University of Mumbai 2007-1st Sem B.E FE Applied Maths Rev - Question Paper
(2) Attempt any fourfluestions out or remaining six questions.
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(a) If y = - 3 - * *ind yn 5
x -2x - 4x + 8
(b) If I z2 - 1 t = I z I2 + 1 prove that z lies on imaginary axis where z is a complex 5 number.
(c) If u = log (x3 + y3-x2y-xy2) prove that 5
-4
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2 | ||||||||||||
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d2u
+2.......+
(a) Prove that cos6 0 - sin60 = ~ [ cos 60 + 15 cos 20 ].
(b) Test the convergence of -i2 \~1
(x + y)
dx 5y dy
dx'
(d) Evaluate lim x -> a
'4* 4 ' 3
23- 2
+
12 1
(c) if f, g and h are continuous on [ a, b ) and differentiable on (at b) prove that there 8 exist c e (a, b) such that
fl(c) g'(c) h(c) f(a) g(a) h(a) =0. f(b) g(b) h(b)
Deduce Cauchys and Lagranges mean value Theorem from this result.
(a) If x + iy = c cot (u + iv) show that 6
x -y c
sin(2u) sinh(2v) cosh(2v) - cos(2u)
1
2 + y2 + xz)2
7l( 2x
show that
(b) If v = log sin
2 ( x2 + xy + 2yz + z2) dv dv 5v 1
<7V OV 0V 1 v / H 2v -1 / v\
x + y + z = -evJl-e* sin 1 (e
dx dy 3z 3 y ' '
(c) If y = x2 +a2 prove that yn = sinn + 1 G cos (n + 1) 0
(a) Considering only principle value, if (1+ i tan a)1 +t,anP is real prove that its value 6
secap
is (.sec a)
x 7
(b) Show that x cosec x = 1 + + - x4 +
6 3oQ
(c) If r = xi + yj + zk and a,b are constant vectors, prove that 8 r r r [ TURN OVER
5. (a) Find the directional derivative of 4> = x2 + y2 + z2 in the direction of the line 6
i-y-iat (1,2,3).
3 4 5
(b) If u = f (2x -3y, 3y - 4z, 4z - 2x) g
_ du . du _ du .
Show that 6' + 4i + 3-
(c) Find a point in the plane x + 2y + 3z = 13 nearest to the point (1, 1, 1) using the q method of Lagranges multipliers.
x 3
6. (a) Show that sin- 1 (x) = x + + x5 +....................6
(b) State and prove Euler's theorem for two variables.
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(c) Prove that nth root of unity are in geometric progression. Also find sum of nth root of unity.
7. (a) Find (1-04)3 01 by using theory of approximation. 6
(b) If yAn + y~%i = 2x prove that 6 (*2-1)yn+2 + (2n + 1)xyn + 1 +(n2 - m2) yn = 0.
(c) If u = log tan ( 4 + %) prove that 8
(i) cos h (u) = sec (0)
(ii) sin h (u) = tan 0
(iii) tan h (u) = sin 0
(iv) tan h (u/2) = tan (/2).
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| Earning: Approval pending. |
