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1. Let X be a countably infinite subset of $\mathbb{R}$ and A be a countably infinite subset of X. Then the set

X\backslash A=\{x\in X|x\not\in A\}
(a) is empty

(b) is a finite set

(c) can be a countably infinite set

(d) can be an uncountable set


2. The Subset A= $\left \{ x \in \mathbb{Q}: x^{2}< 4 \right \}$ of $\mathbb{R}$ is
(a) bounded above but not bounded below.

(b) bounded above and sup A = 2

(c) bounded above but does not have a suprimum.

(d) not bounded above.


3. Let f be a function defined on $\left [ 0,\infty \right ]$ , by f (x) = [x], the greatest Integer less than or equal to x. Then

(a) $f$ is continuous at each point $\mathrm{o}\mathrm{f}\mathbb{N}$

(b) $f$ is continuous on $[0,\ \infty$)

(c) $f$ is continuous on $[0,7].$

(d) $f$ is discontinuous at $x=1,2,3$


4. The series $x+\frac{2^{2}x^{2}}{2!}+\frac{3^{3}x^{3}}{3!}+…$ is convergent if $\mathrm{x}$ belongs to the interval.

(a) $(0, 1/\mathrm{e})$

(b) $(1/\mathrm{e},\ \infty)$

(c) $(2/\mathrm{e},\ 3/\mathrm{e})$

(d) $(3/\mathrm{e},\ 4/\mathrm{e})$



5. The subset $A=\{x\in \mathbb{Q}:-1<x<0\}\cup \mathbb{N}$ of $\mathbb{R}$ is

(a) bounded, infinite set and has a limit point in $\mathbb{R}$

(b) unbounded, finite set and has a limit point in $\mathbb{R}$

(c) unbounded, finite set and does not have a limit point in $\mathbb{R}$

(d) bounded, finite set and does not have a limit point in $\mathbb{R}$



6. Let f be a real-valued monotone non-decreasing function on $\mathbb{R}$. then

(a)$\left\{x_{n}\right\} \text { be a sequence of real numbers such that } \lim _{n \rightarrow \infty}\left(x_{n+1}-x_{n}\right)=c$
For $a\displaystyle \in \mathbb{R},\lim_{x\rightarrow a}f(x)$ exists.

(b) $h({x})=e^{-f(x)}$ is a bounded function.

(c) $f$ is an unbounded function.

(d) if $a < b$ then $\displaystyle \lim_{x\rightarrow a^{+}}f(x)\leq\lim_{x\rightarrow b^{-}}f(x)$


7.Let $X=C[0,1]$ be the space of all real -valued continuous functions on $[0,1]$. Then $({\it X. d})$ is not complete metric space if

(a) $d(f,g)=\displaystyle \int_{0}^{1}|f(x)-g(x)|dx$

(b) $d(f,g)=\underset{a \in [0,1]}{Max}|f(x)-g(x)|$

(d) $d(f,g)=\left\{\begin{array}{ll} 0, & if\ f=g\\ 1, & if\ f\neq g \end{array}\right.$

(c) $d(f,g)=\underset{a \in [0,1]}{Sup}|f(x)-g(x)|$


8. Consider the one dimensional wave equation: $\displaystyle \frac{\partial^{2}u}{\partial t^{2}u}+4\frac{\partial^{2}u}{\partial x^{2}},x>0,$

Subject to the initial conditions: $u(x,0)=|\sin x|,x\geq 0;u_{t}(x,0)=0, x\geq 0$ and

the boundary condition: $u(0,t)=0, t\geq 0$. Then $u(\pi, \pi /4)$ is equal to

(a) 1

(b) 0

(c) $\displaystyle \frac{1}{2}$

(d) – $\displaystyle \frac{1}{2}$

9. The initial value problem $x\displaystyle \frac{dy}{dx}=y+x^{2}, x>0,y(0)=0$ has

(a) infinitely many solutions

(b) exactly two solution

(c) a unique solution

(d) no solution


10. In a tank there is 120 litre of brine (salted water) containing a total of 50 kg of dissolved salt. Pure water is allowed to run into the tank at the rate of 3 litres per minute. Brine runs out of the tank at the rate of 2 litres per minute. The instantaneous concentration in the tank is kept uniformly stirring. How much slat is in the tank at the end of one hour?

(a) 15.45 kg

(b) 19.53 kg

(c) 14.81 kg

(d) 18. $39\mathrm{k}\mathrm{g}.$


11. If the differential equation $2t^{2}\displaystyle \frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}-3y=0$ is associated with the boundary conditions $\mathrm{y}(1)=5, \mathrm{y}(4)=9$, then y(9) $=$

(a) 27.44

(b) 13.2

(c) 19

(d) 11.35


12. The third degree hermite polynomial approximation for the function $\mathrm{y}=\mathrm{y}(\mathrm{x})$ such that

$y(0)=1,y^{‘}(0)=0,y(1)=3$ and $y^{‘}(1)=5$ is given by

(a) $1+\mathrm{x}^{2}+\mathrm{x}^{3}$

(b) $1+\mathrm{x}^{3}+\mathrm{x}$

(c) $x^{2}+x^{3}$

(d) none of the above


13. Let $\mathrm{y}$ be the solution of the initial value problem


Using Runge-Kutta second order method with step size $h=0.1$,the approximate value of $y(0,1)$ correct tom four decimal places is given by

(a) 2.8909

(b) 2.7142

(c) 2.6714

(d) 2.7716

14. Consider the system of linear equations

2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2
\end{bmatrix} \begin{bmatrix}
x1\\ x2
\\ x3
\end{bmatrix} = \begin{bmatrix}

With the initial approximation $[0,0,0]^{t}$, the approximate value of the solution after one iteration by Gauss Seidel method is

(a) $[$3.2, 2.25, $1.5]^{T}$

(b) $[$3.5, 2.25, $1.625]^{T}$

(c) $[$2.25, 3.5, $1.625]^{T}$

(d) $[$2.5, 3.5, $1.6]^{T}$


15. For the wave equation $u_{tt}=16u_{xx}$ the characteristic coordinates are

(a) $\zeta=x+16t, \eta=x-16t$

(b) $\zeta=x+4t, \eta=x-4t$

(c) $\zeta=x+256t, \eta=x-256t$

(d) $\zeta=x+2t, \eta=x-2t$

16. Let $f_{1}$ and $f_{2}$ be two solution of $a_{0}(x)\displaystyle \frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{2}(x)_{y}=0$, where $a_{0}, a_{1}$ and $a_{2}$ are continuous on $[0,1]$ and $a_{0}(x)\neq 0$ for all $x\in[0,1]$. Moreover, let $f_{1}(\displaystyle \frac{1}{2})=f_{2}(\frac{1}{2})=0$ then

(a) One $\mathrm{o}\mathrm{f}f_{1}$ and $f_{2}$ must be identically zero.

(b) $f_{1}(x)=f_{2}(x)=0$ for all $x\in[0, 1]$

(c) $f_{1}(x)=cf_{2}(x)$ for some constant $\mathrm{c}.$

(d) none of the above.


17. The Laplace transformation of $e^{4t}$ is

(a) $1/(s+2)$

(b) $1/(s-2)$

(c) $1/(s+4)$

(d) $1/(s-4)$


18. Let $f(t)=4\sin^{2}\mathrm{t}$ and Let $\displaystyle \sum_{n=0}^{\infty}a_{n}\cos nt$ be the Fourier cosine series of f $({\it t})$ . Which one is true ?

(a) $a_{0}=0, a_{2}=1, a_{4}=2$

(b) $a_{0}=2, a_{2}=0, a_{4}=-2$

(c) $a_{0}=2, a_{2}=-2, a_{4}=0$

(d) $a_{0}=0, a_{2}=-2, a_{4}=2$


19. For $a, b, c\in \mathrm{R}$, if the differential equation $(ax^{2}+bxy+y^{2})dx+(2x^{2}+cxy+y^{2})dy=0$ is exact, then

(a) $b=2, c=2a$

(b) $b=4, c=2$

(c) $b=2, c=4$

(d) $b=2, a=2c$


20. Let $u(x,\ t)$ be the solution of the wave equation

$u_{xx}=u_{tt}, u(x,\ 0)=\cos(5\pi x), u_{t}(x,\ 0)=0$. Then the value of u$(1,\ 1)$ is

(a) $-1$

(b) $0$

(c) 2

(d) 1




July 19, 2019

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