STUPIDSID
1 (a)
If Y=cos(m log x), prove that \[ x^2 y_{n+2}+ (2n+1)xy_{n+1} + (m^2 + n^2)y_s=0 \]
7 M
1 (b)
Find the angle intersection between the curves \[ r=a \log \theta \ and \ r = \dfrac {n}{\log \theta} \]
6 M
1 (c)
Derive an expression to find radius of curvature in Cartesian form.
7 M
2 (a)
\[ If \ \sin^{-1} y=2 \log (x+1) \ prove \ that \ (x^2 +1) y_{n+2} + (2n+1) (x+1)y_{n+1}+ (n^2 + 4)y_n=0 \]
7 M
2 (b)
Find the pedal equation rn=sec hn?
6 M
2 (c)
Show that the radius of curvature of the curve \[ x^3 + y^3 = 3xy \ at \ \left ( \dfrac {3}{2}, \dfrac {3}{2} \right ) \ is \ \dfrac {-3}{8\sqrt{2}} \]
7 M
3 (a)
Find the first four non zero terms in the expansion of \[ f(x) = \dfrac {x}{e^{x-1}} \]
7 M
3 (b)
\[ If \ \cos u = \dfrac {x+y}{\sqrt{x}+ \sqrt{y}} \ show \ that \ x\dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = - \dfrac {\cot u}{2} \]
6 M
3 (c)
\[ Find \ \dfrac {\partial (u,v,w)}{\partial (x,yz)} \ where \ u=x^2+y^2+z^2, \ v=xy+yz + zx \ and \ w=x+y+z \] Hence interpret the result.
7 M
4 (a)
If w=f(x,y), x=r cos &theta, y=r sin ? show that \[ \left ( \dfrac {\partial t}{\partial x} \right )^2 + \left ( \dfrac {\partial t}{\partial y} \right )^2 - \left ( \dfrac {\partial w}{\partial r} \right )^2 = \dfrac {1}{r^2} \left ( \dfrac {\partial w}{\partial \theta} \right )^2 \]
7 M
4 (b)
Evaluate \[ \lim_{x\to 0} \left ( \dfrac {\sin x} {x}\right )^{\frac {1}{x}} \]
6 M
4 (c)
Examine the function f(x,y)=1+sin(x2+y2) for extremum.
7 M
5 (a)
A particle moves along the curve x=2t2, y=t2-4t, z=3t-5. Find the components of velocity and acceleration at t=1 in the direction \[ \widehat{i}-2\widehat{j}+2\widehat{k} \]
7 M
5 (b)
Using differentiation under integral sign, evaluate \[ \int^\infty_0 \dfrac{e^{-\alpha x}\sin x}{x}dx \]
7 M
5 (c)
Use general rules to trace the curve y2(a-x)=x3, a>0
6 M
6 (a)
\[ If \ \overrightarrow{r} = x\widehat{i} + y\widehat{j}+ z\widehat{k} \ and \ |\overrightarrow{r}|= r. \ Find \ grad \ div \left ( \dfrac {\overrightarrow{r}}{r} \right ) \]
7 M
7 (a)
Obtain the reduction formula for \[ \int^{\frac {\pi}{2}}_0 \cos^n xdx \]
7 M
7 (b)
Solve (xy3+y)dx+2(x2y2+x+y4)dy=0
6 M
7 (c)
Show that the orthogonal trajectories of the family of cardioids \[ r= a \cos^2 \left ( \dfrac {\theta}{2} \right ) \] is another family of cardioids \[ r= b \sin^2 \left ( \dfrac {\theta}{2} \right ) \]
7 M
8 (a)
\[ Evaluate \ \int^\pi_0 x \sin^2 x \cos^4 xdx \]
7 M
8 (b)
\[ Solve \ \dfrac {dy}{dx}- y \tan x=y^2 \sec x \]
6 M
8 (c)
If the temperature of the air is 30°C and the substance cools from 100°C to 70°C in 15 minutes, find when the temperature will be 40°C.
7 M
9 (a)
Solve 3x-y+2z=12, x+2y+3z=11, 2x-2y-z=2 by Gauss elimination method.
6 M
9 (b)
Diagonalize the matrix \[ A= \begin{bmatrix}-1 &1 &2 \\0 &-2 &-1 \\0 &0 &-3 \end{bmatrix} \]
7 M
9 (c)
Determine the largest eigen value and the corresponding eigen vector of \[ A= \begin{bmatrix}1 &3 &-1 \\3 &2 &4 \\-1 &4 &10 \end{bmatrix} \] Staring with [0, 0, 1]? as the initial eigenvector. Perform 5 iterations.
7 M
10 (a)
Show that the transformation \[ y_1 = x_1 + 2x_2 +5x_3 , \ y_2 = 2x_3 + 4x_2 + 11 x_3 , \ y_3 = -x_2 + 2x_3 \] is regular and find the inverse transformation.
6 M
10 (b)
Solve by LU decomposition method 2x+y+4z=12, 8x-3y+2z=20, 4x+11y-z=33
7 M
10 (c)
Reduce the quadratic form 2x2+2y2-2xy-2yz-2zx into canonical form. Hence indicate its nature, rank, index and signature.
7 M
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