Given that there are real constants $a, b, c, d$ such that the identity $\lambda x^2 + 2xy + y^2 = (ax + by)^2 + (cx + dy)^2$ holds for all $x, y \in \mathbb{R}$. This implies
a.
$\lambda = -5$
b.
$\lambda \ge 1$
Correct Answer
c.
$0 < \lambda < 1$
d.
there is no such $\lambda \in \mathbb{R}$
Step-by-Step AI Explanation
Verified Method
Expanding the right-hand side gives:
$$(ax + by)^2 + (cx + dy)^2 = (a^2 + c^2)x^2 + 2(ab + cd)xy + (b^2 + d^2)y^2$$
Comparing coefficients with $\lambda x^2 + 2xy + y^2$, we obtain:
1. $\lambda = a^2 + c^2$
2. $ab + cd = 1$
3. $b^2 + d^2 = 1$
By the Cauchy-Schwarz inequality in $\mathbb{R}^2$:
$$(ab + cd)^2 \le (a^2 + c^2)(b^2 + d^2)$$
Substituting the values:
$$1^2 \le \lambda \cdot 1 \implies \lambda \ge 1$$
Hence, $\lambda \ge 1$ must hold.
Q22mcq
Let $\mathbb{R}^n$, $n \ge 2$, be equipped with standard inner product. Let $\{v_1, v_2, ..., v_n\}$ be $n$ columns vectors forming an orthonormal basis of $\mathbb{R}^n$. Let $A$ be the $n \times n$ matrix formed by the column vectors $v_1, ..., v_n$. Then,
a.
$A = A^{-1}$
b.
$A = A^T$
c.
$A^{-1} = A^T$
Correct Answer
d.
$\det(A) = 1$
Step-by-Step AI Explanation
Verified Method
Since the columns $v_1, v_2, \dots, v_n$ form an orthonormal basis of $\mathbb{R}^n$ with respect to the standard inner product, the matrix $A$ is an orthogonal matrix.
For any orthogonal matrix $A$, the inner product of the $i$-th and $j$-th columns is $\delta_{ij}$, which means $A^T A = I$. Since $A$ is square, this is equivalent to $A^{-1} = A^T$.
Q23mcq
Given $\{a_n\}, \{b_n\}$ two monotone sequences of real numbers and that $\sum a_n b_n$ is convergent, which of the following is true ?
a.
$\sum a_n$ is convergent and $\sum b_n$ is convergent.
b.
At least one of $\sum a_n, \sum b_n$ is convergent.
c.
$\{a_n\}$ is bounded and $\{b_n\}$ is bounded.
d.
At least one of $\{a_n\}, \{b_n\}$ is bounded.
No explanation saved for this question yet.
Q24mcq
Let $S = \left\{(x, y) \mid x^2 + y^2 = \frac{1}{n^2}, \text{ where } n \in \mathbb{N} \text{ and either } x \in \mathbb{Q} \text{ or } y \in \mathbb{Q}\right\}$
Here, $\mathbb{Q}$ is the set of rational numbers and $\mathbb{N}$ is the set of positive integers. Which of the following is true ?
a.
$S$ is a finite non empty set
b.
$S$ is countable
Correct Answer
c.
$S$ is uncountable
d.
$S$ is empty
Step-by-Step AI Explanation
Verified Method
For each fixed $n \in \mathbb{N}$, the equation is $x^2 + y^2 = \frac{1}{n^2}$.
- If $x \in \mathbb{Q}$, then $y^2 = \frac{1}{n^2} - x^2$. For each rational $x$, there are at most two real solutions for $y$, namely $y = \pm \sqrt{\frac{1}{n^2} - x^2}$. Since $\mathbb{Q}$ is countable, the set of such pairs $(x, y)$ is at most countable.
- Similarly, if $y \in \mathbb{Q}$, there are at most two real solutions for $x = \pm \sqrt{\frac{1}{n^2} - y^2}$, forming another countable set.
Thus, for each $n \in \mathbb{N}$, the set of points on the circle with at least one rational coordinate is countable. Since $S$ is a countable union (over $n \in \mathbb{N}$) of countable sets, $S$ is countable. Moreover, points like $(1/n, 0) \in S$ guarantee that $S$ is non-empty, so it is an infinite countable set.
Q25mcq
Define the sequence $\{a_n\}$ as follows:
$$a_1 = 1 \text{ and for } n \ge 1, a_{n+1} = (-1)^n \left(\frac{1}{2}\right) \left(|a_n| + \frac{2}{|a_n|}\right)$$
Which of the following is true ?
a.
$\limsup a_n = \sqrt{2}$
b.
$\liminf a_n = -\infty$
c.
$\lim a_n = \sqrt{2}$
d.
$\sup a_n = \sqrt{2}$
No explanation saved for this question yet.
Q26mcq
If $\{x_n\}$ is a convergent sequence in $\mathbb{R}$ and $\{y_n\}$ is a bounded sequence in $\mathbb{R}$, then we can conclude that
a.
$\{x_n + y_n\}$ is convergent
b.
$\{x_n + y_n\}$ is bounded
c.
$\{x_n + y_n\}$ has no convergent subsequence
d.
$\{x_n + y_n\}$ has no bounded subsequence
No explanation saved for this question yet.
Q27mcq
The difference $\log(2) - \sum_{n=1}^{100} \frac{1}{2^n \cdot n}$ is :
a.
less than $0$
b.
greater than $1$
c.
less than $\frac{1}{2^{100} \cdot 101}$
d.
greater than $\frac{1}{2^{100} \cdot 101}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
Using the Taylor series expansion for $-\log(1 - x)$ around $x = 0$:
$$-\log(1 - x) = \sum_{n=1}^{\infty} \frac{x^n}{n} \quad \text{for } |x| < 1$$
Setting $x = \frac{1}{2}$ gives:
$$-\log\left(1 - \frac{1}{2}\right) = \log(2) = \sum_{n=1}^{\infty} \frac{1}{2^n \cdot n}$$
Therefore, the difference is the remainder of the series:
$$R = \log(2) - \sum_{n=1}^{100} \frac{1}{2^n \cdot n} = \sum_{n=101}^{\infty} \frac{1}{2^n \cdot n}$$
Since all terms are strictly positive, the first term of this remainder is $\frac{1}{2^{101} \cdot 101}$, and we have:
$$R > \frac{1}{101} \sum_{n=101}^{\infty} \frac{1}{2^n} = \frac{1}{101} \cdot \frac{1/2^{101}}{1 - 1/2} = \frac{1}{2^{100} \cdot 101}$$
Thus, the difference is strictly greater than $\frac{1}{2^{100} \cdot 101}$.
Q28mcq
Let $f(x, y) = \log\left(\cos^2(e^{x^2})\right) + \sin(x + y)$. Then, $\frac{\partial}{\partial y} \frac{\partial}{\partial x} f(x, y)$ is :
Let $f(x, y) = g(x) + \sin(x + y)$, where $g(x) = \log\left(\cos^2(e^{x^2})\right)$.
First, differentiating with respect to $x$:
$$\frac{\partial f}{\partial x} = g'(x) + \cos(x + y)$$
Next, differentiating the result with respect to $y$, note that $g'(x)$ does not depend on $y$, so its derivative with respect to $y$ is $0$:
$$\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y} [g'(x) + \cos(x + y)] = 0 - \sin(x + y) = -\sin(x + y)$$
Q29mcq
Let $A$ be a $(m \times n)$ matrix and $B$ be a $(n \times m)$ matrix over real numbers with $m < n$. Then,
a.
$AB$ is always non-singular.
b.
$AB$ is always singular.
c.
$BA$ is always non-singular.
d.
$BA$ is always singular.
Correct Answer
Step-by-Step AI Explanation
Verified Method
Here, $A$ is an $m \times n$ matrix and $B$ is an $n \times m$ matrix with $m < n$. The matrix $BA$ is an $n \times n$ matrix.
By rank properties of matrix multiplication:
$$\text{rank}(BA) \le \min(\text{rank}(B), \text{rank}(A)) \le m < n.$$
Since the rank of $BA$ is strictly less than its size $n$, $BA$ cannot have full rank. Hence, $\det(BA) = 0$, which implies that $BA$ is always singular.
On the other hand, $AB$ is of size $m \times m$ and its rank can be $m$ (for instance, if $A = [I_m \mid 0]$ and $B = [I_m \mid 0]^T$, then $AB = I_m$, which is non-singular; but choosing zero matrices makes it singular). Thus, $AB$ is neither always singular nor always non-singular. Therefore, (D) is the correct choice.
Q30mcq
If $A$ is a $(2 \times 2)$ matrix over $\mathbb{R}$ with $\det(A + I) = 1 + \det(A)$, then we can conclude that
a.
$\det(A) = 0$
b.
$A = 0$
c.
$\text{Tr}(A) = 0$
Correct Answer
d.
$A$ is non-singular
Step-by-Step AI Explanation
Verified Method
For any $2 \times 2$ matrix $A$, the characteristic polynomial gives:
$$\det(A - \lambda I) = \lambda^2 - \text{Tr}(A)\lambda + \det(A)$$
Setting $\lambda = -1$:
$$\det(A + I) = (-1)^2 - \text{Tr}(A)(-1) + \det(A) = 1 + \text{Tr}(A) + \det(A)$$
We are given that:
$$\det(A + I) = 1 + \det(A)$$
Equating the two expressions yields:
$$1 + \text{Tr}(A) + \det(A) = 1 + \det(A) \implies \text{Tr}(A) = 0$$
Therefore, we must have $\text{Tr}(A) = 0$.
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Consider the space $S = \{(\alpha, \beta) \mid \alpha, \beta \in \mathbb{Q}\} \subset \mathbb{R}^2$, where $\mathbb{Q}$ is the set of rational numbers. Then,
a.
$S$ is connected in $\mathbb{R}^2$
b.
$S^C$ is connected in $\mathbb{R}^2$
c.
$S$ is closed in $\mathbb{R}^2$
d.
$S^C$ is closed in $\mathbb{R}^2$
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Given that the Lagrangian for the motion of a simple pendulum is :
$$L = \frac{1}{2}ml^2 \dot{\theta}^2 + mgl \cos \theta$$
where $m$ is the mass of the pendulum bob suspended by a string of length $l$, $g$ is the acceleration due to gravity and $\theta$ is the amplitude of the pendulum from the mean position, then a Hamiltonian corresponding to $L$ is :
Consider the ordinary differential equation $y'' + P(x)y' + Q(x)y = 0$, where $P$ and $Q$ are smooth functions. Let $y_1$ and $y_2$ be any two solutions of the ODE. Let $W(x)$ be the corresponding Wronskian. Then which of the following is always true ?
a.
If $y_1$ and $y_2$ are linearly dependent then $\exists x_1, x_2$ such that $W(x_1) = 0$ and $W(x_2) \neq 0$.
b.
If $y_1$ and $y_2$ are linearly independent then $W(x) = 0\ \forall\ x$.
c.
If $y_1$ and $y_2$ are linearly dependent then $W(x) \neq 0\ \forall\ x$.
d.
If $y_1$ and $y_2$ are linearly independent then $W(x) \neq 0\ \forall\ x$.
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The values of a, b, c such that $\int_0^h f(x)dx = h \left\{ af(0) + bf\left(\frac{h}{3}\right) + cf(h) \right\}$ is exact for polynomials $f$ of degree as high as possible are:
A standard fair die is rolled until some face other than 5 or 6 turn up. Let $X$ denote the face value of the last roll, and $A = \{X \text{ is even}\}$ and $B = \{X \text{ is at most 2}\}$. Then,
a.
$P(A \cap B) = 0$
b.
$P(A \cap B) = \frac{1}{6}$
c.
$P(A \cap B) = \frac{1}{4}$
d.
$P(A \cap B) = \frac{1}{3}$
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Let $X$ and $Y$ be i.i.d. uniform $(0, 1)$ random variables. Let $Z = \max(X, Y)$ and $W = \min(X, Y)$. Then $P\left((Z - W) > \frac{1}{2}\right)$ is :
a.
$\frac{1}{2}$
b.
$\frac{3}{4}$
c.
$\frac{1}{4}$
d.
$\frac{2}{3}$
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Suppse that the lifetime of an electric bulb follows an exponential distribution with mean $\theta$ hours. In order to estimate $\theta$, $n$ bulbs are switched on at the same time. After $t$ hours, $n - m(> 0)$ bulbs are found to be in functioning state. If the lifetimes of the other $m(> 0)$ bulbs are noted as $x_1, x_2, \dots, x_m$, respectively, then the maximum likelihood estimate of $\theta$ is given by:
Let $X_1, X_2, \dots, X_n$ be i.i.d. uniform $(\theta_1, \theta_2)$ variables, where $\theta_1 < \theta_2$ are unknown parameters. Which of the following is an ancillary statistic ?
a.
$\frac{X_{(k)}}{X_{(n)}}$ for any $k < n$
b.
$\frac{X_{(n)} - X_{(k)}}{X_{(n)}}$ for any $k > n$
c.
$\frac{X_{(k)}}{X_{(n)} - X_{(k)}}$ for any $k < n$
d.
$\frac{X_{(k)} - X_{(1)}}{X_{(n)} - X_{(k)}}$ for any $k$ where $1 < k < n$
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Consider the problem of estimation of a parameter $\theta$ on the basis of $X$, where $X \sim N(\theta, 1)$ and $-\infty < \theta < \infty$. Under squared error loss, $X$ has uniformly smaller risk than that of $kX$, for
a.
$k < 0$
b.
$0 < k < 1$
c.
$k > 1$
d.
no value of $k$
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To test the equality of effects of 10 schools against all alternatives, we take a random sample of 5 students from each school and note their marks in a common examination. “Between sum of squares” and “total sum of squares” are found to be 180 and 500 respectively. What is the $p$-value for the standard $F$-test ?
a.
$P[F_{4, 45} > 1.5]$
b.
$P[F_{9, 40} \ge 1.6]$
c.
$P[F_{4, 45} \ge 3.6]$
d.
$P[F_{9, 40} \ge 2.5]$
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The covariance matrix of a four dimensional random vector $X$ is of the form:
$$\begin{bmatrix} 1 & \rho & \rho & \rho \\ \rho & 1 & \rho & \rho \\ \rho & \rho & 1 & \rho \\ \rho & \rho & \rho & 1 \end{bmatrix}; \text{ where } \rho < 0.$$
If $v$ is the variance of the first principal component, then
a.
$v$ cannot exceed $5/4$
b.
$v$ can exceed $5/4$, but cannot exceed $4/3$
c.
$v$ can exceed $4/3$, but cannot exceed $3/2$
d.
$v$ can exceed $3/2$
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A simple random sample of size $n$ will be drawn from a class of $125$ students, and the mean mathematics score of the sample will be computed. If the standard error of the sample mean for "with replacement sampling" is twice as much as the standard error of the sample mean for "without replacement" sampling the value of $n$ is:
a.
$32$
b.
$63$
c.
$79$
d.
$94$
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Let $A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & -2 \\ 0 & 0 & 1 \end{pmatrix}$ and define for $x, y, z \in \mathbb{R}$; $Q(x, y, z) = (x\ y\ z) A \begin{pmatrix} x \\ y \\ z \end{pmatrix}$. Which of the following statements are true ?
a.
The matrix of second order partial derivatives of the quadratic form $Q$ is $2A$.
b.
The rank of the quadratic form $Q$ is $2$.
c.
The signature of the quadratic for $Q$ is $(+ + 0)$.
d.
The quadratic form $Q$ takes the value $0$ for some non-zero vector $(x, y, z)$.
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For each $\alpha \in \mathbb{R}$, let $S_\alpha = \left\{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = \alpha^2\right\}$. Let $E = \bigcup_{\alpha \in \mathbb{R} \setminus \mathbb{Q}} S_\alpha$. Which of the following are true ?
a.
The Lebesgue measure of $E$ is infinite.
b.
$E$ contains a non-empty open set.
c.
$E$ is path connected.
d.
Every open set containing $E^C$ has infinite Lebesgue measure.
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Let $C_C(\mathbb{R}) = \{f : \mathbb{R} \to \mathbb{R} \mid f \text{ is continuous and there exists a compact set } K \text{ such that } f(x) = 0 \text{ for all } x \in K^C \}$. Let $g(x) = e^{-x^2}$ for all $x \in \mathbb{R}$. Which of the following statements are true ?
a.
There exists a sequence $\{f_n\}$ in $C_C(\mathbb{R})$ such that $f_n \to g$ uniformly.
b.
There exists a sequence $\{f_n\}$ in $C_C(\mathbb{R})$ such that $f_n \to g$ pointwise.
c.
If a sequence in $C_C(\mathbb{R})$ converges pointwise to $g$ then it must converge uniformly to $g$.
d.
There does not exist any sequence in $C_C(\mathbb{R})$ converging pointwise to $g$.
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Given that :
$$\begin{aligned} a(n) &= \frac{1}{10^{100}} 2^n \\ b(n) &= 10^{100} \log (n) \\ c(n) &= \frac{1}{10^{10}} n^2 \end{aligned}$$
Which of the following statements are true ?
a.
$a(n) > c(n)$ for all sufficiently large $n$.
b.
$b(n) > c(n)$ for all sufficiently large $n$.
c.
$b(n) > n$ for all sufficiently large $n$.
d.
$a(n) > b(n)$ for all sufficiently large $n$.
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Let $M_n(\mathbb{R})$ denote the space of all $n \times n$ real matrices identified with the Euclidean space $\mathbb{R}^{n^2}$. Fix a column vector $x \neq 0$ in $\mathbb{R}^n$. Define $f : M_n(\mathbb{R}) \to \mathbb{R}$ by $f(A) = \langle A^2 x, x \rangle$. Then,
a.
$f$ is linear
b.
$f$ is differentiable
c.
$f$ is continuous but not differentiable
d.
$f$ is unbounded
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Let $V$ denote the vector space of all sequence $a = (a_1, a_2, \dots)$ of real numbers such that $\sum 2^n |a_n|$ converges. Define $\|\cdot\| : V \to \mathbb{R}$ by $\|a\| = \sum 2^n |a_n|$. Which of the following are true ?
a.
$V$ contains only the sequence $(0, 0, \dots)$
b.
$V$ is finite dimensional
c.
$V$ has a countable linear basis
d.
$V$ is a complete normed space
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Let $V$ be a vector space over $\mathbb{C}$ with dimension $n$. Let $T : V \to V$ be a linear transformation with only $1$ as eigenvalue. Then which of the following must be true ?
a.
$T - I = 0$
b.
$(T - I)^{n-1} = 0$
c.
$(T - I)^n = 0$
d.
$(T - I)^{2n} = 0$
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Let $A$ be an $n \times n$ matrix (with $n > 1$) satisfying $A^2 - 7A + 12I_{n\times n} = O_{n\times n}$, where $I_{n\times n}$ and $O_{n\times n}$ denote the identity matrix and zero matrix of order $n$ respectively. Then which of the following statements are true ?
a.
$A$ is invertible
b.
$t^2 - 7t + 12n = 0$, where $t = \text{Tr}\,(A)$
c.
$d^2 - 7d + 12 = 0$, where $d = \det(A)$
d.
$\lambda^2 - 7\lambda + 12 = 0$, where $\lambda$ is an eigenvalue of $A$
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Let $A$ be a $(6 \times 6)$ matrix over $\mathbb{R}$ with characteristic polynomial $= (x - 3)^2 (x - 2)^4$ and minimal polynomial $= (x - 3)(x - 2)^2$. Then Jordan canonical form of $A$ can be :
Let $V$ be an inner product space and $S$ be a subset of $V$. Let $\overline{S}$ denote the closure of $S$ in $V$ with respect to the topology induced by the metric given by the inner product. Which of the following statements are true ?
a.
$S = (S^\perp)^\perp$
b.
$\overline{S} = (S^\perp)^\perp$
c.
$\overline{\text{span}(S)} = (S^\perp)^\perp$
d.
$S^\perp = ((S^\perp)^\perp)^\perp$
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Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying
$$f(x) \equiv 1 \bmod (x - 1)$$
$$f(x) \equiv 0 \bmod (x - 3)$$
Which of the following statements are true ?
a.
$S$ is empty
b.
$S$ is a singleton
c.
$S$ is a finite non-empty set
d.
$S$ is countably infinite
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Let $\Omega$ be an open connected subset of $\mathbb{C}$. Let $E = \{z_1, z_2, ..., z_r\} \subseteq \Omega$. Suppose that $f : \Omega \to \mathbb{C}$ is a function such that $f|_{(\Omega \setminus E)}$ is analytic. Then $f$ is analytic on $\Omega$ if
a.
$f$ is continuous on $\Omega$.
b.
$f$ is bounded on $\Omega$.
c.
For every $j$, if $\sum_{m \in \mathbb{Z}} a_m (z - z_j)^m$ is Laurent series expansion of $f$ at $z_j$, then $a_m = 0$ for $m = -1, -2, -3, ...$
d.
For every $j$, if $\sum_{m \in \mathbb{Z}} a_m(z - z_j)^m$ is Laurent series expansion of $f$ at $z_j$, then $a_{-1} = 0$.
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Let $\mathbb{D}$ be the open unit disk centered at $0$ in $\mathbb{C}$ and $f : \mathbb{D} \to \mathbb{C}$ be an analytic function. Let $f = u + iv$, where $u$, $v$ are the real and imaginary parts of $f$. If $f(z) = \sum a_n z^n$ is the power series of $f$, then $f$ is constant if
a.
$f$ is analytic
b.
$u\left(\frac{1}{2}\right) \ge u(z)$, $\forall\ z \in \mathbb{D}$
c.
The set $\{n \in \mathbb{N} \mid a_n = 0\}$ is infinite
d.
For any closed curve $\gamma$ in $\mathbb{D}$, $\int_\gamma \frac{f(z)\,dz}{(z-2)^2} = 0\ \forall\ a \in \mathbb{D}$ with $|a| \ge \frac{1}{2}$
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If $\{a_k\}$ is bounded then $\sum_{0}^{\infty} a_k z^k$ defines an analytic function on the open unit disk.
b.
If $\sum_{0}^{\infty} a_k z^k$ defines an analytic function on the open unit disk then $\{a_k\}$ must converge to zero.
c.
If $f(z) = \sum_{0}^{\infty} a_k z^k$ and $g(z) = \sum_{0}^{\infty} b_k z^k$ are two power series functions whose radii of convergence are 1, then the product $f \cdot g$ has a power series representation of the form $\sum_{0}^{\infty} c_k z^k$ on the open unit disk.
d.
If $f(z) = \sum_{0}^{\infty} a_k z^k$ has a radius of convergence 1, then $f$ is continuous on $\Omega = \{z \in \mathbb{C} \mid |z| \le 1\}$.
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Assume that a non-singular matrix $A = L + D + U$, where $L$ and $U$ are lower and upper triangular matrices respectively with all diagonal entries are zero, and $D$ is a diagonal matrix. Let $x^*$ be the solution of $Ax = b$. Then the Gauss-Seidel iteration method $x^{(k+1)} = Hx^{(k)} + c$, $k = 0, 1, 2, \dots$ with $\|H\| < 1$ converges $x^*$ provided $H$ is equal to :
a.
$-D^{-1}(L + U)$
b.
$-(D + L)^{-1}U$
c.
$-D(L + U)^{-1}$
d.
$-(L - D)^{-1}U$
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Let $a$ be a fixed real constant. Consider the first order partial differential equation:
$$\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0, \quad x \in \mathbb{R}, \; t > 0$$
with the initial data $u(x, 0) = u_0(x)$, $x \in \mathbb{R}$ where $u_0$ is a continuously differentiable function. Consider the following two statements.
$\text{S}_1$: There exists a bounded function $u_0$ for which the solution $u$ is unbounded.
$\text{S}_2$: If $u_0$ vanishes outside a compact set then for each fixed $T > 0$ there exists a compact set $K_T \subset \mathbb{R}$ such that $u(x, T)$ vanishes for $x \notin K_T$.
Which of the following are true ?
a.
$S_1$ is true and $S_2$ is false.
b.
$S_1$ is true and $S_2$ is also true.
c.
$S_1$ is false and $S_2$ is true.
d.
$S_1$ is false and $S_2$ is also false.
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Assume that $a : [0, \infty) \to \mathbb{R}$ is a continuous function. Consider the ordinary differential equation:
$$y'(x) = a(x) y(x),\ x > 0,\ y(0) = y_0 \neq 0$$
Which of the following statements are true ?
a.
If $\int_0^\infty |a(x)| dx < \infty$, then $y$ is bounded.
b.
If $\int_0^\infty |a(x)| dx < \infty$, then $\lim_{x\to\infty} y(x)$ exists.
c.
If $\lim_{x\to\infty} a(x) = 1$, then $\lim_{x\to\infty} |y(x)| = \infty$.
d.
If $\lim_{x\to\infty} a(x) = 1$, then $y$ is monotone.
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Consider the system of differential equations:
$$\frac{dx}{dt} = 2x - 7y;\ \&\ \frac{dy}{dt} = 3x - 8y$$
Then the critical point $(0, 0)$ of the system is an:
a.
asymptotically stable node.
b.
unstable node.
c.
asymptotically stable spiral.
d.
unstable spiral.
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Consider the Sturm-Liouville problem $y'' + \lambda y = 0,\ y(0) = 0$ and $y(\pi) = 0$. Which of the following statements are true ?
(a) There exists only countably many characteristic values.
(b) There exists uncountably many characteristic values.
(c) Each characteristic function corresponding to the characteristic value $\lambda$ has exactly $[\sqrt{\lambda}] - 1$ zeros in $(0, \pi)$.
(d) Each characteristic function corresponding to the characteristic value $\lambda$ has exactly $[\sqrt{\lambda}]$ zeros in $(0, \pi)$.
a.
There exists only countably many characteristic values.
b.
There exists uncountably many characteristic values.
c.
Each characteristic function corresponding to the characteristic value $\lambda$ has exactly $[\sqrt{\lambda}] - 1$ zeros in $(0, \pi)$.
d.
Each characteristic function corresponding to the characteristic value $\lambda$ has exactly $[\sqrt{\lambda}]$ zeros in $(0, \pi)$.
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The Hamiltonian for a simple harmonic oscillator is $H(p, q) = \frac{p^2}{2m} + \frac{k}{2}q^2$. Then a possible Lagrangian corresponding to $H$ can be:
The values of $\lambda$ for which the following equation has a non-trivial solution
$$\phi(x) = \lambda \int_0^\pi K(x, t)\phi(t)\,dt,\ (0 \le x \le \pi); \text{ where } K(x, t) = \begin{cases} \sin x \cos t & ;\quad 0 \le x \le t \\ \cos x \sin t & ;\quad t \le x \le \pi \end{cases} \text{ are:}$$
a.
$\left(n + \frac{1}{2}\right)^2 - 1,\ n \in \mathbb{N}$
b.
$n^2 - 1,\ n \in \mathbb{N}$
c.
$\frac{1}{2}(n+1)^2 - 1,\ n \in \mathbb{N}$
d.
$\frac{1}{2}(2n+1)^2 - 1,\ n \in \mathbb{N}$
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Consider the integral equations:
$$\phi(x) = \lambda \int_0^\pi [\cos x \cos t - 2\sin x \sin t]\phi(t)\,dt + \cos 7x,\ (0 \le x \le \pi)$$
Which of the following statements are true ?
a.
For every $\lambda \in \mathbb{R}$, a solution exists.
b.
There exists $\lambda \in \mathbb{R}$ such that solution does not exist.
c.
There exists $\lambda \in \mathbb{R}$ such that there are more than one but finitely many solutions.
d.
There exists $\lambda \in \mathbb{R}$ such that there are infinitely many solutions.
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The forward difference operator is defined as, $\Delta U_n = U_{n+1} - U_n$. Then which of the following difference equations has an unbounded general solution ?
a.
$\Delta^2 U_n - 3\Delta U_n + 2U_n = 0$
b.
$\Delta^2 U_n + \Delta U_n + \frac{1}{4}U_n = 0$
c.
$\Delta^2 U_n - 2\Delta U_n + 2U_n = 0$
d.
$\Delta^2 U_{n+1} - \frac{1}{3}\Delta^2 U_n = 0$
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