1 (a)
If α+iβ=tanh (x+iπ4), prove that α2+β2=1.
3 M
1 (b)
If u=x2y+exy2 show that ∂2u∂x∂y=∂2u∂y∂x
3 M
1 (c)
If u=1−x, v=x(1−y), w=xy(1−z)show that∂(u, v, w)∂(x, y, z)=x2y.
3 M
1 (d)
prove thatlog(1−x+x2)=−x+x22+2x33 −−−−
3 M
1 (e)
Express the relation in α , β ,γ , δ for which A=α+iγ−β+iδβ+iδα−iγ 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 x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2+x∂u∂x+y∂u∂yfor u=ex+y+log(x3+y3−x2y−xy2)
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+y−λx=0, 4x−2y−3z=0, 2λx+4y−λz=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(∂x∂u)v+vy(∂y∂v)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(msin−1X) prove that (1−x2)yn+2−(2n+1)xyn+1+(m2−n2)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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