STUPIDSID
1
Find the Eigen values of the inverse of the matrix \[ A=\begin{bmatrix} 2&1 &0 \\0 &3 &4 \\0 &0 &4 \end{bmatrix} \]
2 M
2
If 2, -1, -3 are the Eigen values of the matrix A, then find the Eigen values of the matrix A2-2I
2 M
3
Find the nature of the series 1+2+3+....+n+.
2 M
4
Define Cauchy's integral test.
2 M
5
Find the centre of curvature of y=x2 at the origin.
2 M
6
Define Involutes and Evolutes.
2 M
7
\[ Evaluate: \ \lim_ {x\to \infty,y\to 2} \dfrac {xy+5}{x^2+2y^2} \]
2 M
8
\[ If \ x^y+y^x=c, \ then \ find \ \dfrac {dy}{dx} \]
2 M
9
\[ Evaluate \ \int^{5}_0\int^2_0x^2+y^2 \ dxdy \]
2 M
10
\[ Evaluate \ \int^{\frac {\pi}{2}}_0\int^{\sin \theta}_0 rd\theta dr \]
2 M
Answer any one question from Q11 (a) & Q11 (b)
11.(a) (i)
Verify Cayley Hamilton theorem for the matrix \[ \begin{bmatrix} 1&0 &3 \\2 &1 &-1 \\1 &-1 &1 \end{bmatrix} \] hence find it?s A-1
8 M
11.(a) (ii)
Find the Eigen values and Eigen vectors of \[ \begin{bmatrix} -2&2 &-3 \\2 &1 &-6 \\-1 &-2 &0 \end{bmatrix} \]
8 M
11.(b) (i)
Reduce the quadratic form \[ x^2_1+5x^2_2+x^2_3+2x_1x_2+2x_2x_3+6x_3x_1 \] to the canonical form through orthogonal transformation and find its nature.
10 M
11.(b) (ii)
Prove that the Eigen values of a real symmetric matrix are real.
6 M
Answer any one question from Q12 (a) & Q12 (b)
12.(a) (i)
Prove that the harmonic series is divergent
8 M
12.(a) (ii)
Test the convergence of the series \[ \dfrac {1}{4.7.10}+\dfrac {4}{7.10.13}+\dfrac {9}{10.13.16}+..... \]
8 M
12.(b) (i)
Find the nature of the series \[ \sum^\infty_{n=2}\dfrac {1}{n(\log n)^p} \] by Cauchy's integral test.
8 M
12.(b) (ii)
Test the convergence of the series \[ 1+\dfrac {2^p}{2!}+\dfrac {3^p}{3!}+\dfrac {4^p}{4!}+..... \] by D'Alembert's ratio test.
8 M
Answer any one question from Q13 (a) & Q13 (b)
13.(a) (i)
Find the envelope of \[ \dfrac {x}{a}+\dfrac {y}{b}=1 \] subject to an+bn=cn, where c is constant.
8 M
13.(a) (ii)
Find the Evolute of \[ \sqrt{x}+\sqrt{y}=\sqrt{a} \]
8 M
13.(b) (i)
Find the equation of the circle of curvature of \[ \dfrac {x^2}{4}+\dfrac {y^2}{9}=2 \ at \ (2,3)\]
8 M
13.(b) (ii)
Find the radius of curvature at any point on x=et cos t, y=et sin t.
8 M
Answer any one question from Q14 (a) & Q14 (b)
14.(a) (i)
Find the extreme value of x2+y2+z2 subject to the condition x+y+z=3a
8 M
14.(a) (ii)
\[ If \ u=(x-y)f\left (\dfrac {y}{x} \right ),\ then \ find \ x^2\dfrac {\partial^2u}{\partial x^2}+2xy\dfrac {\partial^2 u}{\partial x \partial y}+ y^2\dfrac {\partial^2 u}{\partial y^2} \]
8 M
14.(b) (i)
\[ If \ u=\dfrac {yz}{x}, \ v=\dfrac {zx}{y}, \ w=\dfrac {xy}{z} \ the \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)} \]
8 M
14.(b) (ii)
Expand ex cos y at \[ \left ( 0, \dfrac {\pi}{2} \right ) \] upto the third terms using Taylor's series.
8 M
Answer any one question from Q15 (a) & Q15 (b)
15.(a) (i)
Find the volume of region bounded by the paraboloid z=x2+y2 and the plane z=4
8 M
15.(a) (i)
Find the surface area of the section of the cylinder x2+y2=a2 made by the plane x+y+z=a
8 M
15.(b) (i)
Change the order of Integration \[ \int^a_0\int^{2a-x}_{\frac {x^2}{a}} xy \ dxdy \] and hence evaluate it.
10 M
15.(b) (ii)
Find the area of the cardioid r=a(1+cos ?)
6 M
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