GATE Engineering Sciences

Suresh said, "I did it yesterday." Which one of the following options is the correct form of this sentence in indirect speech?

To continue the sequence of tiles shown, the tile indicated by the question mark should be

Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the w...

The 2nd of June is a Thursday in a certain year. Which day of the week is the 3rd of July in that year?

A coin with heads facing up is shown as H and a coin with tails facing up is shown as T . Six coins are placed in the Starting Arrangement, as shown in the figure below. A "step" i...

Exacerbate : Mitigate :: _________ : _________ Choose the option with the correct pair of words to fill the blank.

A paper shown in Panel I is folded along the dashed lines (---) to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube. Referring to cubes...

In a population, patients who have high cholesterol also have high blood-pressure (BP). Some patients with high BP also have diabetes. There are no patients who have both high chol...

Four people P, Q, R, and S, of different ages, make the following observations. P - I am younger than S. Q - I am neither the youngest nor the oldest. R - P is older than me. Based...

Circles C1, C2, and C3, with centers O1, O2, and O3, and radii r1, r2, and r3, respectively, touch each other as shown in the following figure. Given r1 = 2 cm, r2 = 1 cm and the a...

For the complex function f(z) = f(x + iy) = x³ – 4xy² + i(4x²y – 2y³), consider the following two statements P: f is not analytic at (0, 0). Q: f does not satisfy the Cauchy-Rieman...

Let C be the circle (x – 1)² + y² = 25 oriented counterclockwise. Then, the value of the line integral $\oint_C [(x^5 – 3y)dx + (−2x + e^{y^2})dy]$ is

Let W(x) denote the Wronskian of two linearly independent solutions y₁(x) and y₂(x) of the differential equation $(x-1)\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + xe^xy = 0, \quad x > 1$....

Which of the following statements is/are true?

Consider the matrix A = $\begin{bmatrix} 1 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Then, which of the following statements is/are true?

Let L be the lamina of the form $x^2 + 4y^2 \le 64$, $0 \le y \le 4$, with density $\rho(x, y) = |x|y$. Then, the mass of L (in integer) is _________

Let X be a random variable having probability density function (pdf) $f_X(x) = \begin{cases} \frac{e}{2(e-1)}e^{-x}, & 0 < x < 1, \\ \frac{1}{2}, & 1 < x < 2, \\ 0, & \text{elsewhe...

Match each entry of List-1 with a suitable entry in List-2 and choose the correct option. List-1 P The sum of the series $\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}$ is equal to Q $\...

Consider the following partial differential equation (PDE) $(y - 1)\frac{\partial^2 u}{\partial x^2} - (x - 3)^2\frac{\partial^2 u}{\partial y^2} + y^2\frac{\partial u}{\partial x}...

Let $\begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & -10 \end{bmatrix} \begin{bmatrix} 1 & u_{12} & u_{13} \\ 0 & 1 & u_{23} \\ 0 & 0 & 1 \end{bmatrix}$...

The initial value problem $\frac{du}{dt} = u^2 + t^2$, $t \ge 0$, with $u(0) = 1$, is solved by using the explicit Euler method with step size $h = 0.2$. Then, the value of $u(0.4)...

For a laminar, incompressible, and fully-developed flow through a circular pipe, the ratio of the maximum velocity to the average velocity of the flow is

Air flows with a freestream velocity U over four different bodies having same frontal area facing the flow direction, as shown in the figure. Which one of the following bodies has...

Consider a steady, and incompressible flow over a body with characteristic length L. The boundary layer thickness at a distance x from the leading edge is δ. Which one of the follo...

Three different siphons steadily discharge water at velocities V₁, V₁₁, and V₁₁₁, as shown in the figure. The tubes of the siphons are of same diameter. If the frictional losses ar...

Consider the following statements: Assertion (a) Surface tension acts along the interface of two fluids. Reason (r) The pressure of the fluid inside a bubble is higher than that of...

The basic dimensions, i.e., mass, length, and time are represented by M, L, and T, respectively. The correct dimension of dynamic viscosity is

The velocity components in x- and y-directions of a two-dimensional, incompressible flow field are $u(x, y) = 2x^2 + y^3$ and $v(x, y) = x^3 – 2xy + f(x, y)$, respectively. Here, $...

Which of the following statements about streamlines, pathlines, and streaklines is/are correct?

A piezometer and a Pitot tube are tapped into a horizontal water pipe, as shown in the figure, where h₁ = 4 cm, h₂ = 6 cm and h₃ = 5 cm. Consider the flow to be steady, laminar, an...

A steady, laminar, and incompressible flow between a pair of infinite parallel plates is driven by a constant pressure gradient ($-dp/dx$). The plates are separated by a distance $...

An incompressible fluid flows between a pair of infinite plates separated by a distance L. The top plate is moving with a constant velocity U, whereas the bottom plate is stationar...

Consider two different cases of water flowing through a smooth pipe of 50 cm diameter. The mass flow rates for the two cases are (i) 0.25 kg.s⁻¹, and (ii) 0.8 kg.s⁻¹. Assume the de...

A two-dimensional source flow (with stream function, $\psi_1 = m \tan^{-1}\frac{y}{x}$) is placed at the origin in a uniform flow (with stream function, $\psi_2 = Uy$). Here, the s...

Consider a steady, laminar, and incompressible flow over a flat plate, as shown in the figure. With freestream velocity U∞ and kinematic viscosity v1, the boundary layer thickness...

A vertical jet of diameter $d_1$ strikes a horizontal plate with a velocity $U$, as shown in the figure. The plate has a hole of diameter $d_2(<d_1)$ concentric to the flow through...

The velocity of a fluid particle in a flow is given as: V = (a – x)î + (b + y)ĵ + (c + z)k where a, b, c are constants, and î, ĵ, k are unit vectors in x-, y-, z-directions, respec...

A gas is pressurized in a vertical frictionless piston-cylinder device, as shown in the figure. The piston has a mass of 4 kg and a cross-sectional area of 40 cm². A metallic block...

A ship is designed to sail at a speed of 8 m.s⁻¹. A designer makes a 1/10 scaled model to test the ship in a water tunnel. The model and the ship satisfy the dynamic similarity. Th...

A rectangular block (density = 600 kg.m⁻³) with base area of 0.06 m² and height 15 cm is partially submerged in water (density = 1000 kg.m⁻³), as shown in the figure. Assume accele...

For a steady, laminar, and incompressible flow over a flat plate, the local skin friction coefficient is given as Cf = $\frac{0.664}{\sqrt{Re_x}}$, where $Re_x$ is the local Reynol...

The axial velocity profile of a laminar, incompressible, and fully-developed flow in a circular pipe of radius R is given as $v_z = \frac{1}{4\mu} \frac{dp}{dz} (r^2 - R^2)$, where...

Air flows through a pipe of diameter D with an average velocity of 3 m.s⁻¹. The Darcy friction factor of the pipe is 0.02. Assume acceleration due to gravity as 10 m.s⁻². If the he...

The band gap of a material is $E_g$ and the energy of an incident photon is $E_p$. Optical absorption will occur in this material if,

Discoloration and loss of mechanical strength of plastic chairs exposed to sunlight for several days is due to

Which one of the following polymer processing techniques involves shaping a heated sheet?

The plot shows three different paths (1, 2, and 3) connecting the initial equilibrium state X to the final equilibrium state Y in a thermodynamic process. Which of the following st...

Which of the following is/are polarization mechanisms in dielectric solids?

Which of these methods is/are used for the synthesis of ceramic powders?

The degrees of freedom of a system with 2 components and 2 phases in equilibrium is ________ (answer in integer).

The interplanar spacing of $(h k l)$ planes is 4.5 nm in a given crystal. For the $(3h 3k 3l)$ planes, the interplanar spacing, in nm, is ________ (rounded off to one decimal place...

The equilibrium vacancy concentration in aluminum at 900 K is 1.1 x 10⁻⁴. The enthalpy of formation of vacancies, in kJ.mol⁻¹, is ________ (rounded off to one decimal place). Given...

Three phases, α, β and L are shown in the binary phase diagram below. Identify the correct G-X (Gibbs free energy vs Composition) plot corresponding to temperature T₀.

In a cubic crystal, the Burgers vector for a mixed dislocation line is $\frac{a}{2}[110]$. The dislocation line lies along the [011] direction. The slip plane of the dislocation is

Match the Thermodynamic laws in Column I with the description given in Column II Column I P Zeroth law Q First law R Second law S Third law Column II 1 Entropy of a perfect crystal...

Match the material property in Column I with the measurement technique in Column II. Column I P Electrical conductivity Q Band gap energy R Young's modulus S Heat of fusion Column...

Which one of the following statements regarding point defects in ionic solids is correct?

Consider 3 crystals (all having FCC lattice) labelled as P, Q and R as described below: Crystal P: Cu crystal with 4 Cu atoms per unit cell Crystal Q: NaCl crystal with 4 Na+ ions...

Which of the following statements is/are applicable to Fick's second law:

According to quantum free electron theory, which of the following statements is/are correct regarding the behavior of valence electrons in a metal?

Fe has a density of 7.87 g.cm⁻³, atomic mass of 55.84 g.mol⁻¹ and net magnetic moment per atom of 2.22 Bohr magnetons (μB). The saturation magnetization of Fe, in A.m⁻¹, is _______...

At the peak (also denoted as UTS) of an engineering stress vs. engineering strain curve for ductile metal, the engineering strain is 0.2. The corresponding true stress (σ) vs. true...

A 1 cm³ Si cube is doped with As with a concentration of 1 atom per 10⁹ Si atoms. The resistance of the doped Si cube, in Ohms, is ________ (rounded off to one decimal place). Give...

A circular disk-shaped ceramic green body has a relative density of 50%. On sintering, both the height and the diameter of the disk shrink by 20%. The sintered disk's relative dens...

The standard electrode potential of Cu²⁺ | Cu is +0.34 V and Zn²⁺ | Zn is -0.76 V. For the Zn | Zn²⁺ || Cu²⁺ | Cu cell, the EMF, in Volts, is ________ (rounded off to two decimal p...

Two axial members, namely OP and OQ, are pin joined at O as shown in the figure. A force F acts at point P along the positive x direction and a force √3F acts at point Q along the...

Two smooth drums each weighing W and radius, r are connected by a stiff rope of length, h as shown in the figure. Force F is applied using a massless lever (RS) of length, l. The f...

Identify the zero-force member in the truss structure as shown in the figure.

A rigid block of mass m on a horizontal surface is connected to three springs each having spring constant k as shown in the figure. Which one of the following is the CORRECT natura...

Which among the following options is/are CORRECT unit(s) of stress?

A car is moving on a horizontal surface in a straight line with a constant velocity of 3 m/s. A ball is thrown vertically upwards at time t = 0 from the top of the moving car with...

A cart of weight (W) 5 kN, stands on an inclined smooth surface at an angle (θ) of 45° as shown in the figure. The cart is maintained in equilibrium by a horizontal force F acting...

A 2D state of stress at a point in a body is given by \sigma_{xx} = -40 MPa, \sigma_{yy} = 100 MPa and \tau_{xy} = 50 MPa. The radius of the Mohr's circle for the given state of st...

Sum of the principal stresses of a 2D stress state $\begin{bmatrix} 11 & 4 \\ 4 & 5 \end{bmatrix}$ is ________ (in integer).

The force (F) acting on the block, as shown in the figure is 2 N. The stiffness of the spring (k) is 100 N/m. Assume that the block is in static equilibrium and its mass is negligi...

Which one of the following options is the CORRECT absolute value of the bending moment at R in the frame as shown in the figure?

One column with square cross section of side r and another column with rectangular cross section of breadth p and width q (< p) are made from the same material. Both the columns ha...

Two prismatic rods of identical lengths are designed for the same strain energy density when subjected to the same axial load. One of the rods is made of steel (Young's modulus = 2...

A spring of stiffness k is connected to the center of a homogeneous right circular cylinder of radius r placed on a horizontal surface as shown in the figure. The cylinder is assum...

A 2D stress state of pure shear is shown in the figure. Which one of the following options is equivalent to the given stress state? Figures are not to scale.