STUPIDSID
1(a)
Solve the equation 7coshx+8sinhx = 1 for real values of x.
3 M
1(b)
If ( z(x+y)=(x-y) ext{find}left ( dfrac{partial z}{partial x} -dfrac{partial z}{partial y}
ight )^2 )
3 M
1(c)
If ( u=r^2cos 2 heta, v=r^2sin 2 heta ext {find}dfrac{partial (u,v)}{partial (r, heta)} )
3 M
1(d)
Prove that ( sec^2x=1+x^2+dfrac{2x^4}{3}+cdots )
3 M
1(e)
find the rank of the Matrix by reducing it to normal form. [egin{bmatrix}
1 & 1 & 1\
1 & -1 & -1\
3 & 1 & 1
end{bmatrix}]
3 M
1(f)
Find nth derivatives of (dfrac{x}{(x-1)(x-2)(x-3)} )
3 M
2(a)
If α, β are the roots of the equation ( x^2-2sqrt{3}.x+4=0 )
find the value of α3+β3
find the value of α3+β3
6 M
2(b)
Examine whether the vectors
X1 = [3 1 1], X2 = [2 0 -1], X3 = [4 2 1] are linearly independent.
X1 = [3 1 1], X2 = [2 0 -1], X3 = [4 2 1] are linearly independent.
6 M
2(c)(i)
State an prove Euler's theorem for a homogeneous function in two variables.
4 M
2(c)(ii)
If y=xcosu
Find the value of x2uxx+2xy uxy + y2uyy
Find the value of x2uxx+2xy uxy + y2uyy
4 M
3(a)
Is the following system has trivial or non trivial solution? Obtain the non trival solution if exist.
3x1 + 4x2 - x3 - 9x4 = 0
2x1 + 3x2 + 2x3 - 3x4 = 0
2x1 + x2 - 14x3 - 12x4 = 0
x1 + 3x2 + 13x3 + 3x4 = 0
3x1 + 4x2 - x3 - 9x4 = 0
2x1 + 3x2 + 2x3 - 3x4 = 0
2x1 + x2 - 14x3 - 12x4 = 0
x1 + 3x2 + 13x3 + 3x4 = 0
6 M
3(b)
Discuss the stationary points for Maxima and Minima of
x3 + xy2 - 12x2 - 2y2 + 21x + 10
x3 + xy2 - 12x2 - 2y2 + 21x + 10
6 M
3(c)(i)
If tan (x+iy) = a+ib prove that ( anh 2y=dfrac{2b}{1+a^2+b^2} )
4 M
3(c)(ii)
Separate into real and imaginary parts of Log (3+4i)
4 M
4(a)
if x = u cosv, y = u sinv
( ext {Prove that}dfrac{partial (u,v)}{partial (x,y)},dfrac{partial (x,y)}{partial (u,v)}=1 )
( ext {Prove that}dfrac{partial (u,v)}{partial (x,y)},dfrac{partial (x,y)}{partial (u,v)}=1 )
6 M
4(b)
Show that ( logleft [ e^{ialpha}+e^{ieta}
ight ]=logleft [ 2cosleft ( dfrac{alpha -eta }{2}
ight )
ight ]+ileft ( dfrac{alpha +eta }{2}
ight ) )
6 M
4(c)(i)
Solve the system of equation by Gauss Jordan Method
x + 2y + 6z = 22, 3x + 4y + z = 26, 6x - y - z = 19
x + 2y + 6z = 22, 3x + 4y + z = 26, 6x - y - z = 19
4 M
4(c)(ii)
Solve the system of equation by Gauss Siedel Method.
2x - 4y + 49z = 49
43x + 2y + 25z = 23
3x + 53y + 3z = 91
2x - 4y + 49z = 49
43x + 2y + 25z = 23
3x + 53y + 3z = 91
4 M
5(a)
Prove that ( cos^6 heta+sin^6 heta=dfrac{1}{8}[3cos 4 heta+5] )
6 M
5(b)
Find the value of a and b [ ext{if}lim_{x
ightarrow 0}dfrac{x(1+acos x)-bsin x}{x^3}=1]
4 M
5(c)(i)
if y = ex cos2x cosx find yn
4 M
5(c)(ii)
If y = etan-1x prove that (1+x2)yn+2+[2 (n+1) x-1]yn+1+n (n+1)yn = 0
4 M
6(a)
Find non-Singular Matrices P & Q such that, ( A=egin{bmatrix}
1 & 2 & 3 & 4\
2 & 1 & 4 & 3\
3 & 0 & 5 & -10
end{bmatrix} ) is reduced to normal form. Also find rank.
6 M
6(b)
( u=f(e^{y-z},e^{z-x},e^{x-y}) ext{find} dfrac{partial u}{partial x}+dfrac{partial u}{partial y}+dfrac{partial u}{partial z} )
6 M
6(c)(i)
Fit a straight line to the following data :
| Year x: | 1951 | 1961 | 1971 | 1981 | 1991 |
| Production y: | 10 | 12 | 8 | 10 | 15 |
4 M
6(c)(ii)
Fit a second degree parabolic curve to the following data :
| x : | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| y : | 2 | 6 | 7 | 8 | 10 | 11 | 11 | 10 | 9 |
4 M
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