STUPIDSID
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 2x2+8z2+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=ex 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 3x2+5y2+3z2-2xy-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 x2+y2+z2-4x+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 x2=4ay
8 M
13.(a) (ii)
Find the radius of curvature of the curve x3+xy2-6y2=0 at (3,3).
8 M
13.(b) (i)
Find the centre of curvature of the curve y=x3-6x2+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)=x3+y3-12x-3y+20
8 M
14.(b) (i)
Expand ex sin y in power of x and y as far as the terms of the 3rd 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 y2=4ax and x2=4ay.
8 M
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