac{\sum_1^n X_i^2 - \mu^2}{n\chi_{n,\alpha/2}^2}, \frac{\sum_1^n X_i^2 - \mu^2}{n\chi_{n,1-\alpha/2}^2}\right)$
b.
$\left(\frac{\sum_1^n (X_i - \mu)^2}{(n-1)\chi_{(n-1),\alpha/2}^2}, \frac{\sum_1^n (X_i - \mu)^2}{(n-1)\chi_{(n-1),1-\alpha/2}^2}\right)$
c.
$\left(\frac{\sum_1^n (X_i - \bar{X})^2}{n\chi_{n,\alpha/2}^2}, \frac{\sum_1^n (X_i - \bar{X})^2}{n\chi_{n,1-\alpha/2}^2}\right)$
d.
$\left(\frac{\sum_1^n (X_i - \mu)^2}{n\chi_{n,\alpha/2}^2}, \frac{\sum_1^n (X_i - \mu)^2}{n\chi_{n,1-\alpha/2}^2}\right)$
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Q35
mcq
In the context of testing of statistical hypotheses, which one of the following statements is true?
a.
When testing a simple hypothesis $\text{H}_0$ against an alternative simple hypothesis $\text{H}_1$, the likelihood ratio principle leads to the most powerful test.
b.
When testing a simple hypothesis against an alternative simple hypothesis $\text{H}_0$, $\text{H}_1$, $\text{P}[\text{rejecting } \text{H}_0 | \text{ H}_0 \text{ is true}] + \text{P}[\text{accepting } \text{H}_0 | \text{ H}_1 \text{ is true}] = 1$.
c.
For testing a simple hypothesis $\text{H}_0$ against an alternative simple hypothesis $\text{H}_1$, randomized test is used to achieve the desired level of the power of the test.
d.
UMP tests for testing a simple hypothesis $\text{H}_0$ against an alternative composite $\text{H}_1$, always exist.
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Q36
mcq
Let $Y_1, Y_2, Y_3$ uncorrelated observations with common variance $\sigma^2$ and expectations given by
$$\mathbb{E}(Y_1) = \beta_1, \mathbb{E}(Y_2) = \beta_2 \text{ and } \mathbb{E}(Y_3) = \beta_1 + \beta_2$$
where $\beta_1, \beta_2$ are unknown parameters. The best linear unbiased estimator of $\beta_1 + \beta_2$
a.
$Y_3$
b.
$Y_1 + Y_2$
c.
$\frac{1}{2}(Y_1 + Y_2 + 2Y_3)$
d.
$\frac{1}{2}(Y_1 + Y_2 + Y_3)$
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Q37
mcq
Let $\underline{X} \sim N_3(\underline{\mu}, \Sigma)$ where $\underline{\mu} = (1, 1, 1)$ and $\Sigma = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 3 & c \\ 1 & c & 2 \end{pmatrix}$. The value of $c$ such that $X_2$ and $-X_1 + X_2 - X_3$ are independent is
a.
$-2$
b.
$0$
c.
$2$
d.
$1$
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Q38
mcq
A sample of size $n (\ge 2)$ is drawn without replacement from a finite population of size $N$, using an arbitrary sampling scheme. Let $\pi_i$ denote the inclusion probability of the $i$-th unit and $\pi_{ij}$, the joint inclusion probability of units $i$ and $j, 1 \le i < j \le N$. Which of the following statements is always true?
a.
$\sum_{i=1}^N \pi_i = n$
b.
$\sum_{j=1}^N \pi_{ij} = n \pi_i, 1 \le i \le N$
c.
$\pi_{ij} > 0$ for all $i, j, 1 \le i < j \le N$
d.
$\pi_i \pi_j - \pi_{ij} > 0$ for all $i, j, 1 \le i < j \le N$
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Q39
mcq
Consider a series system with two independent components. Let the component lifespan have exponential distribution with density
$$f(x) = \begin{cases} \lambda e^{-\lambda x}, & \lambda > 0, x > 0 \\ 0, & \text{otherwise} \end{cases}$$
If $n$ observations $X_1, X_2, \dots, X_n$ on lifespan of this component are available and
$$\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i,$$
then the maximum likelihood estimator of the reliability of the system is given by
a.
$\left(1 - e^{-t/\bar{X}}\right)^2$
b.
$1 - \left(1 - e^{-t/\bar{X}}\right)^2$
c.
$e^{-2t/\bar{X}}$
d.
$1 - e^{-2t/\bar{X}}$
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Q40
mcq
Customers arrive at an ice cream parlour according to a Poisson process with rate 2. Service time distribution has density function
$$f(x) = \begin{cases} 3e^{-3x}, & x > 0 \\ 0, & x \le 0 \end{cases}$$
Upon being served a customer may rejoin the queue with probability 0.4, independently of new arrivals; also a returning customer's service time is the same as that of a new arriving customer. Customers behave independently of each other. Let $X(t) =$ number of customers in the queue at time $t$. Which among the following is correct?
a.
$\{X(t)\}$ grows without bound with probability 1.
b.
$\{X(t)\}$ has stationary distribution given by $\pi_k = \left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^k, k = 0, 1, 2, \dots$
c.
$\{X(t)\}$ has stationary distribution given by $\pi_k = (0.1)(0.9)^k, k = 0, 1, 2, \dots$
d.
$\{X(t)\}$ has stationary distribution given by $\pi_k = (0.4)(0.6)^k, k = 0, 1, 2, \dots$
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Q41
mcq
Let $x_1 = 0, x_2 = 1$, and for $n \ge 3$, define $x_n = \frac{x_{n-1} + x_{n-2}}{2}$. Which of the following is/are true?
a.
$\{x_n\}$ is a monotone sequence
b.
$\lim_{n \to \infty} x_n = \frac{1}{2}$
c.
$\{x_n\}$ is a Cauchy sequence
d.
$\lim_{n \to \infty} x_n = \frac{2}{3}$
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Q42
mcq
Let $\{x_n\}$ be an arbitrary sequence of real numbers. Then
a.
$\sum_{n=1}^\infty |x_n|^p < \infty$ for some $1 < p < \infty$ implies $\sum_{n=1}^\infty |x_n|^q < \infty$ for any $q > p$.
b.
$\sum_{n=1}^\infty |x_n|^p < \infty$ for some $1 < p < \infty$ implies $\sum_{n=1}^\infty |x_n|^q < \infty$ for any $1 \le q < p$.
c.
Given any $1 < p < q < \infty$, there is a real sequence $\{x_n\}$ such that $\sum_{n=1}^\infty |x_n|^p < \infty$ but $\sum_{n=1}^\infty |x_n|^q = \infty$
d.
Given any $1 < q < p < \infty$, there is a real sequence $\{x_n\}$ such that $\sum_{n=1}^\infty |x_n|^p < \infty$ but $\sum_{n=1}^\infty |x_n|^q = \infty$
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Q43
mcq
Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $f(x+1) = f(x)$ for all $x \in \mathbb{R}$. Then
a.
$f$ is bounded above, but not bounded below
b.
$f$ is bounded above and below, but may not attain its bounds
c.
$f$ is bounded above and below and $f$ attains its bounds
d.
$f$ is uniformly continuous
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Q44
mcq
Take the closed interval $[0,1]$ and open interval $(1/3, 2/3)$. Let $K = [0,1] \setminus (1/3, 2/3)$. For $x \in [0,1]$ define $$f(x) = d(x, K) \text{ where } d(x, K) = \inf \{|x - y| \mid y \in K\}.$$ Then
a.
$f : [0,1] \to \mathbb{R}$ is differentiable at all points of $(0,1)$
b.
$f : [0,1] \to \mathbb{R}$ is not differentiable at $1/3$ and $2/3$
c.
$f : [0,1] \to \mathbb{R}$ is not differentiable at $1/2$
d.
$f : [0,1] \to \mathbb{R}$ is not continuous
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Q45
mcq
Which of the following is/are true?
a.
$(0, 1)$ with the usual topology admits a metric which is complete
b.
$(0, 1)$ with the usual topology admits a metric which is not complete
c.
$[0, 1]$ with the usual topology admits a metric which is not complete
d.
$[0, 1]$ with the usual topology admits a metric which is complete
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Q46
mcq
Let $V$ be the span of $(1, 1, 1)$ and $(0, 1, 1) \in \mathbb{R}^3$. Let $u_1 = (0,0,1), u_2 = (1,1,0)$ and $u_3 = (1,0,1)$. Which of the following are correct?
a.
$(\mathbb{R}^3 \setminus V) \cup \{(0,0,0)\}$ is not connected.
b.
$(\mathbb{R}^3 \setminus V) \cup \{tu_1 + (1-t)u_3 : 0 \le t \le 1\}$ is connected.
c.
$(\mathbb{R}^3 \setminus V) \cup \{tu_1 + (1-t)u_2 : 0 \le t \le 1\}$ is connected.
d.
$(\mathbb{R}^3 \setminus V) \cup \{(t, 2t, 2t) : t \in \mathbb{R}\}$ is connected
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Q47
mcq
Let $A$ be any set. Let $\mathbb{P}(A)$ be the power set of $A$, that is, the set of all subsets of $A$; $\mathbb{P}(A) = \{B : B \subseteq A\}$. Then which of the following is/are true about the set $\mathbb{P}(A)$?
a.
$\mathbb{P}(A) = \Phi$ for some $A$
b.
$\mathbb{P}(A)$ is a finite set for some $A$
c.
$\mathbb{P}(A)$ is a countable set for some $A$
d.
$\mathbb{P}(A)$ is a uncountable set for some $A$
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Q48
mcq
Which of the following functions is/are uniformly continuous on the interval $(0,1)$ ?
a.
$\frac{1}{x}$
b.
$\sin \frac{1}{x}$
c.
$x \sin \frac{1}{x}$
d.
$\frac{\sin x}{x}$
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Q49
mcq
Define $f$ on $[0, 1]$ by $$f(x) = \begin{cases} x^2 & \text{if } x \text{ is rational} \\ x^3 & \text{if } x \text{ is irrational} \end{cases}.$$ Then
a.
$f$ is not Riemann integrable on $[0, 1]$.
b.
$f$ is Riemann integrable and $\int_0^1 f(x) dx = \frac{1}{4}$
c.
$f$ is Riemann integrable and $\int_0^1 f(x) dx = \frac{1}{3}$
d.
$\frac{1}{4} = \int_{\underline{0}}^1 f(x) dx < \int_0^{\overline{1}} f(x) dx = \frac{1}{3}$, where $\int_{\underline{0}}^1 f(x) dx$ and $\int_0^{\overline{1}} f(x) dx$ are the lower and upper Riemann integrals of $f$.
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Q50
mcq
Let $V$ be the vector space of all complex polynomials $p$ with $\deg p \le n$. Let $T\colon V \to V$ be the map $(Tp)(x) = p'(1), x \in \mathbb{C}$. Which of the following are correct?
a.
$\dim \operatorname{Ker} T = n$
b.
$\dim \operatorname{range} T = 1$
c.
$\dim \operatorname{Ker} T = 1$
d.
$\dim \operatorname{range} T = n + 1$
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Q51
mcq
Consider the real vector space $V$ of polynomials of degree less than or equal to $d$. For $p \in V$ define $$\|p\|_k = \max \left\{ |p(0)|, \left|p^{(1)}(0)\right|, \ldots, \left|p^{(k)}(0)\right| \right\}$$ where $p^{(i)}(0)$ is the $i^{\text{th}}$ derivative of $p$ evaluated at $0$. Then $\|p\|_k$ defines a norm on $V$ if and only if
a.
$k \ge d - 1$
b.
$k < d$
c.
$k \ge d$
d.
$k < d - 1$
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Q52
mcq
Let $A, B$ be $n \times n$ real matrices such that $\det A > 0$ and $\det B < 0$. For $0 \le t \le 1$ consider $C(t) = tA + (1-t)B$. Then
a.
$C(t)$ is invertible for each $t \in [0, 1]$
b.
There is a $t_0 \in (0, 1)$ such that $C(t_0)$ is not invertible.
c.
$C(t)$ is not invertible for each $t \in [0, 1]$.
d.
$C(t)$ is invertible for only finitely many $t \in [0, 1]$.
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Q53
mcq
Let $A$ be an $n \times n$ real matrix. Pick the correct answer(s) from the following
a.
$A$ has at least one real eigenvalue.
b.
For all nonzero vectors $v, w \in \mathbb{R}^n$, $(Aw)^T (Av) > 0$
c.
Every eigenvalue of $A^T A$ is a nonnegative real number
d.
$I + A^T A$ is invertible
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Q54
mcq
Let $\{a_1, \dots, a_n\}$ and $\{b_1, \dots, b_n\}$ be two bases of $\mathbb{R}^n$. Let $P$ be an $n \times n$ matrix with real entries such that $Pa_i = b_i$, $i = 1, 2, \dots, n$. Suppose that every eigenvalue of $P$ is either $-1$ or $1$. Let $Q = I + 2P$. Then which of the following statements are true?
a.
$\{a_i + 2b_i \mid i = 1, 2, \dots, n\}$ is also a basis of $V$
b.
$Q$ is invertible.
c.
Every eigenvalue of $Q$ is either $3$ or $-1$.
d.
$\det Q > 0$ if $\det P > 0$.
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Q55
mcq
Let $T$ be a $n \times n$ matrix with the property $T^n = 0$. Which of the following is/are true?
a.
$T$ has $n$ distinct eigenvalues.
b.
$T$ has one eigenvalue of multiplicity $n$.
c.
$0$ is an eigenvalue of $T$.
d.
$T$ is similar to a diagonal matrix
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Q56
mcq
Let $A$ be an $n \times n$ matrix with real entries. Define $\langle x, y \rangle_A := \langle Ax, Ay \rangle$, $x, y \in \mathbb{R}^n$. Then $\langle x, y \rangle_A$ defines an inner-product if and only if
a.
$\operatorname{Ker} A = 0$
b.
$\operatorname{Rank} A = n$
c.
All eigenvalues of $A$ are positive
d.
All eigenvalues of $A$ are non-negative
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Q57
mcq
Suppose $\{v_1, \dots, v_n\}$ are unit vectors in $\mathbb{R}^n$ such that
$$\|v\|^2 = \sum_{i=1}^n |\langle v_i, v \rangle|^2, \quad \forall v \in \mathbb{R}^n$$
Then decide the correct statements in the following
a.
$v_1, \dots, v_n$ are mutually orthogonal.
b.
$\{v_1, \dots, v_n\}$ is a basis for $\mathbb{R}^n$
c.
$v_1, \dots, v_n$ are not mutually orthogonal.
d.
Atmost $n-1$ of the elements in the set $\{v_1, \dots, v_n\}$ can be orthogonal
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Q58
mcq
Let $V = \{f : [0, 1] \to \mathbb{R} \mid f \text{ is a polynomial of degree less than or equal to } n\}$
Let $f_j(x) = x^j$ for $0 \le j \le n$ and let $A$ be the $(n+1) \times (n+1)$ matrix given by $a_{ij} = \int_0^1 f_i(x) f_j(x) dx$. Then which of the following is/are true ?
a.
$\dim V = n$
b.
$\dim V > n$
c.
$A$ is nonnegative definite, i.e., for all $v \in \mathbb{R}^n, \langle Av, v \rangle \geq 0$.
d.
$\det A > 0$
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Q59
mcq
Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function. Suppose that $f = u + iv$ where $u, v$ are the real and imaginary parts of $f$ respectively. Then $f$ is constant if
a.
$\{u(x, y) : z = x + iy \in \mathbb{C}\}$ is bounded.
b.
$\{v(x, y) : z = x + iy \in \mathbb{C}\}$ is bounded.
c.
$\{u(x, y) + v(x, y) : z = x + iy \in \mathbb{C}\}$ is bounded.
d.
$\{u^2(x, y) + v^2(x, y) : z = x + iy \in \mathbb{C}\}$ is bounded.
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Q60
mcq
Let $A = \{z \in \mathbb{C} \mid |z| > 1\}$, $B = \{z \in \mathbb{C} \mid z \neq 0\}$. Which of the following are true?
a.
There is a continuous onto function $f : A \to B$
b.
There is a continuous one to one function $f : B \to A$
c.
There is a nonconstant analytic function $f : B \to A$
d.
There is a nonconstant analytic function $f : A \to B$
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Q61
mcq
Let $H = \{z = x + iy \in \mathbb{C} : y > 0\}$ be the upper half plane and $D = \{z \in \mathbb{C} : |z| < 1\}$ be the open unit disc. Suppose that $f$ is a Mobius transformation, which maps $H$ conformally onto $D$. Suppose that $f(2i) = 0$. Pick each correct statement from below.
a.
$f$ has a simple pole at $z = -2i$
b.
$f$ satisfies $f(i)\overline{f(-i)} = 1$
c.
$f$ has an essential singularity at $z = -2i$
d.
$|f(2 + 2i)| = \frac{1}{\sqrt{5}}$
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Q62
mcq
Consider the function
$$F(z) = \int_{1}^{2} \frac{1}{(x-z)^2} dx, \operatorname{Im}(z) > 0$$
Then there is a meromorphic function $G(z)$ on $\mathbb{C}$ that agrees with $F(z)$ when $\operatorname{Im}(z) > 0$, such that
a.
$1, \infty$ are poles of $G(z)$
b.
$0, 1, \infty$ are poles of $G(z)$
c.
$1, 2$ are poles of $G(z)$
d.
$1, 2$ are simple poles of $G(z)$
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Q63
mcq
Consider the integral $A = \int_0^1 x^n (1-x)^n dx$. Pick each correct statement from below.
a.
$A$ is not a rational number
b.
$0 < A \le 4^{-n}$
c.
$A$ is a natural number.
d.
$A^{-1}$ is a natural number
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Q64
mcq
Let $G$ be a finite abelian group of order $n$. Pick each correct statement from below.
a.
If $d$ divides $n$, there exists a subgroup of $G$ of order $d$.
b.
If $d$ divides $n$, there exists an element of order $d$ in $G$.
c.
If every proper subgroup of $G$ is cyclic, then $G$ is cyclic.
d.
If $H$ is a subgroup of $G$, there exists a subgroup $N$ of $G$ such that $G/N \cong H$.
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Q65
mcq
Consider the symmetric group $S_{20}$ and its subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a 7-Sylow subgroup of $A_{20}$. Pick each correct statement from below:
a.
$|H| = 49$
b.
$H$ must be cyclic
c.
$H$ is a normal subgroup of $A_{20}$
d.
Any 7-Sylow subgroup of $S_{20}$ is a subset of $A_{20}$
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Q66
mcq
Let $p$ be a prime. Pick each correct statement from below. Up to isomorphism,
a.
there are exactly two abelian groups of order $p^2$
b.
there are exactly two groups of order $p^2$
c.
there are exactly two commutative rings of order $p^2$
d.
there is exactly one integral domain of order $p^2$
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Q67
mcq
Let $R$ be a commutative ring with unity, such that $R[X]$ is a UFD. Denote the ideal $(X)$ of $R[X]$ by $I$. Pick each correct statement from below:
a.
$I$ is prime
b.
If $I$ is maximal, then $R[X]$ is a PID
c.
If $R[X]$ is a Euclidean domain, then $I$ is maximal
d.
If $R[X]$ is a PID, then it is a Euclidean domain
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Q68
mcq
Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $\ge 2$. Pick each correct statement from below:
a.
If $f(x)$ is irreducible in $\mathbb{Z}[x]$, then it is irreducible in $\mathbb{Q}[x]$
b.
If $f(x)$ is irreducible in $\mathbb{Q}[x]$, then it is irreducible in $\mathbb{Z}[x]$
c.
If $f(x)$ is irreducible in $\mathbb{Z}[x]$, then for all primes the reduction $\overline{f(x)}$ of $f(x)$ modulo $p$ is irreducible in $\mathbb{F}_p[x]$.
d.
If $f(x)$ is irreducible in $\mathbb{Z}[x]$, then it is irreducible in $\mathbb{R}[x]$
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Q69
mcq
Consider the smallest topology $\tau$ on $\mathbb{C}$ in which all the singleton sets are closed. Pick each correct statement from below:
a.
$(\mathbb{C}, \tau)$ is Hausdorff.
b.
$(\mathbb{C}, \tau)$ is compact
c.
$(\mathbb{C}, \tau)$ is connected.
d.
$\mathbb{Z}$ is dense in $(\mathbb{C}, \tau)$
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Q70
mcq
Let $\{X_\alpha\}_{\alpha \in I}$ be discrete topological spaces and let $X = \prod_{\alpha \in I} X_\alpha$. From the statements given below, pick each statement that implies that the product topology on $X$ equals the discrete topology on $X$
a.
$I$ is finite.
b.
$I$ is countably infinite and $X_\alpha$ are singletons for all but finitely many $\alpha$
c.
$I$ is uncountably infinite and $X_\alpha$ are singletons for all but finitely many $\alpha$
d.
$I$ is infinite and $X_\alpha$ are infinite for all $\alpha$
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Q71
mcq
Let $y : \mathbb{R} \to \mathbb{R}$ be a solution of the ordinary differential equation, $2y'' + 3y' + y = e^{-3x}$, $x \in \mathbb{R}$ satisfying $\lim_{x\to\infty} e^x y(x) = 0$. Then
a.
$\lim_{x\to\infty} e^{2x} y(x) = 0$
b.
$y(0) = \frac{1}{10}$
c.
$y$ is a bounded function on $\mathbb{R}$
d.
$y(1) = 0$
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Q72
mcq
For $\lambda \in \mathbb{R}$, consider the differential equation
$$y'(x) = \lambda \sin(x + y(x)), y(0) = 1$$
Then this initial value problem has :
a.
no solution in any neighbourhood of $0$.
b.
a solution in $\mathbb{R}$ if $|\lambda| < 1$
c.
a solution in a neighbourhood of $0$.
d.
a solution in $\mathbb{R}$ only if $|\lambda| > 1$
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Q73
mcq
The problem
$$\begin{aligned} -y'' + (1+x) y &= \lambda y, \quad x \in (0,1) \\ y(0) &= y(1) = 0 \end{aligned}$$
has a non zero solution
a.
for all $\lambda < 0$
b.
for all $\lambda \in [0,1]$
c.
for some $\lambda \in (2,\infty)$
d.
for a countable number $\lambda$'s
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Q74
mcq
Let $u : \mathbb{R} \times [0,\infty) \to \mathbb{R}$ be a solution of the initial value problem
$$\left.\begin{aligned} u_{tt} - u_{xx} &= 0, & \text{for } (x,t) \in \mathbb{R} \times (0,\infty) \\ u(x,0) &= f(x), & x \in \mathbb{R} \\ u_t(x,0) &= g(x), & x \in \mathbb{R} \end{aligned}\right\}$$
Suppose $f(x) = g(x) = 0$ for $x \notin [0,1]$, then we always have
a.
$u(x,t) = 0$ for all $(x,t) \in (-\infty,0) \times (0,\infty)$
b.
$u(x,t) = 0$ for all $(x,t) \in (1,\infty) \times (0,\infty)$
c.
$u(x,t) = 0$ for all $(x,t)$ satisfying $x+t < 0$
d.
$u(x,t) = 0$ for all $(x,t)$ satisfying $x-t > 1$
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Q75
mcq
Consider the Cauchy problem for the Eikonal equation
$$p^2+q^2=1;\quad p\equiv\frac{\partial u}{\partial x},\quad q\equiv\frac{\partial u}{\partial y}$$
$u(x, y)=0$ on $x+y=1$, $(x, y) \in \mathbb{R}^2$. Then
a.
The Charpit's equations for the differential equation are
$$\frac{dx}{dt}=2p;\quad \frac{dy}{dt}=2q;\quad \frac{du}{dt}=2;\quad \frac{dp}{dt}=-p;\quad \frac{dq}{dt}=-q$$
b.
The Charpit's equations for the differential equation are
$$\frac{dx}{dt}=2p;\quad \frac{dy}{dt}=2q;\quad \frac{du}{dt}=2;\quad \frac{dp}{dt}=0;\quad \frac{dq}{dt}=0$$
c.
$u(1, \sqrt{2})=\sqrt{2}$
d.
$u(1, \sqrt{2})=1$
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Q76
mcq
Let $H(x)$ be the cubic Hermite interpolation of $f(x)=x^4+1$ on the interval $I=[0,1]$ interpolating at $x=0$ and $x=1$. Then
a.
$\max_{x \in I}|f(x)-H(x)|=\frac{1}{16}$
b.
The maximum of $|f(x)-H(x)|$ is attained at $x=\frac{1}{2}$
c.
$\max_{x \in I}|f(x)-H(x)|=\frac{1}{21}$
d.
The maximum of $|f(x)-H(x)|$ is attained at $x=\frac{1}{4}$
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Q77
mcq
Let $f:[0,3] \rightarrow \mathbb{R}$ be defined by $f(x)=|1-|x-2||$ where $|.|$ denotes the absolute value. Then for the numerical approximation of $\int_{0}^{3} f(x)\,dx$, which of the following statements are true?
a.
The composite trapezoid rule with three equal subintervals is exact
b.
The composite midpoint rule with three equal subintervals is exact
c.
The composite trapezoid rule with four equal subintervals is exact
d.
The composite midpoint rule with four equal subintervals is exact
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Q78
mcq
Let $u$ be the solution of the boundary value problem
$$u_{xx}+u_{yy}=0 \text{ for } 0<x, y<\pi$$
$$u(x,0)=0=u(x,\pi) \text{ for } 0\le x\le \pi$$
$$u(0,y)=0,\, u(\pi, y)=\sin y+\sin 2y \text{ for } 0\le y\le \pi$$ then
a.
$u\left(1, \frac{\pi}{2}\right) = (\sinh(\pi))^{-1} \sinh(1)$
b.
$u\left(1, \frac{\pi}{2}\right) = (\sinh(1))^{-1} \sinh(\pi)$
c.
$u\left(1, \frac{\pi}{4}\right) = (\sinh(\pi))^{-1} (\sinh(1)) \frac{1}{\sqrt{2}} + (\sinh(2\pi))^{-1} \sinh(2)$
d.
$u\left(1, \frac{\pi}{4}\right) = (\sinh(1))^{-1} (\sinh(\pi)) \frac{1}{\sqrt{2}} + (\sinh(2))^{-1} \sinh(2\pi)$
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Q79
mcq
Consider the Runge-Kutta method of the form
$$y_{n+1} = y_n + ak_1 + bk_2$$
$$k_1 = hf(x_n, y_n)$$
$$k_2 = hf(x_n + \alpha h, y_n + \beta k_1)$$
to approximate the solution of the initial value problem
$$y'(x) = f(x, y(x)), y(x_0) = y_0$$
Which of the following choices of a, b, $\alpha$ and $\beta$ yield a second order method
a.
$a = \frac{1}{2}, b = \frac{1}{2}, \alpha = 1, \beta = 1$
b.
$a = 1, b = 1, \alpha = \frac{1}{2}, \beta = \frac{1}{2}$
c.
$a = \frac{1}{4}, b = \frac{3}{4}, \alpha = \frac{2}{3}, \beta = \frac{2}{3}$
d.
$a = \frac{3}{4}, b = \frac{1}{4}, \alpha = 1, \beta = 1$
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Q80
mcq
The curve $y = y(x)$ passing through the point $(\sqrt{3}, 1)$ and defined by the following property(Voltera integral equation of the first kind) $\int_0^y \frac{f(v)dv}{\sqrt{y-v}} = 4\sqrt{y}$, where $f(y) = \sqrt{1 + \frac{1}{y'^2}}$, is the part of a
a.
straight line
b.
circle
c.
parabola
d.
cycloid
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Q81
mcq
Let $y = y(x)$ be the extremal of the functional $I[y(x)] = \int_{x_1}^{x_2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$, subject to the condition that the left end of the extremal moves along $y = x^2$ while the right end moves along $x - y = 5$. Then the
a.
shortest distance between the parabola and the straight line is $\left(\frac{19\sqrt{2}}{8}\right)$
b.
slope of the extremal at $(x, y)$ is $\left(-\frac{3}{2}\right)$
c.
point $\left(\frac{3}{4}, 0\right)$ lies on the extremal
d.
extremal is orthogonal to the curve $y = \frac{x}{2}$
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Q82
mcq
A particle of unit mass moves in the direction of -axis such that it has the Lagrangian
$$L = \frac{1}{12} \dot{x}^4 + \frac{1}{2} x\dot{x}^2 - x^2$$
Let $Q = \dot{x}^2 \ddot{x}$ represent a force (not arising from a potential) acting on the particle in the x-direction. If $x(0) = 1$ and $\dot{x}(0) = 1$, then the value of $\dot{x}$ is
a.
some non-zero finite value at $x = 0$.
b.
$1$ at $x = 1$
c.
$\sqrt{5}$ at $x = \frac{1}{2}$
d.
$0$ at $x = \sqrt{\frac{3}{2}}$
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