## TMA Questions & Answers: PHY313 - Mathematical Methods For Physics

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### TMA Questions & Answers: PHY313 - Mathematical Methods For Physics

TMA Questions & Answers: PHY313 - Mathematical Methods For Physics

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Whatsapp: 08155572788

1 A line segment is drawn in the z-plane joining the points $$z=2+3i$$ and $$z=4+5i$$. Find the images
$$A^{􀳦??}$$ and $$B^{􀳦??}$$ in the w-plane of the points $$A\left ( 2,3 \right )$$ and $$B\left ( 4,5 \right )$$ in the z-plane.

$$A{'}\left ( 1,\frac{5}{2} \right )$$ and $$B{'}\left ( 2,\frac{7}{2} \right )$$

2 Identify the functions u and v if the function $$f\left ( z \right )=z^{2}-z+2$$ is expressed in the polar
form.

$$u\left ( r,\theta \right )=r^{2}cos2\theta-rcos\theta{+}2$$ and $$v\left ( r,\theta \right )=r^{2}sin2\theta{-}rsin\theta$$

3 Identify the functions u and v if the function $$f\left ( z \right )=z^{2}-z+2$$ is expressed in the
cartesian form.

$$u\left ( x,y \right )=x^{2}-y^{2}-x+2$$ and $$v\left ( x,y \right )=2xy-y$$

4 Determine the image of the point $$z=-2+3i$$ on the w-plane under the transformation $$w=z+3-i$$1+2i$$5 If$$w=\frac{1}{z}$$is expressed in the form$$u+iv$$, then$$u\left ( x,y \right )=\frac{x}{x^{2}+y^{2}}$$and$$v\left ( x,y \right )=\frac{-y}{x^{2}+y^{2}}$$6 Given that$$w=f\left ( z\right )$$to find$$w=z^{2}$$, which of the following is INCORRECT$$u\left ( x,y \right )=x^{2}+y^{2}$$7 Given that$$w=u+iv$$, such that for every point z in a certain region R of the z-plane, there corresponds one or more values of w; which of the following is NOT a correct representation of function of complex variable?$$w=f\left ( x,y \right )$$8 Given that$$z=-2\sqrt{3}-2i$$, find Arg z$$-5\pi/6$$9 If$$z=3+7i$$, then$$\left | z \right |=?\sqrt{58}$$10 Find$$\frac{z_{1}}{z_{2}}$$given that$$z_{1}=(3+2i)$$and$$z_{2}=(1+3i)\frac{9}{10}-\frac{7i}{10}$$11 Given$$z_{1}=a+ib$$and$$z_{2}=c+id$$, then, for$$z_{2}\neq{0}$$,$$\frac{z_{1}}{z_{2}}=\frac{(ac+bd)+i(bc-ad)}{c^2+d^2}$$12 Given$$z=x+iy$$, which of the following is INCORRECT?$$z-\bar{z}=-2iIm(z)$$13 Which of the following is correct with respect to complex numbers?$$(1,0)=1$$qnd$$(0,1)=-i$$14 Given$$z=x+iy$$, then$$z\bar{z}$$= --------$$

$$x^2+y^2$$

15 The name of the equation that gives the wave dynamics of a particle is called?

Schrödinger

16 The equation for adiabatic of the a system is of the form:____

PV4/3 = constant

17 What is the value of [x,p]?

i􀳦?

18 What is the number of permutations of N different objects when all are taken at a time?

NPN

19 Calculations of occupancy of states involves

classical statistics

20 Which of the following is correct about a black body radiation?

The radiation in the interior must be the same as that of spectral distribution

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Jed
Posts: 1449
Joined: Tue Oct 10, 2017 6:37 pm
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### Re: TMA Questions & Answers: PHY313 - Mathematical Methods For Physics

Email: Solutions2tma@gmail.com
Whatsapp: 08155572788

21 The energy of a system with vibrational degrees of freedom is:________.

Both b and c

22 What is the probability that a student who bought a copy of Liboff also bought a copy of Jackson?

72.3%

23 The energy of a system with translational degrees of freedom is:________.

Mv2/2

24 If $$f\left ( z \right )=e^{x}$$ is analytic, find \frac{\partial {v}}{\partial x}e^{x}sinsy$$25 If$$f\left ( z \right )=e^{x}$$is analytic, find \frac{\partial u}{\partial x}$$

$$e^{x}cosy$$

26 Find the map of the circle $$\left | z \right |=\lambda$$ by the transformation $$w=\sqrt{2}\left ( 1+i \right )z$$. Hint $$w=u+iv=\sqrt{2}\left ( 1+i \right )\left ( x+iy \right )=\sqrt{2}\left (x-y \right )+i\sqrt{2}\left ( x+y \right )$$

$$\left | w \right |=2\lambda$$

27 Find the image of the circle $$\left | z \right |=\lambda$$ under the transformation $$w=5z$$

$$\left | w \right |=5\lambda$$

28 Find $$log\left ( 1+i\sqrt{3} \right )$$. Hint: $$Log\left (z\right )=ln\left | z \right |+i\left [Arg\left ( z \right ) \right ]$$ with $$z\neq 0$$ and $$-\pi < Arg\left ( z \right )\leq \pi$$.

$$ln2+i\left ( \frac{\pi}{3}+2n\pi \right )$$

29 For whch of the following are Cauchy-Riemann equations NOT satisfied?

$$w=cos\bar{z}=cos\left ( x-iy \right )$$

30 Find $$F{'}$$ given that $$F\left ( z \right )\frac{z}{z^{2}{-}1}$$.

$$\frac{\left ( z^{2}+1 \right )}{\left ( z^{2}-1 \right )^{2}}$$ for $$z \neq \pm 1$$

31 Express v in polar form, given that $$v=\sqrt{3}-i$$

$$v=2e^{-i\pi/6}$$

32 Express u in polar form, given that $$u=1+i$$

$$u=\sqrt{2}e^{i\frac{\pi}{4}}$$

33 $$e^{-i\frac{\pi}{2}}=?$$

-i

34 $$e^{i\pi}=?$$

-1

35 $$e^{i\frac{\pi}{2}}=?$$

i

36 $$e^{0}=?$$

1

37 Given $$z=re^{i\theta}$$, for$$-\pi <\theta < \pi$$, find z for $$\theta{=}0$$

r

38 Given $$z=re^{i\theta}$$, for$$-\pi <\theta < \pi$$, find z for $$\theta{=}\frac{\pi}{2}$$

ir

39 Given $$z=x+iy$$, which of the following is INCORRECT?

$$z=r(cos\theta{-}isin\theta)$$

40 A sequence $$\left \{ z \right \}_{n}$$ is called a --------------- sequence if for any arbitrarily small
number $$\varepsilon > 0$$ it is always possible to find an integer N, usually depending on
$$\varepsilon$$, such that $$\left | z_{m}-z_{n} \right | < \varepsilon$$ for all $$m> n> N$$.

Cauchy

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Jed
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Joined: Tue Oct 10, 2017 6:37 pm
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### Re: TMA Questions & Answers: PHY313 - Mathematical Methods For Physics

Email: Solutions2tma@gmail.com
Whatsapp: 08155572788

41 If $$f\left ( z \right )=e^{x}$$ is analytic, find \frac{\partial u}{\partial y}-e^{x}sinsy$$42 If$$f\left ( z \right )=e^{x}$$is analytic, find \frac{\partial {v}}{\partial x}$$

$$e^{x}sinsy$$

43 If $$f\left ( z \right )=e^{x}$$ is analytic, find \frac{\partial u}{\partial x}e^{x}cosy$$44 Find the map of the circle$$\left | z \right |=\lambda$$by the transformation$$w=\sqrt{2}\left ( 1+i
\right )z$$. Hint$$w=u+iv=\sqrt{2}\left ( 1+i \right )\left ( x+iy \right )=\sqrt{2}\left (x-y \right
)+i\sqrt{2}\left ( x+y \right )\left | w \right |=2\lambda$$45 Find the image of the circle$$\left | z \right |=\lambda$$under the transformation$$w=5z\left | w \right |=5\lambda$$46 Find$$log\left ( 1+i\sqrt{3} \right )$$. Hint:$$Log\left (z\right )=ln\left | z \right |+i\left [Arg\left ( z
\right ) \right ]$$with$$z\neq 0$$and$$-\pi < Arg\left ( z \right )\leq \pi$$.$$ln2+i\left ( \frac{\pi}{3}+2n\pi \right )$$47 For whch of the following are Cauchy-Riemann equations NOT satisfied?$$w=cos\bar{z}=cos\left ( x-iy \right )$$48 Find$$F{'}$$given that$$F\left ( z \right )\frac{z}{z^{2}{-}1}$$.$$\frac{\left ( z^{2}+1 \right )}{\left ( z^{2}-1 \right )^{2}}$$for$$z \neq \pm 1$$49 At what point does the function$$f\left ( z \right )=\frac{z}{z-1}$$have discontinuity?$$\left ( 1,0 \right )$$50 A line segment is drawn in the z-plane joining the points$$z=2+3i$$and$$z=4+5i$$. Find the images$$A^{􀳦??}$$and$$B^{􀳦??}$$in the w-plane of the points$$A\left ( 2,3 \right )$$and$$B\left (
4,5 \right )$$in the z-plane.$$A{'}\left ( 1,\frac{5}{2} \right )$$and$$B{'}\left ( 2,\frac{7}{2} \right )$$51 Identify the functions u and v if the function$$f\left ( z \right )=z^{2}-z+2$$is expressed in the polar form.$$u\left ( r,\theta \right )=r^{2}cos2\theta-rcos\theta{+}2$$and$$v\left ( r,\theta \right
)=r^{2}sin2\theta{-}rsin\theta$$52 Identify the functions u and v if the function$$f\left ( z \right )=z^{2}-z+2$$is expressed in the cartesian form.$$u\left ( x,y \right )=x^{2}-y^{2}-x+2$$and$$v\left ( x,y \right )=2xy-y$$53 Determine the image of the point$$z=-2+3i$$on the w-plane under the transformation$$w=z+3-i

$$1+2i$$

54 If $$w=\frac{1}{z}$$ is expressed in the form $$u+iv$$, then

$$u\left ( x,y \right )=\frac{x}{x^{2}+y^{2}}$$ and $$v\left ( x,y \right )=\frac{-y}{x^{2}+y^{2}}$$

55 Given that $$w=f\left ( z\right )$$ to find $$w=z^{2}$$, which of the following is INCORRECT

$$u\left ( x,y \right )=x^{2}+y^{2}$$

56 Given that $$w=u+iv$$, such that for every point z in a certain region R of the z-plane, there
corresponds one or more values of w; which of the following is NOT a correct representation of function
of complex variable?

$$w=f\left ( x,y \right )$$

57 Given that $$z=-2\sqrt{3}-2i$$, find Arg z

$$-5\pi/6$$

58 Given that $$z=-2\sqrt{3}-2i$$, find $$r=\left | z \right |$$

4

59 Which of the following is the correct expansion of $$sin4\theta$$? Hint: $$\left ( cos\theta{+isin\theta} \right )^{n}=\left ( cosn\theta{+}isinn\theta \right )$$, for n a natural number. This
is the De Moivre's theorem.

$$4cos^{3}\theta\sin\theta{-}4cos\theta{sin^{3}\theta}$$

60 Which of the following is the correct expansion of $$cos4\theta$$? Hint: $$\left ( cos\theta{+isin\theta} \right )^{n}=\left ( cosn\theta{+}isinn\theta \right )$$, for n a natural number. This
is the De Moivre's theorem.

$$cos^{4}\theta{-}6cos^{2}\theta{sin}^{2}\theta{+}sin^{4}\theta$$

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Jed
Posts: 1449
Joined: Tue Oct 10, 2017 6:37 pm
Contact:

### Re: TMA Questions & Answers: PHY313 - Mathematical Methods For Physics

Email: Solutions2tma@gmail.com
Whatsapp: 08155572788

61 Find $$uv$$ given that $$u=(1+i)$$ and $$v=(\sqrt{3}-i)$$

$$v=2\sqrt{2}e^{i\pi/12}$$

62 Express v in polar form, given that $$v=\sqrt{3}-i$$

$$v=2e^{-i\pi/6}$$

63 Express u in polar form, given that $$u=1+i$$

$$u=\sqrt{2}e^{i\frac{\pi}{4}}$$

64 $$e^{-i\frac{\pi}{2}}=?$$

-i

65 $$e^{i\pi}=?$$

-1

66 $$e^{i\frac{\pi}{2}}=?$$

i

67 $$e^{0}=?$$
1

68 Given $$z=re^{i\theta}$$, for$$-\pi <\theta < \pi$$, find z for $$\theta{=}0$$

r

69 Given $$z=re^{i\theta}$$, for$$-\pi <\theta < \pi$$, find z for $$\theta{=}\frac{\pi}{2}$$

ir

70 Given $$z=x+iy$$, which of the following is INCORRECT?

$$z=r(cos\theta{-}isin\theta)$$

71 A sequence $$\left \{ z \right \}_{n}$$ is called a --------------- sequence if for any arbitrarily small
number $$\varepsilon > 0$$ it is always possible to find an integer N, usually depending on
$$\varepsilon$$, such that $$\left | z_{m}-z_{n} \right | < \varepsilon$$ for all $$m> n> N$$.

Cauchy

72 If $$z=3+7i$$, then $$\left | z \right |=?$$

$$\sqrt{58}$$

73 Find $$\frac{z_{1}}{z_{2}}$$ given that $$z_{1}=(3+2i)$$ and $$z_{2}=(1+3i)$$

$$\frac{9}{10}-\frac{7i}{10}$$

74 Given $$z_{1}=a+ib$$ and $$z_{2}=c+id$$, then, for $$z_{2}\neq{0}$$, $$\frac{z_{1}}{z_{2}}=$$

$$\frac{(ac+bd)+i(bc-ad)}{c^2+d^2}$$

75 Given $$z=x+iy$$, which of the following is INCORRECT?

$$z-\bar{z}=-2iIm(z)$$

76 Which of the following is correct with respect to complex numbers?

$$(1,0)=1$$ qnd $$(0,1)=-i$$

77 Given $$z=x+iy$$, then $$z\bar{z}$$= --------x^2+y^2$$78 If$$f\left ( z \right )=e^{x}$$is analytic, find \frac{\partial u}{\partial y}$$

$$-e^{x}sinsy$$

79 Given that $$z=x+iy$$, then $$\bar{z}== -------$$

(x,-y)

80 Given $$z=x+iy$$, where $$i=\sqrt{-1}$$, which of the following is INCORRECT?

y is the complex part of z

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