1
Code No: R7100102
I B.Tech (R07) Regular & Supplementary Examinations, June 2009 MATHEMATICS-I (Common to Civil Engineering, Electrical & Electronics Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? dy = y. 1. (a) Solve (x + 2y 3 ) dx
(b) If radioactive carbon-14 has a half-life of 5750 years, what will remain of one gram after 3000 years. [8+8] 2. (a) Solve (D2 + D +1) y = sin2x (b) Solve (D2 + 9) y = cos3x.
[8+8]
3. (a) Verify Rolle’s theorem for f(x) = x(x + 3) e
−x 2
in the interval (-3 , 0 ).
(b) Prove that u = x + y + z , v = xy + yz + zx , w = x2 + y2 + z2 are functionally dependent and find the relation between them. [8+8] 4. (a) If
√
r=
√
a cos 2θ
prove that ρ =
2√ ar. 3
(b) Trace the curve x= a(θ + sin θ), y = a(1+cos θ). [8+8] RR 5. (a) Evaluate xy dxdy where R is the region bounded by the line x + 2y = 2, lying in the R
first quadrant. (b) By changing the order of integration, evaluate
R1
√ 1−x R 2
0
0
[8+8]
P
3n + 1 / (n + 1) 2n P 2 n+1 (b) Examine the convergence or divergence of n x
6. (a) Examine the convergence of
y 2 dydx.
, ( x>0 )
[6+10]
7. (a) Find the angle between the normals to the surface x2 = yz at the points (1,1,1) and (2,4,1) (b) Find the work done by the force F~ = z ~i + x~j + y ~k, when it moves a particle along the arc of the curve ~r = cos t ~i + sin t ~j − t ~k fromt = 0 to t = 2π. [8+8] 8. (a) Find L [ e - 3t
Rt
sin t /t dt ].
0
(b) Find L - 1 [ s/ (s2 - 25 )2 ].
[8+8] ?????
2
Code No: R7100102
I B.Tech (R07) Regular & Supplementary Examinations, June 2009 MATHEMATICS-I (Common to Civil Engineering, Electrical & Electronics Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Solve
dy dx
y cos x+sin y+y + sin = 0. x+x cos y+x
(b) A tank initially contains 4000 gallons of brine in which 100 pounds of salt is dissolved. Pure water is running into the tank at the rate of 20 gallons per minute and the mixture (which is kept uniform by stirring) is drained off at the same rate. How many pounds of salt remain in the tank after 30 minutes? [8+8] 2. (a) Solve (D2 + 4D +4) y = e−x sin2x (b) Solve (D2 + 9) y = (x2 + 1) e3x .
[8+8]
3. (a) Using Mean Value Theorem prove that tanx > x in 0 < x < π/2. (b) Find the maximum value of x2 y3 z4 subject to the condition 2x + 3y + 4z = a.
[8+8]
4. (a) Find the centre of curvature 2 2 2 x3 + y 3 = a3 . (b) Trace the curve x = a ( t + sint ), y = a ( 1 - cost ). √ 2 2 √ 2 1−x −y 1 1−x R R R √ 1 dzdydx. 5. (a) Evaluate 2 2 2 0
0
0
[8+8]
1−x −y −z
(b) Find the perimeter of the circle with radius α.
[8+8]
6. (a) Examine the convergence or divergence of P 2n - 2 x / ( n + 1 )n1/2 , x > 0. (b) Examine the convergence or divergence of P (n!)2 n x , x > 0. (n+1)!
[8+8]
7. Verify Stoke’s theorem for F~ = (2x − y) ~i − yz 2~j − y 2 z ~k where S is the upper half surface x2 + y 2 + z 2 = 1 of the sphere and C is its boundary. [16] 8. (a) Find L - 1 [ 1/ (s3 +1) s3 ]. (b) Find L - 1 [ 1/ (s ( s2 -1) ( s2 +1 ) ].
[8+8] ?????
3
Code No: R7100102
I B.Tech (R07) Regular & Supplementary Examinations, June 2009 MATHEMATICS-I (Common to Civil Engineering, Electrical & Electronics Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Solve 2 xy dy - (x2 + y2 + 1) dx = 0. (b) Find the equation of Orthogonal trajectories of circles r = a cosθ. 2. Solve
d4 y dx4
[8+8]
- y = cosx coshx + 2x4 + x - 1.
3. (a) Examine the extrema of f( x, y) = x2 + xy + y 2 +
[16] 1 x
+ y1 .
(b) If f 0 (x) = 0 throughout an interval [a, b], Prove using Mean value theorem that f(x) is a constant in that interval. [8+8] ¡ ¢ 4. (a) Prove that the evolute of the curve x = a cos θ + log tan 2θ , y = a sin θ is y = a cosh xa . (b) Show that if ρ is the radius of curvature for the curve r = a ( 1 + cosθ) then constant. √ √ 2 2 R1 1−x R 2 1−x R −y √ 21 2 2 dzdydx. 5. (a) Evaluate 0
0
0
ρ2 r
is a [8+8]
1−x −y −z
(b) Find the perimeter of the circle with radius α.
[8+8]
6. (a) Examine the convergence of ¡ ¢ ¡ ¢2 ¡ ¢3 3 12 −4 21 + 5 12 + .... (b) Discuss the convergence of P n ( n + 1 ) xn + 1 , ( x > 0 ) [6+10] RRR 7. Evaluate (2x + y) dv where V is the region bounded by the cylinder z = 4 − x2 and the V
planes x = 0, y = 0, z = 2 and z = 0.
[16]
8. (a) Find L - 1 [( 2s2 +3s+5 )/ ( s3 -s ) ]. (b) Find L - 1 [ log ( 1+s ) / s2 ].
[8+8] ?????
4
Code No: R7100102
I B.Tech (R07) Regular & Supplementary Examinations, June 2009 MATHEMATICS-I (Common to Civil Engineering, Electrical & Electronics Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Find the differential equation of the family of curves y = ex (A cosx + B sinx) where A and B are arbitrary constants. (b) According to Newton’s law of cooling the rate at which the substance cools in moving air is proportional to the difference between the temperature of the substance and that of air. If the temperature of the air is 300 C and the substance cools from 1000 C to 700 C in 15 minutes find when the temperature will be 400 C. [8+8] 2. (a) Solve (D2 -4)y = x sin λ x (b) Solve the equation (D2 +a2 ) y = tan ax. 3. (a) Verify Rolle’s theorem for f(x) =
x2 −x−6 x−1
[8+8] in the interval (-2 , 3 ).
(b) Verify if u = 2x - y + 3z , v = 2x y - z , w = 2x - y + z are functionally dependent and if so, find the relation between them. [8+8] 4. (a) Find the radius of curvature of xy2 = a3 - x3 at ( a, 0). (b) Trace the curve r = ( 1 + cosθ). [8+8] RRR 5. (a) Evaluate dxdydz, where V is the finite region of space formed by the planes x = V
0, y = 0, z = 0 and 2x + 3y + 4z = 12. (b) By changing the order of integration evaluate
R∞ Ry
y2
ye− x dxdy.
[8+8]
0 1
6. (a) Examine the convergence of ¡ ¢2 ¡ ¢3 ¡ ¢ 3 12 −4 21 + 5 12 + .... (b) Discuss the convergence of P n ( n + 1 ) xn + 1 , ( x > 0 )
[6+10]
7. (a) Find the work done in moving a particle by the force F~ = 3x2 ~i + (2xz − y)~j + z ~k along the line joining (0,0,0) to (2,1,3) R (b) Using Green’s theorem evaluate (2xy − x2 ) dx + (x2 + y 2 ) dy where C is the closed curve C
of the region bounded y = x2 and y 2 = x.
[8+8]
8. (a) Find L - 1 [ ( 1+e - πs )/ ( s2 +1 ) ]. (b) Using convolution theorem find L - 1 [ 1/ (s2 -1)(s2 +25) ]. ?????
[8+8]