Free Printable Model exam

1.

When solving a 1-Dimensional wave equation using variable separable method, we get the solution if _____________

a)

k is positive

b)

k is negative

c)

k is 0

d)

k can be anything

2.

When solving a 1-Dimensional heat equation using a variable separable method, we get the solution if ______________

a)

k is positive

b)

k is negative

c)

k is 0

d)

k can be anything

3.

The partial differential equation of 1-Dimensional heat equation is ___________

a)

u_t = c²u_xx

b)

u_t = pu_xx

c)

u_tt = c²u_xx

d)

u_t = – c²u_xx

4.

Howmany possible solutions are derived for one dimensional wave equation

a)

0

b)

1

c)

2

d)

3

5.

Find the nature of one dimensional heat equation

a)

Hyperbolic

b)

Parabolic

c)

Eliptic

d)

None of these

6.

Howmany different solutions are derived for Laplace's equation in cartesian co-ordinates

a)

0

b)

1

c)

2

d)

3

7.

What is the nature of Lagrange’s linear partial differential equation?

a)

First-order, Third-degree

b)

Second-order, First-degree

c)

First-order, Second-degree

d)

First-order, First-degree

8.

Find the general solution of the linear partial differential equation, yzp+zxq=xy.

a)

$φ(x^2-y^2,\ x^2–z^2)=0$

b)

$φ(x^2-y^2,\ y^2-z^2)=0$

c)

$φ(x^2-y^2,\ y^2-x^2)=0$

d)

$φ(x^2-z^2,\ z^2-x^2)=0$

9.

Which of the following is true with respect to formation of differential equation by elimination of arbitrary constants?

a)

The given equation should be differentiated with respect to independent variable

b)

Elimination of the arbitrary constant by replacing it using derivative

c)

If ‘n’ arbitrary constant is present, the given equation should be differentiated ‘n’ number of times

d)

To eliminate the arbitrary constants, the given equation must be integrated with respect to the dependent variable

10.

Find the singular solution of z=px+qy+p²

a)

4z+x²

b)

4x+z²

c)

4x=z²

d)

x=4

11.

Choose the trial solution of the given PDE f(p,q)=0

a)

z=ax

b)

z=ax+by+c

c)

z=ax+b

d)

z=ax-b

12.

A linear PDE with constant coefficients in which all the partial derivatives are of the same order is called --------

a)

Homogeneous

b)

NonHomogeneous

c)

Lagarange's equation

d)

None of these

13.

If the function f(x) is even, then which of the following is zero?

a)

$a_n$

b)

$b_n$

c)

$a_0$

d)

nothing is zero

14.

If the function f(x) is odd, then which of the only coefficient is present?

a)

a_n

b)

b_n

c)

a₀

d)

everything is present

15.

Which of the following is not Dirichlet’s condition for the Fourier series expansion?

a)

f(x) is periodic, single valued, finite

b)

f(x) has finite number of discontinuities in only one period

c)

f(x) has finite number of maxima and minima

d)

f(x) is a periodic, single valued, finite

16.

Find a_n if the function f(x) = x – x^3.

a)

finite value

b)

infinite value

c)

zero

d)

Infinity

17.

Find b_n if the function f(x) = x².

a)

finite value

b)

infinite value

c)

zero

d)

Infinity

18.

The function tanx cannot be expressed as fourier series Why?

a)

Infinite no.of infinite discontinuity

b)

finite

c)

infinite

d)

finite no.of discontinuity

19.

What is the fourier sine transform of $e^{-ax}$ ?

a)

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

b)

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

c)

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

d)

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

20.

Find the fourier transform of F(x) = 1, |x|<a

0 , otherwise

a)

$2\ \sin\ \frac{as}{s}$

b)

$2a\ \sin\ \frac{as}{s}$

c)

$4\ \sin\ \frac{as}{s}$

d)

$4a\ \sin\ \frac{as}{s}$

21.

f(x)=1 cannot be represented by a fourier integral. why?

a)

not convergent

b)

convergent

c)

absolutely convergent

d)

none of these

22.

F(e^iaxf(x)) = _____

a)

F(s+a)

b)

F(s-a)

c)

F(sa)

d)

F(a/s)

23.

$F\left(x\right)=x\left(^{\frac{-1}{2}}\right)$

is self reciprocal under Fourier cosine transform.

a)

true

b)

false

24.

For any nonzero real a, F(f(ax)) = ____________

a)

$\frac{1}{\left|a\right|}F\left(sa\right)$

b)

$\frac{1}{\left|a\right|}F\left(\frac{s}{a}\right)$

c)

$\frac{1}{\left|a\right|}F\left(s\right)$

d)

$\frac{1}{\left|a\right|}F\left(\frac{a}{s}\right)$

25.

Digital control systems can be analyzed and designed using ________________transform

a)

fourier

b)

inverse fourier

c)

Z

d)

Laplace

26.

If $\left|z\right|>1$ find z(1)

a)

$\frac{z}{z-1}$

b)

$\frac{z}{z+1}$

c)

$\frac{1}{z-1}$

d)

$\frac{1}{z+1}$

27.

If $\left|z\right|>\left|a\right|$ find $z\left(a^n\right)$

a)

$\frac{z}{\left(z+1\right)^2}$

b)

$\frac{z}{\left(z-1\right)^2}$

c)

$\frac{1}{\left(z+1\right)^2}$

d)

$\frac{1}{\left(z-1\right)^2}$

28.

Z transform of unit impulse sequence is ____________

a)

0

b)

1

c)

z

d)

-z

29.

If $F\left(z\right)=\frac{5z}{\left(z-2\right)\left(z-3\right)}$ find f(0)

a)

0

b)

5

c)

infinity

d)

1

30.

find the Z transform of (n+1)(n+2)

a)

$\left(\frac{z}{z-1}\right)^3$

b)

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

c)

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

d)

$2\left(\frac{z}{z+1}\right)^3$

31.

Which of the following is an example for first order linear partial differential equation?

a)

Lagrange’s Partial Differential Equation

b)

Clairaut’s Partial Differential Equation

c)

One-dimensional Wave Equation

d)

One-dimensional Heat Equation

32.

What is the degree of the homogeneous partial differential equation,

\frac{\partial^2u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0

a)

Second-degree

b)

First-degree

c)

Third-degree

d)

Zero-degree

33.

Find the solution of

px^2+qy^2=z^2

a)

$f\left(xy,yz\right)=0$

b)

$f\left(x,z\right)=0$

c)

$f\left(\frac{1}{x}-\frac{1}{y},\ \frac{1}{y}-\frac{1}{z}\right)=0$

d)

$f\left(\frac{1}{x},\frac{1}{z}\right)=0$

34.

Find the nature of PDE

4U_{xx}+4U_{xy}+U_{yy}+2U_x-U_y=0

a)

Parabolic

b)

Elliptic

c)

Hyperbolic

d)

None of these

35.

Find the nature of PDE

$U_{xx}+U_{xy}=U_x^2+U_y^2$

a)

Parabolic

b)

Elliptic

c)

Hyperbolic

d)

None of these

36.

The PDE of a vibrating string is

\frac{\partial^2u}{\partial t^2}-a^2\frac{\partial^2u}{\partial x^2}=0

What is

a^2

a)

$\frac{T}{m}$

b)

$\frac{m}{T}$

c)

Tm

d)

0

37.

Find $a_0$ for Half range cosine series of $f\left(x\right)=x\ \ in\ \left(0,\pi\right)$

a)

0

b)

$\pi$

c)

$-\pi$

d)

1

38.

Find the Half range sine series for $f\left(x\right)=x\left(\pi-x\right)\ in\ \left(0,\pi\right)$

a)

$\frac{2}{n^3\pi}\left(-1\right)^n$

b)

$\frac{4}{n^3\pi}\left(1-\left(1\right)^n\right)^{ }$

c)

0

d)

$\frac{4}{n^3\pi}$

39.

Find the root mean square (RMS) value of the function

f\left(x\right)=x

in the interval

\left(0,l\right)

a)

$\frac{l}{\sqrt{3}}$

b)

$\frac{1}{\sqrt{3}}$

c)

0

d)

$l$

40.

Find the fourier sine transform of $3e^{-5x}$

a)

$\sqrt{\frac{2}{\pi}}\ \left(\frac{3s}{s^2+25}\right)$

b)

$\ \left(\frac{3s}{s^2+25}\right)$

c)

$\sqrt{\frac{2}{\pi}}\ \left(\frac{3}{s^2+25}\right)$

d)

$\sqrt{\frac{2}{\pi}}\ \left(\frac{3s}{s^2-25}\right)$

41.

Find the fourier cosine transform of $e^{-5x}$

a)

$\sqrt{\frac{2}{\pi}}\ \left(\frac{5}{s^2+25^{ }}\right)$

b)

$\sqrt{\frac{2}{\pi}}\ \left(\frac{5}{s^2+1}\right)$

c)

$\ \left(\frac{5}{s^2+25^{ }}\right)$

d)

$\sqrt{\frac{2}{\pi}}\ \left(\frac{5}{s^2-25^{ }}\right)$

42.

Find $Z^{-1}\left(\frac{z^2}{\left(z-a\right)^2}\right)$

a)

$a^n$

b)

$na^n$

c)

$\left(n+1\right)\ a^n$

d)

$\left(n-1\right)\ a^n$

43.

What is the fourier transform of $e^{-ax}\cdot\ e^{-bx}$

a)

$\frac{4ab}{\left(s^2+a^2\right)\left(s^2+b^2\right)}$

b)

$\frac{2ab}{\left(s^2+a^2\right)\left(s^2+b^2\right)}$

c)

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

d)

$\frac{a^2b^2}{\left(s^2+a^2\right)\left(s^2+b^2\right)}$

44.

From $y_n=a2^n+b\left(-2\right)^n$ derive a difference equation by eliminating the constants

a)

$y_{n+2}-4y_n=0$

b)

$y_{n+2}+4y_n=0$

c)

$y_{n+2}=0$

d)

$y_n=0$

45.

Solve $y_{n+1}-2y_n=0\ \ \ \ \ given\ \ \ y_0=3$

a)

$2^n$

b)

$3\left(-1\right)^n$

c)

$3\left(2^n\right)$

d)

$3^n$

Model exam

Total questions: 45

Worksheet time: 23mins

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