STUPIDSID
Answer any one question from Q1 and Q2
1 (a)
Examine the following system of equations for consistency and solve it, if consistent.
4x-2y+6z=8
x+y-3z=-1, 15x-3y+9z=21
4x-2y+6z=8
x+y-3z=-1, 15x-3y+9z=21
4 M
1 (b)
Examine the following vectors for Linear dependence. Find the relation between them, if dependent.
(2, -1, 3, 2), (1, 3, 4, 2) and (3, -5, 2, 2)
(2, -1, 3, 2), (1, 3, 4, 2) and (3, -5, 2, 2)
4 M
1 (c)
If 2 cos ϕ=x+1/x, 2cosψ=y+1/y
prove that, xpyq+1/xpyq=2 cos (pϕ + qψ)
prove that, xpyq+1/xpyq=2 cos (pϕ + qψ)
4 M
2 (a)
Use de Moivre's theorem, to solve the equation
x7+x4+I (x3+1)=0
x7+x4+I (x3+1)=0
4 M
2 (b)
If (1+ai)(1+bi)=p+iq, then prove that,
i) p tan [tan-1 + tan-1b]=q
ii) (1+a2)(1+b2)=p2+q2
i) p tan [tan-1 + tan-1b]=q
ii) (1+a2)(1+b2)=p2+q2
4 M
2 (c)
Reduce the following matrix A to its normal form and hence find its rank, where \[ A=\begin{vmatrix}
2 &-3 &4 &4 \\1
&1 &1 &2 \\3
&-2 &3 &6
\end{vmatrix} \]
4 M
Answer any one question from Q3 and Q4
3 (a)
Test convergence of the series (any one) \[ i) \ \ \sum^{\infty}_{n=1} \dfrac {2^n + 1}{3^n +1} \\ ii) \ \ \dfrac {1* 2}{3^2 * 4^2} + \dfrac {3 * 4}{5^2 *6^2}+ \dfrac {5* 6}{7^2 * 8^2} + \cdots \ \cdots \]
4 M
3 (b)
Expand 40+53(x-2)+19(x-2)2+2(x-2)3 in ascending powers of x
4 M
3 (c)
If y=xn log x then, prove that \[ Y_{n+1} = \dfrac {n!}{x} \]
4 M
4 (a)
Solve any one: \[ i) \ \ Evaluate \ \lim_{x\to \infty} (\cot x)^{\sin x} \] ii) Find the values of a and b such that, \[ \lim_{x\to 0}\dfrac {a \cos x - a + b \ x^2}{x^4}= \dfrac {1}{12} \]
4 M
4 (b)
prove that, \[ e^x \tan x=x +x^2 + \dfrac {5x^2}{6} + \dfrac {x^4}{2}+ \cdots \ \cdots \]
4 M
4 (c)
\[ if \ Y=\dfrac {x} {(x+1)^4} \ find \ Y_n \]
4 M
Solve any two of the following:
5 (a)
\[ Verify \ \dfrac {\partial^2 u}{\partial x \partial y} = \dfrac {\partial ^2 u}{\partial y \partial x} \ for \ u =\tan ^{-1}\left [ \dfrac {y}{x} \right ] \]
7 M
5 (b)
if x=u tan v, y=u sec v prove that \[ \left ( \dfrac {\partial u} {\partial x} \right )_y \left ( \dfrac {\partial v}{\partial x} \right )_y = \left ( \dfrac {\partial u}{\partial y} \right )_x \left ( \dfrac {\partial v}{\partial y} \right )_x \]
7 M
Answer any one question from Q5 and Q6
5 (c)
\[ If \ u = \dfrac {x^3 +y^3}{y \sqrt{x}} + \dfrac {1}{x^7} \sin ^{-1} \left ( \dfrac {x^2 + y^2}{2xy} \right ) \] Then, find the value of \[ x^2 \dfrac {\partial^2 u}{\partial x^2}+ 2xy \dfrac {\partial ^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^ 2 u}{\partial y^2} \ At \ point \ (1,1) \]
7 M
Solve any two of the following:
6 (a)
If u=(x2-y2) f(xy) then show that
uxx+uyy=(x4-y4) f''(xy)
uxx+uyy=(x4-y
7 M
6 (b)
Verify Euler's theorem for homogeneous functions
F(x,y,z)=3x2yz+5xy2z+4z4
F(x,y,z)=3x2yz+5xy
7 M
6 (c)
If x=u+v+w, y=uv+uw+vw, z=uvw and F is function of x,y,z then prove that, \[ x \dfrac {\partial F}{\partial x} + 2y \dfrac {\partial F}{\partial y} +3z \dfrac {\partial F}{\partial z} = u \dfrac {\partial F} {\partial u} + v \dfrac {\partial F}{\partial v}+ w \dfrac {\partial F}{\partial w} \]
7 M
Answer any one question from Q7 and Q8
7 (a)
If x=v2+w2, y=w2+u2, z=u2+v2 Find \[ \dfrac {\partial (u,v,w)} {\partial (x,y,z)} \]
4 M
7 (b)
Examine for functional dependence for u=x+y+z, v=x2+y2+z3, w=x3+y3+z3-3xyz.
4 M
7 (c)
Find the extreme values of f(x,y)=x3+y-3axy, a>0
5 M
8 (a)
If u2+xv2=x+y and v2+yu2=x-y find \[ \dfrac {\partial v} {\partial y} \]
4 M
8 (b)
The resistance R of a circuit was calculated using the formula I=E/R. If there is an error of 0.1 Amp in reading I and 0.5 Volts in E, find the corresponding percentage error in R when I=15 Amp and E=100 Volts
4 M
8 (c)
Divide 24 into three parts such that, the continued product of the first, square of the second and cube of the third may be maximum. Use Lagrange's method.
5 M
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