Mathematical Physics MCQs Question and Answer in Engineering Physics

11

For the function $$\phi = {x^2}y + xy,$$ the value of $$\left| {\overrightarrow \nabla \phi } \right|$$ at x = y = 1 is

A) 5 B) $$\sqrt 5 $$ C) 13 D) $$\sqrt {13} $$

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Answer & Solution

Answer: Option A

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12

The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&i \\ { - i}&1 \end{array}} \right]\] are

A) +1 and +1 B) zero and +1 C) zero and +2 D) -1 and +1

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Answer: Option C

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13

If the Fourier transform F[δ(x - a)] = exp (-i2πv a), then F^-1(cos 2π av) will correspond to

A) δ(x - a) - δ(x + a) B) a constant C) \[\frac{1}{2}\][δ(x - a) + iδ(x + a)] D) \[\frac{1}{2}\][δ(x - a) + δ(x + a)]

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Answer: Option D

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14

The solutions to the differential equation \[\frac{{dy}}{{dx}} = - \frac{x}{{y + 1}}\] are a family of

A) circles with different radii B) circles with different centres C) straight lines with different slopes D) straight lines with different intercepts on the Y-axis

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Answer: Option A

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15

The Laplace transform of f(t) = sin πt is $$F\left( s \right) = \frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}},\,s > 0.$$ Therefore, the Laplace transform of t sin πt is

A) $$\frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}}$$ B) $$\frac{{2\pi }}{{{s^2}{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$ C) $$\frac{{2\pi s}}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$ D) $$\frac{{2\pi }}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$

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Answer & Solution

Answer: Option C

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16

A finite wave train, of an unspecified nature, propagates along the positive X-axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is

A) $$\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right)\psi \left( {x,\,t} \right) = 0$$ B) $$\left( {{\nabla ^2} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right)\psi \left( {\overrightarrow r ,\,t} \right) = 0$$ C) $$\left( { - \frac{h}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} - ih\frac{\partial }{{\partial t}}} \right)\psi \left( {x,\,t} \right) = 0$$ D) $$\left( {{\nabla ^2} + a\frac{\partial }{{\partial t}}} \right)\psi \left( {\overrightarrow r ,\,t} \right) = 0$$

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Answer & Solution

Answer: Option A

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17

The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 2&3&0 \\ 3&2&0 \\ 0&0&1 \end{array}} \right]\] are

A) 5, 2, -2 B) -5, -1, 1 C) 5, 1, -1 D) -5, 1, 1

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Answer & Solution

Answer: Option C

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18

The value of the integral \[\int_C {\frac{{dz}}{{{z^2}}},} \] where z is a complex variable and C is the unit circle with the origin as its centre, is

A) zero B) 2πi C) 4πi D) -4πi

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Answer: Option A

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19

Which of the following vectors is orthogonal to the vector \[\left( {a\hat i + b\hat j} \right),\] where a and b (a ≠ b) are constants, and \[{\hat i}\] and \[{\hat j}\] are unit orthogonal vectors?

A) \[ - b\hat i + a\hat j\] B) \[ - a\hat i + b\hat j\] C) \[ - a\hat i - b\hat j\] D) \[ - b\hat i - a\hat j\]

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Answer & Solution

Answer: Option D

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20

The two vectors \[\overrightarrow P = i,\,\overrightarrow Q = \frac{{i + j}}{{\sqrt 2 }}\] are

A) related by a rotation B) related by a reflection through the XY-plane C) related by an inversion D) not linearly independent

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Answer & Solution

Answer: Option A

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