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# Multiple choice question for engineering

## Set 1

1. Gradient of a function is a constant. State True/False.
a) True
b) False

Answer: b [Reason:] Gradient of any scalar function may be defined as a vector. The vector’s magnitude and direction are those of the maximum space rate of change of φ.

2. The mathematical perception of the gradient is said to be
a) Tangent
b) Chord
c) Slope
d) Arc

Answer: c [Reason:] The gradient is the rate of change of space of flux in electromagnetics. This is analogous to the slope in mathematics.

3. Divergence of gradient of a vector function is equivalent to
a) Laplacian operation
b) Curl operation
d) Null vector

Answer: a [Reason:] Div (Grad V) = (Del)2V, which is the Laplacian operation. A function is said to be harmonic in nature, when its Laplacian tends to zero.

4. The gradient of xi + yj + zk is
a) 0
b) 1
c) 2
d) 3

Answer: d [Reason:] Grad (xi + yj + zk) = 1 + 1 + 1 = 3. In other words, the gradient of any position vector is 3.

5. Find the gradient of t = x2y+ ez at the point p(1,5,-2)
a) i + 10j + 0.135k
b) 10i + j + 0.135k
c) i + 0.135j + 10k
d) 10i + 0.135j + k

Answer: b [Reason:] Grad(t) = 2xy i + x2 j + ez k. On substituting p(1,5,-2), we get 10i + j + 0.135k.

6. Curl of gradient of a vector is
a) Unity
b) Zero
c) Null vector
d) Depends on the constants of the vector

Answer: c [Reason:] Gradient of any function leads to a vector. Similarly curl of that vector gives another vector, which is always zero for all constants of the vector. A zero value in vector is always termed as null vector(not simply a zero).

7. Find the gradient of the function given by, x2 + y2 + z2 at (1,1,1)
a) i + j + k
b) 2i + 2j + 2k
c) 2xi + 2yj + 2zk
d) 4xi + 2yj + 4zk

Answer: b [Reason:] Grad(x2+y2+z2) = 2xi + 2yj + 2zk. Put x=1, y=1, z=1, the gradient will be 2i + 2j + 2k.

8. The gradient can be replaced by which of the following?
a) Maxwell equation
b) Volume integral
c) Differential equation
d) Surface integral

Answer: c [Reason:] Since gradient is the maximum space rate of change of flux, it can be replaced by differential equations.

9. When gradient of a function is zero, the function lies parallel to the x-axis. State True/False.
a) True
b) False

Answer: a [Reason:] Gradient of a function is zero implies slope is zero. When slope is zero, the function will be parallel to x-axis or y value is constant.

10. Find the gradient of the function sin x + cos y.
a) cos x i – sin y j
b) cos x i + sin y j
c) sin x i – cos y j
d) sin x i + cos y j

Answer: a [Reason:] Grad (sin x + cos y) gives partial differentiation of sin x+ cos y with respect to x and partial differentiation of sin x + cos y with respect to y and similarly with respect to z. This gives cos x i – sin y j + 0 k = cos x i – sin y j.

## Set 2

1. Mathematically, the functions in Green’s theorem will be
a) Continuous derivatives
b) Discrete derivatives
c) Continuous partial derivatives
d) Discrete partial derivatives

Answer: c [Reason:] The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then, ∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise.

2. Find the value of Green’s theorem for F = x2 and G = y2 is
a) 0
b) 1
c) 2
d) 3

Answer: a [Reason:] ∫∫(dG/dx – dF/dy)dx dy = ∫∫(0 – 0)dx dy = 0. The value of Green’s theorem gives zero for the functions given.

3. Which of the following is not an application of Green’s theorem?
a) Solving two dimensional flow integrals
b) Area surveying
c) Volume of plane figures
d) Centroid of plane figures

Answer: c [Reason:] In physics, Green’s theorem is used to find the two dimensional flow integrals. In plane geometry, it is used to find the area and centroid of plane figures.

4. The path traversal in calculating the Green’s theorem is
a) Clockwise
b) Anticlockwise
c) Inwards
d) Outwards

Answer: b [Reason:] The Green’s theorem calculates the area traversed by the functions in the region in the anticlockwise direction. This converts the line integral to surface integral.

5. Calculate the Green’s value for the functions F = y2 and G = x2 for the region x = 1 and y = 2 from origin.
a) 0
b) 2
c) -2
d) 1

Answer: c [Reason:] ∫∫(dG/dx – dF/dy)dx dy = ∫∫(2x – 2y)dx dy. On integrating for x = 0->1 and y = 0->2, we get Green’s value as -2.

6. If two functions A and B are discrete, their Green’s value for a region of circle of radius a in the positive quadrant is
a) ∞
b) -∞
c) 0
d) Does not exist

Answer: d [Reason:] Green’s theorem is valid only for continuous functions. Since the given functions are discrete, the theorem is invalid or does not exist.

7. Applications of Green’s theorem are meant to be in
a) One dimensional
b) Two dimensional
c) Three dimensional
d) Four dimensional

Answer: b [Reason:] Since Green’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with respect to x,y, which is two dimensional.

8. The Green’s theorem can be related to which of the following theorems mathematically?
a) Gauss divergence theorem
b) Stoke’s theorem
c) Euler’s theorem
d) Leibnitz’s theorem

Answer: b [Reason:] The Green’s theorem is a special case of the Kelvin- Stokes theorem, when applied to a region in the x-y plane. It is a widely used theorem in mathematics and physics.

9. The Shoelace formula is a shortcut for the Green’s theorem. State True/False.
a) True
b) False

Answer: a [Reason:] The Shoelace theorem is used to find the area of polygon using cross multiples. This can be verified by dividing the polygon into triangles. It is a special case of Green’s theorem.

10. Find the area of a right angled triangle with sides of 90 degree unit and the functions described by L = cos y and M = sin x.
a) 0
b) 45
c) 90
d) 180

Answer: d [Reason:] dM/dx = cos x and dL/dy = -sin y ∫∫(dM/dx – dL/dy)dx dy = ∫∫ (cos x + sin y)dx dy. On integrating with x = 0->90 and y = 0->90, we get area of right angled triangle as -180 units (taken in clockwise direction). Since area cannot be negative, we take 180 units.

## Set 3

1. Calculate the emf of a coil with turns 100 and flux rate 5 units.
a) 20
b) -20
c) 500
d) -500

Answer: d [Reason:] The emf is the product of the turns of the coil and the flux rate. Thus e = -N dφ/dt, where the negative sign indicates that the emf induced is opposing the flux. Thus e = -100 x 5 = -500 units.

2. The equivalent inductances of two coils 2H and 5H in series aiding flux with mutual inductance of 3H is
a) 10
b) 30
c) 1
d) 13

Answer: d [Reason:] The equivalent inductance of two coils in series is given by L = L1 + L2 + 2M, where L1 and L2 are the self inductances and M is the mutual inductance. Thus L = 2 + 5 + 2(3) = 13H.

3. The expression for the inductance in terms of turns, flux and current is given by
a) L = N dφ/di
b) L = -N dφ/di
c) L = Niφ
d) L = Nφ/i

Answer: a [Reason:] We know that e = -N dφ/dt and also e = -L di/dt. On equating both we get, L = Ndφ/di is the expression for inductance.

4. The equivalent inductance of two coils with series opposing flux having inductances 7H and 2H with a mutual inductance of 1H.
a) 10
b) 7
c) 11
d) 13

Answer: b [Reason:] The equivalent inductance of two coils in series with opposing flux is L = L1 + L2 – 2M, where L1 and L2 are the self inductances and M is the mutual inductance. Thus L = 7 + 2 – 2(1) = 7H.

5. A coil is said to be loosely coupled with which of the following conditions?
a) K>1
b) K<1
c) K>0.5
d) K<0.5

Answer: d [Reason:] k is the coefficient of coupling. It lies between 0 and 1. For loosely coupled coil, the coefficient of coupling will be very less. Thus the condition K<0.5 is true.

6. With unity coupling, the mutual inductance will be
a) L1 x L2
b) L1/L2
c) √(L1 x L2)
d) L2/L1

Answer: c [Reason:] The expression for mutual inductance is given by M = k √(L1 x L2), where k is the coefficient of coupling. For unity coupling, k = 1, then M = √(L1 x L2).

7. The inductance is proportional to the ratio of flux to current. State True/False.
a) True
b) False

Answer: a [Reason:] The expression is given by L = Ndφ/di. It can be seen that L is proportional to the ratio of flux to current. Thus the statement is true.

8. Calculate the mutual inductance of two tightly coupled coils with inductances 49H and 9H.
a) 21
b) 58
c) 40
d) 49/9

Answer: a [Reason:] For tightly coupled coils, the coefficient of coupling is unity. Then the mutual inductance will be M = √(L1 x L2)= √(49 x 9) = 21 units.

9. Find the inductance of a coil with turns 50, flux 3 units and a current of 0.5A
a) 150
b) 300
c) 450
d) 75

Answer: b [Reason:] The self inductance of a coil is given by L = Nφ/I, where N = 50, φ = 3 and I = 0.5. Thus L = 50 x 3/0.5 = 300 units.

10. The inductance of a coaxial cable with inner radius a and outer radius b, from a distance d, is given by
a) L = μd ln(b/a)/2π
b) L = 2π μd ln(b/a)
c) L = πd/ln(b/a)
d) L = 0

Answer: a [Reason:] The inductance of a coaxial cable with inner radius a and outer radius b, from a distance d, is a standard formula derived from the definition of the inductance. This is given by L = μd ln(b/a)/2π.

## Set 4

1. The characteristic impedance of a quarter wave transformer with load and input impedances given by 30 and 75 respectively is
a) 47.43
b) 37.34
c) 73.23
d) 67.45

Answer: a [Reason:] In quarter wave transformer, the characteristic impedance will be the geometric mean of the input impedance and the load impedance. Thus Zo2 = ZIN ZL. On substituting for ZIN = 75 and ZL = 30, we get the characteristic impedance as 47.43 units.

2. The input impedance of a quarter wave line 50 ohm and load impedance of 20 ohm is
a) 50
b) 20
c) 1000
d) 125

Answer: d [Reason:] The characteristic impedance will be the geometric mean of the input impedance and the load impedance. Thus Zo2 = Zin ZL. On substituting for Zo = 50 and ZL = 20, we get the input impedance as 502/20 = 125 ohm.

3. For a matched line, the input impedance will be equal to
b) Characteristic impedance
c) Output impedance
d) Zero

Answer: b [Reason:] A matched line refers to the input and characteristic impedance being the same. In such condition, maximum transmission will occur with minimal losses. The reflection will be very low.

4. The reflection coefficient lies in the range of
a) 0 < τ < 1
b) -1 < τ < 1
c) 1 < τ < ∞
d) 0 < τ < ∞

Answer: a [Reason:] The reflection coefficient lies in the range of 0 < τ < 1. For full transmission, the reflection will be zero. For no transmission, the reflection will be unity.

5. When the ratio of load voltage to input voltage is 5, the ratio of the characteristic impedance to the input impedance is
a) 1/5
b) 5
c) 10
d) 25

Answer: b [Reason:] From the transmission line equation, the ratio of the load voltage to the input voltage is same as the ratio of the characteristic impedance to the input impedance. Thus the required ratio is 5.

6. The power of the transmitter with a radiation resistance of 12 ohm and an antenna current of 3.5A is
a) 147
b) 741
c) 174
d) 471

7. The group delay of the wave with phase constant of 62.5 units and frequency of 4.5 radian/sec is
a) 13.88
b) 31.88
c) 88.13
d) 88.31

Answer: a [Reason:] The group delay is given by td = β/ω. Given that β = 62.5 and ω = 4.5, we get the group delay as td = 62.5/4.5 = 13.88 units.

8. The maximum impedance of a transmission line 50 ohm and the standing wave ratio of 2.5 is
a) 20
b) 125
c) 200
d) 75

Answer: b [Reason:] The maximum impedance of a line is given by Zmax = SZo. On substituting for S = 2.5 and Zo = 50, we get Zmax = 2.5 x 50 = 125 ohm.

9. The minimum impedance of a transmission line 75 ohm with a standing wave ratio of 4 is
a) 75
b) 300
c) 18.75
d) 150

Answer: c [Reason:] The minimum impedance of a line is given by Zmin = Zo/S. On substituting for Zo = 75 and S = 4, we get Zmin = 75/4 = 18.75 units.

10. The average power in an electromagnetic wave is given by
a) propagation constant
b) poynting vector
c) phase constant
d) attenuation constant

Answer: b [Reason:] The Poynting vector is the cross product of the electric field and magnetic field intensities. It gives the total power of an electromagnetic wave.

11. The characteristic impedance of a transmission line is normally chosen to be
a) 50
b) 75
c) 50 or 75
d) 100

Answer: c [Reason:] The characteristic impedance is always 50 ohm or 75 ohm for a transmission line. This is because of the GHz range of operation and the load impedences employed.

12. Identify the material which is not present in a transmission line setup.
a) waveguides
b) cavity resonator
c) antenna
d) oscillator

Answer: d [Reason:] The transmission line setup consists of antennae for transmitting and receiving power. It consists of waveguides and cavity resonator for guided transmission of electromagnetic waves. Thus oscillator is the odd one out.

## Set 5

1. An electric field is given as E = 6y2z i + 12xyz j + 6xy2 k. An incremental path is given by dl = -3 i + 5 j – 2 k mm. The work done in moving a 2mC charge along the path if the location of the path is at p(0,2,5) is (in Joule)
a) 0.64
b) 0.72
c) 0.78
d) 0.80

Answer: b [Reason:] W = -Q E.dl W = -2 X 10-3 X (6y2z i + 12xyz j + 6xy2 k) . (-3 i + 5 j -2 k) At p(0,2,5), W = -2(-18.22.5) X 10-3 = 0.72 J.

2. The integral form of potential and field relation is given by line integral. State True/False
a) True
b) False

Answer: a [Reason:] Vab = -∫ E.dl is the relation between potential and field. It is clear that it is given by line integral.

3. If V = 2x2y – 5z, find its electric field at point (-4,3,6)
a) 47.905
b) 57.905
c) 67.905
d) 77.905

Answer: b [Reason:] E = -Grad (V) = -4xy i – 2×2 j + 5k At (-4,3,6), E = 48 i – 32 j + 5 k, |E| = √3353 = 57.905 units.

4. Find the potential between two points p(1,-1,0) and q(2,1,3) with E = 40xy i + 20x2 j + 2 k
a) 104
b) 105
c) 106
d) 107

Answer: c [Reason:] V = -∫ E.dl = -∫ (40xy dx + 20x2 dy + 2 dz) , from q to p. On integrating, we get 106 volts.

5. Find the potential between a(-7,2,1) and b(4,1,2). Given E = (-6y/x2 )i + ( 6/x) j + 5 k.
a) -8.014
b) -8.114
c) -8.214
d) -8.314

Answer: c [Reason:] V = -∫ E.dl = -∫ (-6y/x2 )dx + ( 6/x)dy + 5 dz, from b to a. On integrating, we get -8.214 volts.

6. The potential of a uniformly charged line with density λ is given by,
λ/(2πε) ln(b/a). State True/False.
a) True
b) False

Answer: a [Reason:] The electric field intensity is given by, E = λ/(2πεr) Vab = -∫ E.dr = -∫ λ/(2πεr). On integrating from b to a, we get λ/(2πε) ln(b/a).

7. A field in which a test charge around any closed surface in static path is zero is called
a) Solenoidal
b) Rotational
c) Irrotational
d) Conservative

Answer: d [Reason:] Work done in moving a charge in a closed path is zero. It is expressed as, ∫ E.dl = 0. The field having this property is called conservative or lamellar field.

8. The potential in a lamellar field is
a) 1
b) 0
c) -1
d) ∞

Answer: b [Reason:] Work done in a lamellar field is zero. ∫ E.dl = 0,thus ∑V = 0. The potential will be zero.

9. Line integral is used to calculate
a) Force
b) Area
c) Volume
d) Length