STUPIDSID
Answer any one question from Q1 and Q2
1 (a)
If y1/m + y-1/m=2x prove that \[ (x^2 - 1) y_{n+2} + (2n+1)xy_{n+1} + (n^2 - m^3) y_n = 0 \]
7 M
1 (b)
Find the pedal equation for the curve
rinfin;=a∞ sin mθ+b∞ cos mθ.
rinfin;=a∞ sin mθ+b∞ cos mθ.
6 M
1 (c)
Derive an expression to find radius of curvature in Cartesian form.
7 M
2 (a)
Find the nth derivative of sin2x cos3x.
7 M
2 (b)
Show that the curves r=a(1+cos θ) and r=b (1-cos &theta) intersect at right angles.
6 M
2 (c)
Find the radius of curvature when x=a log (sec t + tan t) y=a sect.
7 M
Answer any one question from Q3 and Q4
3 (a)
Using McLaurin's series expand tan x upto the term containing x5.
7 M
3 (b)
Show that \[ x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = 2u \log u \ where \log u = \dfrac {x^3 + y^3}{3x+4y} \]
6 M
3 (c)
Find the extreme values of x4+y4-2(x-y)2.
7 M
4 (a)
\[ Evaluate \ \lim_{x \to \inty} \left \{ \dfrac {e^x \sin x-x-x^2}{x^2 + x \log (1-x)} \right \} \]
7 M
4 (b)
If u=x log xy where x3+y33xy=1 Find fu/dx.
6 M
4 (c)
\[ If \ u=\dfrac {yz}{x}, \ v= \dfrac {xz}{y}, \ w=\dfrac {xy}{z}, \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)} \]
7 M
Answer any one question from Q5 and Q6
5 (a)
Find div \[ \overrightarrow {F} \] and Curl \[ \overrightarrow {F} \] where \[overrightarrow {F} = grad (x^3 + y^3 + z^3 - 3xyz) \]
7 M
5 (b)
Using differentiation under integral sign,
Evaluate \[ \int^1_0 \dfrac {x^\alpha - } {\log x} dx (\alpha \ge 0) \] Hence find \[ \int^1_0 \dfrac {x^3 -1} {\log x} dx \]
Evaluate \[ \int^1_0 \dfrac {x^\alpha - } {\log x} dx (\alpha \ge 0) \] Hence find \[ \int^1_0 \dfrac {x^3 -1} {\log x} dx \]
6 M
5 (c)
Trace the curve y2(a-x)=x3, a>0 use general rules.
7 M
6 (a)
\[ if \ \overrightarrow {r} = xi + yj + zk \ and \ r=|\overrightarrow {r} | \] then prove that \[ \nabla r^n = nr^{n-2} \overrightarrow{r} \]
7 M
6 (b)
Find the constants a, b, c such that \[ \overrightarrow {F} = (x+y+az)i + (bx+2y-z)j + (x+cy + 2z)k \] is irrotational. Also find ϕ such that \[ \overrightarrow {F} = \nabla \phi \]
6 M
6 (c)
Using differentiation under integral sign, evaluate \[ int^\infty _0 e^{-\alpha x } \dfrac {\sin x } {x } dx \]
7 M
Answer any one question from Q7 and Q8
7 (a)
Obtain reduction formula for \[ int^{1/2}_0 \cos^n x \ dx \]
7 M
7 (b)
Solve: (1+2xy \cos x^2 - 2xy) dx + (\sin x^2 - x^2) dy = 0.
6 M
7 (c)
A body originally at 80°C cools down to 60°C in 20 minutes, the temperature of the air being 40°C. What will be temperature of the body after 40 minutes from the original?
7 M
8 (a)
Evaluate \[ \int^{2a}_0 x^2 \sqrt { 2ax - x^2 } dx \]
7 M
8 (b)
Solve: \[ xy \left (1+x \ y^2 \right ) \dfrac {dy}{dx}= 1 \]
6 M
8 (c)
Fid the orthogonal trajectories of the family of confocal conics \[ \dfrac {x^2}{b^2}+ \dfrac {y^2}{b^2 + \lambda}= 1 \] where λ is parameter.
7 M
Answer any one question from Q9 and Q10
9 (a)
Solve by Gauss elimination method.
\[ 5x_1 + x_2 + x_3 + x_4 =4, \ x_1+7x_2 + x_3 + x_4 =12, \ x_1 + x_2 + 6x_3 + x_4 = -5, \ x_1 + x_2 + x_3 + 4x_4 = 6 \]
\[ 5x_1 + x_2 + x_3 + x_4 =4, \ x_1+7x_2 + x_3 + x_4 =12, \ x_1 + x_2 + 6x_3 + x_4 = -5, \ x_1 + x_2 + x_3 + 4x_4 = 6 \]
7 M
9 (b)
Diagonalize the matrix \[ A= \begin{bmatrix}-19 &7 \\-42 &16 \end{bmatrix} \]
6 M
9 (c)
Find the dominant Eigen value and the corresponding Eigen vector of the matrix \[ A= \begin{bmatrix}6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix} \] by power method taking the initial Eigen vector (1,1,1)1.
7 M
10 (a)
Solve by L U decomposition method
x+5y+z=14, 2x+y+3z=14, 3x+y+4z=17
x+5y+z=14, 2x+y+3z=14, 3x+y+4z=17
7 M
10 (b)
Show that the transformation y1=2x1- 2x2-x3, y2=-4x1 + 5x2 + 3x3, y3= x1-x2-x3 is regular and find the inverse transformation.
6 M
10 (c)
Reduce the quadratic form \[ 2x^2_1 + 2x^2 _2 + 2x^2_3 + 2x_1x_3 \] into canonical form by orthogonal transformation.
7 M
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