STUPIDSID
1(a)
If cosh x = sec θ, prove that x = log (sec θ + tan θ)
3 M
1(b)
If u = log (x2+y2), prove that
\[\dfrac{\partial^2 u}{\partial x \ \partial y} = \dfrac{\partial^2 u}{\partial y \ \partial x}\]
\[\dfrac{\partial^2 u}{\partial x \ \partial y} = \dfrac{\partial^2 u}{\partial y \ \partial x}\]
3 M
1(c)
If x = r cosθ, y = r sinθ;
\[ \dfrac{\partial(x,y)}{\partial(r, \theta)} \]
\[ \dfrac{\partial(x,y)}{\partial(r, \theta)} \]
3 M
1(d)
Expand log(1 + x + x2 + x3) in powers of x upto x8.
3 M
1(e)
Show that every square matrix can be uniquely expressed as sum of a symmetric and Skew-symmetric matrix.
4 M
1(f)
If y = cos x.cos 2x.cos 3x then find its nth order derivative
4 M
2(a)
Solve the equation x6-i=0.
6 M
2(b)
Reduce matrix A to normal form and find its rank where
\[A={ \left[ \begin{array}{ccc} 1 & 2 & 3 &2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 &5 \end{array} \right]}\]
\[A={ \left[ \begin{array}{ccc} 1 & 2 & 3 &2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 &5 \end{array} \right]}\]
6 M
2(c)
State and prove Euler's theorem for a homogeneous function in two variable. And hence find
\[x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} \ \ \ where \ \ u = \dfrac{\sqrt{x}+\sqrt{y}}{x+y}\]
\[x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} \ \ \ where \ \ u = \dfrac{\sqrt{x}+\sqrt{y}}{x+y}\]
8 M
3(a)
Determine the values of λ so that the equations
x+y+z=1,
x+2y+4z= λ ,
x+4y+10z=λ2
have a solution and solve them completely in each case.
x+y+z=1,
x+2y+4z= λ ,
x+4y+10z=λ2
have a solution and solve them completely in each case.
6 M
3(b)
Find the stationary values of
x3 + y3 - 3axy, a > 0
x3 + y3 - 3axy, a > 0
6 M
3(c)
Separate into real and imaginary parts
tan-1(eiθ)
tan-1(eiθ)
8 M
4(a)
If x = u cos v and y = u sin v
\[\dfrac{\partial(x,y)}{\partial(u,v)}.\dfrac{\partial(u,v)}{\partial(x,y)} =1\]
\[\dfrac{\partial(x,y)}{\partial(u,v)}.\dfrac{\partial(u,v)}{\partial(x,y)} =1\]
6 M
4(b)
If tan[log(x + iy)] = a+ib,
\[prove \ that \ tan[log(x^2+y^2)]=\dfrac{2a}{1-a^2-b^2}\]
where a2 + b2 ≠ 1.
\[prove \ that \ tan[log(x^2+y^2)]=\dfrac{2a}{1-a^2-b^2}\]
where a2 + b2 ≠ 1.
6 M
4(c)
Using Gauss- Seidel iteration method solve,
10x1 + x2 + x3 = 12,
2x1 + 10x2 + x3=13,
2x1 + 2x2 + 10x3 = 14
Upto three iterations.
10x1 + x2 + x3 = 12,
2x1 + 10x2 + x3=13,
2x1 + 2x2 + 10x3 = 14
Upto three iterations.
8 M
5(a)
In a series of sines of multiple of θ, expand sin7 θ
6 M
5(b)
Evaluate the following:
\[\lim_{x\rightarrow 1}\dfrac{x^x-x}{x-1-logx}\]
\[\lim_{x\rightarrow 1}\dfrac{x^x-x}{x-1-logx}\]
6 M
5(c)
Prove the following if y1/m + y-1/m = 2x;
(x2 -1) y(n+2) + (2n+1)xy(n+1) + (n2-m2)yn = 0
(x2 -1) y(n+2) + (2n+1)xy(n+1) + (n2-m2)yn = 0
8 M
6(a)
Examine the following vectors for linear dependence/independence
X1 = (a,b,c), X2 = (b,c,a), X3 = (c,a,b)
where a+b+c ≠ to zero.
X1 = (a,b,c), X2 = (b,c,a), X3 = (c,a,b)
where a+b+c ≠ to zero.
6 M
6(b)
If z = f(x,y) , x=(eu + e-v), y=(e-u - ev)
\[ \dfrac{\partial z}{\partial u}-\dfrac{\partial z}{\partial v} = x\dfrac{\partial z}{\partial x}-y\dfrac{\partial z}{\partial y}\]
\[ \dfrac{\partial z}{\partial u}-\dfrac{\partial z}{\partial v} = x\dfrac{\partial z}{\partial x}-y\dfrac{\partial z}{\partial y}\]
6 M
6(c)
Fit a straight line to following data and also estimate the production in 1957.
| Year | 1951 | 1961 | 1971 | 1981 | 1991 |
| Production in Thousand Tones | 10 | 12 | 8 | 10 | 13 |
8 M
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