Punjab Technical University 2008-1st Sem B.Tech <>EnggMathematics -II (AMA-102) (_ 2nd) << >> - Question Paper

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Total No. of Questions : 09]    [Total No. of Pages : 04

B.Tech. (Sem. - 1^st / 2^,,d)

ENGINEERING MATHEMATICS - II SUBJECT CODE : AM - 102 (New)

Paper ID : [AOll'J]

[Note : Please fill subject code and paper ID on OMR]

Maximum Marks: 60

Time : 03 Hours Instruction to Candidates:

1)    Section - A is Compulsory.

2)    Attempt any Five questions from Section - B & C.

3)    Select atleast Two questions from Section - B & C.

Section - A

(Marks : 2 each)

	"2	0	o'		'1	2	3'
a) If A-	0	2	0	and B =	0	1	3
	_0	0	2_		0	0	2.

(iii) 16, (iv) 32

c)    Two balls of m_i and m₂ gms are projected vertically upward such that the velocity of projection of m_l is double that of m_r If the maximum height to which m, and m₂ rise, be and h₂ respectively, then

(i) h_l = 2h₂ . (ii) 2h_l = h₂ (iii) h_x ~ 4h₂ (iv) 4h_l = h₂

d)    The complementary part of the differential equation

x²y" - xy' + y - log x is-----.

e) The particular integral of (D² + a²) y = sin ax is

f)    If u = (x² + y²) then v-(Vh)

(i) 0 (ii) 1 (iii) -1

g)    Maximum value of the directional derivative of

/'= x² - 2y² + 4z² at point (1, 1, -1) is----

h)    Average scores of three batsman A, B, C are respectively 40,45, 55 and their standard deviations are respectively 9, 11, 16. Which batsman is more consistant?

i)    If the correlation coefficient is zero, then regression lines are (i) parallel    (ii) perpendicular

(iii) coincident    (iv) intersect at 45.

j) The probability that a leap year should have 53 sundays is

J.

1

(iii) 0.3    (iv) 0.5

Section - B

Q2) (a) Find the values of a, b, c if the matrix

	'0	2b	c
A =	a	b	-c
	a	-b	+c

is orthogonal.

Show that A is a Hermitian matrix and /A is a skew - Hermitian matrix.

Q3) Solve the following:

~{^x2 +6xy²) (h) __L = -\

dx 6 x²y + 4 y³

(c) (px - y) (x + py) = 2p.

Q4) Solve the following:

(a)    (D - 2)²y = Sie²* + sin 2x + X²}.

(b)    xY' + 2xY + 2y = lo(x + \.

Q5) (a) Solve

(D² - 1) y - e^3x cos 2x-e^2x sin 3x using method of undetermined coefficients.

(b) Two particles each of mass m gms are suspended from two springs of same stiffness coefficient L After the system comes to rest, the lower mass is pulled / cms downwards and released. Discuss their motion.

Section - C

(Marks : 8 each)

Q6) (a) What is conservative field? Show that

F = (j² cosx + z³) i + (2>>sinx~4)j + (3xz² + 2

is conservative. Find its scalar potential.

(b) Use Divergence theorem to evaluate

JF-dS ^where

s

F = x³i + y³] + z³k ' and S is the surface of the sphere x² + y² + z² = a².

Q7) (a) Show that the function (fi - a cos nix is not a valid velocity potential flow function of liquid.

(b) Test whether the motion specified by

q - k²[xj - y/) j (x² + y²) {k is constant)

is a possible motion of a liquid.

Q8) (a) Discuss Binomial frequency distribution. The probability that a bomb dropped from a plane hits the target is A. If 6 bombs are dropped, find

the probability that atleast two will hit the target.

(b) The pressure and volume of a gas are related by the equation pv^a - k, a and k being constants. Find the equation to the following set of values.

p (kg/cm²) 0.5 1.0 1.5 2.0 2.5 3.0

v (litres) 1.62 1.00 0.75 0.62 0.52 0.46

Q9) (a) Discuss Chi-square test and its properties. Use this to test the hypothesis that data follows a binomial distribution for the problem in which a set of five similar coins is tossed 320 times and the result is

No. of heads: 0 1 2 3 4 5

Frequency: 6 27 72 112 71 32

(b) Two independent samples of size 7 and 6 have the following values:

Sample A: 28 30 32 33 33 29 34

Sample B: 29 30 30 24 27 29

Examine whether the samples have been drawn from normal populations having the same variance. Given the values of F at 5% level for 16, 57 degrees of freedom is 4.95 and for 15, 67 degrees of freedom is 4.39.

M-467    4

Attachment: punjab-technical-university-2008-b-tech-23814-1.pdf

Earning:   Approval pending.