Free Printable Engg.Mathematics II

1.

Find the Eigen values for the following 2×2 matrix. $A=\left|\frac{1}{2}\frac{8}{1}\right|$

a)

-3

b)

2

c)

6

d)

4

2.

The sum of the eigen values of a matrix is the sum of the elements of the

a)

principal diagonal

b)

diagonal element

c)

equal

3.

A square matrix A and its transpose have the same

a)

Eigen vectors

b)

Eigen values

c)

same values

4.

Use of cayley hamilton theorem

a)

+ve integral powers of A , invers of a non singular square matrix

b)

-ve integral powers of A , invers of a non singular square matrix

c)

+ve integral powers of A , invers of a singular square matrix

5.

Find the nature of $2x_1x_2+2x_1x_3-2x_2x_3$

a)

Definite

b)

semi definite

c)

indefinite

6.

$Find\ rank\ and\ signature\ 2x_1x_2+2x_1x_3+2x_2x_3$

a)

3,1

b)

3,-1

c)

2,1

d)

2,-1

7.

A vector field which has a vanishing divergence is called as ____________

a)

Solenoidal field

b)

Rotational field

c)

Hemispheroidal field

d)

Irrotational field

8.

A vector field with a vanishing curl is called as __________

a)

a) Irrotational

b)

b) Solenoidal

c)

c) Rotational

d)

d) Cycloidal

9.

$find\ \nabla r$

a)

$\frac{\overrightarrow{r}}{r}$

b)

$\frac{r}{\overrightarrow{r}}$

c)

$\overrightarrow{r}$

d)

$r$

10.

$if\ \nabla\phi=2xy\overrightarrow{i}+x^2z\overrightarrow{j}+x^2y\overrightarrow{k},\ find\ \phi$

a)

xyz

b)

$x^2yz$

c)

$xy^2z$

d)

$xyz^2$

11.

$if\ \overrightarrow{f}\ =\ x^2\overrightarrow{i}+y^2\overrightarrow{j}+z^2\overrightarrow{k,}\ \ find\ \nabla X\overrightarrow{f}$

a)

1

b)

0

c)

2

12.

$Show\ that\ r^n\overrightarrow{r}\ is\ an\ irrotational\ vector\ for\ any\ value\ of\ n$

a)

$\left(3+n\right)r$

b)

$\left(3+n\right)r^n$

c)

$\left(3+n\right)$

13.

$C\ R\ equation\ is\$

a)

$u_x=v_{y\ \ \ \ \ }u_y=-v_x$

b)

$u_x=v_{y\ \ \ \ \ }u_y=v_x$

c)

$u_x=-v_{y\ \ \ \ \ }u_y=v_x$

14.

$Show\ that\ f\left(z\right)=\left|z\right|^2\ is\ differential\ at$

a)

z=1

b)

z=0

c)

z=2

d)

z=-1

15.

$shown\ that\ the\ function\ w=e^zis\ analytic\$

a)

everywhere

b)

anywhere

c)

almost

d)

atleast

16.

$whether\ the\ function\ 2xy+i\left(x^2-y^2\right)\ is\$

a)

analytic

b)

not analytic

17.

An analytic function with constant modulus is

a)

constant

b)

equal

c)

same

18.

$if\ f\left(z\right)\ -u+iv\ is\ regular\ function\ of\ z\ in\ a\ domain\ D\ then\ \nabla^2\left|f\left(z\right)\right|$

a)

4(f '(z))

b)

(f '(z))

c)

4(f '(z))^2

19.

contour integral is

a)

integral with closed curve

b)

closed curve

c)

integral with open curve

d)

open curve

20.

cauchy's integral theorem is

a)

$\int_c^{ }f\left(z\right)dz=0$

b)

$\int_c^{ }f\left(z\right)dz=1$

c)

$\int_c^{ }f\left(z\right)dz=0$

21.

$Evaluate\ \int_{ }^{ }e^{\frac{1}{z}}dz\ where\ c\ is\ \left|z\right|=1$

a)

Analytic inside c

b)

analytic outside c

22.

Evaluate $\int_{ }^{ }\frac{dz}{ze^z}\ where\ c\ is\ \left|z\right|=1$

a)

$2\pi i$

b)

$2\pi$

c)

$2i$

23.

$find\ pole\ \ \frac{1}{\left(z-4\right)^2\left(z-3\right)^4}$

a)

2,4

b)

4,2

24.

contour integration is

a)

evaluating definite integral

b)

evaluating indefinite integral

c)

evaluating integral

25.

State condition for laplace transform

(a)

26.

$find\ L\left(t^n\right)$

a)

$\frac{n!}{s^{\left(n+1\right)}}$

b)

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

27.

$find\ L\left(\cos at\right)$

a)

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

b)

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

c)

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

28.

$find\ L\left(\sin at\right)$

a)

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

b)

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

c)

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

29.

$find\ L\left(\cosh at\right)$

a)

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

b)

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

c)

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

30.

$find\ L\left(\sinh at\right)$

a)

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

b)

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

c)

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

31.

$Reduce\ the\ quadratic\ form\ Q=6x^2+3y^2+3z^2-4xy-2yz+4zx\ into\ canonical\ form\ by\ orthoganal\ transformation.$

a)

$8y_1^2+2y_2^2+2y_3^2$

b)

$8y_1^2-2y_2^2-2y_3^2$

c)

$2y_1^2+2y_2^2+2y_3^2$

32.

Reduce the quadratic form

x^2+y^2+z^2-2xy-2yz-2zx

to canonical form an orthoganal transformation.

a)

$-y_1^2+2y_2^2+2y_3^2$

b)

$y_1^2+2y_2^2+2y_3^2$

c)

$-2y_1^2+2y_2^2+2y_3^2$

33.

Reduce the quadratic form Q to its canonical form using orthoganal transformation

Q=x^2+3y^2+3z^2-2yz

a)

$y_1^2+2y_2^2+4y_3^2$

b)

$y_1^2+2y_2^2+y_3^2$

c)

$y_1^2+y_2^2+4y_3^2$

34.

Find greens theorem in the XY plane for

\int_c^{ }\left\{\left(3x-8y^2\right)dx+\left(4y-6xy\right)dy\right\}

where C is the boundary of the region given by x=0,y=0,x+y=1

a)

$\frac{5}{3}$

b)

$\frac{3}{5}$

c)

$\frac{1}{5}$

35.

Find the G.D.T for

\overrightarrow{F}\ =4xz\overrightarrow{i\ }\ -y^2\overrightarrow{j}\ +yz\overrightarrow{k}\

over the cube bounded by x=0,x=1,y=0,y=1,z=0,z=1

a)

$\frac{3}{2}$

b)

$\frac{2}{3}$

c)

$\frac{1}{3}$

d)

$\frac{1}{2}$

36.

Find stokes theorem for a vector field defined by

\overrightarrow{F}\ =\left(x^2-y^2\right)\overrightarrow{i}\ +2xy\overrightarrow{j}\

in the rectangular region in the XOY plane bounded by the line x=0,x=a,y=0,y=b

a)

$2ab^2$

b)

$2ab$

c)

$2a^2b$

d)

$2a^2b^2$

37.

Find the image of

\left|z-2i\right|=2

under the transformation

w=\frac{1}{z}

a)

$v=\frac{1}{4}$

b)

$v=\frac{-1}{4}$

c)

$v=\frac{1}{2}$

38.

Find the bilinear transformation that maps the point $\infty,i,0\ onto\ 0,i,\infty$ respectively.

a)

$w=\frac{1}{z}$

b)

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

c)

$w=z$

39.

Find the image of the circle $\left|z-1\right|=1$ in the complex plane under the mapping w=1/z

a)

$u=\frac{1}{2}$

b)

$v=\frac{1}{2}$

c)

$u=\frac{-1}{2}$

d)

$v=\frac{-1}{2}$

40.

Evaluate using cauchys residue theorem $\int_{ }^{ }\frac{\sin\pi z^2+\cos\pi z^2}{\left(z-1\right)\left(z-2\right)}dz,\ c:\left|z\right|=3$

a)

$4\pi i$

b)

$2\pi i$

c)

$\pi i$

41.

Using contour integration , eveluate $\int_0^{2\pi}\frac{d\theta}{13+5\cos\theta}$

a)

$\frac{\pi}{6}$

b)

$\frac{\pi}{3}$

c)

$\frac{\pi}{2}$

42.

Evaluate $\int_0^{\infty}\ \frac{x^2\ dx}{\left(x^2+1\right)\left(x^2+4\right)}$ using contour integration

a)

$\frac{\pi}{3}$

b)

$\frac{\pi}{2}$

c)

$\pi$

Engg.Mathematics II

Total questions: 42

Worksheet time: 54mins

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