1
Find the eigen values of A

^{-1}where \[ A=\begin{bmatrix}3 &1 &4 \\0 &2 &6 \\0 &0 &5 \end{bmatrix}\]
2 M

2
Write down the matrix of the quadratic form 2x

^{2}+8z^{2}+4xy+10xz-2yz
2 M

3
Find the equation of the sphere on the line joining the points (2, -3, 1) and (1, -2, -1) as diameter

2 M

4
Define right circular cone.

2 M

5
Find the readius of curvature of the curve y=e

^{x}at x=0
2 M

6
Find the envelope of the lines x/t+yt=2c, 't' being a parameter.

2 M

7
\[ Find \ \dfrac {\partial u}{\partial x} \ and \ \dfrac {\partial u}{\partial y} \ if \ u=y^x \]

2 M

8
\[ if \ x=r \cos \theta, \ y=r\sin \theta \ find \ \dfrac {\partial (r, \theta)}{\partial (x,y)} \]

2 M

9
\[ Evaluate \ \int^b_1 \int^a_1 \dfrac {dxdy}{xy} \]

2 M

10
Change the order of Integration in \[ \int^a_0 \int^y_0 f(x, y)dxdy\]

2 M

Answer any one question form Q11 (a) or Q11 (b)

11.(a) (i)
Find the eigen values and eigen vectors of the matrix \[ A=\begin{bmatrix} 2&0 &1 \\0 &2 &0 \\1 &0 &2 \end{bmatrix} \]

8 M

11.(a)(ii)
Show that the matrix \[ A=\begin {bmatrix}2&-1&2\\-1&2&-1\\1&-1&2 \end{bmatrix} \] satisfies its own characteristics equation. Find also its inverse.

8 M

11.(b)
Reduce the quadratic form 3x

^{2}+5y^{2}+3z^{2}-2xy-2yz+2zx canonical form
16 M

Answer any one question from Q12 (a) or Q12 (b)

12.(a) (i)
Find the equation of the sphere passing through the points (4, -1, 2), (0, -2, 3), (1, 5, -1), (2, 0, 1)

8 M

12.(a) (ii)
Find the equation of the right circular cylinder whose axis is \[ \dfrac {x-1}{2}=\dfrac {y}{3}=\dfrac {z-3}{1} \] and radius '2'.

8 M

12.(b) (i)
Find the two tangent planes to the sphere x

^{2}+y^{2}+z^{2}-4x+2y-6z+5=0 which are parallel to the plane 2x+2y=z. Find their points of contacts
8 M

12.(b) (ii)
Find the equation of the cone formed by rotating the line 2x+3y=5, z=0 about the y-axis.

8 M

Answer any one question from Q13(a) or Q13 (b)

13.(a) (i)
Find the evolute of the parabola x

^{2}=4ay
8 M

13.(a) (ii)
Find the radius of curvature of the curve x

^{3}+xy^{2}-6y^{2}=0 at (3,3).
8 M

13.(b) (i)
Find the centre of curvature of the curve y=x

^{3}-6x^{2}+3z+1 at the point (1, -1).
8 M

13.(b) (ii)
Find the readius of curvature of the curve x=a(cost+ t sin t); y=a(sin t - t cos t) at 't'.

8 M

Answer any one question from Q14 (a) & Q14 (b)

14.(a) (i)
\[ If \ u=xy+yz+zx \ where \ x= \dfrac{1}{t}, \ y=e^t \ and \ z=e^{-t} \ find \ \dfrac {du}{dt} \]

8 M

14.(a) (ii)
Test for maxima and minima of the function f(x,y)=x

^{3}+y^{3}-12x-3y+20
8 M

14.(b) (i)
Expand e

^{x}sin y in power of x and y as far as the terms of the 3^{rd}degree using Taylor's expansion.
8 M

14.(b) (ii)
Find the dimensions of the rectangular box, open at the top, of maximum capacity whose surface area is 432 square meter.

8 M

Answer any one question from Q15 (a) & Q15 (b)

15.(a) (i)
Change the order of integration in \[ \int^a_0\int^a_y \dfrac {x}{x^2+y^2}dx \ dy \] and hence evaluate it.

8 M

15.(a) (ii)
Using double integral find the area of the ellipse \[ \dfrac {x^2}{a^2}+\dfrac{y^2}{b^2}=1 \]

8 M

15.(b) (i)
\[ Evaluate \ \int^{\log 2}_0 \int^{x}_0\int^{x+\log y}_0 e^{x+y+z}dzdydx \]

8 M

15.(b) ii)
Using double integral find the area bounded by the parabolas y

^{2}=4ax and x^{2}=4ay.
8 M

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