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1
If the eigen values of the matrix a of order 3×3 are 2, 3 and 1, then find the eigen values of adjoint of A.
2 M
2
If λ is the eigen value of the matrix A, then prove that λ2 is the eigen value of A2
2 M
3
Give an example for conditionally convergent series.
2 M
4
Test the convergence of the series \[ 1-\dfrac {1}{2^2}-\dfrac {1}{3^2}+\dfrac {1}{4^2}+\dfrac {1}{5^2}-\dfrac {1}{7^2}-\dfrac {1}{8^2}..... \ to \infty \]
2 M
5
What is the curvature of the circle (x-1)2+(y+2)2=16 at any point on it?
2 M
6
Find the envelope of the family of curves \[ y=mx+\dfrac {1}{m} \] where m is the parameter.
2 M
7
\[ If \ x^y+y^x=1 \ then \ find \ \dfrac{dy}{dx} \]
2 M
8
\[ If \ x=r \cos \theta, \ y=r\sin\theta, \ then \ find \ \dfrac {\partial (r, \theta)}{\partial (x,y)} \]
2 M
9
Find the area bounded by the lines x=0, y=1 and y=x, using double integration.
2 M
10
\[ Evaluate \ \int^\pi_0\int^a_0 r \ drd\theta \]
2 M
Answer any one question from Q11 (a) & Q11 (b)
11.(a) (i)
Find the eigen values and the eigen vectors of the matrix \[ \begin{bmatrix}2 &2 &1 \\1 &3 &1 \\1 &2 &2 \end{bmatrix} \]
8 M
11.(a) (ii)
Using Cayley-Hamilton theorem find A-1 and A4, if \[ A=\begin{bmatrix} 1&2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix} \]
8 M
11.(b)
Reduce the quadratic form 6x2+3y2+3z2-4xy-2yz+4xz into a canonical form by an orthogonal reduction. Hence find its rank and nature.
16 M
Answer any one question from Q12 (a) & Q12 (b)
12.(a) (i)
Examine the convergence and the divergence of the following series \[ 1+\dfrac {2}{5}x+\dfrac {6}{9}x^2+\dfrac {14}{17}x^3+.....+\dfrac {2^n-2}{2^n+1}(x^{n-1})+....(x>0). \]
8 M
12.(a) (ii)
Discuss the convergence and the divergence of the following series \[ \dfrac {1}{2^3}-\dfrac {1}{3^3}(1+2)+\dfrac {1}{4^3}(1+2+3)-\dfrac {1}{5^3}(1+2+3+4)+..... \]
8 M
12.(b) (i)
Test the convergence of the series \[ \sum^\infty_{n=0}ne^{-n^2} \]
8 M
12.(b) (ii)
Test the convergence of the series \[ \dfrac {x}{1+x}-\dfrac {x^2}{1+x^2}+\dfrac {x^3}{1+x^3}-\dfrac {x^4}{1+x^4}+\cdots \cdots (0<x<1) \]
8 M
Answer any one question form Q13 (a) & Q13 (b)
13.(a) (i)
Find the radius of curvature of the cycloid x=a(θ+sin θ), y=a(1-cos θ)
8 M
13.(a) (ii)
Find the equation of the evolutes of the parabola y2=4ax.
8 M
13.(b) (i)
Find the equation of circle of curvature at \[ \left ( \dfrac {a}{4},\dfrac {a}{4} \right ) \ on \ \sqrt{x}+\sqrt{y}=\sqrt{a} \]
8 M
13.(b) (ii)
Find the envelope of the family of straight lines y=mx-2am-am3, where m is the parameter.
8 M
Answer any one question from Q14 (a) & Q14 (b)
14.(a) (i)
Expand ex log (+1+y) in powers of x and y upto the third degree terms using Taylor's theorem.
8 M
14.(a) (ii)
\[ If \ u=\dfrac {yz}{x}, \ v=\dfrac {zx}{y}, \ w=\dfrac {xy}{z}, \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)} \]
8 M
14.(b) (i)
Discuss the maxima and minima of f(x,y)=x3y2(1-x-y)
8 M
14.(b) (ii)
If w=f(y-z, z-x, x-y), then show that \[ \dfrac {\partial w}{\partial x}+\dfrac {\partial w}{\partial y}+\dfrac {\partial w}{\partial z}=0 \]
8 M
Answer any one question from Q15 (a) & Q15 (b)
15.(a) (i)
By changing the order of integration evaluate \[ \int^1_0\int^{2-x}_{x^2}xy \ dydx \]
8 M
15.(a) (ii)
By changing to polar coordinates, evaluate\[ \int^\infty_0\int^{\infty}_0e^{-(x^2+y^2)}dxdy \]
8 M
15.(b) (i)
Evaluate ∬ xy dxdy over the positive quadrant of the circle x2+y2=a2
8 M
15.(b) (ii)
\[ Evaluate \ \iiint_v \dfrac {dzdydx}{(x+y+z+1)^3}, \] where V is the region bounded by x=0, y=0, z=0 and x+y+z=1
8 M
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