Questions by admin

Let\[f(x)=\left\{\begin{array}{cr}\pi+x, & -\pi \leq x<0, \\0, & 0 \leq x<\pi,\end{array}\right.\]with $f(x+2 \pi)=f(x)$. If $F(x)$ represents the Fourier series of $f(x)...

Let $y$ be a non-zero quadratic polynomial satisfying the differential equation\[\left(2+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-k y=0,\]where $k$ is a rea...

There are four cities, namely, $C_{1}, C_{2}, C_{3}$ and $C_{4}$. The cities are directly connected by four roads as shown in the picture given below, that is, $C_{1}$ is...

Assume that $f:[0,1] \rightarrow \mathbb{R}$ is continuous on $[0,1]$ and differentiable on $(0,1)$ such that $f(x+h)=f(x)+h f^{\prime}(x+\theta h)$ for some $0<\theta<1$...

Let $A$ be a $3 \times 3$ matrix whose eigenvalues are $2,3,4$ and let $I$ be the identity matrix of order 3 . If\[A^{-1}=\frac{1}{2 k}\left(A^{2}-9 A\right)+\frac{13}{k}...

For some integer $k$, the differential equation\[x^{2} \frac{d^{2} y}{d x^{2}}-3 x \frac{d y}{d x}+(k+2) y=0\]is transformed into $(D-2)^{2} y=0$, where $D=\frac{d}{d t}$...

The approximate value $\text{(rounded off to two decimal places)}$ of the integral\[\int_{0}^{1 / 2} e^{-x^{2}} d x\]using the Trapezoidal rule with step-size $h=\frac{1}...

Consider $f(z)=e^{z}$, where $z=x+i y$ and $i=\sqrt{-1}$. Which of the following statements is correct?$f$ is periodic$f$ is not periodic$|f|=1$$\arg (f)=y \pm n \pi$ for...

Let $P$ and $Q$ be two square matrices of the same order. Then, which of the following matrices is/are necessarily equal to $(P+2 Q)^{2}$ ?$P^{2}+4 P Q+4 Q^{2}$$P(P+2 Q)+...

If\[\int_{0}^{\alpha} \int_{\sqrt{x / \alpha}}^{1} e^{y^{3}} d y d x=e-1, \quad \alpha>0,\]then the value $\text{(in integer)}$ of $\alpha$ is __________.

Consider the vector field $\vec{F}=\left(2 x+y^{2}\right) \hat{\imath}+(2 x y+3 y) \hat{\jmath}$ and $\alpha_{m}=\int_{C_{m}} \vec{F} \cdot d \vec{r}$, $m=1,2$, where $C_...

Which one of the following figures shows the CORRECT dependence of apparent viscosity $(\eta)$ on rate of shear strain $(d u / d y)$ for pseudoplastic fluids?

The locus of temporary locations of all particles that have passed through a fixed point in the flow field at a particular instant is known asstreamline.streakline.pathli...

Consider the velocities $u, v$, and $w$ in $x$-, $y$-, and $z$-directions, respectively. The vorticity expression in the $y-z$ plane is$\frac{\partial v}{\partial x}-\fra...

For the laminar, incompressible flow over a flat plate with uniform free stream velocity, the axial pressure gradient within the boundary layer isgreater than zero.less t...

Let $\vec{r}, \vec{V}$, and $m$ be position vector, velocity vector, and mass, respectively in a control mass system. Which one of the following properties is considered ...

The hydraulic diameter for a circular pipe of radius $R$ is$0.5 R$$R$$2 R$$4 R$

For incompressible, laminar, fully-developed flow through a circular pipe, Darcy friction factor and Fanning friction factor are represented as $f$ and $C_{f}$, respectiv...

For an immersed neutrally buoyant body to be in stable equilibrium, the center of gravity of the body is directlyabove the metacenter.below the metacenter.above the cente...

The absolute pressure in a chamber is measured as $400$ mm Hg at a location where the atmospheric pressure is $700$ mm Hg . A vacuum gauge connected to the chamber reads ...

A thin film of an incompressible, Newtonian liquid $\text{(density $\rho$, viscosity $\mu$)}$ with an uniform thickness $(h)$ is flowing down on a vertical plate. The flo...

A one-eighth scale model of a car is to be tested in a wind tunnel. If the air velocity over the car is $16 \mathrm{~m} / \mathrm{s}$, what should be the air velocity $\t...

A set of basic dimensions, mass, length, and time are represented by $\mathrm{M}, \mathrm{L}$, and $\text{T}$, respectively. What will be the dimensions of pressure in $\...

Consider a fluid flow around an airfoil as shown in figure.The directions of drag force and lift force, respectively are alongOA and OC.OA and OD.OB and OC .OB and OD .

A vessel which contains a volatile liquid and its vapour is connected with a mercury manometer as shown in figure. Both the liquid and vapour phases are at equilibrium. T...

The velocity in a one-dimensional flow is given by $u(x)=\frac{a}{(b-x)^{2}} \mathrm{~m} / \mathrm{s}$, where $a=8 \mathrm{~m}^{3} / \mathrm{s}$ and $b=4 \mathrm{~m}$. Th...

Consider two parallel plates separated by a distance of $1$ cm filled with a Newtonian fluid of viscosity $10^{-3} \mathrm{~Pa}$.s. The top plate is moving with a velocit...

A circular water jet of diameter $50$ mm impinges with a velocity of $18 \mathrm{~m} / \mathrm{s}$ normal to a plate. The density of water is $1000 \mathrm{~kg} / \mathrm...

Consider the steady, incompressible flow of water in a horizontal pipe of constant diameter $1$ m with an inlet velocity of $12 \mathrm{~m} / \mathrm{s}$.As shown in figu...

The axial velocity profile of a laminar, incompressible and fully-developed flow in a circular pipe of radius $(R)$ is given as $u_{z}=-\frac{1}{4 \mu} \frac{\partial p}{...

The velocity potential function in a two-dimensional flow field is given by $\phi(x, y)=-\left(a x y+b x^{2}-b y^{2}\right) \mathrm{m}^{2} / \mathrm{s}$ where $\mathrm{a}...

Consider the incompressible, steady and irrotational flow through a concentric reducer in a horizontal pipeline. The pipe diameter reduces from $d_{1}=12 \mathrm{~cm}$ to...

Consider the incompressible fluid flow over a flat plate with a free stream velocity, $U_{\infty}=1 \mathrm{~m} / \mathrm{s}$. The fluid kinematic viscosity is $10^{-6} \...

The correct combination of phases in the one-component $\mathrm{H}_{2} \mathrm{O}$ phase diagram, as given below, is $\alpha$-water; $\beta$-vapour; $\gamma$-ice$\alpha-$...

Mechanical behaviour of a crystalline ceramic material is best described asductilebrittleviscoelasticViscoplastic

Differential scanning calorimetry involves measurement ofweight changeentropyheatvapour pressure

In ball milling of ceramic powder, selection of grinding media depends on the __________ difference between grinding media and powder particles.thermal conductivitydielec...

Which one of the following unit cell parameters represents a tetragonal crystal system?$\mathrm{a}=\mathrm{b}=\mathrm{c} ; \quad \alpha=\beta=\gamma \neq 90^{\circ}$$\mat...

Which of the following types of materials exhibit(s) positive magnetic susceptibility?ParamagneticDiamagneticFerrimagneticFerromagnetic

Which of the following is/are responsible for pitting corrosion in a metal?Rough surfaceGrain boundariesPolished surfacePolymer coated metal surface