Free Printable Mathematics 2 mid sem EC

1.

The order and degree of differential equation $\left(\frac{(d^2y)}{(dx^2)}\right)^2-\left[1-\left(\frac{dy}{dx}\right)^2\right]^3=0$ is

a)

2 and 2

b)

2 and 3

c)

2 and 1

d)

None of above

2.

The general solution of differential equation $\frac{d^2y}{dx^2}-2\ \left(\frac{dy}{dx}\right)-3y=0$ is

a)

$y=c_1e^{3x}+c_2e^x$

b)

$y=c_1e^{(-3x)}+c_2e^x$

c)

$y=c_1e^{3x}+c_2e^{(-x)}$

d)

$y=c_1e^{(-3x)}+c_2e^{(-x)}$

3.

$L^{-1}\left\{\frac{1}{(s+a)^2}\right\}$

a)

$e^{(-at)}$

b)

$te^{(-at)}$

c)

$t^2e^{(-at)}$

d)

$te^{at}$

4.

The complete solution of Linear Differential equations involves

a)

complete function + particular integral

b)

complementary function + particular integral

c)

complementary function + definite integral

d)

complete function + indefinite integral

5.

The complementary function of the equation $\frac{d^2y}{dx^2}+2\frac{\text{d}y}{\text{d}x}+y=0$ is

a)

$c_1e^x+c_2e^{-x}$

b)

$c_1e^{-x}+c_2e^{-x}$

c)

$c_1e^x+c_2e^x$

d)

$\left(c_1+c_2x\right)e^{-x}$

6.

If r = xi + yj + zk then div (r) is

a)

3

b)

1

c)

r

d)

0

7.

Laplace Transform transforms a function from

a)

$f\left(t\right)\ to\ F\left(s\right)$

b)

$F\left(s\right)\ to\ f\left(t\right)$

c)

$f\left(t\right)\ to\ f'\left(t\right)$

d)

$f"\left(t\right)\ to\ f'\left(t\right)$

8.

Define this property: $L\left(f\left(t\right)e^{at}\right)=F\left(s-a\right)$

a)

Linearity Property

b)

First Shifting Property

c)

Convolution Theorem

d)

Second Shifting Property

9.

Find $L\left(e^{3t}\left(t^3-3t^2+5t\right)\right)$

a)

$\frac{2}{\left(s-3\right)^4}-\left(\frac{3}{\left(\left(s-3\right)^3\right)}\right)+\frac{5}{\left(s-3\right)^2}$

b)

$\frac{6}{\left(s-3\right)^4}-\left(\frac{6}{\left(s-3\right)^3}\right)+\frac{5}{\left(s-3\right)^2}$

c)

$\frac{2}{\left(s-3\right)^3}-\left(\frac{3}{\left(s-3\right)^{\left(2\right)}}\right)+\frac{5}{s-3}$

d)

None of the above

10.

$Find\ L\left(t^{ }\sin\ 4t\right)$

a)

$\frac{2}{s^2+4}$

b)

$-\frac{4s}{\left(s^2+4\right)^2}$

c)

$-\frac{8s}{\left(s^2+16\right)^2}$

d)

None of the above

11.

Find $L^{-1}\left(\frac{s}{s^2+25}\right)$

a)

cos 25t

b)

sin 5t

c)

cos 5t

d)

sin 25t

12.

$L^{-1\ }\left(\frac{4s}{s^2+25}+\frac{2}{3\left(s^2+1\right)}\right)$

a)

$4\ \cos\ 25t+\frac{2}{3}\sin\ t$

b)

$\cos\ 5t+\ \sin\ t$

c)

$4\ \cos\ 5t\ +\frac{2}{3}\sin\ t$

d)

None of the above

13.

$L^{-1}\left(F\left(s-5\right)\right)=e^{5t}f\left(t\right)$ Based on first shifting property for Inverse Laplace transform, what is $a$ value?

a)

4

b)

5

c)

3

d)

2

14.

$L^{-1}\left(\frac{1}{\left(s-3\right)^2}\right)$

a)

$e^{2t}t$

b)

$e^tt^2$

c)

$e^{3t}t$

d)

$t^2e^{5t}$

15.

$L^{-1}\left(\frac{s}{\left(s-4\right)^2+16}\right)$

a)

$e^{4t}\left(\cos\ 4t+\sin\ 4t\right)$

b)

$e^{2t}\left(\cos\ 2t+\sin\ 2t\right)$

c)

$e^{2t}\left(4\cos\ t+4\sin\ t\right)$

d)

None of the above

16.

$L^{-1}\left(\frac{\left(s+2\right)}{s^2+2s+5}\right)$

a)

$f(t)=e^{-t}\sin(2t)+e^{-t}\cos(2t)$

b)

$f(t)=\frac{1}{2}e^{-t}\sin(2t)+e^{-t}\cos(2t)$

c)

$f(t)=e^{-t}\sin(2t)+\frac{1}{2}e^{-t}\cos(2t)$

d)

None of the above

17.

By convolution theorem, find Laplace Transform for $X\left(s\right)=\frac{1}{s\left(s^2+4\right)}$

a)

$\frac{1}{2\ }\sin\ 2t$

b)

$\frac{1}{4}\cos\ 2t-4$

c)

$\frac{1}{4}\left(1-\cos\ 2t\right)$

d)

$\frac{1}{2\ }\cos\ 2t$

18.

Given $g\left(t\right)=\frac{f\left(t\right)}{t}$ , what is G(s)?

a)

$G\left(s\right)=\int_s^{\infty}F\left(u\right)du$

b)

$G\left(s\right)=\left(-1\right)\frac{\text{d}F\left(s\right)}{\text{d}s}$

c)

$G\left(s\right)=\int_{\infty}^sF\left(u\right)du$

d)

$G\left(s\right)=^{\left(-1\right)^{-1}}\frac{\text{d}F\left(s\right)}{\text{d}s}$

19.

The particular integral of the equation $\left(D^3-1\right)y=\left(e^x+1\right)^2$ is

a)

$\frac{e^{-2x}}{7}+\frac{2x}{3}-1$

b)

$\frac{e^{3x}}{7}+\frac{2x}{3}-1$

c)

$\frac{e^{2x}}{7}+\frac{5x}{3}-1$

d)

$\frac{e^{2x}}{7}+\frac{2x}{3}-1$

20.

The P.I. of the equation $\left(D^3+D\right)y=\cos x\$ is

a)

$-\frac{x\cos x}{2}$

b)

$\frac{x\cos x}{2}$

c)

$-\frac{\cos x}{2}$

d)

$-\frac{x\cos x}{5}$

21.

The C.F. of the equation $\left(D^2-9\right)y=e^{-3x}+1+e^{3x}$ is

a)

$c_1e^{-3x}+c_2e^{-x}$

b)

$c_1e^{3x}+c_2e^{3x}$

c)

$c_1e^{3x}+c_2e^{-x}$

d)

$c_1e^{3x}+c_2e^{-3x}$

22.

$\frac{\text{d}^2x}{\text{d}y^2}+4\frac{\text{d}x}{\text{d}y}-6y=0$

Which of the following options are TRUE about the above DE?

a)

The roots of the auxiliary equations are two complex roots.

b)

The auxiliary equation has two different roots.

c)

The auxiliary equation has two equal roots.

d)

The equation is non-homogeneous.

23.

Determine the general solution of $y"-2y'+5y=0$

a)

$y=C_1e^x+C_2xe^{2x}$

b)

$y=C_1e^x+C_2e^{2x}$

c)

$y=C_1e^x\cos2x+C_2e^x\sin2x$

d)

$y=C_1e^{2x}\cos x+C_2e^{2x}\sin x$

24.

What is the integrating factor for the differential equation: $\frac{1}{x}\frac{\text{d}x}{\text{d}y}-\frac{1}{1+x^2}y=x^3$

a)

$\ln\left(x^2\right)$

b)

$-\frac{1}{x^2}$

c)

$-\frac{1}{\sqrt{x}}$

d)

$\frac{1}{\sqrt{x}}$

25.

Which of the following statement is TRUE.

a)

$\frac{\text{d}x}{\text{d}y}-\frac{2}{x}=e^x$ is linear and its integrating factor is $-\frac{1}{x^2}.$

b)

$\frac{\text{d}x}{\text{d}y}-4x=0\$ with condition $y\left(1\right)=2$ is linear and its solution is $y=2x^2.$

c)

$\frac{\text{d}x}{\text{d}y}-\frac{2}{x}=e^x\$ is linear and its solution is $y=\frac{1}{x^2}.$

d)

$y^3dx-4x^2dy=0$ is not exact.

26.

By separation of variables, solve the resulting equations $\int\ \frac{v}{v+1}dv=\int_{ }^{ }\ \frac{1}{y}dy\text{}$

a)

$\frac{x}{y}-\ln\left|\frac{x}{y}+1\right|=\ln y+C$

b)

$\frac{x}{y}+\ln\left|\frac{x}{y}+1\right|=\ln y+C$

c)

$\frac{x}{y}+\frac{x^2}{2y^2}=\ln y+C$

d)

$\frac{x}{y}-\frac{x^2}{2y^2}=\ln y+C$

27.

State the order and degree for differential equation below: $\left(\frac{\text{d}^2y}{\text{d}x^2}\right)^3+5\left(\frac{\text{d}y}{\text{d}x}\right)^4=\cos3x$

a)

Order:2
Degree: 4

b)

Order:3
Degree: 2

c)

Order:2
Degree: 3

d)

Order:4
Degree: 2

28.

Form the differential equation of the curve:
$y=Ae^x+Be^{-x}$

a)

$\frac{\text{d}^2y}{\text{d}x^2}=y$

b)

$\frac{\text{d}^2y}{\text{d}x^2}+y=0$

c)

$\frac{\text{d}^2y}{\text{d}x^2}=Ae^x-Be^{-x}$

d)

$\frac{\text{d}^2y}{\text{d}x^2}=Axe^x-Bxe^{-x}$

29.

$Mdx+Ndy=0\$ is the standard form of exact equation.

Which one is the TRUE statement for this exact equation: $e^ydx+\left(xe^y+y\right)dy=0$

a)

$\frac{\partial N}{\partial x}=xe^y+1$

b)

$\frac{\partial M}{\partial y}=ye^y$

c)

$\frac{\partial M}{\partial y}=e^y$

d)

None of the above

30.

$L^{-1}\left(\frac{s}{\left(s-4\right)^2+16}\right)$

a)

$e^{4t}\left(\cos\ 4t+\sin\ 4t\right)$

b)

$e^{2t}\left(\cos\ 2t+\sin\ 2t\right)$

c)

$e^{2t}\left(4\cos\ t+4\sin\ t\right)$

d)

None of the above

31.

$L^{-1}\left(\frac{\left(s+2\right)}{s^2+2s+5}\right)$

a)

$f(t)=e^{-t}\sin(2t)+e^{-t}\cos(2t)$

b)

$f(t)=\frac{1}{2}e^{-t}\sin(2t)+e^{-t}\cos(2t)$

c)

$f(t)=e^{-t}\sin(2t)+\frac{1}{2}e^{-t}\cos(2t)$

d)

None of the above

32.

Find the partial fraction for $\frac{\left(2s-8\right)}{\left(s-2\right)\left(s-3\right)}$

a)

$\frac{2}{s-2}-\frac{4}{s-3}$

b)

$\frac{3}{s-2}-\frac{2}{s-3}$

c)

$\frac{4}{s-2}-\frac{2}{s-3}$

d)

None of the above

33.

By convolution theorem, find F(s) and G(s) $X\left(s\right)=\frac{1}{s\left(s^2+4\right)}$

a)

$F\left(s\right)=\frac{1}{s};G\left(s\right)=\frac{1}{s^2+4}$

b)

$F\left(s\right)=\frac{1}{s^{ }};G\left(s\right)=\frac{1}{s^3+4s}$

c)

$F\left(s\right)=\frac{1}{s^2+4};G\left(s\right)=\frac{1}{s^{ }}$

d)

None of the above

34.

Find curl $\overrightarrow{F}$ : $\overrightarrow{F}\left(x,y,z\right)=<x^2,\ y-z,\ xe^y>$

a)

$<xe^y+1,\ -e^y,\ 0>$

b)

$<xe^y+1,\ e^y,\ 0>$

c)

$<xe^y-1,-e^y,\ 0>$

d)

$<xe^y-1,\ e^y,\ 0>$

35.

Find div $\overrightarrow{F}$ : $\overrightarrow{F}\left(x,y,z\right)=<x^2,\ y-z,\ xe^y>$

a)

2x + 1

b)

2x - 1

c)

$2x+xe^y$

d)

$2x+1+xe^y$

36.

Determine the validity of the statements.

a) The divergence of a scalar valued function is vector valued function.

b) The curl of a vector valued function is vector valued function.

c) A vector f is solenoidal if div f=o.

d) A vector is called harmonic if curl f=o vector.

a)

a and b are true

b)

only d is true

c)

b and c are true

d)

only c is true

37.

The vector $f=\left(x^2-yz\right)i+\left(y-zx\right)j+\left(z^2-xy\right)k$ is

a)

irrotational

b)

solenoidal

c)

linear

d)

harmonic

38.

$If\ \overrightarrow{F\ \ }\ \ \ is\ solenoidal\ vector\ then$

a)

$\nabla\times\overrightarrow{F\ }$ $=0$

b)

$\nabla\cdot\overrightarrow{F\ }\ \ =0$

c)

$\nabla\overrightarrow{F}\ \ \ =0$

d)

None of the above

39.

Laplace transform is an

a)

differential transform

b)

power series transform

c)

integral transform

d)

exponential transform

40.

The divergence of $F=xyzî+zx^2yĵ+(3z^2-y^2z)k$ is

a)

yz+zx+2xz

b)

$yz+3x^2+(2xz-y^2)$

c)

yz+xy

d)

xy-yz

41.

The value of $\lambda$ so that the vector $(x+3y)î+(y-2z)ĵ+(x+λz)k̂$ is solenoidal vector is

a)

-2

b)

3

c)

1

d)

none

42.

A differential equation of the form $\frac{dy}{dx}=\ \frac{M(x,y)}{N(x,y)}$
is called homogeneous if $M(x,y)\&N(x,y)$ are

a)

Homogeneous functions of different degree

b)

Homogeneous functions of same degree

c)

Non-Homogeneous functions of different degree

d)

Non-Homogeneous functions of same degree

43.

The solution of the differential equation $xdx+ydy=x^2ydy-y^2xdx$ is

a)

$x^2-1=c(1+y^2)$

b)

$x^2+1=c(1-y^2)$

c)

$x^3-1=c(1+y^3)$

d)

$x^3+1=c(1-y^3)$

44.

Which of the following function can be used as an integrating factor to turn the following non-exact D.E. into an exact D.E.? $(3y\cos x-xy\sin x)+2x\cos xdy/dx=0$

a)

$x^2$

b)

$x^2y$

c)

$y^2$

d)

$xy^2$

45.

The solution of the differential equation $(\cos x+y\sin x)dx=\cos xdy,y(π)=0$ is

a)

$y=\tan x+c$

b)

$y=\sin x^2$

c)

$y=\sin x$

d)

$y=\tan x$

46.

Which of the following D.E. are non-exact D.E.?

a)

(y^2-x^2)dx+2xydy=0

b)

$ye^xdx+(2y+e^x)dy=0$

c)

$dy/dx=(x^2-x-y^2)/2xy$

d)

$(y^4+2y)dx+(xy^3+2y^4-4x)dy=0$

47.

After converting the D.E. $(y^4+2y)dx+(xy^3+2y^4-4x)dy=0$ into an exact D.E., the value of

a)

$y+\frac{2}{y}$

b)

$y+\frac{2}{y^2}$

c)

$y-\frac{2}{y}$

d)

$y^2+\frac{2}{y^2}$

48.

Integrating factor of the differential equation $(xy-2y^2)dx-(x^2-3xy)dy=0$ is

a)

$\frac{1}{xy^2}$

b)

$\frac{-1}{xy^2}$

c)

$\frac{1}{x^2y^2}$

d)

$\frac{1}{x^2y}$

49.

The solution of D.E. $(y-xp)(p-1)=p$ is

a)

$y=cx+\frac{c}{c-1}$

b)

$y^2=cx+\frac{c}{c-1}$

c)

$y=cx^2+\frac{c}{c-1}$

d)

$y=cx-\frac{c}{c+1}$

50.

$The\ solution\ of\ D.E.\ p=\log(px-y)\ is$

a)

$y=cx+e^x$

b)

$y=cx^2+c$

c)

$y=cx-e^c$

d)

y=cx

51.

$The\ solution\ of\ D.E.\ y=px+p-p^2\ is$

a)

$y=cx^2$

b)

$y=cx+c^2$

c)

$y=cx-c+c^2$

d)

$y=cx+c-c^2$

52.

An equation of the form $y=px+f(p)$ is known as

a)

Bernoulli’s equation

b)

Lagrange’s equation

c)

Clairaut’s equation

d)

None of these

53.

Laplace transform if cos(at) u(t) is

a)

$\frac{s}{a^2+s^2}$

b)

$\frac{a}{a^{2\ }+\ s^2}$

c)

$\frac{s^2}{s^{2\ }+a^2}$

d)

$\frac{a^2}{s^2+a^2}$

54.

The solution of D.E. $\left(x^2D^2+xD-4\right)y=0$ is

a)

$y=\ c_1x^2+\ c_2x^{-2}$

b)

$y=\ \left(c_1+c_2\right)x^{-2}$

c)

$y\ =\ c_!e^{2x}+c_2e^{-2x}$

d)

$y=\left(c_!+c_2x\right)e^{-2x}$

55.

The general solution of $\left(4D^2+12D+9\right)y=144e^{-3x\ }$ is

a)

$y=\left(c_1+c_2x\right)e^{-\frac{3x}{2}}-16e^{-3x}$

b)

$y=\left(c_1+c_2x\right)e^{-\frac{3x}{2}}+\ 8e^{3x}$

c)

$y=\left(c_1+c_2x\right)e^{-\frac{3x}{2}}-8e^{-3x}$

d)

$y=\left(c_1+c_2x\right)e^{-\frac{3x}{2}}+16e^{-3x}$

56.

The P.I. of $\left(x^2D^2-3xD+5\right)y\ =\ \sin\left(\log\ x\right)\$ is

a)

$y=\frac{1}{8}\left(\sin\left(\log x\right)+\cos\left(\log x\right)\right)$

b)

$y=\frac{1}{8}\left(\sin\left(\log x\right)+\cos\ x\right)$

c)

$y=\frac{1}{8}\left(\sin x+\cos\left(\log x\right)\right)$

d)

None

57.

The solution of $\left(D^4-D^3-9D^2-11D-4\right)y=0$ is

a)

$y=c_1e^{2x}+c_2e^{-2x}+c_3e^x+c_4e^{-x}$

b)

$y=c_1e^{4x}+\left(c_2+c_3x+c_4x^2\right)e^{-x}$

c)

$y=c_1e^x+c_2e^{-2x}+c_3e^{3x}+c_4e^{-4x}$

d)

$y=\left(c_1+c_2x\right)e^{-2x}+c_3e^x+c_4e^{-x}$

58.

The general solution of $\left(D^2+4D+4\right)y=2\ \sinh\ 2x$ is

a)

$y=\left(c_1+c_2x\right)e^{-2x}+\frac{e^{2x}}{16}-\frac{x^2e^{-2x}}{2}$

b)

$y=\left(c_1+c_2x\right)e^{-6x}+\frac{e^{2x}}{136}-\frac{x^2e^{-2x}}{12}$

c)

$y=\left(c_1+c_2x\right)e^{-2x}+\frac{e^{2x}}{16}-x^2$

d)

$y=\left(c_1+c_2x\right)e^{-2x}-\frac{x^2e^{-2x}}{2}$

59.

$L^{-1}\left(\frac{24}{s^5}\right)$ is $t^4$ . What is $L^{-1}\left(\frac{1}{s^{\left(5\right)}}\right)$ ?

a)

$\frac{24}{s^5}$

b)

$\frac{1}{24}t^4$

c)

$\frac{1}{24}t^5$

d)

None

60.

if $∅=\frac{x^2}{2}+\frac{y^2}{3}$ then the value of

$\left|grad\ \phi\right|$ at (1, 3) is

a)

$\sqrt{\frac{13}{2}}$

b)

$5$

c)

$\sqrt{5}$

d)

$\frac{13}{2}$

61.

Write your fullname

4 lines

62.

WRITE YOUR

ENROLLMENT NO.

4 lines

63.

WRITE YOUR EMAIL ID

4 lines

64.

WRITE YOUR MOBILE NUMBER

4 lines

65.

WRITE YOUR PARENT'S MOBILE NUMBER

4 lines

Mathematics 2 mid sem EC

Total questions: 65

Worksheet time: 2hrs 10mins

Similar Resources on Wayground