Engineering Mathematics - II - B.Tech 2nd Semester Examination, 2025

If \( f(z)=u(x,y)+iv(x,y) \) is analytic function, then \( f'(z)= \)

1. \( \frac{u}{x} - i \frac{v}{x} \)
2. \( \frac{u}{x} - i \frac{v}{y} \)
3. \( \frac{v}{y} - i \frac{v}{x} \)
4. \( \frac{u}{x} - i \frac{v}{y} \)

For an analytic function \( f(z)=u(x,y)+i v(x,y) \), if the real part \( u(x,y)=x^{2}-y^{2} \) is given, what is the imaginary part \( v(x,y) \)?

1. \( 2xy+c \)
2. \( xy+c \)
3. \( x^{2}+y^{2}=c \)
4. \( -2xy+c \)

The value of the integral \( \oint_C |z|dz \), where C is the left half of the unit circle \( |z|=1 \) from \( z=-i \) to \( z=i \)

1. \( 2\pi i \)
2. \( \pi i \)
3. \( 0 \)
4. \( 2i \)

The residue of \( e^{z}/\cos \pi z \) at its pole \( z=-1/2 \) is

1. \( -e^{1/2}/\pi \)
2. \( e^{1/2}/\pi \)
3. \( 0 \)
4. \( -e^{1/2} \)

The particular integral \( y_p \) of the differential equation \( (D^{2}+4)y=\cos 2x \), \( D \equiv \frac{d}{dx} \) is

1. \( \frac{1}{2}\sin 2x \)
2. \( \frac{1}{2}x \sin 2x \)
3. \( \frac{1}{4}x \sin 2x \)
4. \( \frac{1}{2}x \cos 2x \)

By the method of undetermined coefficients, the particular integral \( y_p \) of the differential equation \( (D^{4}+2D^{3}+D^{2}) y=12x^{2} \), \( D \equiv \frac{d}{dy} \) is

1. \( a+bx+cx^{2} \)
2. \( ax+bx^{2}+cx^{3} \)
3. \( x(a+bx+cx^{2}) \)
4. \( x^{2}(a+bx+cx^{2}) \)

Let \( \lim_{n\rightarrow\infty}(u_{n})^{1/n}=l \), then the series \( \sum_{n=1}^{\infty}u_{n} \) converges if

1. \( l > 1 \)
2. \( l < 1 \)
3. \( l = 1 \)
4. \( l \ge 1 \)

\( L(\sin at)=\frac{a}{s^{2}+a^{2}} \) when

1. \( s > a \)
2. \( s < a \)
3. \( s < 0 \)
4. \( s > 0 \)

If \( L^{-1}(\frac{3s+8}{s^{2}+4s+25})=e^{kt}[3 \cos\sqrt{21}t+\frac{2}{\sqrt{21}}\sin\sqrt{21}t] \) then the value of k is

1. -1
2. 1
3. 2
4. -2

If \( f(x)=\sqrt{\frac{1-\cos x}{2}}, 0 < x < 2\pi \) then the value of \( a_0 \) is

1. \( \frac{4}{\pi} \)
2. \( \frac{2}{\pi} \)
3. \( \frac{\pi}{4} \)
4. \( \frac{\pi}{2} \)

Find \( \lim_{z\rightarrow 0} [\frac{1}{1-e^{1/x}} + iy^2] \) if it exists.

Check the continuity of the following function at \( z=0 \):
\( f(z)=\begin{cases} \frac{Re(z^{2})}{|z|^{2}}, & z \ne 0 \\ 0, & z=0 \end{cases} \)

Examine whether the Cauchy-Riemann equations are satisfied at the origin for the function given below or not:
\( f(z)=\begin{cases} \frac{x^{3}(1+i)-y^{3}(1-i)}{x^{2}+y^{2}}, & z \ne 0 \\ 0, & z=0 \end{cases} \)
Also check, whether \( f(z) \) is analytic at \( z=0 \) or not.

If \( f(\xi)=\int_{C}\frac{3z^{2}+7z+1}{z-\xi}dz, \) where C is an ellipse \( 9x^{2}+16y^{2}=144 \). Find the values of \( f(3) \), \( f'(1-i) \) and \( f''(1-i) \).

Evaluate the integral \( \int_{0}^{\infty}\frac{\sin ax}{x(x^{2}+b^{2})}dx \), \( a>0, b>0 \) by the use of Cauchy residue theorem.

Find the Laurent's expansion for \( f(z)=\frac{7z-2}{z^{3}-z^{2}-2z} \) in the region \( 1 <|z+1|<3 \).

Find the complete solution of \( \frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=x+e^{x}\cos x \).

Find the solution of \( x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+y=\log x \cdot \sin(\log x) \).

Find the solution of \( \frac{d^{2}y}{dx^{2}}-6\frac{dy}{dx}+9y=\frac{e^{3x}}{x^{2}} \) by the method of variation of parameters.

Test the convergence of the series \( \Sigma(1+\frac{1}{n})^{-n^2} \).

Test the convergence of the series \( \frac{a}{b}+\frac{a+x}{b+x}+\frac{a+2x}{b+2x}+\frac{a+3x}{b+3x}+...... \)

Find Laplace transform of \( t \cos at \).

Evaluate the integral \( \int_{0}^{\infty}te^{-3t}\sin t dt \), by the use of Laplace transform.

Find the inverse Laplace transform of \( \tan^{-1}(\frac{2}{s^{2}}) \).

Find the solution of the following initial value problem by the use of Laplace transform:
\( \frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}-y=0, \) if \( y(0)=0 \) and \( (\frac{dy}{dt})_{t=0}=1. \)

Find the Fourier series expansion of the following function:
\( f(x)=\begin{cases} 0, & -\pi \le x \le 0 \\ \sin x, & 0 \le x < \pi \end{cases} \)
Hence deduce that \( \frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{5.7}-\frac{1}{7.9}+......=\frac{\pi-2}{4} \).

Find the Fourier expansion of \( x\sin x \) as Fourier cosine series in the interval \( [0,\pi] \).