{\text{The}}\,\,{\text{power}}\,\,{\text{series}},\,\,\sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{n!}},\,\,\_\_\_\_\_\_\_\_\_\,\,\,x = 1\,\,{\text{to}}\,\,{\text{the}}\,\,{\text{number}}\,\,e.\,}

• A) {\text{diverges}}
• B) {\text{converges}}
• C)
• D)

The gravitational force exerted by the earth on a body of mass m is called ---------- of the body.

• A) Force
• B) weight
• C)
• D)

The solution of x\frac{{dy}}{{dx}} = 0 is ________.

• A) y = {c_1}
• B) none of them
• C) y = {c_1} + {c_2}{x^2}
• D) y = {c_1} + {c_2}x

The displacement measured below the equilibrium position are ________.

• A) negative
• B) positive
• C)
• D)

The coefficient A{e^{ - \lambda t}} is called the damped ________ of vibration.

• A) period
• B) frequency
• C) amplitude
• D) All of these

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

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

Any linear differential equation of the form {a_{n - 1}}{x^{n - 1}}\frac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}} + .... + \frac{{dy}}{{dx}} + {a_0}y = g(x) where {a_n},{a_{n - 1}},...,{a_0} are constants, is said to be a ____________ equation.

• A) Homogeneous
• B) Cauchy-Euler
• C) Non homogeneous
• D) Cauchy-Euler

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

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

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

The nature of the roots of the differential equation {x^2}\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} - 4y = 0 is __________.

• A) none of these
• B) Real and repeated
• C) Conjugate complex
• D) Real and distinct

Consider a power series x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} - .... represents _______.

• A) sin x
• B) cos x
• C) e
• D) ln (1+x)

For a system in simple harmonic motion which of the following is the time required to complete a cycle of motion?

• A) Period
• B) Amplitude
• C) Frequency
• D) Revolution

Auxiliary equation of the differential equation f{x^2}\frac{{{d^2}y}}{{d{x^2}}} + gx\frac{{dy}}{{dx}} + hy\, = \,k(x) is

• A) fm + (g - f){m^2} + h\, = 0
• B) f{m^2} - (g - f)m + h\, = 0
• C) f{m^2} + (g - f)m + h\, = 0
• D) none of them

{\text{The}}\,\,{\text{quantity}}\,\,X = L\gamma - \frac{1}{{C\gamma }}\,\,{\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/y
• B) -1/x
• C) -y
• D) -x

Any linear differential equation of the form \[{a_{n - 1}}{x^{n - 1}}\frac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}} + .... + \frac{{dy}}{{dx}} + {a_0}y = g(x)\] where \[{a_n},{a_{n - 1}},...,{a_0}\] are constants, is said to be a ____________ equation.

• A) Homogeneous
• B) Cauchy-Euler
• C) Non homogeneous
• D) Cauchy-Euler

The coefficient \[A{e^{ - \lambda t}}\] is called the damped ________ of vibration.

• A) amplitude
• B) All of these
• C) period
• D) frequency

To solve a differential equation {a_2}(x){y^{''}} + {a_1}(x)y' + {a_0}(x)y = 0 about a regular singular point we employ the __________ theorem.

• A) Legendre
• B) Bessel
• C) none of them
• D) Frobenius

If determinant \[\left| {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {D + 1} \\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 2 \\\ {D - 1} \end{array}} \end{array}} \right| = 0\] then __________.

• A) \[{D^2} + 1\]
• B) \[(D + 1)(D - 1) - 2\]
• C) \[{D^2} - 1\]
• D) \[(D + 1)(D - 1) + 2\]

The regular singular point of the differential equation {({x^2} - 4)^2}y + (x - 2)y' + y = 0 is ________.

• A) 1
• B) -2
• C) 2
• D) -1