Firstranker's choice
Set No. 1
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Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours
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Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
- (a) Find the differential equations of all parabolas with x-axis as its axis and (a, 0) as its focus.
(b) Find the orthogonal trajectories of coaxial circles x2 + y2 + 2?x + c = 0, where ? is the parameter. [7M + 8M] - (a) Solve (D2 - 2) y = e2x + cos x
(b) Solve (D2 + 4D + 5) y = 2 sinh x subject to y=0 and dy/dx = 1 at x=0. [7M + 8M] - (a) If u = xy + yz + zx, v = x2 + y2 + z2 and w = x + y + z, verify whether there exists a possible relationship between u, v and w. If so find the relation.
(b) Find the minimum value of x2 + y2 + z2 on the plane x + y + z = 3a [7M + 8M] - (a) Trace the curve x(x2+y2)= 4(x2 - y2)
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(b) Trace the polar curve r = 2 + 3 cos ? . [7M + 8M] - (a) Find the perimeter of one loop of the curve 3a y2 = x ( x - a)2 .
(b) Find the volume generated by revolving the area bounded by one loop of the curve r = a (1 + cos ?) about the initial line. [7M + 8M] - (a) Evaluate ?e-v(x2+y2) dx dy by changing the order of integration. Limits are from 0 to x.
(b) Evaluate ? v(x + y) dy dx by changing into polar coordinates. Limits are from 0 to v2x-x2 and 0 to 0. [7M + 8M] - (a) Find the directional derivative of f ( x, y, z ) = x2 y + y z3 at the point (2,-1,1) in the direction of the vector i + 2 j + 2k .
(b) Find curl [r f (r)] where r = xi + yj + zk , r = |r| [7M + 8M] - (a) Compute the line integral ? (y2dx-x2dy) round the triangle whose vertices are (1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? (x3 dy dz + x2 y dz dx + x2 z dx dy) using divergence theorem, where S is the surface consisting of the cylinder x2 + y2 = a2 (0 = z = b) and the circular disks z=0 and z = b (x2 + y2 = a2). [7M + 8M]
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Firstranker's choice
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Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
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(Common to all branches)
Time: 3 hours
Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
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- (a) Find the solution of the differential equation dy/dx = xey-x2 and y(0) = 0 .
(b) A body initially at 80° C cools down to 50° C in 10 minutes, the temperature of the air being 40° C. What will be the temperature of the body after 20 minutes? [7M + 8M] - (a) Solve (D2 +9) y = e2x + x2
(b) Find the general solution of (D2-2D+1)y = ex sin 2x [7M + 8M] - (a) Verify whether the functions u = sin-1x + sin-1y and v = xv(1 - y2) + yv(1 - x2) are functionally dependent. If so, find the relation between them. [7M + 8M]
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(b) Prove that the rectangular solid of maximum volume that can be inscribed into a sphere of radius 'a' is a cube. - (a) Trace the parametric curve x = a (cos t + log tan(t/2)) and y = a sin t .
(b) Trace the lemniscate r2 = a2 cos 2?. [7M + 8M] - (a) Find the surface area generated by revolving the arc of the curve y = a cosh (x / c) from x=0 to x=c about the x-axis.
(b) Find the total length of the lemniscate r2 = a2 cos 2?. [7M + 8M] - (a) Find the area of the region which is outside the circle r=1 and inside the cardioid r = (1 + cos ?)
(b) Evaluate ? (1-(x2+y2))/(1+x2+y2) dy dx over the positive coordinate of the circle x2 + y2 = 1 by changing into polar coordinates. [7M + 8M] - (a) Find the directional derivative of the divergence of F = x2 y i + y z j + z2k at the point (2,1,2) in the direction of the outer normal to the sphere x2 + y2 + z2 = 9.
(b) Find the value of a,b and c such that (x + y + a z) i + ( b x + 2 y - z ) j + ( - x + cy+2z)k is irrotational. [7M + 8M] - (a) If F = ( x2 + y - 4) i + 3 x y j + (2 xz + z 2 )k and S is the upper half of the sphere x2 + y2 + z2 = 16. Show by using Stokes theorem that ? Curl F .n ds = 2p a3.
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(b) If S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and ax + by + cz = 1. Show that ? F.n ds = 1 / 2abc. [7M + 8M]
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Firstranker's choice
Set No. 3
Code No: R10102 / R10
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I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours
Max Marks: 75
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Answer any FIVE Questions
All Questions carry equal marks
- (a) Solve dy/dx = (x2 + y2)/xy
(b) A colony of bacteria is grown under ideal condition in laboratory so that the population increases exponentially with time. At the end of 3 hours there are 10000 bacteria. At the end of 5 hours there are 40000. How many bacteria were present initially? [7M + 8M] - (a) Solve (D3 - 6 D2 + 11 D - 6) y = e-2x + x3
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(b) Solve (D2 + 1) y = x2 e2x + x cos x [7M + 8M] - (a) If u = x + y + z, u2v = y + z and u3 w = z, then find ?(u, v, w)/?(x, y, z)
(b) Find the minimum and maximum distances of a point on the curve 2 x 2 + 4 xy + 4 y 2 -8=0. [7M + 8M] - (a) Trace the parametric curve x = a ( t - sint) and y = a (1 + cost) , a > 0
(b) Trace the curve y2 (x - a ) = x 2 ( x + a) . [7M + 8M] - (a) Find the volume of the solid formed by revolving the area bounded by the curve 27 a y 2 = 4 (x - 2 a)3 about x-axis
(b) Find the length of the loop of the curve r = a (1 - cos ?). [7M + 8M] - (a) Find the area of the loop of the curve x3 + y3 = 3a xy, by transforming it into polar coordinates.
(b) Change the order of integration and evaluate I = ? xy dy dx. Limits are from x to 1 and 0 to x. [7M + 8M] - (a) In what direction from the point (1, 3, 2) is the directional derivative of f = 2 x z - y 2 is maximum and what is its magnitude.
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(b) Show that F = (y2 cos x + z 3 )i + (2 y sin x - 4) j + (3 x z 2 + 2)k is a conservative force field and find its scalar potential. [7M + 8M] - (a) Show that F = (2 xy + z 3 ) i + x 2 j + 3xz2k is a conservative force field. Find the scalar potential and the work done in moving an object in this field from (1,-2,1) to (3,1,4).
(b) Verify Green's theorem,if ? Mdx + Ndy = ?(xy + y2)dx + x2 dy with c: closed curve of the region bounded by y = x and y = x2 . [7M + 8M]
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Firstranker's choice
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Set No. 4
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
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Time: 3 hours
Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
- (a) Solve dy/dx = xv(x2 + y2)/y
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(b) A body is heated to 110° C is placed in air at 10º C. After 1 hour its temperature is 80° C. When will the temperature be 30° C? [7M + 8M] - (a) Solve (D2 + 3 D + 2) y = sin x sin 2x
(b) Solve (D2 + 2 D - 3) y = x 3 e-2x [7M + 8M] - (a) Verify whether the functions u = x-y and v= (x+z)/(y+z) are functionally dependent. If so, find the relation in between them.
(b) The temperature T at any point (x, y, z) in the space is given as T = 400 x y z2. Find the highest temperature on the surface of the sphere x2 + y2 + z2 = 1 [7M + 8M]. - (a) Trace the curve x3 + y3 = 3a x y
(b) Trace the polar curve r = a (1 - sin ? ). [7M + 8M] - (a) Find the surface area generated by revolving the arc x2/3 + y2/3 = a2/3 about x-axis.
(b) Find the volume of the solid generated by revolving the cardioid r = a (1 + cos ?) about the initial line. [7M + 8M] - (a) Find the area of a plate in the form of a quadrant of an ellipse x2 / a2 + y2 / b2 = 1 by changing into polar coordinates.
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(b) By changing the order of integration, evaluate the integral ? v(4a2 - y2) dx dy. Limits are from 0 to 2a and 0 to y2/4a . [7M + 8M] - (a) Find the constants a and b so that the surface a x 2 - b y z = (a + 2) x will be orthogonal to the surface 4 x 2 y + z3 = 4 at the point (1, -1, 2).
(b) Determine the constant b such that A = (b x2y + y z )i + (x y 2 - x z 2 ) j + (2 x y z - 2 x2 y2)k has zero divergence. [7M + 8M] - (a) Evaluate ? F.dr where F = x2 i+ y2 j and curve c is the arc of the parabola y=x2 in the xy-plane from (0,0) to (1,1).
(b) Evaluate by Stokes theorem ? (x + y) dx + (2 x - z) dy + (y + z)dz, where C is the boundary of the triangle vertices (0,0,0), (1,0,0) and (1,1,0). [7M + 8M]
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