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Test: Solution Integrals - Engineering Mathematics MCQ


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5 Questions MCQ Test Engineering Mathematics - Test: Solution Integrals

Test: Solution Integrals for Engineering Mathematics 2025 is part of Engineering Mathematics preparation. The Test: Solution Integrals questions and answers have been prepared according to the Engineering Mathematics exam syllabus.The Test: Solution Integrals MCQs are made for Engineering Mathematics 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Solution Integrals below.
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Test: Solution Integrals - Question 1

The number of integral solutions of  is

Detailed Solution for Test: Solution Integrals - Question 1


4x + 8 ≥ x2 + 8

∴ x2 – 4x ≤ 0

x(x – 4) ≤ 0 → (1)

Clearly the integral solution of (1) are 0, 1, 2, 3 and 4

∴ Total 5 values of x satisfies (1)

Test: Solution Integrals - Question 2

 where c is the upper half of the circle |z| = 1.

Detailed Solution for Test: Solution Integrals - Question 2

Given counter c is the circle, |z| = 1

⇒  z = e ⇒ dz = ie

Now, for upper half of the circle, 0 ≤ θ ≤ π

Test: Solution Integrals - Question 3

Evaluate the line integral ⁡(x + 4iy2)dz where c is the line x = 2y and x varies from 0 to 1 and z = x + iy

Detailed Solution for Test: Solution Integrals - Question 3

Calculation:

z = x + iy ⇒ dz = dx + i dy

given line is x = 2y ⇒ dy/dx = 1/2

lets substitute y in terms of x

I = ∫ x + i x2 (dx + i/2 dx) 

I = ∫ x dx + i/2 x dx + i x2 dx - x2/2 dx

Test: Solution Integrals - Question 4

Evaluate along the straight line joining the points (0, 0) and (3, 1)

Detailed Solution for Test: Solution Integrals - Question 4

Concept:

Integral of a complex function f(z) is given

∫ f(z) dz = ∫ (udx -vdy) + i ∫ (vdx + udy)

Noting f(z) = u(x, y) + i v(x, y) and dz = dx + i dy;

Calculation:

Given Along the straight line joining the points (0, 0) and (3, 1);

The equation of straight line will be x = 3y

⇒ dx = 3 dy ⇒ dz = (3 + i) dy;

Along the line x = 3y, the complex number z will be

z = x + iy = 3y + iy = (3 + i) y

Substituting both in the integral,

Test: Solution Integrals - Question 5

The value of where contour D is |z| = 2

Detailed Solution for Test: Solution Integrals - Question 5

 has its poles at z = -1

and contour |z| = 2 is a circle of radius 2, centre (0, 0)

So

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