UPSC ESE Electrical Engineering Solved Papers

A box contains 10 screws, three of which are defective. If two screws are drawn at random without replacement, then what is the probability that neither of the two screws is defective?

If the random variable X, which is the sum of the two numbers of two fair dice turned up, is discrete, then what is the probability of sum of at least 4 and at most 8?

A bag contains 7 green and 5 yellow balls. Two balls are selected at random, one at a time. What is the probability that the two balls are of different colours, if the selection is made with replacement?

If the events A and B are such that $P(A)=0.6$, $P(B)=0.2$ and $P(A\cup B)=0.7$ then $P(B|A)$ is

If $X \sim \text{Binomial}(16, \frac{1}{2})$ then what is the variance of the random variable?

If $X\sim N(15, 6.25)$, then what is the value of X, which is 1-2 standard deviations below the mean?

If the height h in metres of a swing above the ground at time t seconds is given by $h=2-1.5\sin t$ for $0<t<3$, then the maximum height of the swing is

Consider the following statements regarding rank in terms of column vectors:

1. The rank r of a matrix A equals the maximum number of linearly independent column vectors of A.
2. The matrix A and its transpose $A^T$ have the same rank.
3. Consider p vectors each having n components; if $n<p$, then these vectors are linearly independent.

Which of the above statements is/are not correct?

The eigenvector $x_i$ of the $n\times n$ matrix A, corresponding to the eigenvalues $\lambda=\lambda_i$, is a solution of the homogeneous equation, if

What is the value of the Fourier coefficient $a_0$ for the function given below?

$$f(x)=\begin{cases}\sin 2x, & -\pi<x<-\frac{\pi}{2} \\ 0 & -\frac{\pi}{2}\le x\le 0 \\ \sin 2x, & 0<x\le\pi\end{cases}$$

The straight line $r(t)=ati+btj+ctk+C$, with a, b and c as scalar constants and C as a constant vector, has

Consider the following statements regarding properties of eigenvalues:

1. Any square matrix A and its transpose $A^T$ have the same eigenvalues.
2. The eigenvalues of a triangular matrix are just the diagonal elements of the matrix.
3. The sum of the eigenvalues of a matrix is the product of the elements of the principal diagonal.

Which of the above statements are correct?

Let r be the rank for the coefficient matrix A of the system of homogeneous linear equations having m number of equations and n number of variables.

1. If $r<n$, then the equations have $(n-r)$ linearly independent solutions.
2. If $m<n$ the number of solutions is finite.
3. If $m=n$, then such solution is non-trivial solution.

Which of the above statements is/are not correct?

Consider the following statements regarding atomic bonds:

1. If two ions with same ionization potential form a bond, then this bond will most probably be ionic.
2. With the increase in the difference of the ionization potentials, the bond will be more and more ionic.
3. If the atom or molecule has completely filled outer shell, then the bonding between the atoms or molecules will be a primary bond at the time of solidification.

Which of the above statements is/are correct?

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What is the drift velocity of electrons knowing that in metals, the free electron concentration is about $n_0=10^{28}\text{ electrons/m}^3$? (Take maximum admitted value of the density of electric current for metals as $J=10^7\text{ A/m}^2$ and electrical charge of the electron as $q_0=1.602\times 10^{-19}\text{ C}$)

Which one of the following is generated by the supplementary (precession) movements of the conduction electrons that appear when the material is introduced in a magnetic field?

The cermet of Au/SiO is obtained by

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A transformer core is wound with a coil carrying an alternating current at a frequency of 50 Hz. The hysteresis loop has an area of 70000 units, when the axes are drawn in units of $10^{-4}\text{ Wb-m}^{-2}$ and $10^2\text{ A-m}^{-1}$. What is the hysteresis loss by assuming the magnetization to be uniform throughout the core volume of $0.02\text{ m}^3$?

L is measured value of inductance with magnetic core, $L_0$ is measured value of inductance without magnetic core, $\mu_0$ is magnetic permeability of free space, h is height of the material under test, c is outer diameter of the material under test, d is inner diameter of the material under test. What is the formula for relative magnetic permeability ($\mu_r$)?

Consider the following statements regarding ferrites:

1. Ferrites are linear materials and present hysteresis cycle.
2. The values of saturation induction and remanent induction are lower.
3. Magnetic permeability is lower and more sensitive at the action of external magnetic fields.

Which of the above statements is/are correct?

What is the conductivity of copper at 300 K, if the collision time t for electron scattering is $2\times 10^{-14}\text{ s}$ at this temperature? (Take the number of free electrons per $\text{m}^3$ of copper as $8.5\times 10^{28}$, electron charge as $1.602\times 10^{-19}\text{ C}$ and mass of free electron as $9.1\times 10^{-31}\text{ kg}$)

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The resistivity of pure copper at room temperature is $1.8\times 10^{-8}\text{ ohm-m}$. The resistivity of Cu-4% Ni alloy at room temperature is $7.0\times 10^{-8}\text{ ohm-m}$. What is the resistivity due to impurity scattering by 1% of nickel in the copper lattice?