Objective Questions:
Problem 1: Let $A=\left(\begin{array}{rrr}1 & 1 & 1 \\ 3 & -1 & 1 \\ 1 & 5 & 3\end{array}\right)$ and $V$ be the vector space of all $X \in \mathbb{R}^{3}$ such that $A X=0$. Then ${dim}(V)$ is
(A) 0
(B) 1
(C) 2
(D) 3
Problem 2. The value of $n$ for which the divergence of the function $\vec{F}$=$\frac{\vec{r}}{|\vec{r}|^{n}}, \vec{r}=x \hat{i}+y \hat{j}+z \hat{k},|\vec{r}| \neq 0$, vanishes is
(A) 1
(B) $-1$
(C) 3
(D) $-3$
Problem 3. Let $A$ and $B$ be subsets of $\mathbb{R}$. Which of the following is NOT necessarily true?
(A) $(A \cap B)^{o} \subseteq A^{o} \cap B^{o}$
(B) $\quad A^{o} \cup B^{o} \subseteq(A \cup B)^{o}$
(C) $\bar{A} \cup \bar{B} \subseteq \overline{A \cup B}$
(D) $\bar{A} \cap \bar{B} \subseteq \overline{A \cap B}$
Problem 4. Let $[x]$ denote the greatest integer function of $x$. The value of $\alpha$ for which the function
$$
f(x)= \begin{cases}\frac{\sin \left[-x^{2}\right]}{\left[-x^{2}\right]}, & x \neq 0 \ \alpha, & x=0\end{cases}
$$
is continuous at $x=0$ is
(A) 0
(B) $\sin (-1)$
(C) $\sin 1$
(D) 1
Problem 5. Let the function $f(x)$ be defined by
$f(x)= \begin{cases}e^{x}, & x \text { is rational } \ e^{1-x}, & x \text { is irrational }\end{cases}$
for $x$ in $(0,1)$. Then
(A) $f$ is continuous at every point in $(0,1)$
(B) $f$ is discontinuous at every point in $(0,1)$
(C) $f$ is discontinuous only at one point in $(0,1)$
(D) $f$ is continuous only at one point in $(0,1)$
Problem 6. The value of the integral
$\iint_{D} \sqrt{x^{2}+y^{2}} d x d y, \quad D$=${(x, y) \in \mathbb{R}^{2}: x \leq x^{2}+y^{2} \leq 2 x}$
is
(A) 0
(B) $\frac{7}{9}$
(C) $\frac{14}{9}$
(D) $\frac{28}{9}$
Problem 7. Let $x_{n}$=
$$\left(1-\frac{1}{3}\right)^{2}\left(1-\frac{1}{6}\right)^{2}\\\left(1-\frac{1}{10}\right)^{2} \ldots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^{2}$$, $$\quad n \geq 2 \text {. }$$
Then $lim_{n \to \infty} x_{n}$ is
(A) $\frac{1}{3}$
(B) $\frac{1}{9}$
(C) $\frac{1}{81}$
(D) 0
Problem 8. Let $p$ be a prime number. Let $G$ be the group of all $2 \times 2$ matrices over $\mathbb{Z}_{p}$ with determinant 1 under matrix multiplication. Then the order of $G$ is
(A) $(p-1) p(p+1)$
(B) $p^{2}(p-1)$
(C) $p^{3}$
(D) $p^{2}(p-1)+p$
Problem 9. Let $V$ be the vector space of all $2 \times 2$ matrices over $\mathbb{R}$. Consider the subspaces $W_{1}={\left(\begin{array}{rr}a & -a \\ c & d\end{array}\right): a, c, d \in \mathbb{R}}$ and $W_{2}={\left(\begin{array}{rr}a & b \\-a & d\end{array}\right): a, b, d \in \mathbb{R}}$.
If $m={dim}\left(W_{1} \cap W_{2}\right)$ and $n={dim}\left(W_{1}+W_{2}\right)$, then the pair $(m, n)$ is
(A) $(2,3)$
(B) $(2,4)$
(C) $(3,4)$
(D) $(1,3)$
Problem 10. Let $\wp_{n}$ be the real vector space of all polynomials of degree at most $n$. Let $D: \& \rho_{n} \rightarrow \wp_{n-1}$ and $T: \wp \wp_{n} \rightarrow \wp \wp_{n+1}$ be the linear transformations defined by
$D\left(a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}\right)$=$a_{1}+2 a_{2} x+\ldots+n a_{n} x^{n-1}$,
$T\left(a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}\right)$ = $a_{0} x+a_{1} x^{2}+a_{2} x^{3}+\ldots+a_{n} x^{n+1}$
respectively. If $A$ is the matrix representation of the transformation $D T-T D: \wp o_{n} \rightarrow \wp \wp_{n}$ with respect to the standard basis of $\wp_{n}$, then the trace of $A$ is
(A) $-n$
(B) $n$
(C) $n+1$
(D) $-(n+1)$
Fill in the blanks:
Problem 11. The equation of the curve satisfying $\sin y \frac{d y}{d x}=\cos y(1-x \cos y)$ and passing through the origin is _____.
Problem 12. Let $f$ be a continuously differentiable function such that $\int_{0}^{2 x^{2}} f(t) d t$=$e^{\cos x^{2}}$ for all $x \in(0, \infty)$. The value of $f^{\prime}(\pi)$ is _____.
Problem 13. Let $u=\frac{y^{2}-x^{2}}{x^{2} y^{2}}$, $v=\frac{z^{2}-y^{2}}{y^{2} z^{2}}$ for $x \neq 0, y \neq 0, z \neq 0$. Let $w=f(u, v)$, where $f$ is a real valued function defined on $\mathbb{R}^{2}$ having continuous first order partial derivatives. The value of $x^{3} \frac{\partial w}{\partial x}+y^{3} \frac{\partial w}{\partial y}+z^{3} \frac{\partial w}{\partial z}$ at the point $(1,2,3)$ is _____.
Problem 14. The set of points at which the function $f(x, y)=x^{4}+y^{4}-x^{2}-y^{2}+1,(x, y) \in \mathbb{R}^{2}$ attains local maximum is _____.
Problem 15. Let $C$ be the boundary of the region in the first quadrant bounded by $y=1-x^{2}, x=0$ and $y=0$, oriented counter-clockwise. The value of $\oint\left(x y^{2} d x-x^{2} y d y\right)$ is _____.
Problem 16. Let $f(x)$={${\begin{array}{lc}0, & -1 \leq x \leq 0 \\ x^{4}, & 0<x \leq 1\end{array}}$. If $f(x)=\sum_{k=0}^{n} \frac{f^{(k)}(0)}{k !} x^{k}+\frac{f^{(n+1)}(\xi)}{(n+1) !} x^{n+1}$ is the Taylor's formula for $f$ about $x=0$ with maximum possible value of $n$, then the value of $\xi$ for $0<x \leq 1$ is _____.
Problem 17. Let $\vec{F}=2 z \hat{i}+4 x \hat{j}+5 y \hat{k}$, and let $C$ be the curve of intersection of the plane $z=x+4$ and the cylinder $x^{2}+y^{2}=4$, oriented counter-clockwise. The value of $\oint_{C} \vec{F} \cdot d \vec{r}$ is _____.
Problem 18. Let $f$ and $g$ be the functions from $\mathbb{R} \backslash{0,1}$ to $\mathbb{R}$ defined by $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-1}{x}$ for $x \in \mathbb{R} \backslash{0,1}$. The smallest group of functions from $\mathbb{R} \backslash{0,1}$ to $\mathbb{R}$ containing $f$ and $g$ under composition of functions is isomorphic to _____.
Problem 19. The orthogonal trajectory of the family of curves $\frac{x^{2}}{2}+y^{2}=c$, which passes through $(1,1)$ is _____.
Problem 20. The function to which the power series $\sum_{n=1}^{\infty}(-1)^{n+1} n x^{2 n-2}$ converges is _____.
Descriptive Questions:
Problem 21. Let $0<a \leq 1, s_{1}=\frac{a}{2}$ and for $n \in \mathbb{N}$, let $s_{n+1}=\frac{1}{2}\left(s_{n}^{2}+a\right)$. Show that the sequence ${s_{n}}$ is convergent, and find its limit.
Problem 22. Evaluate
$$
\int_{1 / 4}^{1} \int_{\sqrt{x-x^{2}}}^{\sqrt{x}} \frac{x^{2}-y^{2}}{x^{2}} d y d x
$$
by changing the order of integration.
Problem 23. Find the general solution of the differential equation $x^{2} \frac{d^{3} y}{d x^{3}}+x \frac{d^{2} y}{d x^{2}}-6 \frac{d y}{d x}+6 \frac{y}{x}$=$\frac{x \ln x+1}{x^{2}}$, $\quad x>0$.
Problem 24. Let $S_{1}$ be the hemisphere $x^{2}+y^{2}+z^{2}=1, z>0$ and $S_{2}$ be the closed disc $x^{2}+y^{2} \leq 1$ in the $x y$ plane. Using Gauss' divergence theorem, evaluate $\iint_{S} \vec{F} \cdot d \vec{S}$, where $\vec{F}=z^{2} x \hat{i}+\left(\frac{y^{3}}{3}+\tan z\right) \hat{j}+\left(x^{2} z+y^{2}\right) \hat{k}$ and $S=S_{1} \cup S_{2}$.
Also evaluate $\iint_{S_{1}} \vec{F} \cdot d \vec{S}$
Problem 25. Let
$f(x, y)= \begin{cases}\frac{2\left(x^{3}+y^{3}\right)}{x^{2}+2 y}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{cases}$
Show that the first order partial derivatives of $f$ with respect to $x$ and $y$ exist at $(0,0)$. Also show that $f$ is not continuous at $(0,0)$.
Problem 26. Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{q}$, where $a$ and $b$ are distinct real numbers. Let $V$ be the real vector space of all $n \times n$ matrices $B$ such that $A B=B A$. Determine the dimension of $V$.
Problem 27. Let $A$ be an $n \times n$ real symmetric matrix with $n$ distinct eigenvalues. Prove that there exists an orthogonal matrix $P$ such that $A P=P D$, where $D$ is a real diagonal matrix.
Problem 28. Let $K$ be a compact subset of $\mathbb{R}$ with nonempty interior. Prove that $K$ is of the form $[a, b]$ or of the form $[a, b] \backslash \cup I_{n}$, where ${I_{n}}$ is a countable disjoint family of open intervals with end points in $K$.
Problem 29. Let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function such that $f$ is differentiable in $(a, c)$ and $(c, b)$, $a<c<b$. If $lim_{x \to c} f'(x)$ exists, then prove that $f$ is differentiable at $c$ and $f'(c)=lim_{x \to c}f'(x)$.
Problem 30. Let $G$ be a finite group, and let $\varphi$ be an automorphism of $G$ such that $\varphi(x)=x$ if and only if $x=e$, where $e$ is the identity element in $G$. Prove that every $g \in G$ can be represented as $g=x^{-1} \varphi(x)$ for some $x \in G$. Moreover, if $\varphi(\varphi(x))=x$ for every $x \in G$, then show that $G$ is abelian.
