CSVTU Exams Questions Papers – Ist Year – Applied Mathematics-I –Nov-Dec- 2007

BE (1st Semester)

Examination Nov-Dec, 2007

Applied Mathematics-II

UNIT- I

1. (a) State De Moire’s theorem.

(b) cos a + cos ? + cos ? = sin a + sin ? + sin ? = 0 Prove that

(i) cos2 a + cos2 ? + cos2 ? = sin2 a + sin2 ? + sin2 ?

(ii)cos2a + cos2 ? + cos 2 ? = sin2a + sin2 ? + sin 2 ? = 0

(c) If tan (? + i?) = eiaShow that:

? – (n +1)  and                 log tan(?  +  a)

2    2                                        4       2

(d) Sum the series: ? – 1

2

sin2 ? 1 sin 2? sin2? + 1 sin3? – sin 4?sin4 ?+……….. ?

2                         3              4

UNIT- II

2. (a) Solve d2y  – 3dy – 4y =0

dx2       dx

(b) Solve d2y  – 5dy + 6y =sin3x

dx2       dx

(c) Solve by method of variation of parameters d2+ 4y=tan2x

  dx2

(d) Solve the simultaneous equations: dx + 2x +5y=tt

dt

dx + 4x +3y=t

dt

UNIT- III

3. (a)Define Bet function .

a   b    a   

(b) Evaluate:   ?    ?     ?(x2 + y2 + z2)dzdydx

a   b     a

     (c) Evaluate:   01?xm(log)ndx =     (-1)n n!

(m+1)n+1

Where n is a positive integer and m>-1.

(d) Find the area included between the parabola y= 4x – x2 and the line y = x.

UNIT- IV

4. (a) Define Gradient.

(b) Find the constants a and b so that the surface ax2 – byz= (a+2)x is orthogonal to the surface

ax2y + z3 = 4 at the point (1, -1, 2)

(c) If F = (x2 + y2)i + 2xyj + (y2-xy)k, then find div F and Curl F.

(d) Verify stake’s theorem for :

           F = (x2 + y2)I + 2xyj  Taken round the rectangle bounded by  x= ±a, y = y = b

 UNIT- V

5. (a)Find the number of roots of the equation x3 – x24x + 4 = 0

(b) Find the condition that the equation x3 + px24x + 4 = 0 had roots a, ? which satisfy a? + 1 = 0

(c) If a, ?, ? are roots of the equation .x3 + qx + r = 0 find the equation whose roots are

(a – ?)2, (? – ? )2, (? – a)2

(d) Solve the equation x2  -15x – 126 = 0 Cardoon’s method.

 

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