List of top Questions asked in GATE Engineering Sciences

In a steady two-dimensional {compressible} flow, $u$ and $v$ are the $x$- and $y$-components of velocity, and $\rho$ is the fluid density. Among the following pairs of relations, which one(s) perfectly satisfies/satisfy the definition of stream function, $\psi$, for this flow?

Which of the following statements are true? (i) Conservation of mass for an unsteady incompressible flow can be represented as $\nabla\! . \vec V = 0$, where $\vec V$ denotes velocity vector.
(ii) Circulation is defined as the line integral of {vorticity} about a closed curve.
(iii) For some fluids, shear stress can be a nonlinear function of the shear strain rate.
(iv) Integration of the Bernoulli’s equation along a streamline under steady-state leads to the Euler’s equation.

For a two-dimensional flow field given as $\vec V = -x\,\hat{\imath} + y\,\hat{\jmath}$, a streamline passes through points $(2,1)$ and $(5,p)$. The value of $p$ is

A stationary object is fully submerged in a static fluid, as shown in the figure. Here, CG and CB stand for the center of gravity and the center of buoyancy, respectively. Which one(s) among the following statements is/are true?

Consider steady fully-developed incompressible flow of a Newtonian fluid between two infinite parallel flat plates. The plates move in the opposite directions, as shown in the figure. In the absence of body force and pressure gradient, the ratio of shear stress at the top surface ($y=H$) to that at the bottom surface ($y=0$) is

A two-dimensional incompressible flow field is defined as \[ \vec V(x,y)=(Axy)\,\hat{\imath}+(By^2)\,\hat{\jmath}, \] where $A$ and $B$ are constants. The dynamic viscosity is $\mu$. In the absence of body force, which expression represents the {pressure gradient} at the location $(5,0)$ in the concerned flow field?

Let $u(x,t)$ be the solution of the initial–boundary value problem \[ \frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}=0, x\in(0,2),\ t>0, \] \[ u(x,0)=\sin(\pi x), x\in(0,2),\qquad u(0,t)=u(2,t)=0. \] Then the value of $e^{\pi^2}\!\left(u\!\left(\tfrac{1}{2},1\right)-u\!\left(\tfrac{3}{2},1\right)\right)$ is ________________ (in integer).

Match the following measuring instruments with the appropriate figures: I – Pitot probe II – Pitot-static probe III – Piezometer

Figures: (P), (Q), (R)

Among the following non-dimensional numbers, which one characterizes periodicity present in a transient flow?

For an incompressible boundary layer flow over a flat plate (figure shown), the momentum thickness is expressed as

Among the shear stress versus shear strain rate curves shown in the figure, which one corresponds to a shear thinning fluid?

Consider steady incompressible flow over a flat plate, where the dashed line represents the edge of the boundary layer, as shown in the figure. Which one among the following statements is true?

An inviscid steady incompressible flow is formed by combining a uniform flow with velocity $U_\infty$ and a clockwise vortex of strength $K$ at the origin, as shown in the figure. Velocity potential ($\phi$) for the combined flow in polar coordinates $(r,\theta)$ is

Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$?

If the polynomial \[ P(x)=a_0+a_1x+a_2\,x(x-1)+a_3\,x(x-1)(x-2) \] interpolates the points $(0,2)$, $(1,3)$, $(2,2)$, and $(3,5)$, then the value of $P\!\left(\tfrac{5}{2}\right)$ is ____________ (round off to 2 decimal places).

The value of $m$ for which the vector field \[ \vec F(x,y)=\big(4x^{m}y^{2}-2xy^{m}\big)\,\mathbf{i}+\big(2x^{4}y-3x^{2}y^{2}\big)\,\mathbf{j} \] is a conservative vector field, is ____________________ (in integer).

Let \[ P=\begin{bmatrix}4 & -2 & 2\\ 6 & -3 & 4\\ 3 & -2 & 3\end{bmatrix}, \qquad Q=\begin{bmatrix}3 & -2 & 2\\ 4 & -4 & 6\\ 2 & -3 & 5\end{bmatrix}. \] The eigenvalues of both $P$ and $Q$ are $1,1,2$. Which one of the following statements is TRUE?

The surface area of the portion of the paraboloid \[ z=x^2+y^2 \] that lies between the planes $z=0$ and $z=\tfrac{1}{4}$ is

The probability of a person telling the truth is $\dfrac{4}{6}$. An unbiased die is thrown by the same person twice and the person reports that the numbers appeared in both the throws are the same. Then the probability that actually the numbers appeared in both the throws are same is ______________ (rounded off to 2 decimal places).

The information contained in DNA is used to synthesize proteins necessary for life. DNA is composed of four nucleotides: Adenine (A), Thymine (T), Cytosine (C), and Guanine (G). Coding regions—where nucleotide triplets code for amino acids—constitute only about 2% of human DNA. Based only on the information above, which of the following statements can be inferred with {certainty}? (i) The majority of human DNA has no role in the synthesis of proteins. (ii) The function of about 98% of human DNA is not understood.

Which one of the given figures P, Q, R and S represents the graph of the following function? \[ f(x)=\left|\,|x+2|-|x-1|\,\right| \]

An opaque cylinder (shown below) is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The cylinder can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is {NOT} possible?

Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,3$. If \[ B = A^2 + 2A + I, \] where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ are:

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by \[ f(x,y)= \begin{cases} \dfrac{xy}{|x|+y}, & y \neq -|x|, \\[6pt] 0, & \text{otherwise}. \end{cases} \] Then which one of the following statements is TRUE?

If the quadrature formula \[ \int_{-1}^{1} f(x)\,dx \;\approx\; \frac{1}{9}\Big(c_1 f(-1) + c_2 f\!\left(\tfrac{1}{2}\right) + c_3 f(1)\Big) \] is exact for all polynomials of degree less than or equal to $2$, then