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MU First Year Engineering (Semester 1)
Applied Mathematics 1
December 2013
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


1 (a) If α+iβ=tanh (x+iπ4), prove that α2+β2=1.
3 M
1 (b) If u=x2y+exy2 show that 2uxy=2uyx
3 M
1 (c) If u=1x, v=x(1y), w=xy(1z)show that(u, v, w)(x, y, z)=x2y.
3 M
1 (d) prove thatlog(1x+x2)=x+x22+2x33  
3 M
1 (e) Express the relation in α , β ,γ , δ for which A=α+iγβ+iδβ+iδαiγ is unitary.
4 M
1 (f) Find nth derivative of 2xcos2x sin x.
4 M

2 (a) Z3=(z+1)3, then show that z=12+i2cotθ2 where θ= 20π3.
6 M
2 (b) Find the non-singular matrices P and Q such that PAQ is in Normal Form. Also find rank of A. A= [431624221214516]
6 M
2 (c) State and Prove Euler's theorem for homogeneous function in two variables anf hence find the value of x22ux2+2xy2uxy+y22uy2+xux+yuyfor u=ex+y+log(x3+y3x2yxy2)
8 M

3 (a) For what values of λ the system of equations have non-trivial solution? Obtain the solution for real values of λ where 3x+yλx=0, 4x2y3z=0, 2λx+4yλz=0}
6 M
3 (b) Find the stationary values of sin x sin (x+y).
6 M
3 (c) If cos(x+iy) cos(u+iv)=1, where x,y,u, v are real then show that tanh2 cosh2 v=sin2u.
8 M

4 (a) if ux+vy=a,ux+vy=1, show that ux(xu)v+vy(yv)u=0
6 M
4 (b) If (1+i(tanα)(1+i tanβ) is real then one of the principle values is (secα)sec2β
6 M
4 (c) Solve by Crout's Method the system of equaition 2x+3y+z=-1, 5x+y+z=9, 3x+2y+4z=11
8 M

5 (a) If sin4θ cos3θ = a cosθ- bcos3θ + ccos 5θ + dcos 7θ then find a,b,c,d
6 M
5 (b) Use Taylor theorem and arrange the equaition in power of x.7+(x+2)+3(x+2)3+(x+2)4-(x+2)5
6 M
5 (c) If y=cos(msin1X) prove that (1x2)yn+2(2n+1)xyn+1+(m2n2)yn=0
8 M

6 (a) Solve correctly upto three iterations the following equaition by Gauss-Seidel method. 10x-5y-2z=3, 4x-10y+3z=-3, x+6y+10z=-3.
6 M
6 (b) If u=sin(x2+y2)and a2x2+b2y2=c2 finddudx.
6 M
6 (c) Fit a curve y=ax+bx2 for the data;
x : 1 2 3 4 5 6
y : 2.51 5.82 9.93 14.84 20.55 27.06
8 M



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