Calculus

Subtopic: Multiple Integrals & Vector Calculus

Consider the contour C shown in the figure below. For the vector F⃗ = (x + 2y)êx + (2x + 4y)êy, the integral ∮ F⃗ ⋅ dl⃗ C =________. Here dl⃗ represents an infinitesimal length along the contour C.

Subtopic: Limits, Continuity & Differentiability

f(x) = {A + x , if x < 2 1 + x^2 , if x ≥ 2 If the function f(x) is continuous at x = 2, the value of A is _____.

Subtopic: Limits, Continuity & Differentiability

Two fair dice with numbered faces are rolled together. The faces are numbered from 1 to 6. The probability of getting odd numbers on both the dice is ________________ (rounded off to 2 decimal places).

Subtopic: Multiple Integrals & Vector Calculus

The volume of the solid formed by a complete rotation of the shaded portion of the circle of radius R about the y-axis is k$\pi$R^3. The value of k is:

Subtopic: Limits, Continuity & Differentiability

Consider the function f(x) = {x2 for x < 0 x for x ≥ 0 where x is real. Which of the following statements is/are correct?

Subtopic: Limits, Continuity & Differentiability

The direction in which a scalar field ( , , )x y z has the largest rate of change at any point with position vector ˆˆˆr xi yj zk= + + is the same as that of the vector

Subtopic: Limits, Continuity & Differentiability

Consider the one-dimensional wave equation 0uu tx  += for x-    , 0t  . For an initial condition 2 ( ,0) xu x e -= , the solution at 1t = is

Subtopic: Limits, Continuity & Differentiability

Consider the International Standard Atmosphere (ISA) with h being the geopotential altitude (in km) and dT dh being the temperature gradient (in K/m). Which of the following combination(s) of , dTh dh   is/are as per ISA?

Subtopic: Multiple Integrals & Vector Calculus

$\vec{v} = x^3 \hat{i} + y^3 \hat{j} + z^3 \hat{k}$ is a vector field where $\hat{i}, \hat{j}, \hat{k}$ are the base vectors of a Cartesian coordinate system. Using the Gauss divergence theorem, the value of the outward flux of the vector field over the surface of a sphere of unit radius centered at the origin is ______ (rounded off to one decimal place).

Subtopic: Multiple Integrals & Vector Calculus

Consider the 2D velocity field $\vec{V} = (2x + 3y)\hat{i} + (3x + 2y)\hat{j}$. The field $\vec{V}$ is:

Subtopic: Limits, Continuity & Differentiability

For the function $f(x) = |x|$ defined on $[-1, 1]$, with $f'$ denoting the derivative, Rolle's / Lagrange's Mean Value Theorem is not applicable because:

Subtopic: Limits, Continuity & Differentiability

§ (4i - 37) Q.4 The dimensions of kinematic viscosity of a fluid (where L is length, T is time) are

Subtopic: Limits, Continuity & Differentiability

dy de + 6y = f(t); y y(0) = 2,1 = 1 dt t=0 If Y(s) = Jo° y (t)e-st dt and F(s) = Jo° f(t)e-st dt are the Laplace transforms of y(t) and f(t) respectively, then Y (s) is given by: F(s)

Subtopic: Limits, Continuity & Differentiability

The Laplace transform L(u(t))=U(s), for the solution u(t) of the problem 0 , 1 22 2 > = + +t udt du dt u d with initial conditions . 5) 0 (, 0 ) 0 ( = =dt duu is given by:

Subtopic: Limits, Continuity & Differentiability

Two overflowing water reservoirs are connected with a100m long pipe of circular cross-section (of radius, R = 0.02m), such that height difference h remains constant as shown in the figure below. The centerline velocity in the pipe is 10m/s. The velocity profile inside the pipe over the entire length is uu = - RR2 4$\mu \mu$ dddd dd xx �1 - $\pi \pi$2 RR2�, where, dddd dd xx is a constant pressure gradient along the pipe length, x is measured from theleft end of the pipe along its central axis and r is radial location inside the pipe with respect to its axis.(Given data: Density and kinematic viscosity of water are 1000kg/m3 and 1x10-6m2/srespectively; acceleration due to gravity is 10m/s^2). If all other losses except the frictional losses at the pipe wall are neglected, the value of h in meter is ____. AE 7/11 AE (GATE 2014) - AE

Subtopic: Limits, Continuity & Differentiability

In a factory, two machines M1 and M2 manufacture 60% and 40% of the autocomponents respectively. Out of the total production, 2% of M1 and 3% of M2 are found to be defective. If a randomly drawn autocomponent from the combined lo t is found defective, what is the probability that it was manufactured by M^2?

Subtopic: Limits, Continuity & Differentiability

Two policemen, A and B, fire once each at the same time at an escaping convict. The probability that A hits the convict is three times the probability that B hits the convict. If the probability of the convict not getting injured is 0.5, the probability that B hits the convict is:

Subtopic: Limits, Continuity & Differentiability

An aircraft with a mass of 5000 kg takes off from sea level with a forward speed of 50 m/s and reaches a climb gradient of 10%. The excess thrust required is approximately:

Subtopic: Limits, Continuity & Differentiability

Consider a steady two dimensional zero-pressure gradient laminar flow of air over a flat plate. The boundary layer thickness delta scales with distance x from leading edge as:

Subtopic: Limits, Continuity & Differentiability

For a critically damped single degree of freedom spring - mass - damper system with a damping constant c of 4 Ns/m and spring constant k of 16 N/m, the system mass m is

Subtopic: Limits, Continuity & Differentiability

A turbulent boundary layer remains attached over a longer distance on the upper surface of an airfoil than does a laminar boundary layer, because

Subtopic: Multiple Integrals & Vector Calculus

At a stationary point of a multi-variable function, which of the following is true?