I{oll No. ...........................

Total No. of Questions: 091-"""""

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ITotal No. of Pages : 03

Paper In [AMI02] ;:I.~(l'lrasr fill this I'a(lrr II) ill (nm Shl'l'O ~~,

B.Te~h. (Sem. - pt/2"d) . ~MATHEMATlCS - II (AMA - 102)

Time: 03 Hours Maximum Marks: .60

Instruction to Candidates:

1) Section -A is Cnmpulsory.

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

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

S~ction - A

Ql) , (2 Marks each), -

a) Reduce quadratic form 3x2+ 5y +3Z2- 2yz + 2zx - 2xy to canonicalform. .

b) One eigen value of matrix A is zero. Explain whether Ajs invertibl~ ornot.

c) Find particular integral of (IY -. 2D + 1)y = xe'rsin x.

d) State Stokes theorem.

e) Find differential equation of all circles having centre on x-axis.

f) State Lagrange's interpolation formula.

g) State the condition under which given system of m-equations inn-variables is consistent and haveintinite many solutions.

h) Can I(x) ={ fX :~~~ oe a probability density function. Explain.

i) Find Binomi~ldistribution with mean 4 and variance 4 .- '3

j) Find the complementary function of (lj - 2IY + 1)y = sin x.

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Section - B(8 Marks each)

. ,- . ,..Q2) (a) Find A"Ll such that system of equationsx + y + z = 6, x + 2y + 3z= 10,

and x + 2y+ AZ= I-i have (i) no solution (ii) unique solution.

[

21 I

](b) Find characteristics equation of A = 0 I O. Hence find the matrix

1 '1 2

A8- 5A7 + 7A6- 3A5 + A4- 5A3 + 8A2 - 2A + I. - (

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Q3) (a) If F=(2x2_3z)/-2xy]-4xk then evaluate ffJV'FdV, where V isv

the volume bounded by x = 0, y = 0, z = 0 and 2x + 2y + Z.=4.

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(b) Show that Vx(AxB) =(V'B)A-(V.A)B+(B.V)A-(A.V)B.

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Q4) Verify divergence theorem for F =4x/ - 2y2] + Z2k taken over region of

cylinder x? + Y = 4, Z = 0, Z =-3.

Q5) (a) In a sample of 100battery cells tested to find the length oflife producedfollowing data x = 12 hours and a = 3 hours. Assuming data to be

normally distributed what percentage of cells are expe~ted to have life of(i) more than 1~ hours (ii) between 10 to 14 hours.(Given P (0 < Z < I) = 0.3413, P (0 < Z < 0.67) = 0~2487).

(b) In 800 families each with 5 children, how many would you expect to. have (i) 2 boys and 3 girls (ii) no girls.

Section - C

(8 Mttrks eac")

Q6) (a) In two random samples of size 16and 25 respectively varfance is 40 and42 respectively. Test whether two samples come from same population.

(Given FOO5= 2.1 for 15,24degree of freedom and Ito.051 = 1.96for 39dt).

R:..24 2

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(b) Verify whether Poisson distribution can be assumed for fo1lowing data

.. I" Number of defects 0 1 2 3 4 5Frequency 6 13 13 8 ' 4 3

Given that e-2= 0.] 36, and %,;.U5 = 9.49 for 4 df.

Q7) (a) Solve (4x - 6y - I) dx + (3y - 2x -2) dy = O.

dy vcosx+sin v + v(b) Find the solution of -+ "'. "'., -0.

dx smx+xcosy+x

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Q8) (a) Solve (lY - IY) y = xl + e>;.

(b) Find the solution ofy logy d" + (x -logy) dy = O.

Q9) (a) Find sin 52° from the following data using Newton~s interpolation formula

angle0 0 45 50 55 60

sino 0.7071 0.7660 0.8192 0.8660

(b) Define shift and averaging operator.Also find the relation between two.

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