List of top Questions asked in GATE XE- 2022

Consider an inviscid flow through a smooth pipe which has a pitot-static tube arrangement as shown. Find the centre-line velocity in the pipe. Consider that the density of the fluid is 1000 kg/m\(^3\), acceleration due to gravity is 10 m/s\(^2\), and the specific gravity of the manometric fluid is 11.
The speed of propagation, \(c\), of a capillary wave depends on the density of the fluid, \(\rho\), the wavelength of the wave, \(\lambda\), and the surface tension, \(\sigma\). The dimensions are: \(\rho = ML^{-3}\), \(\lambda = L\), and \(\sigma = MT^{-2}\). Which expression is dimensionally correct?
A two-dimensional flow field is described by a combination of a source of strength \( m \) at the origin and a uniform flow \( U \) in the positive x-direction such that the velocity potential is given by \[ \phi = Ux + \frac{m}{2\pi} \ln \sqrt{x^2 + y^2} \] The stagnation streamline is shown in the figure. Find the distance \( aa' \).
A typical boundary layer over a flat plate has a linear velocity profile with zero velocity at the wall and freestream velocity \(U_{\infty}\) at the outer edge of the boundary layer. What is the ratio of the momentum thickness to the thickness of the boundary layer?
Identify the configuration(s) in which steady two-dimensional internal flow may show boundary layer separation if the flow direction is left to right.
Consider steady fully developed flow of a liquid through two large horizontal flat parallel plates separated by a distance of 2 mm. One of the plates is fixed and the other plate moves at a speed of 0.5 m/s. What is the magnitude of the pressure gradient (in Pa/m) in the direction of the flow required to ensure that the net flow through the plates is zero? Dynamic viscosity of the liquid is $5 \times 10^{-4}$ Ns/m$^2$ (Round off to the nearest integer)
The left-hand column lists some non-dimensional numbers and the right-hand column lists some physical phenomena. Indicate the correct combination.
As temperature increases:
Which of the following statement(s) regarding a venturimeter is/are correct?
Which of the following statement(s) is/are true for streamlines in a steady incompressible flow?
A flow has a velocity potential given by \( \phi = Ax^3 \) where ‘A’ is a non-zero constant. Which of the following statement(s) is/are true about the flow?
A boundary layer develops due to a two-dimensional steady flow over a horizontal flat plate. Consider a vertical line away from the leading edge which extends from the wall to the edge of the boundary layer. Which of the following quantity/quantities is/are not constant along the vertical line? \( u \) and \( v \) represent the components of velocity in the direction along the plate and normal to it, respectively, and \( x \) is taken along the length of the plate while \( p \) is the pressure.
Let \[ A = \begin{pmatrix} 2 & 0 & 1 \\ 1 & 2 & 5 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix} \] . Then the sum of the geometric multiplicities of the distinct eigenvalues of \( A \) is equal to ________.
In a cosmopolitan city, the population comprises of 30% female and 70% male. Suppose that 5% of females and 30% of males in the population are foreigners. A person is selected at random from this population. Given that the selected person is a foreigner, the probability that the person is a female is ________ (round off to three decimal places).

Let \( f : (0, \infty) \to \mathbb{R} \) be the continuous function such that:

\[ f(x) = 2 + \frac{g(x)}{x} \quad \text{for all} \ x > 0, \quad g(x) = \int_1^x f(t) \, dt \quad \text{for all} \ x > 0. \]

Then \( f(2) \) is equal to:

Let \( A \) and \( B \) be \( n \times n \) matrices with real entries. Consider the following statements:

  • P: If \( A \) is symmetric, then rank(\( A \)) = Number of nonzero eigenvalues (counting multiplicity) of \( A \)
  • Q: If \( AB = 0 \), then rank(\( A \)) + rank(\( B \)) \( \leq n \)

Then:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x, y) = 4xy - 2x^2 - y^4 + 1. \] The number of critical points where \( f \) has local maximum is equal to ________.
If the quadrature rule \[ \int_{-1}^1 f(x) \, dx \approx f(\alpha) + \gamma f(\beta), \] where \( \alpha, \beta \) and \( \gamma \) are real constants, is exact for all polynomials of degree \( \leq 3 \), then \( \gamma + 3 (\alpha^2 + \beta^2) + (\alpha^3 + \beta^3) \) is equal to ________.
A heavy horizontal cylinder of diameter D supports a mass of liquid having density ρ as shown in the figure. Find out the vertical component of force exerted by the liquid per unit length of the cylinder if g is the acceleration due to gravity.
The figure shows the developing zone and the fully developed region in a pipe flow where the steady flow takes place from left to right. The wall shear stress in the sections A, B, C, and D are given by \( \tau_A, \tau_B, \tau_C, \) and \( \tau_D \), respectively. Select the correct statement.
The value of \[ \lim_{x \to 0} \frac{1}{x} \int_{2}^{2+x} \left(t + \sqrt{t^2 + 5}\right) dt \]
Let \( \mathbb{C} = \{ z = x + iy : x \text{ and } y \text{ are real numbers}, i = \sqrt{-1} \} \) be the set of complex numbers. Let the function \( f(z) = u(x, y) + i v(x, y) \) for \( z = x + iy \in \mathbb{C} \) be analytic in \( \mathbb{C} \), where \[ u(x, y) = x^3 y - y x^3 \quad \text{and} \quad v(x, y) = \frac{x^4}{4} + \frac{y^4}{4} - \frac{3}{2} x^2 y^2. \] If \( f'(z) \) denotes the derivative of \( f(z) \), then:
If the partial differential equation \[ (x + 2) \frac{\partial^2 u}{\partial x^2} + 2(x + y) \frac{\partial^2 u}{\partial x \partial y} + 2(y - 1) \frac{\partial^2 u}{\partial y^2} - 3y^2 \frac{\partial u}{\partial y} = 0 \] is parabolic on the circle \( (x - a)^2 + (y - b)^2 = r^2 \), then the values of \( a \), \( b \), and \( r \) are given by:
Let \( \Gamma \) be the positively oriented circle \( x^2 + y^2 = 9 \) in the xy-plane. If \[ \oint_{\Gamma} (3y + e^x \sin x) \, dx + \left( 7x + \sqrt{e^y + 2} \right) \, dy = \alpha \pi, \] where \( \alpha \) is a real constant, then \( \alpha \) is equal to ________.
Let \( y_1(x) \) and \( y_2(x) \) be two linearly independent solutions of the differential equation: \[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0, \quad x>0. \] Let \( W(y_1, y_2)(x) \) denote the Wronskian of \( y_1(x) \) and \( y_2(x) \) at \( x \). If \( W(y_1, y_2)(1) = 1 \), then \( W(y_1, y_2)(2) \) is equal to ________.