(Revised)

**[Time: Three Hours] **

**[Max. Marks: 80]**

“Please check whether you have got the right question paper.”

N.B i) Q.1 and Q.6 of are compulsory.

ii) Solve any two questions from Q.2, Q.3, Q.4 and Q.5.

iii) Solve any two questions from Q.7, Q.8, Q.9 and Q.10.

iv) Figures to the right indicate full marks.

v) Assume suitable data, if necessary.

_________________________________________________________________________________________________________

**SECTION- A**

Q.1 Solve any five questions from the following. 10

a) Find the amplitude and magnitude of \[ \frac{(1+i)(2+i)}{(3-i)} \]

b) Find the locus of z if| 2z-1|= |z-2 |.

c) Find the n^{th }derivative of 3^{x} sin( 2x+1)

d) State comparison test of series.

e) Expand sin x in powers of (x-π/2)

f) Evaluate : \[ lim_{x\rightarrow\infty}\lbrack\cos h^{-1}x-\log x\rbrack \]

g) Find the solution of exact differential equation:

[3x^{2}+6xy^{2}]dx+[6x^{2}y+4y^{3}]dy=0

h) Reduce the Bernoulli differential equation cosx dy=y(sinx- y)dx to linear differential equation.

Q.2 A) If z_{1} and z_{2}and be two complex numbers then prove that: **[04]**

B) If y=tan^{-1}[1+x/1-x]then prove that \[ y_n=(-1)^{n-1}\left(n-1\right)!\sin^n\theta\;\sin\;n\theta \] **[06]**

C) Solve : ** [05]**

\[ \frac{dy}{dx}+\frac{y\log y}{x-\log\;y}=0 \]

Q.3 A) Separate into real and imaginary parts the expression sin^{-1}( e^{i}^{θ}) **[05]**

B) Expand \[ \left[\frac{1+e^x}{ze^x}\right]^\frac12 \]up to the terms containing x^{2}.**[05]**

C) Solve : **[05]**

[sinx siny- xe^{y}]dy=[e^{y }+cosx cosy]dx

Q.4 A) Prove that **[05]**\[ \frac{1+\cos6\theta}{1+\cos2\theta}=16\cos^4\theta-24\;\cos^2\theta\;+9 \]

B) If is finite, find the value of p and hence, the limit. **[05]**

C) Find the current in series in series R-C circuit with R=10Ω, C=0.1F, E=110sin(314t) ,i(0)=0. **[05]**

Q.5 A) If (a+ib)^{p} =m ^{(x+iy)}, prove that one of the values of y/x is \[ \frac{2\tan^{-1}{\displaystyle\frac ba}}{\log(a^2+b^2)} \]

B) Test the convergence of the series **[04]**

\[ 1+\frac x2+\frac{x^2}{3^2}+\frac{x^3}{4^3}+………..x>0 \]

C) Find the orthogonal trajectories of the family of curve e^{x}+e^{-y}=c **[04]**

**SECTION -B**

Q.6 Solve any five questions from the following.** [10]**

a) Find the points of intersection of the curve x ( x^{2}+y^{2})=a (x^{2}-y^{2} ).

b) Write the symmetry of the curve x=(θ+sinθ ) ,y=a( 1+cosθ) with justification.

c) Find the equations of asymptote to the curve ( ) .

d) The length of curve x=f( t) and y=g(t ) from t=A to t=B is given by the Formula…….

e) If u=x^{3}+y^{3}-3axy then prove that \[ \frac{\partial^2u}{\partial x\;\partial y}=\frac{\partial^2u}{\partial y\partial x \]

f) If \[ u=\tan^{-1}\left[\frac{x^3+y^3}{x+y}\right] \] then find the value of \[ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} \]

g) Find the stationary points of the function

f(x,y ) =x^{3}+y^{3}-63( x+y)+12xy .

h) Find \frac{\partial(x,y)}{\partial(u,v)},if u=x^{2}-y^{2},v=2xy

Q.7 A) Trace the curve a^{2}x^{2}=y^{2}(x^{2}+a^{2})with full justification. **[05]**

B) Prove that at a point of the surface x^{x }y^{y }z^{z}=c where x=y=z,\frac{\partial^2z}{\partial x\partial y}=-(x\log ex)^{-1}

C) If, then prove that

( )

( )

05

Q.8 A) Trace the curve with full justification. 05

B) If

(

) then find the value of

05

C) Find the length of an arc of the curve

( )

05

Q.9 A)

Trace the curve

with full justification.

05

B) If ( ) then prove that

05

C) Find the extreme values of ( ). 05

Q.10 A) Find the length of the cardioid ( ) lying outside the circle . 05

B) Divide 24 into three parts such that the continued product of first, square of second and cube of

third may be maximum.

05

C) Show that in the catenary (

) the length of arc from vertex to any point is

(

)

05

93b0165d1abe61613196ddd85acf8fe5

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