Free Printable MA8353- TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

1.

Pick out the even function.

a)

$\sin x$

b)

$\tan x$

c)

$x^2$

d)

$x^3$

2.

The period of $\sin x$ is

a)

$2π$

b)

$π$

c)

$π/2$

d)

$3π/2$

3.

The value of $a_0$ in the Fourier series of $f(x)=x$ in $(0,2π)$ is

a)

$2π$

b)

$π$

c)

$0$

d)

$-π$

4.

if $f\left(x\right)$ is an even function, then the value of $b_n$ in the Fourier series for $f\left(x\right)$ in $(-π,π)$ is

a)

$\left(2/π\right)∫_0^πf(x)\sin xdx$

b)

$π$

c)

$\left(1/π\right)\int_{-\pi}^{\pi}f(x)\sin xdx$

d)

$0$

5.

Which of the following function doesnot have a Fourier series in $(0,2π)$ ?

a)

$\sin x$

b)

$\tan x$

c)

$\cos x$

d)

$x^2$

6.

Pick out one of the conditions of Dirichlet’s condition.

a)

f\left(x\right)

is periodic, single-valued and finite

b)

$f\left(x\right)$ has infinite number of infinite discontinuous in any one period

c)

$f\left(x\right)$ is infinite valued function

d)

$f\left(x\right)$ has infinite number of maxima and minima

7.

if $|f(x)|$ is even whenever $f(x)$ is

a)

Even only

b)

Odd only

c)

Neither even nor odd

d)

Either even or odd

8.

Which of the following function is neither even nor odd?

a)

$\sin x$

b)

$x^2+x$

c)

$\cos x$

d)

$x^2$

9.

if $f\left(x\right)$ is an odd function, then the value of $a_n$ in the Fourier series for $f\left(x\right)$ in $(-π,π)$ is

a)

$\left(2/π\right)∫_0^πf(x)\sin xdx$

b)

$π$

c)

$\left(1/π\right)\int_{-\pi}^{\pi}f(x)\sin xdx$

d)

$0$

10.

if $f\left(x\right)$ is an even function, then $a_n$ in the Fourier series for $f\left(x\right)$ in $(-π,π)$ is given by

a)

$\left(2/π\right)∫_0^πf(x)\sin xdx$

b)

$π$

c)

$\left(2/π\right)∫_0^πf(x)\cos xdx$

d)

$0$

11.

The Fourier Transform of f(x) is $F[f(x)]=$

a)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F\left[s\right]$

b)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{isx}\ dx\ =\ F\left[s\right]$

c)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ ds\ =\ F\left[s\right]$

d)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F\left[s\right]$

12.

The Fourier sine Transform of f(x) is $F_s[f(x)]=$

a)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F_s\left[s\right]$

b)

$\frac{1}{\sqrt{2\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$

c)

$\sqrt{\frac{2}{\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$

d)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F_s\left[s\right]$

13.

The Fourier Sine Transform of

e-^{ax}

a)

$F_s\left[e-^{ax}\right]=\sqrt{\frac{2}{\pi}}\ \frac{a}{a^2+s^2}$

b)

$F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\ \frac{s}{a^2+s^2}$

c)

$F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\ \frac{s}{a^2+s^2}$

d)

none of the above

14.

The Fourier cosine Transform of

e-^{ax}

a)

$F_s\left[e-^{ax}\right]=\sqrt{\frac{2}{\pi}}\ \frac{a}{a^2+s^2}$

b)

$F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\ \frac{s}{a^2+s^2}$

c)

$F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\ \frac{a}{a^2+s^2}$

d)

none of the above

15.

$F_c\left[f\left(x\right)\ \cos ax\right]\ is$

a)

$\frac{1}{2}\left[F_s\left(A+S\right)+F_s\left(A-S\right)\right]$

b)

$\frac{1}{2}\left[F_s\left(A+S\right)+F_s\left(A+S\right)\right]$

c)

$\frac{1}{2}\left[F_c\left(A+s\right)+F_c\left(A-s\right)\right]$

d)

$\frac{1}{2}\left[F_c\left(A+s\right)-F_c\left(A-s\right)\right]$

16.

$F_s\left[f\left(x\right)\ \cos ax\right]\ is$

a)

$\frac{1}{2}\left[F_s\left(a+S\right)-F_s\left(a-S\right)\right]$

b)

$\frac{1}{2}\left[F_s\left(a+S\right)+F_s\left(a-S\right)\right]$

c)

$\frac{1}{2}\left[F_c\left(a+s\right)+F_c\left(a-s\right)\right]$

d)

$\frac{1}{2}\left[F_c\left(a+s\right)-F_c\left(a-s\right)\right]$

17.

The linear Fourier transform is

a)

$F\left[af+bg\right]\ =aF\left[s\right]\ \ +bG\left[s\right]$

b)

$F\left[f\cdot g\right]\ =F\left[s\right]\ \ \cdot G\left[s\right]$

c)

$F\left[af\right]\ =\frac{1}{\left|a\right|}F\left[\frac{s}{a}\right]\ \$

d)

$F\left[f'\left(x\right)\right]\ =-isF\left[s\right]\ \$

18.

The Fourier Sine Transform of

e^{-5x}

a)

$F_s\left[e^{-5x}\right]=\sqrt{\frac{2}{\pi}}\ \frac{5}{5^2+s^2}$

b)

$F_s\left[e^{-5x}\right]=\sqrt{\frac{2}{\pi}}\ \frac{s}{5^2+s^2}$

c)

$F_s\left[e^{-5x}\right]=\sqrt{\frac{2}{\pi}}\ \frac{s}{a^2+s^2}$

d)

none of the above

19.

The Fourier Cosine Transform of

e^{-5x}

a)

$F_c\left[e^{-5x}\right]=\sqrt{\frac{2}{\pi}}\ \frac{5}{5^2+s^2}$

b)

$F_c\left[e^{-5x}\right]=\sqrt{\frac{2}{\pi}}\ \frac{s}{5^2+s^2}$

c)

$F_c\left[e^{-5x}\right]=\sqrt{\frac{2}{\pi}}\ \frac{5}{a^2+s^2}$

d)

none of the above

20.

$F_s\left(xf\left(x\right)\right)\ =$

a)

$\frac{d}{ds}\ F_c\left[s\right]$

b)

$-\frac{d}{ds}\ F_c\left[s\right]$

c)

$\frac{d}{ds}\ F_s\left[s\right]$

d)

$\frac{d}{ds}\ F_s\left[s\right]$

21.

The value of cos $2\pi$ and cos $\pi$

a)

0 and 0

b)

1 and 1

c)

1 and -1

d)

-1 and -1

22.

The Fourier series is a combination of -----and ---- trigonometric terms

a)

sine and co secant

b)

tangent and cotangent

c)

sine and cosine

d)

cosine and secant

23.

Identify Even function from the following

a)

x+x²

b)

x² + x³

c)

x²

d)

1+x

24.

Identify odd function

a)

1+x+x²

b)

x+2

c)

x-x³

d)

x² +x⁴

25.

The Fourier constant b_nfor f(x)= x sinx in (-2,2) is

a)

2

b)

1

c)

0

d)

4

26.

The trigonometric Fourier series of an even function does not have

a)

Sine terms

b)

Cosine terms

c)

Constant term

d)

harmonic terms

27.

If f(x) = sinh(x) defined in -π<x<π then a₀and a_n values are

a)

a₀ = 0 , a_n= 1

b)

a₀ = 1 , a_n= 1

c)

a₀ = 0 , a_n= 0

d)

a₀ = 1 , a_n= 1

28.

Which of the following functions cannot be expressed as a Fourier series ?

a)

tanx

b)

x

c)

xsinx

d)

cosx

29.

If\ b_n\ =\frac{4}{n\pi}\left\{1-\left(-1\right)^n\right\}\ ,\ What\ is\ the\ value\ of\ b_3\ ?

a)

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

b)

$\frac{8}{n\pi}$

c)

$\frac{-8}{n\pi}$

d)

0

30.

The incorrect statement from the following choices is

a)

The product of two even functions is an even function

b)

The product of two odd functions is an even function

c)

The product of an odd function and an even function is an even function

d)

The product of an odd function and an even function is an odd function

31.

The incorrect statement from the following choices is

a)

sinx is an odd function

b)

sin2x is an odd function

c)

sinx² is an odd function

d)

sinx² is an even function

32.

What will be the value of

a_0

in the Fourier Series of the above function in

\left(-4,\ 4\right)

?

a)

1/4

b)

4

c)

-4

d)

0

33.

A half range series consists of

a)

sine terms only

b)

cosine terms only

c)

both sine and cosine terms

d)

either only sine terms or only cosine terms

34.

sinx+cosx is

a)

an odd function

b)

an even function

c)

neither an odd function nor an even function

d)

both odd and even function

35.

For any integer n the value of cosnπ is

a)

1

b)

0

c)

-1

d)

(-1)ⁿ

36.

The Real part of $e^{isx\ }$ is

a)

sinsx

b)

i sinsx

c)

cossx

d)

cosx

37.

$f(x)=a^2-\ x^2\ in\ the\ interval\ \left|x\right|\ \le a\ ,\ where\ a>0\$ means what and find the value of f(0) ?

a)

$f(x)=\ a^2-x^2\ in\ the\ interval\ \ x\ \le a\ \ and\ f\left(0\right)\ =1$

b)

$f(x)=\ a^2-x^2\ in\ the\ interval\ -a\le x\ \le a\ \ and\ f\left(0\right)\ =\ a^2$

c)

$f(x)=a^2-\ x^2\ in\ the\ interval\ \ \ 0\le x\ \le1\ \ and\ f\left(0\right)\ =\ a^2$

d)

$f(x)=\ a^2-x^2\ in\ the\ interval\ -a\le x\ \le a\ \ and\ f\left(0\right)\ =\ 0$

38.

$f(x)=\ x^2\ in\ the\ interval\ \left|x\right|\ \le1\$ means what and find the value of f(0) ?

a)

$f(x)=\ x^2\ in\ the\ interval\ \ x\ \le1\ \ and\ f\left(0\right)\ =1$

b)

$f(x)=\ x^2\ in\ the\ interval\ -1\le x\ \le1\ \ and\ f\left(0\right)\ =\ 1$

c)

$f(x)=\ x^2\ in\ the\ interval\ \ \ 0\le x\ \le1\ \ and\ f\left(0\right)\ =\ 0$

d)

$f(x)=\ x^2\ in\ the\ interval\ -1\le x\ \le1\ \ and\ f\left(0\right)\ =\ 0$

39.

Infinite Fourier Transform pair is also called

a)

complex form

b)

Complex pair

c)

complex transform pair

d)

none of the above

40.

\int e^{ax}\ \cos bx\ dx\ =

a)

$\frac{e^{ax}}{a^2+b^2}\left[a\cos bx-b\sin bx\right]$

b)

$\frac{e^{ax}}{a^2+b^2}\left[a\cos bx+b\sin bx\right]$

c)

$\frac{e^{ax}}{a^2+b^2}\left[b\cos bx+a\sin bx\right]$

d)

none of the above

41.

\int e^{-ax}\ \sin sx\ dx\ =

a)

$\frac{e^{ax}}{a^2+s^2}\left[a\cos sx-s\sin sx\right]$

b)

$\frac{e^{ax}}{a^2+b^2}\left[a\cos bx+b\sin bx\right]$

c)

$\frac{e^{-ax}}{a^2+s^2}\left[a\cos sx-s\sin sx\right]$

d)

$\frac{e^{-ax}}{a^2+s^2}\left[-a\cos sx-s\sin sx\right]$

42.

SinA sin B=

a)

$\frac{1}{2}\left[\sin\left(A+B\right)+\sin\left(A-B\right)\right]$

b)

$\frac{1}{2}\left[\sin\left(A+B\right)-\sin\left(A-B\right)\right]$

c)

$\frac{1}{2}\left[\cos\left(A+B\right)+\cos\left(A-B\right)\right]$

d)

$\frac{1}{2}\left[\cos\left(A-B\right)-\cos\left(A+B\right)\right]$

43.

CosA sin B=

a)

$\frac{1}{2}\left[\sin\left(A+B\right)+\sin\left(A-B\right)\right]$

b)

$\frac{1}{2}\left[\sin\left(A+B\right)-\sin\left(A-B\right)\right]$

c)

$\frac{1}{2}\left[\cos\left(A+B\right)+\cos\left(A-B\right)\right]$

d)

$\frac{1}{2}\left[\cos\left(A-B\right)-\cos\left(A+B\right)\right]$

44.

CosA Cos B=

a)

$\frac{1}{2}\left[\sin\left(A+B\right)+\sin\left(A-B\right)\right]$

b)

$\frac{1}{2}\left[\sin\left(A+B\right)-\sin\left(A-B\right)\right]$

c)

$\frac{1}{2}\left[\cos\left(A+B\right)+\cos\left(A-B\right)\right]$

d)

$\frac{1}{2}\left[\cos\left(A-B\right)-\cos\left(A+B\right)\right]$

45.

SinA Cos B=

a)

$\frac{1}{2}\left[\sin\left(A+B\right)+\sin\left(A-B\right)\right]$

b)

$\frac{1}{2}\left[\sin\left(A+B\right)-\sin\left(A-B\right)\right]$

c)

$\frac{1}{2}\left[\cos\left(A+B\right)+\cos\left(A-B\right)\right]$

d)

$\frac{1}{2}\left[\cos\left(A-B\right)-\cos\left(A+B\right)\right]$

46.

The inverse Fourier Transform of $F[f(x)]\$ is $f\left(x\right)\ =$

a)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F\left[s\right]\ dx\ =\ f\left(x\right)$

b)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]e^{isx}\ ds\ =f\left(x\right)$

c)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]\ e^{-isx}ds\ =f\left(x\right)$

d)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]\ \ \cos sx\ ds\ =\ f\left(x\right)$

47.

The complex form $e^{isx\ }$ and $e^{-isx}$ is

a)

$e^{isx\ }\ =\ \cos sx\ +\ \sin sx\ \ and\ e^{-isx\ }\ =\ \cos sx\ -\ \sin sx\ \$

b)

$e^{isx\ }\ =\ \cos sx\ +i\sin sx\ \ and\ e^{-isx\ }\ =\ \cos sx\ -i\sin sx\ \$

c)

$e^{isx\ }\ =\ \cos sx\ \ and\ e^{-isx\ }\ =\ \ \sin sx\ \$

d)

None of the above

48.

The Imaginary part of $e^{isx\ }$ is

a)

sinsx

b)

i sinsx

c)

cossx

d)

cosx

49.

The Fourier sine Transform of f(x) is $F_s[f(x)]=$

a)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F_s\left[s\right]$

b)

$\frac{1}{\sqrt{2\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$

c)

$\sqrt{\frac{2}{\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$

d)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F_s\left[s\right]$

50.

The inverse Fourier sine Transform of $F^{-1}\left\{F_s[f(x)]\right\}\ =\ f\left(x\right)$

a)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ ds=\ F_s\left[s\right]$

b)

$\frac{1}{\sqrt{2\pi}}\int_0^{\infty}\ F_s\left[s\right]\sin sx\ ds=f\left(x\right)\$

c)

$\sqrt{\frac{2}{\pi}}\int_0^{\infty}F_s\left[s\right]\sin sx\ ds\ =\ f\left(x\right)\$

d)

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F_s\left[s\right]\ \ \cos sx\ dx\ =f\left(x\right)$

51.

1. Find the nature of the one-dimensional wave equation.

a)

Hyperbolic

b)

Parabolic

c)

Elliptic

d)

None of These

52.

The nature of the one-dimensional heat equation is

a)

Circular

b)

Elliptic

c)

Parabolic

d)

Hyperbolic

53.

The nature of PDE 4u_xx+3 u_xy+3 u_yy=0

a)

Parabolic

b)

Hyperbolic

c)

Elliptic

d)

Laplace

54.

The PDE u_xx+ u_yy= 0, is known as

a)

1-D heat equation

b)

1-D wave equation

c)

Laplace equation

d)

None of these

55.

Let PDE u_xx+ u_yy= u_t, By method separation, we consider the solution

a)

u(x,y)=X(x)Y(y)

b)

u(x,y,t)=X(x)Y(y)T(t)

c)

u(x,t)=X(x)T(t)

d)

None of these

56.

Let PDE c²(u_xx+ u_yy)= u_t, By is known as

a)

2-D heat equation

b)

2-D wave equation

c)

Laplace equation

d)

None of these

57.

Let PDE c²(u_xx+ u_yy)= u_tt, By is known as

a)

2-D heat equation

b)

2-D wave equation

c)

Laplace equation

d)

None of these

58.

The nature of PDE 4u_xx+3 u_xy=0

a)

Circular

b)

Elliptic

c)

Hyperbolic

d)

Parabolic

59.

The nature of PDE u_xx+4 u_xy+3 u_yy,=0

a)

Circular

b)

Elliptic

c)

Hyperbolic

d)

Parabolic

60.

A rod of length 10 m has temperature 30⁰C and 40⁰C at end points. What is the temperature gradient

a)

1⁰C per cm

b)

3⁰C per cm

c)

2⁰C per cm

d)

None of these

61.

In the steady state, 2-D heat equation reduces to

a)

1-D heat equation

b)

1-D wave equation

c)

Laplace equation

d)

None of these

62.

In the steady state, the change in temperature at any point with respect to time is

a)

0⁰C

b)

>0⁰C

c)

<0⁰C

d)

None of these

63.

The ends A and B of a rod of length10cm are at 30⁰C and 80⁰C at end points until steady state prevails. Then Initial temperature distribution in the rod

a)

10+(5x)/2

b)

30+(5x)/2

c)

30+5x

d)

None of these

64.

The ends A and B of a rod of length20cm are at 30⁰C and 80⁰C at end points until steady state prevails. Then the temperature of the rod at ends are changed to 40⁰C and 60⁰C respectively. Final temperature distribution (i.e. in Steady state) is

a)

40+x

b)

50+x

c)

60+x

d)

70+x

65.

A partial differential equation requires

a)

exactly one independent variable

b)

two or more independent variables

c)

more than one dependent variable

d)

equal number of dependent and independent variables

66.

Using substitution, which of the following equations are solutions to the partial differential equation?

a)

b)

c)

d)

67.

The partial differential equation

is classified as

a)

elliptic

b)

parabolic

c)

hyperbolic

d)

none of the above

68.

The partial differential equation

is classified as

a)

elliptic

b)

parabolic

c)

hyperbolic

d)

none of the above

69.

The complete integral of $z=px+qy+f\left(p,q\right)$

a)

$z=px+qy$

b)

$z=ax+by+f\left(a,b\right)$

c)

$z=f\left(a,b\right)$

d)

none of the above

70.

What is the order & degree of the p.d.e $x^2\left(\frac{\partial z}{\partial x}\right)^2=z\left(x-y\frac{\partial z}{\partial y}\right)$

a)

1 & 2

b)

2 & 1

c)

2 & 2

d)

none of the above

71.

Singular solution of Partial differential equation can be obtained by

a)

General solution

b)

Complete solution

c)

both a & b

d)

none of the above

72.

The condition of compatibility of partial differential equations $f\left(x,y,z,p,q\right)=0\ ,\ \ g\left(x,y,z,p,q\right)=0$ is

a)

$\frac{\partial\left(f,g\right)}{\partial\left(p,q\right)}=0$

b)

$\frac{\partial\left(f,g\right)}{\partial\left(p,q\right)}\ne0$

c)

$\frac{\partial\left(f,p\right)}{\partial\left(g,q\right)}=0$

d)

$\frac{\partial\left(f,q\right)}{\partial\left(g,p\right)}\ne0$

73.

If u is homogeneous function of order n then $\frac{\partial u}{\partial x}\ ,\ \frac{\partial u}{\partial y}$ both are homogeneous function of order

a)

n

b)

n-1

c)

n+1

d)

n-2

74.

If $f\left(x,y\right)=0\$ then what is the value of $\frac{dy}{dx}$ is

a)

$\frac{f_x}{f_y}$

b)

$-\frac{f_x}{f_y}$

c)

$\frac{f_y}{f_x}$

d)

$-\frac{f_y}{f_x}$

75.

An equation containing partial derivatives of one or more dependent variables of two or more independent variables

a)

Ordinary Partial Differential Equation

b)

Partial Differential Equation

c)

Ordinary Differential Equation

d)

Partially Ordinary Differential Equation

76.

Find the Partial derivative of f with respect to y for $f\left(x,y\right)=\frac{y^2}{x+y}$

a)

$\frac{2x+y}{\left(x+y\right)^2}$

b)

$\frac{2xy+y^2}{\left(x+y\right)^2}$

c)

$\frac{2}{\left(x+y\right)^{ }}$

d)

$\frac{2x^2+y^2}{\left(x+y\right)^2}$

77.

Find $\frac{\partial f}{\partial x}$ of $f\left(x,y\right)=\sin\left(\frac{x}{1+y}\right)$

a)

$\cos\left(\frac{x}{1+y}\right)$

b)

$\cos\left(\frac{x}{1+y}\right)\left(\frac{1}{1+y}\right)$

c)

$-\cos\left(\frac{x}{1+y}\right)\left(\frac{x}{\left(1+y\right)^2}\right)$

d)

$\sin\left(\frac{x}{1+y}\right)\left(\frac{1}{1+y}\right)$

78.

$Find\ \frac{\partial f}{\partial x}for\ f\left(x,y\right)=3e^{2x}+7xy-3x$

a)

$6e^{2x}+7y-3$

b)

$6e^{2x}+7x-3$

c)

$6e^x+7y-3$

d)

$6e^{2x}+7y+7x-3$

79.

$find\ \frac{\partial f}{\partial y},\ given\ f\left(x,y\right)=e^x+2xy^2-4y\ at\ \left(0,3\right)$

a)

$8$

b)

$-4$

c)

$-12$

80.

Find the partial derivative of $f_y$ for the function $f\left(x,y\right)=\left(3x^2+y^2\right)^3$

a)

$3y\left(3x^2+y^2\right)^2$

b)

$3x^2\left(3x^2+y^2\right)^2$

c)

$6\left(3x^2+y^2\right)^2$

d)

$6y\left(3x^2+y^2\right)^2$

81.

$Find\ \frac{\partial f}{\partial y}\ if\ f\left(x,y\right)=3\cos\left(2y\right)-\sin\left(x+y\right)$

a)

$f_y\left(x,y\right)=-6\sin\left(2y\right)-\cos\left(x+y\right)$

b)

$f_y\left(x,y\right)=-\cos\left(x+y\right)$

c)

$f_y\left(x,y\right)=6\sin\left(2y\right)+\cos\left(x+y\right)$

82.

$Find\ \frac{\partial^2f}{\partial x^2}\ if\ f\left(x,y\right)=x^3+2x^2y-3y+2x+5$

a)

$3x^2+4xy+2$

b)

$6x+4$

c)

$6x+4y$

d)

$x^2+3xy$

83.

if $f(x,y)=x^3-2xy+xy^3+3y^2$ which of the following is true?

a)

$f_x(x,y)=3x^2-2y+y^3$ $f_y(x,y)=-2x+6y+3xy^2$ $f_{xy}(x,y)=6x$

b)

$f_x(x,y)=-2x+6y+3xy^2$ $f_y(x,y)=3x^2-2y+y^3$ $f_{xy}(x,y)=-2+3y^2$

c)

$f_{xx}(x,y)=6x$ $f_{yy}(x,y)=6+6xy$
$f_{xy}(x,y)=-2+3y^2$

d)

$f_{xx}(x,y)=6+6xy$ $f_{yy}(x,y)=6x$
$f_{xy}(x,y)=-2+3y^2$

84.

Find the first order partial derivative with respect to y
$f(x,y)=x^3y^2+3xe^y$
.

a)

^{$f_y(x,y)=3x^2y^2+3e^y$}

b)

$f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$

c)

$f_y(x,y)=2x^3y+3xe^y$

d)

$f_y(x,y)=6x^2+3e^y$

85.

Find the Partial derivative of f with respect to y for
$f\left(x,y\right)=\frac{y^2}{x+y}$

a)

$\frac{2x+y}{\left(x+y\right)^2}$

b)

$\frac{2xy+y^2}{\left(x+y\right)^2}$

c)

$\frac{2}{\left(x+y\right)}$

d)

$\frac{2x^2+y^2}{\left(x+y\right)^2}$

86.

Given a contour plot of $f\left(x,y\right)$ . What would be a good estimate of $\frac{\partial f}{\partial x}$ at the origin?

a)

0

b)

1

c)

1.5

d)

2

87.

Given a contour plot of $f\left(x,y\right)$ . What would be a good estimate of $\frac{\partial f}{\partial y}$ at the origin?

a)

$-\frac{2}{3}$

b)

-2

c)

0

d)

$\frac{3}{2}$

88.

Find the gradient vector of $f\left(x,y,z\right)=z^2\ln\left(xy\right)$

a)

$<\frac{z^2}{y},\frac{z^2}{x},2z>$

b)

$<\frac{z^2}{x},\frac{z^2}{y},2z>$

c)

$<\frac{z^2}{y},\ \frac{z^2}{x},2z\ln\left(xy\right)>$

d)

$<\frac{z^2}{x},\frac{z^2}{y},2z\ln\left(xy\right)>$

89.

Find the second derivative of the function:
f (x) = 2x - 5x⁶

a)

f ''(x)= 2 - 30x

b)

f ''(x) = 2-30x⁵

c)

f ''(x) = -30x⁵

d)

f ''(x) = -150x⁴

90.

If z is a function which depends on variables x and y, the change of z with respect to one of its variables is called as:

a)

Total derivative.

b)

Partial derivative.

c)

Inexact differential.

d)

I don't know :)

91.

A partial differential equation has:

a)

one independent variable.

b)

equal number of dependent and independent variables.

c)

more than one dependent variable.

d)

two or more independent variables

92.

If z is a function which depends on variables x and y, the change of z with respect to variable x can be expressed as

a)

$\left(\frac{\partial z}{\partial x}\right)_y$

b)

$\left(\frac{\partial z}{\partial x}\right)_x$

c)

$\left(\frac{\partial x}{\partial z}\right)_x$

d)

I don't know :)

93.

If z = f (x, y) ; then its total differential can be expressed as

a)

$dz=\left(\frac{\partial z}{\partial x}\right)_ydx\ +\left(\frac{\partial z}{\partial y}\right)_xdy$

b)

$dz=\left(\frac{\partial z}{\partial x}\right)_ydy\ +\left(\frac{\partial z}{\partial y}\right)_xdx$

c)

$dz=\left(\frac{\partial z}{\partial x}\right)_y\ +\left(\frac{\partial z}{\partial y}\right)_x\$

d)

$dz=\left(\frac{\partial z}{\partial x}\right)_x\ +\left(\frac{\partial z}{\partial y}\right)_y\$

94.

a)

b)

c)

d)

95.

a)

b)

c)

d)

96.

a)

b)

c)

d)

97.

a)

b)

c)

d)

98.

a)

b)

c)

d)

99.

Find the second order partial derivatives of

f(x,y)=(3x+2y)⁴

a)

f_xx(x,y)=12(3x+2y)³, f_yy(x,y)=24(3x+2y)²

b)

f_xx(x,y)=36(3x+2y)², f_yy(x,y)=8(3x+2y)³

c)

f_xx(x,y)=24(3x+2y)², f_xy(x,y)=32(3x+2y)

d)

f_xx(x,y)=108(3x+2y)², f_yy(x,y)=48(3x+2y)²

100.

For an ideal gas, evaluate the product of partial derivatives:

\left(\frac{\partial P}{\partial V}\right)_T\left(\frac{\partial T}{\partial P}\right)_V\left(\frac{\partial V}{\partial T}\right)_P

a)

1

b)

-1

c)

R

d)

None of them

101.

The z-transform of a signal X(n) whose definition is given by X(z)=

\sum_{n=0}^{\infty}x\left(n\right)z^{-n}

is known as

a)

Unilateral Z-transform

b)

Bilateral Z-transform

c)

Rational Z-transform

d)

None of the above

102.

For what kind of signals one sided Z-transfrom is unique

a)

All signals

b)

Anti-causal signal

c)

Causal signal

d)

None of the above

103.

Z-transform is linear

a)

True

b)

False

104.

The Z-transform of

\delta\left(n+k\right)>0

is

a)

$z^{-k,\ }z\ne0$

b)

$z^k,z\ne0$

c)

$z^{-k},\ all\ z$

d)

$z^k,all\ z$

105.

Z-transform of the sequence $\left\{2^k\right\},k\ge0\ is\ \frac{z}{z-2}$

a)

True

b)

False

106.

$Z-transform\ of\ the\ sequence\ \left\{\frac{a^k}{k!}\right\},k\ge0=e^{\frac{a}{z}}$

a)

True

b)

False

107.

$Z-transform\ of\ the\ sequence\ \left\{nc_r\right\},\left(0\le r\le n\right)\ is\ \left(1+z\right)^n$

a)

True

b)

False

108.

$The\ value\ of\ the\ Radius\ of\ convergence\ of\ f\left(n\right)=2^n,n<0\ is$

a)

0< $\left|z\right|$ <1

b)

-2< $\left|z\right|$

c)

$\left|z\right|$ <2

d)

z-plane

109.

$The\ ROC\ of\ u\left(n\right)=4^n,for\ n<0;2^n,for\ n\ge0\ is$

a)

0<z<1

b)

z<4

c)

2<z

d)

2<z<4

110.

$If\ Z\left(u_n\right)=u\left(z\right),\ then\ \lim_{n\rightarrow\infty}u_n=\lim_{n\rightarrow\infty}\left(z-1\right)u\left(z\right)$

a)

True

b)

False

111.

$The\ Inverse\ Z-transform\ of\frac{z}{\left(z+1\right)^2}is$

a)

$\left(-1\right)^{n+1}$

b)

$n\left(-1\right)^{n-1}$

c)

$\left(-1\right)^{n-1}$

d)

$n\left(-1\right)^{n+1}$

112.

The value of sin $n\pi$ is

a)

1

b)

0

c)

-1

d)

n

113.

The value of cos $n\pi=$ ------ and Cos 0=-------

a)

0 and 0

b)

$\left(-1\right)^n$ and 0

c)

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

d)

1 and 1

114.

The value of $\int_{ }^{ }\cos5x\ dx$ =

a)

5 sin5x

b)

-5sin5x

c)

$\frac{\sin5x}{5}$

d)

$-\frac{\sin5x}{5}$

115.

The value of $\int_0^{\pi}\ \sin3x\ dx$

a)

$\frac{2}{3}$

b)

$-\frac{2}{3}$

c)

0

d)

$\frac{1}{3}$

116.

$\int_0^{2\pi}\ \ x\ \sin x\ dx$ =

a)

$\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|$

b)

$\left|x\left(\cos x\right)-\left(\sin x\right)\right|$

c)

$\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|_0^{2\pi}\ =\ 0$

d)

$\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|_0^{2\pi}\ =\ -2\pi$

117.

$\int_0^{2\pi}\ x\ \cos nx\ dx\ =$

a)

$\left|x\left(\sin x\right)\left(-\cos x\right)\right|$

b)

$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|$

c)

$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\ \ =0$

d)

$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\ =2$

118.

$\int_0^{2\pi}\ x\ \cos\left(\frac{n\pi}{l}\ \right)x\ dx\ =$

a)

$\left|x\left(\sin\left(\frac{n\pi}{l}\ \right)x\right)\left(-\cos\left(\frac{n\pi}{l}\ \right)x\right)\right|$

b)

$\left|x\left(\frac{\sin\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)}\right)-\left(-\frac{\cos\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)^2}\right)\right|$

c)

$\left|x\left(\frac{\sin\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)}\right)-\left(-\frac{\cos\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)^2}\right)\right|_0^{2\pi}\ \$

= 0

d)

$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\$
=2

119.

The Fourier coefficient a_o for f(x) = e^x in (-π,π ) is

a)

sinhπ/π

b)

e^π+e^-π

c)

e^π-e^-π

d)

2sinhπ/π

120.

The Fourier coefficient a₀for f(x) = x sinx in (-π,π) is

a)

2π

b)

2

c)

π

d)

-π

121.

The Fourier constant b_nfor f(x)= x sinx in (-2,2) is

a)

2

b)

1

c)

0

d)

4

122.

A Periodic function is given by a function which

a)

has a period T= 2π

b)

has a period T= π

c)

satisfies f(t+T) = f(t)

d)

satisfies f(t+T) = - f(t)

123.

Find the sum of 1

1/2+1/3^2+1/5^2+……… using Fourier series expansion if f(x) = a when [0,π] and 2 π – x when [ π, 2 π].

a)

π^2/8

b)

π^2/4

c)

π^2/16

d)

π^2/2

124.

A periodic function is defined as:

a)

f(x+P) = f(x)

b)

f(x) = -f(x)

c)

f(x+nP) = f(x)

d)

f(x) = f(-x)

125.

The particular conditions that a function f(x) must fulfill in order that it may be expanded as a Fourier series is:

a)

Gibbs phenomenon

b)

Fourier–Mellin theorem

c)

Dirichlet conditions

d)

Convolution theorem

MA8353- TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Total questions: 125

Worksheet time: 4hrs 14mins

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