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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, Bio-Technology and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Test the convergence of the following series
sin2 1 n [5]
(b) Prove that the series
∑ (^) (−1)n n(logn)^3 converges absolutely.^ [5] (c) Verify cauchy’s mean value theorem for f(x) = ex^ and g(x) = e−x^ in [a,b] [6]
- (a) If xxyyzz^ = c, show that at x = y = z, ∂
(^2) z ∂x∂y =^ − {x^ log (cx)}
− 1
(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) Trace the curve 3ay^2 = x^2 (a – x). (b) Show that the upper half of the cardiod r = a (1 + cosθ) is bisected by the line θ = π/3. [8+8]
- (a) Form the differential equation by eliminating the arbitrary constant xy = x log x – x + c. [3] (b) Solve the differential equation: (2y sinx + cosy ) dx = (x sin y + 2 cos x + tan y ) dy [7] (c) Radium decomposes at a rate proportional to the amount present at that time. If a fraction p of the original amount disappears in 1 year how much Radium will remain at the end of 21 years. [6]
- (a) Solve the differential equation: y′′^ + 4y′^ + 4y = 4cosx + 3sinx, y(0) = 1, y′(0) = 0. (b) Solve the differential equation:
(2x + 3)^2 D^2 − (2x + 3) D − 12
y = 6x. [8+8]
- (a) State and prove second shifting theorem. [5]
(b) Find the inverse Laplace Transformation of
[
s + 3 (s^2 +6s + 13)^2
]
[5]
(c) Evaluate ∫ ∫ ∫ z^2 dxdydz taken over the volume bounded by x^2 + y^2 = a^2 , x^2 + y^2 = z and z = 0. [6]
- (a) Prove that curl(A×B)=AdivB–BdivA +(B.∇)A–(A.∇)B.
(b) Find the directional derivative of φ (x,y,z) = x^2 yz + 4xz^2 at the point (1, –2, –1) in the direction of the normal to the surface f(x,y,z) = x logz –y^2 at (–1, 2,–1). [8+8]
- Verify divergence theorem for F = 6zi + (2x + y)j – xk, taken over the region bounded by the surface of the cylinder x^2 + y^2 = 9 included in z = 0, z = 8, x = 0 and y = 0. [16]
(b) Find L−^1
[
s^2 +2s− 4 (s^2 +9)(s−5)
]
[6]
(c) Change the order of integration and evaluate
∫^1
0
(^2) ∫−x
x^2
xy dx dy [5]
- (a) Prove that div(AxB)=B.curlA - A.curlB.
(b) Find the directional derivative of the scalar point function φ (x,y,z) = 4xy^2 + 2x^2 yz at the point A(1, 2, 3) in the direction of the line AB where B = (5,0,4). [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, Bio-Technology 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=
1 2 n+3n^ [5]
(b) Find the interval of convergence of the series x
2 2 +^
x^3 3 +^
x^4 4 +^ .....∞^ [5] (c) Verify Rolle’s theorem for f(x) = 2x^3 + x^2 - 4x- 2 in
[
]
[6]
- (a) Given (cos x)y^ = (sin y)x^ , find dy dx. (b) Find the volume of the largest rectangular parallelopiped that can be insibed in the ellipsoid of revolution 4x^2 + 4y^2 + 9z^2 = 36. [8+8]
- (a) Trace the curve r = a + b cos θ. (a > b). (b) Find the surface area got by rotating the ellipse x
2 a^2 +^
y^2 b^2 = 1 about the minor axis. [8+8]
- (a) Form the differential equation by eliminating the arbitrary constant secy + secx = c. [3] (b) Solve the differential equation: dy dx − (^2) xy = 5 x
2 (2 + x)(3 − 2 x)
[7]
(c) Find the orthogonal trajectories of the family of the parabolas y^2 = 4ax. [6]
- (a) Solve the differential equation: y′′^ + 4y′^ + 20y = 23 sint - 15cost, y(0) = 0, y′(0) = - (b) Solve the differential equation: (2x + 5)^2 d
(^2) y dx^2 + 6 (2x^ + 5)^
dy dx + 8y^ = 4(2x^ + 5) [8+8]
- (a) Evaluate
∫^ ∞
0
t^3 e−t^ sin t dt using the Laplace transforms.
(b) Evaluate
∫∫ (^) r dr dθ √ a^2 + r^2 over one loop of the lemniscate r
(^2) = a (^2) cos2θ. [8+8]
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, Bio-Technology 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 ( 22 12 −^
2 1
33 23 −^
3 2
44 34 −^
4 3
+ ........ [5]
(b) Find the interval of convergence of the following series 1 1 −x +^
1 2(1−x)^2 +^
1 3(1−x)^3 +^ .....^ [5] (c) If 0 < a < b < 1, using Lagrange’s mean value theorem, prove that √^ b−a 1 −a^2 <^ sin
− (^1) b − sin− (^1) a < √b−a 1 −b^2 [6]
- (a) Locate the stationary points and examine their nature of the following func- tions: u = x^3 y^2 (12-x-y); (x>0, y>0) (b) Find the envelope of the family of ellipses x 2 a^2 +^
y^2 b^2 = 1 where the two parameters are connected by the relation a + b = c where c is a constant. [8+8]
- (a) Trace the curve : x = a ( θ + sinθ) ; y = a ( 1 + cosθ). Obtain the length of one arch of the curve. (b) A sphere of radius ‘a’ units is divided into two parts by a plane distant (a/2) from the centre. Show that the ratio of the volumes of the two parts is 5 : 27. [8+8]
- (a) Form the differential equation by eliminating the parameter ‘a’: x^2 +y^2 + 2ax + 4 = 0. [3] (b) Solve the differential equation: x dy dx + y = x^2 y^6. [7]
(c) Find the orthogonal trajectories of the coaxial curves x
2 a^2 +^
y^2 b^2 +λ = 1,^ λ^ being a parameter [6]
- (a) Solve the differential equation: (D^2 − 2 D + 2)y = 2excosx. [6] (b) Solve the differential equation: (D^3 + 2D^2 + D)y = e^2 x^ + x^2 + x + sin 2 x. [10]
- (a) Find L [ t^2 sin2t ] [5]
(b) Find L−^1
[
(s+5) s^2 − 6 s+
]
[5]
(c) Evaluate
R
(x^2 + y^2 ) dxdy over the triangular region R with vertices (0,0)
(1,0) and (0,1). [6]
- (a) If a is a constant vector, evaluate curl((a x r)/r^3 ), where r=xi+yj+zk and r=|r|. (b) Evaluate
s
A.n ds for A=(x+y^2 )i – 2xj + 2yzk and S is the surface of the
plane 2x+y+2z=6 in the first octant. [8+8]
- Verify Stokes theorem for the function F = x^2 i + xyi integrated round the square whose sides are x = 0, y = 0, x = a and y = a in the plane z = 0. [16]
