The conversion of Cauchy Euler equation $${x^2}\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + y = \ln x$$ after putting $$x = {e^t}$$ becomes

• A) $$({\Delta ^2} - \Delta + 1)y$$
• B) $$({\Delta ^2} - 2\Delta + 1)y$$
• C) $$({\Delta ^2} - 2\Delta - 1)y$$
• D) $$(2{\Delta ^2} - \Delta - 1)y$$

$${\text{The}}\,\,{\text{power}}\,\,{\text{series}},\,\,1 + x + \frac{x}{{2!}} + \frac{{{x^3}}}{{3!}} + ...\, = \_\_\_\_\_\_\_\_\_.$$

• A) $$\cos x$$
• B) $${e^x}$$
• C) $$\sin x$$
• D) $$\ln (1 + x)$$

$$\begin{gathered} {\text{In}}\,\,{\text{the}}\,\,{\text{study}}\,\,{\text{of}}\,\,{\text{mechanics,}}\,\,{\text{cosider}}\,\,{\text{the}}\,\,{\text{damping}}\,\,{\text{force}}\,\,{\text{acting}}\,\,{\text{on}}\,\,{\text{a}}\,\,{\text{body}}\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,\, - \beta \,{\left( {\frac{{dx}}{{dt}}} \right)^2},\,{\text{where}}\,\,\beta \,\,{\text{is}}\,\,{\text{a}}\,\, \hfill \\ {\text{damping}}\,\,{\text{constant}}\,\,{\text{and}}\,\,{\text{negative}}\,\,{\text{sign}}\,\,{\text{indicates}}\,\,{\text{that}}\,\,{\text{the}}\,\,{\text{damping}}\,\,{\text{force}}\,{\text{acts}}\,\,{\text{in}}\,\,{\text{a}}\,{\text{direction}}\,\,\_\_\_\_\_\_\_\,\, \hfill \\ {\text{to}}\,\,{\text{the}}\,\,{\text{direction}}\,\,{\text{of}}\,{\text{motion}}{\text{.}}\,\,\,\, \hfill \\\ \end{gathered}$$

• A) $${\text{opposite}}$$
• B) $${\text{same}}$$
• C)
• D)

If E(t)=0, R=0 (E(t) is the source voltage & R is the resistance) then the electric vibration can be called free ____________.

• A) damped oscillation
• B) un-damped oscillation
• C)
• D)

Ordinary differential equation $\left( {\frac{{dy}} {{dx}}} \right)^3 + \frac{{d^2 y}} {{dx^2 }} + y = 9$ , is of order ------.

• A) 0
• B) 1
• C) 3
• D) 2

A / An _______ is a passive electronic component that stores energy in the form of magnetic
field.

• A) inductor
• B) voltage
• C) capacitor
• D) resister

The harmonic series of constant $$\sum\limits_{n = 1}^\infty {\frac{1}{n}}$$ always _______.

• A) convergent
• B) divergent
• C)
• D)

To reduce any Cauchy –Euler differential equation into a differential equation with ________ coefficients we often use substitution $$x = {e^t}$$.

• A) constant
• B) variable
• C)
• D)

$$\begin{gathered} {\text{Consider}}\,\,{\text{the}}\,\,{\text{equation}}\,\,{\text{of}}\,{\text{the}}\,\,{\text{free}}\,\,{\text{damped}}\,\,{\text{motion,}}\,\,\frac{{{d^2}x}}{{d{t^2}}} + 2\lambda \frac{{dx}}{{dt}} + {\omega ^2}x = 0,\,\,{\text{has}}\,\,{\text{the}}\,\,{\text{roots}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{auxiliary}}\,\, \hfill \\ {\text{equations}},\,\,m = - \lambda \pm \sqrt {{\lambda ^2} - {\omega ^2}} .\,\,{\text{If}}\,{\text{the}}\,\,{\text{roots}}\,\,{\text{are}}\,\,{\text{complex}}\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,{\lambda ^2} - {\omega ^2} < 0,\,\,\,{\text{then}}\,\,\beta < k\,\,{\text{and}}\,\,\,{\text{the}}\,\,{\text{system}}\,\,{\text{is}}\,\, \hfill \\ {\text{called}}\,\,\_\_\_\_\_\_\_\,\,damped.\,\, \hfill \\\ \end{gathered}$$

• A) $${\text{non - critically}}$$
• B) $${\text{under}}$$
• C) $${\text{critically}}$$
• D) $${\text{over}}$$

Consider a mathematical statement, V = IR, where V be the constant of proportionality and it represents the voltage. The equation is called _________.

• A) Ohm’s law
• B) Hooke’s law
• C) Newtown’s law
• D) Coulomb’s law

Consider a power series $$\sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^{n + 1}}}}{{{n^2}}}} {x^n} = x - \frac{{{x^2}}}{{{2^2}}} + \frac{{{x^3}}}{{{3^2}}} + ....$$ , then the center of series is _______.

• A) $${x^2}$$
• B) x
• C) 1
• D) 0

The damping force is __________to the instantaneous velocity $$\frac{{dx}}{{dt}}$$.

• A) Proportional
• B) Constant
• C) Inverse proportional
• D) None of these

If $${\lambda ^2} - {\omega ^2} = 0$$ and $$\beta = k$$ then the system is said to be _______ damped.

• A) over
• B) under
• C) critically
• D) none of these

$${\text{The}}\,\,{\text{quantity}}\,\,Z = \sqrt {{X^2} + {R^2}} \,\,{\text{is}}\,\,{\text{called}}\,\,\_\_\_\_\_\_\_\_\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{circuit}}{\text{.}}$$

• A) $${\text{impedance}}$$
• B) $${\text{reactance}}$$
• C)
• D)

For $$\frac{{dy}} {{dx}} - \frac{y} {x} = - \frac{{\ln x}} {x}$$ the integrating factor is

• A) -1/x
• B) -x
• C) -y
• D) -1/y

For solving non homogeneous differential equation, we use either the method of ___________.

• A) Variation of parameter
• B) None of them
• C) Undetermined coefficient
• D) a&b both

Consider a power series $$1 - \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}} - ....$$ represents _______.

• A) sin x
• B) e
• C) cos x
• D) ln x

The ________ force is proportional to the instantaneous velocity $$\frac{{dx}}{{dt}}$$ .

• A) retarding
• B) umdamped
• C) damped
• D) restoring

If the system is impressed upon by a _________ force and there is no damping force then there is no transient term in the solution.

• A) periodic
• B) applied
• C) friction
• D) gravitational

The standard unit for measurement of reactance and the impedance is __________.

• A) Ohms
• B) Volt
• C) Coulombs
• D) Henry