Prepare for UPSC Indian Engineering Services (IES/ESE) with official previous years' questions, detailed derivations, phasor analysis, and circuit solutions across Electrical Machines, Power Systems, Control Systems, Power Electronics, Network Theory, and Analog & Digital Circuits.
Marking pattern: +2.00 positive, -0.66 negative (Official UPSC ESE 1/3rd Negative Marking Scheme; 150 Questions, 300 Marks).
What is the value of b such that the system of homogeneous equations
$$2x+y+2z=0$$
$$x+y+3z=0$$
$$4x+3y+bz=0$$
has non-trivial solution?
The orthogonal trajectories of the hyperbolas
$$x^2 - y^2 = c$$
are
The value of the integral $\int_C x^2 y\,dS$, where C is the curve defined by
$$x = 3\cos t, \quad y = 3\sin t, \quad 0 \le t \le \frac{\pi}{2}$$
is
Let D be the region bounded by the closed cylinder $x^2+y^2=16$, $z=0$, $z=4$, and $v=3x^2 i + 6y^2 j + z k$, then by divergence theorem
$$\iiint_D (\nabla\cdot v)\,dV$$
is
In the Fourier series expansion of the function
$$f(x)=x\sin x, \quad -\pi\le x\le\pi$$
the value of the Fourier coefficient $a_1$ is
The partial differential equation formed by the elimination of arbitrary function from
$$z = f(x^2 - y^2)$$
is
If one solution of the differential equation
$$(x^2 - y^2 - z^2)p + 2xyq = 2xz$$
is $\frac{y}{z}=a$, the other solution is
The solution of the differential equation
$$p - x^2 = q + y^2$$
is
The solution of the differential equation
$$p^2 + q^2 = x + y$$
is
The value of the integral $\int_C z^2\,dz$, where C is the arc of the circle $|z|=2$ from $\theta=0\text{ (zero)}$ to $\theta=\frac{\pi}{3}$, is
The value of the integral $\oint_{C}\frac{dz}{z(z^{2}+4)}$, where C is $|z|=1$, is
The mode and modal ordinate of normal distribution respectively are
The theoretical density of a metallic solid crystal structure is given by
$$\rho=\frac{nA}{V_{C}N_{A}},$$
where n is number of atoms associated with each unit cell, A is atomic weight, $V_{C}$ is volume of the unit cell. $N_{A}$ will represent
The value of packing fraction for face-centred cubic (FCC) structure will be nearly
The total core loss of a specimen of silicon steel is found to be 1500 W at 50 Hz. Keeping the flux density constant, the loss becomes 3000 W, when the frequency is raised to 75 Hz. The eddy current loss at 75 Hz will be
A very weak form of magnetism that is non-permanent persists only while an external field is being applied. It is induced by a change in the orbital motion of electrons due to an applied magnetic field. This form of magnetism is called
The magnetic moments of diamagnetic material are mainly due to
The magnetic flux density on the surface of an iron face is 1.6 T, which is a typical saturation level value for ferromagnetic material. If the value of $\mu_0$ is $4\pi\times 10^{-7}\text{ H/m}$, the force density on the iron face will be nearly
When a ferromagnetic substance is magnetized, there are small changes in its dimensions. The phenomenon is known as
In $\text{Fe}_3\text{O}_4$ each cubic unit cell contains $8\text{ Fe}^{2+}$ and $16\text{ Fe}^{3+}$ ions, the unit cell edge length is 0-839 nm and Bohr magnetons/unit cell is 32 and the value of Bohr magneton is $9\cdot 27\times 10^{-24}\text{ A-m}^2/\text{Bohr magneton}$. The saturation magnetization will be
Which one of the following statements is not correct for Fermi level?
Electrons are emitted with zero velocity for a certain metal surface for $c=3\times 10^8\text{ m/s}$ and $1\text{ \AA}=10^{-10}\text{ m}$, when it is exposed to radiations of $\lambda=6800\text{ \AA}$. The threshold frequency of the metal will be nearly
The materials which possess a permanent magnetic moment in the absence of an external field and manifest very large and permanent magnetizations are
The magnetic inductance becomes zero inside a superconductor when it is cooled below $T_c$ in a weak external field. The magnetic flux is expelled from the interior of the superconductor. This effect is called
In a network, the number of linearly independent mesh equations (M) is