Uttar Pradesh Technical University (UPTU) 2007-2nd Sem B.Tech Electronics and Communications Engineering 2006- mathematics-II - Question Paper

Printed Pages : 4    AG-121

(Following Paper ID and Roll No. to be filled in your Answer Book)

Roll No.

[Total Marks : 100

Note : Attempt all the questions. Internal choice is mentioned for each question.

1 Attempt any four parts of the following : 5x4=20

(a)    Find the directional derivative of

f(x, y, z) = 2x² + 3y² + z² at the point (1, 2, 3) in the direction of the vector

a = i + 2j + 3k

(b)    Define gradient, divergence and curl. Assuming

> >

necessary condition(s) on v and u prove that

div (u x v) = v curl u- u. curl v

(c)    Find the work done by a force F(x, y, z) applied at a point P( 1, 2, 3) to displace it to point Q(5, 1, 7)

V-9920]    1    [Contd...

(d) If / and g are two scalars functions prove that div(fVg) = fV²g + VfVg ^where

r3 "id f d

V = l-+ 7-+ k-

dx dy dz '

(e) Show that the vector function

V(x, y, z) = (x + 3y)i + (y - 3z)j + (x - 2z)k

is solenoidal.

(1) Show that div(grad rⁿ) = n(n + l)r^{n l}

V2 2 2 x + y + z

2 Attempt any four parts of following :    5x4=20

(a)    Find the area between the parabolas y = 4ax

o

and x = 4 by .

a x x+y

(b)    Evaluate j j j    dz dy dx

0 0 0

(c)    Find the volume of the ellipsoid

x² ;y² z² ,

^{2 +} i ^{2 + 2} a b c

J" (xy + y²)dx + x²dy

o

Where C is bounded by y = x and y = x .

(e)    State the Gauss divergence and verify for

F = (x² - yz) i + (y² - zx) j + (z² - xy) k taken over the cube 0 < x < 1, 0 <y < 1 and 0<z<l

(f)    State and verify the Stoke's theorem for the vector field F = (x² - y²)i + 2xy j by integrating round the rectangle in plane z = 0 and bounded by the lines x = 0, = 0, x = a and y = b.

Attempt any two parts of following :    10x2=20

(a) Solve the differential equations

(1)    (hx + by + f)dy + (ax + hy + g)dx = 0

(2)    ye^ydx = (j³ +2 xe^y)dy

2 dy    3

(b) Solve sec y + xtanj = x . dx

(c) Solve (D + 4D + 5)y = e^x cos x + x .

(b) For Legendre polynomial P_fl (x) prove that

l

J" P¹ (x)dx

-l

(c) Solve (p² +q²)y = qz.

Attempt any two parts of the following :

(a) Find the inverse of the matrix

(b) Solve the system of linear equations :

x + 2y + z = 3 2x + 3y + 2z = 4 3x + 4y + 3z = 17

(c) State and prove Cayley-Hamilton theorem.

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Attachment: uttar-pradesh-technical-university--uptu--2007-b-tech-electronics-and-communications-engineering-20696-1.pdf

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