Enrolment No.
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- I & II (NEW) EXAMINATION — WINTER 2019
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Subject Code: 3110014Subject Name: Mathematics — I
Time: 10:30 AM TO 01:30 PM
Date: 17/01/2020
Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q.1 Find the equations of the tangent plane and normal line to the surface x2 + y2 + z2 =3 at the point (1,1,1) [03]
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Q.1 Evaluate lim x?0 (x ex - log(1+x)) / x2 [04]
Q.1 Using Gauss Elimination method solve the following system [07]
-x+3y+4z =30
3x+2y-z =9
2x-y+2z =10
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Q.2 Test the convergence of the series S (2/3)n (3/5)n (4/7)n ... (n/(2n+1)) [03]
Q.2 Discuss the Maxima and Minima of the function 3x2y — y3 + x3 [04]
Q.2 Find the Fourier series of f(x) = (p-x)/2 in the interval (0,2p) [07]
OR
Change the order of integration and evaluate ?01 ?x1 x sin(y2) dy dx
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Q.3 Find the value of G(1/2) [03]
Q.3 Obtain the Fourier cosine series of the function f(x) = ex in the range (0,l) [04]
Q.3 Find the maximum and minimum distance from the point (1,2,2) to the sphere x2 + y2 + z2 =36 [07]
OR
Test the convergence of the series 1/12 + 1/22 + 1/32 + 1/42 + ...
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Q.4 Evaluate ? (x2 —y2) dx dy over the triangle with the vertices (0,1), (1,1), (1,2) [03]
Q.4 Find the volume of the solid generated by rotating the plane region bounded by y = vx, x=1 and x=3 about the X axis. [04]
Q.4 Evaluate ?0p/2 ?01 r2 sin ? dr d? [07]
Q.5 Express f(x) = 2x3 + 3x2 — 8x + 7 in terms of (x-2) [03]
Q.5 Using Gauss-Jordan method solve [04]
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Q.5 Using Cayley-Hamilton Theorem find A-1 for A= { {1, 0}, {2, 3} } [07]
OR
Evaluate ?08 dx / (1+x4)
Q.5 Test the convergence of the series x/1.2 + x2/3.4 + x3/5.6 + x4/7.8 +... [03]
Q.5 Evaluate ?01 ?01 (12 xy) dy dx [04]
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Q.5 Find the eigen values and eigenvectors of the matrix { {1, 0, 0}, {0, 0, 1}, {1, -3, 3} } [07]
If u = f (x-y, y-z, z-x) then show that ?u/?x + ?u/?y + ?u/?z = 0
OR
Find the directional derivatives of f=xy2 + yz2 at the point (2,-1,1) ,in the direction of i+2j+2k.
Q.5 Test the convergence of the series S (n3 + 1) / (n4 + 1) [03]
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Q.5 Evaluate ? xyz dx dy dz over the positive octant of the sphere x2 + y2+z2=4 [04]
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