STUPIDSID
1(a)
Prove the following:
11−11−11−cosh2x=cosh2x
11−11−11−cosh2x=cosh2x
3 M
1(b)
If u = log (tanx + tany)
2x∂u∂x+sin2y∂u∂y=2
2x∂u∂x+sin2y∂u∂y=2
3 M
1(c)
If the following expression is true,
u=x+y1−xy , v=tan−1x+tna−1yFind ∂(u,v)∂(x,y)
u=x+y1−xy , v=tan−1x+tna−1yFind ∂(u,v)∂(x,y)
3 M
1(d)
Expand log (1+sinx) = (x - x2/2 + x3/6 +...)
3 M
1(e)
Show that every square matrix can be uniquely expressed as P+iQ where P and Q are Hermitian Matrices.
4 M
1(f)
Find nth order derivative of
y=x2+4(2x+3)(x−1)2
y=x2+4(2x+3)(x−1)2
4 M
2(a)
Show that roots of the equation (x+1)6 + (x-1)6 = 0 are given by
−icot[(2k+1)π12] , k=0,1,2,3,4,5
−icot[(2k+1)π12] , k=0,1,2,3,4,5
6 M
2(b)
Reduce the following matrix into normal form and find its rank
[2−111101233310−4−12]
[2−111101233310−4−12]
6 M
2(c)
State and prove Euler's theorem for a homogeneous function in two variables. And hence
Find x∂u∂x+y∂u∂y where u=√xy√x+√y
Find x∂u∂x+y∂u∂y where u=√xy√x+√y
8 M
3(a)
Test for consistency and solve if consistent -
x1-2x2+x3-x4=2;
x1+2x2+2x4=1;
4x2-x3+3x4=-1.
x1-2x2+x3-x4=2;
x1+2x2+2x4=1;
4x2-x3+3x4=-1.
6 M
3(b)
Find all stationary value of x2 + 3xy - 15x2 - 15y2 + 72x.
6 M
3(c)
If tan[(π/4)+iv] = reiθ show that
(i) r=1
(ii) tanθ = sinh 2v
(iii) tanhv = tan(θ/2)
(i) r=1
(ii) tanθ = sinh 2v
(iii) tanhv = tan(θ/2)
8 M
4(a)
If x = u+e(-v)sin u, and y = v+e(-u)cos u,
Find ∂u∂y,∂v∂x using Jacobian
Find ∂u∂y,∂v∂x using Jacobian
6 M
4(b)
Considering only the principal value,
if (1 + i tanθ)(1+i tanθ) is real, prove that its value is (sec?)(sec2θ).
if (1 + i tanθ)(1+i tanθ) is real, prove that its value is (sec?)(sec2θ).
6 M
4(c)
Solve the system of linear equation by Crout's method
x - y + 2z = 2;
3x + 2y - 3z = 2;
4x - 4y + 2z = 2
x - y + 2z = 2;
3x + 2y - 3z = 2;
4x - 4y + 2z = 2
8 M
5(a)
Expand cos7θ in a series of cosines of multiple of θ .
6 M
5(b)
Evaluate the following:
limx→0[1x2−cot2x]
limx→0[1x2−cot2x]
6 M
5(c)
If y = (sin-1x)2, obtain yn(0).
8 M
6(a)
Show that the vectors are linearly dependent and find the relation between them
X1=[1,2,-1,0],
X2=[1,3,1,2],
X3=[4,2,1,0],
X4=[6,1,0,1].
X1=[1,2,-1,0],
X2=[1,3,1,2],
X3=[4,2,1,0],
X4=[6,1,0,1].
6 M
6(b)
If the expression is
x2(1+u)+y2(2+u)+z2(3=1+u)
prove that
[(∂u∂x)2+(∂u∂y)2+(∂u∂z)2]=2[x∂u∂x+y∂u∂y+z∂u∂z]
x2(1+u)+y2(2+u)+z2(3=1+u)
prove that
[(∂u∂x)2+(∂u∂y)2+(∂u∂z)2]=2[x∂u∂x+y∂u∂y+z∂u∂z]
6 M
6(c)
Fit a second degree parabolic curve to following data:-
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Y | 2 | 6 | 7 | 8 | 10 | 11 | 11 | 10 | 9 |
8 M
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