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I B.Tech Semester Supplimentary Examinations, June 2009 MATHEMATICS-I ( Common to Civil Engineering, Electrical & Electronic Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Chemical Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Mechatronics, Computer Science & Systems Engineering, Electronics & Telematics, Metallurgy & Material Technology, Electronics & Computer Engineering, Production Engineering, Aeronautical Engineering, Instrumentation & Control Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Test the convergence of the series 21 + 21 ..^55 ..^89 + 21 ..^55 ..^89 ..^1113 + ... ∞. [5] (b) Find whether the following series converges absolutely / condtionally 1 6 −^
1
3 8 +^
- 5
- 10 −^
- 7
- 12.^ [5] (c) Prove that π 6 +
√ 3 5 <^ sin
−1 3 5 <^ π/ 6 +^
1 8.^ [6]
- (a) Find the evolute of the curve x2/3^ + y2/3^ = a2/ (b) if u = yz x , v = zx y , w = xy z find ∂ ∂((u, v, wx, y, z)) [8+8]
- Show that the total length of the curve
(x a
(y b
= 1 is 4( a^2 +ab+b^2 ) ab. Hence or otherwise find the whole length of the curve x2/3^ + y2/3^ = a2/3. Show also that in the last curve s^3 α x^2. [16]
- (a) Find the differential equation of the family of cardiods r = a ( 1 + cos θ). (b) Solve the differential equation: dy dx + (^) x logy x = sin 2 log xx (c) A copper ball is heated to 100 0 C temperature. Then at time t = 0 it is placed in water that is maintained at a temperature of 30 0 C. At the end of 3 minutes the temperature of the ball is reduced to 70 0 C. Find the time at which the temperature of the ball is reduced to 31^0 C. [3+7+6]
- (a) Solve the differential equation: y′′^ + 4y′^ + 20y = 23 sint - 15cost, y(0) = 0, y′(0) = -1. (b) Using variation of parameters method solve y′′^ + 4y = 4 sec^2 2x. [8+8]
- (a) Find the Laplace Transformation of the rectified sine-wave function defined by f (t) =
sin , ωt 0 < t < π ω O , π ω < t < (^2) ωπ
(b) Find L − 1
h (^) S (^2) +2S− 4 (S^2 +9)(S−5)
i
(c) Change the order of integration and evaluate
∫^1
0
(^2) ∫−x
x^2
xy dxdy. [5+6+5]
- (a) Find A.∇φ at (1, –1, 1) if A =3xyz^2 i + 2xy^3 j − x^2 yzk and φ = 3x^2 − yz.
(b) Show that F=(2xy + z^3 )i + x^2 j + 3xz^2 k is a conservative force field. Find the scalar potential. Find the work done in moving an object in this field from (1, -2, 1) to (3, 1, 4). [8+8]
- Verify Stoke’s theorem for F = (y-z+2) i+(yz+4) j - xzk where S is the surface of the cube x=0, y=0, z=0, x=2, y=2, z=2 above the xy-plane. [16]
(b) Find L − 1
h (^) S (^2) +2S− 4 (S^2 +9)(S−5)
i
(c) Change the order of integration and evaluate
∫^1
0
(^2) ∫−x
x^2
xy dxdy. [5+6+5]
- (a) Find the angle between the tangent planes to the surface x log z = y^2 – 1, x^2 y = 2 – z at the point (1, 1, 1). (b) Evaluate
S
F. N ds where F = (x + y^2 ) i – 2x j + 2 yz k and S is the surface
of the plane 2x + y + 2z = 6 in the first octant. [8+8]
- Verify divergence theorem for F = 4xi – 2y^2 j + z^2 k taken over the surface bounded by the region x^2 +y^2 =4, z = 0 and z = 3. [16]
I B.Tech Semester Supplimentary Examinations, June 2009 MATHEMATICS-I ( Common to Civil Engineering, Electrical & Electronic Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Chemical Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Mechatronics, Computer Science & Systems Engineering, Electronics & Telematics, Metallurgy & Material Technology, Electronics & Computer Engineering, Production Engineering, Aeronautical Engineering, Instrumentation & Control Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Test the convergence of the series
∑ (^) n 4 n!.^ [5] (b) Find the whether the series 16 − 16 ..^38 + (^61) .. 83 .. 105 − (^61). 8.^3. 10.^5 ..^712 + ... converges absolutely or conditionally. [6] (c) Verify Rolle’s theorem for f (x) = x^2 n−^1 (a - x)^2 n^ in (0,a). [5]
- (a) Find the shortest distance from origin to the surface xyz^2 = 2. (b) Find the evolute of the hyperbola x^2 /a^2 –y^2 /b^2 = 1. Deduce the evolute of a rectangular hyperbola. [8+8]
- (a) Prove that the surface area generated by the revolution of the tractrix x = a cos t+ 12 a log tan 2 t , y=a sint about its asymptote is equal to the surface area of a sphere of radius ‘a’. (b) For the cycloid x=a (θ-sinθ), y=a(1-cosθ). Find the volume of the solid gen- erated about the tangent at the vertex. [8+8]
- (a) Form the differential equation by eliminating the parameter ‘a’ from the equa- tion: x^2 +y^2 + 2ax + 4 = 0. (b) Solve the differential equation: (x^2 – 2xy + 3y^2 ) dx + (y^2 + 6xy – x^2 ) dy = 0. (c) The number N of bacteria in a culture grew at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What is the value of N after 1 1 / 2 hour. [3+7+6]
- (a) Solve the differential equation: (D^2 + 1)y = e−x^ + x^3 + ex^ sinx. (b) Solve (D^2 + 4)y = sec 2 x by the method of variation of parameters. [8+8]
- (a) Find the Laplace Transformation of the rectified sine-wave function defined by f (t) =
sin , ωt 0 < t < π ω O , π ω < t < (^2) ωπ
I B.Tech Semester Supplimentary Examinations, June 2009 MATHEMATICS-I ( Common to Civil Engineering, Electrical & Electronic Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Chemical Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Mechatronics, Computer Science & Systems Engineering, Electronics & Telematics, Metallurgy & Material Technology, Electronics & Computer Engineering, Production Engineering, Aeronautical Engineering, Instrumentation & Control Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Test the convergence of the series
√ 2 − 1 32 − 1 +^
√ 3 − 1 42 − 1 +^
√ 4 − 1 52 − 1 +^ .....^ [5] (b) Examine whether the following series is absolutely convergent or conditionally convergent 1 − (^) 3!^1 + (^) 5!^1 − (^) 7!^1 +.... .. [5]
(c) Verify Rolle’s theorem for f(x) = log
[
x^2 +ab x(a+b)
]
in [a,b] (x 6 = 0). [6]
- (a) Find the shortest distance from origin to the surface xyz^2 = 2. (b) Find the evolute of the hyperbola x^2 /a^2 –y^2 /b^2 = 1. Deduce the evolute of a rectangular hyperbola. [8+8]
- (a) Prove that the surface area generated by the revolution of the tractrix x = a cos t+ 12 a log tan 2 t , y=a sint about its asymptote is equal to the surface area of a sphere of radius ‘a’. (b) For the cycloid x=a (θ-sinθ), y=a(1-cosθ). Find the volume of the solid gen- erated about the tangent at the vertex. [8+8]
- (a) Obtain the differential equation of the co-axial circles of the system x^2 +y^2 +2ax + c^2 = 0 where c is a constant and a is a variable parameter. (b) Solve the differential equation: dy dx = x 1+sin−y^ cos x^ x
(c) Find the orthogonal trajectories of the co-axial curves x
2 a^2 +^
y^2 b^2 +λ = 1,^ λ^ being a parameter [3+7+6]
- (a) Solve the differential equation: (D^2 +2D–3) y=x^2 e−^3 x. (b) Solve the differential equation: (D^2 +4)=sec2x by the method of variation of parameters. [8+8]
- (a) Find L
[sinh 2t t
]
(b) Find L−^1
[
s (s−3)(s^2 +4)
]
(c) Evaluate
(x^2 + y^2 ) dx dy over the region bounded by the ellipse x^2 a^2 +^
y^2 b^2 = 1 in first quadrant.^ [5+5+6]
- Prove that F=(y^2 cos x + z^3 )i + (2y sin x − 4)j + (3xz^2 + 2)k is a conservative
force field. Find the work done in moving an object in this field from (0, 1, –1) to (π/2, –1, 2). [16]
- (a) Apply Stoke’s theorem to evaluate
C
((x + y)dx + (2x − 3) dy + (y + z)dz)
where C is the boundary of the triangle with vertices (2,0,0), (0,3,0) and (0,0,6). (b) Evaluate by Green’s theorem
C
[(cos x sin y − 2 x y ) dx + sinx cosy dy]
where ‘C’ is the circle x^2 + y^2 = 1. [8+8]
