Category: Engineering Mathematics Objective Questions

Fourier Analysis and Partial Differential Equations Multiple Choice Questions on “First Order Non-Linear PDE”.

1. Which of the following is an example of non-linear differential equation?
a) y=mx+c
b) x+x’=0
c) x+x²=0
d) x”+2x=0
View Answer

Answer: c
Explanation: For a differential equation to be linear the dependent variable should be of first degree. Since in equation x+x²=0, x² is not a first power, it is not an example of linear differential equation.

2. Which of the following is not a standard method for finding the solutions for differential equations?
a) Variable Separable
b) Homogenous Equation
c) Orthogonal Method
d) Bernoulli’s Equation
View Answer

3. Solution of a differential equation is any function which satisfies the equation.
a) True
b) False
View Answer

Answer: a
Explanation: A solution of a differential equation is any function which satisfies the equation, i.e., reduces it to an identity. A solution is also known as integral or primitive.

4. A solution which does not contain any arbitrary constants is called a general solution.
a) True
b) False
View Answer

Answer: a
Explanation: The solution of a partial differential equation obtained by eliminating the arbitrary constants is called a general solution.

5. Which of the following is a type of Iterative method of solving non-linear equations?
a) Graphical method
b) Interpolation method
c) Trial and Error methods
d) Direct Analytical methods
View Answer

Answer: b
Explanation: There are 2 types of Iterative methods, (i) Interpolation methods (or Bracketing methods) and (ii) Extrapolation methods (or Open-end methods).

6. A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
a) True
b) False
View Answer

Answer: a
Explanation: A solution which does not contain any arbitrary constants is called a general solution whereas a particular solution is derived by substituting particular values to the arbitrary constants in this solution.

7. Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.
a) True
b) False
View Answer

Answer: a
Explanation: A differential equation is said to have a singular solution if in all points in the domain of the equation the uniqueness of the solution is violated. Hence, this solution cannot be obtained from the general solution.

8. Which of the following equations represents Clairaut’s partial differential equation?
a) z=px+f(p,q)
b) z=f(p,q)
c) z=p+q+f(p,q)
d) z=px+qy+f(p,q)
View Answer

Answer: d
Explanation: Equations of the form, z=px+qy+f(p,q) are known as Clairaut’s partial differential equations, named after the Swiss mathematician, A. C. Clairaut (1713-1765).

9. Which of the following represents Lagrange’s linear equation?
a) P+Q=R
b) Pp+Qq=R
c) p+q=R
d) Pp+Qq=P+Q
View Answer

Answer: b
Explanation: Equations of the form, Pp+Qq=R are known as Lagrange’s linear equations, named after Franco-Italian mathematician, Joseph-Louis Lagrange (1736-1813).

10. A partial differential equation is one in which a dependent variable (say ‘x’) depends on an independent variable (say ’y’).
a) False
b) True
View Answer

Answer: a
Explanation: An ordinary differential equation is divided into two types, ordinary and partial differential equations.
A partial differential equation is one in which a dependent variable depends on one or more independent variables.
Example: (F(x,t,y,frac{∂y}{∂x},frac{∂y}{∂t},……)= 0. )

11. What is the complete solution of the equation, (q= e^frac{-p}{α})?
a) (z=ae^frac{-a}{α}y)
b) (z=x+e^frac{-a}{α}y)
c) (z=ax+e^frac{-a}{α} y+c)
d) (z=e^frac{-a}{α}y)
View Answer

Answer: c
Explanation: Given: (q= e^frac{-p}{α})
The given equation does not contain x, y and z explicitly.
Setting p = a and q = b in the equation, we get (b= e^frac{-a}{α}.)
Hence, a complete solution of the given equation is,
(z=ax+by+c,,with , b= e^frac{-a}{α})
(z=ax+e^frac{-a}{α} y+c.)

12. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False
View Answer

Answer: b
Explanation: A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution thereby eliminating any arbitrary constants present in the solution. Such solution represents a particular member of the family of surfaces given by the complete solution.

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Complex Analysis Multiple Choice Questions on “Regions in the Complex Plane”.

1. What is the shape of the region formed by the set of complex numbers z satisfying |z-ω|≤ α?
a) circle of radius ω
b) circle with center ω
c) disk of radius α
d) disk with center α
View Answer

Answer: d
Explanation: The equation |z-ω|≤ α implies that the distance of z from ω is less than or equal to α. This means that a disk is formed with center ω and radius α.

2. The complex number given by [(√3/2)+i/2]⁵+[(√3/2)-i/2]⁵ lies, on which of the following regions?
a) imaginary axis
b) real axis
c) first quadrant
d) fourth quadrant
View Answer

Answer: b
Explanation: Let z=[(√3/2)+i/2]⁵+[(√3/2)-i/2]⁵ and ω=[(√3/2)+i/2]⁵ ⇒ (overline{omega})=[(√3/2)-i/2]⁵
⇒ z=ω+(overline{omega}) ⇒ z is real (sum of conjugates is real) ⇒ z lies on real axis.

3. Find the area of the region given by 11≤|z| ≤ 19.
a) 120π sq. units
b) 180π sq. units
c) 240π sq. units
d) 320π sq. units
View Answer

Answer: c
Explanation: The region formed is an annulus of inner radius 11 units and outer radius 19 units. Therefore, the required area=π(19²–11²)=240π.

4. Find the largest angle of the triangle formed by thevertices z₁=8(1-i), z₂=8(i-1) and
Z₃=10+2√7i.
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a) π/3 radians
b) 2π/3 radians
c) π/2 radians
d) 3π/4 radians
View Answer

Answer: c
Explanation: Note that z₁ and z₂ are the opposite ends of a diameter of a circle of radius 8√2 units, centered at the origin. Also note that z₃ lies on this circle (distance of z₃ from origin = 8√2). Hence, angle corresponding to z₃=π/2 radians.

5. Find the equation of the circle passing through the origin and having intercepts a and b on real and imaginary axes, respectively, on the arg and plane.
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a) zz̅=a(Im z)–b(Re z)
b) zz̅=a(Im z)+b(Re z)
c) zz̅=a(Re z)–b(Im z)
d) zz̅=a(Re z)+b(Im z)
View Answer

Answer: d
Explanation: Consider a point z on the circle. Therefore, arg [(z-a)/(z-ib)]=±π/2
⇒ (z-a)/(z-ib)+(z̅-a)/(z̅+ib)=0 ⇒ zz̅-a(z+z̅)/2–b(z-z̅)/2i=0
⇒ zz̅=a(Re z)+b(Im z).

6. On the arg and plane, the complex numbers z₁, z₂, z₃, z₄ are the vertices of a parallelogram. Evaluate (z₄–z₁+z₂)/z₃ .
a) 1
b) 2
c) 3
d) 4
View Answer

Answer: a
Explanation: For a parallelogram, the diagonals bisect each other. Hence, their midpoints coincide.
This implies that (z₁+z₃)/2=(z₂+z₄)/2 ⇒ z₁+z₃=z₂+ z₄ ⇒ ( z₄– z₁+ z₂)/ z₃=1.

7. Consider the shape formed by the set of points z=ω-1/ω, where |ω|=2. Which of the following is incorrect?
a) eccentricity=4/5
b) |z|≤3
c) shape is an ellipse
d) major axis is of length=5/2
View Answer

Answer: d
Explanation: | ω|=2 ⇒ ω=2(cosθ+isinθ) ⇒ z=x+iy=2(cosθ+isinθ)-1/2(cosθ-isinθ)(⇒ |z|≤3)
=3/2cosθ+i5/2sinθ⇒x²/(3/2)²+y²/(5/2)²=1 ⇒ ellipse ⇒ e²=1-(9/4)/(25/4)=16/25
⇒ e=4/5.

8. Find the area enclosed by the curve formed by iz³+z²–z+i=0.
a) π/2
b) π
c) 3π/4
d) 2π
View Answer

Answer: b
Explanation: Dividing the equation by i on both sides, z³-iz²+iz+1=0
⇒ z²(z-i)+i(z-i)=0 ⇒ (z-i)(z²+i)=0 ⇒ z=i or z²=-i ⇒ |z|=|i|=1 or |z²|=|z|²=|-i|=1
⇒ |z|=1 ⇒ circle of radius 1 is formed. Hence, area=π(1²)=π.

9. Given a vertex of the square circumscribing the circle |z-1|=√2 as 2+√3i, which of the following is not a vertex of this square ?
a) (1-√3)+i
b) –i√3
c) (√3+i)-i
d) i√3
View Answer

Answer: d
Explanation: The given circle has z₀=1 as its center and √2 as radius. Let z₁=2+i√3. Now, obtain z₂ by rotating z₁ anticlockwise by 90⁰ about z₀ ⇒ z₂=(1-√3)+i. Now, z₀ is midpoint of z₁ and z₃ and z₂ and z₄.
؞(z₁+z₃)/2 ⇒ (2+i√3+z₃)/2=1 ⇒ z₃=-i√3 and(z₂+z₄)/2=z₀ ⇒ z₄=(√3+i)-i.

10. Find the area of the region bounded by arg|z|≤π/4 and |z-1|<|z-3|.
a) 1 sq. units
b) 2 sq. units
c) 3 sq. units
d) 4 sq. units
View Answer

Answer: d
Explanation: |z-1|<|z-3| ⇒ (x-1)²+y²<(x-3)²+y² ⇒ x<2.
Therefore, a triangle is formed with base length 4 and height 2 (along x-axis). Hence, the required area=1/2×4×2=4.

11. Find the locus of z/(1-z²), where z lies on the circle of radius 1 centered at origin and z≠±1.
a) line not passing through origin
b) |z|=√2
c) real axis
d) imaginary axis
View Answer

Answer: d
Explanation: Given |z|=1 and z≠±1, write ω=z/(1-z²)=z/(zz̅-z²)=1/(z̅-z).
Hence ω is a purely imaginary number and lies on imaginary axis.

12. Describe the region given by |z-i|z||-|z+i|z||=0.
a) real axis
b) imaginary axis
c) circle centered at origin
d) quadrant 2
View Answer

Answer: a
Explanation: |z/|z|-i|=|z/|z|+i|, z≠0 ⇒ z/|z| is unimodular complex number and lies on the perpendicular bisector of i and –i ⇒ z/|z|=±1 ⇒ z=±|z| ⇒ z is real.

13. The area of the region enclosed by the curve zz̅+a(z̅+z)+a=0 is 2π. If a²–7a+10=0, find the area of the region enclosed by the curve zz̅+2a(z̅+z)+a=0.
a) 4π sq. units
b) 10π sq. units
c) 14π sq. units
d) 22π sq. units
View Answer

Answer: c
Explanation: The curve represents a circle with center–a and radius (a²–a)^1/2. Therefore, Area=π(a²–a)=2π ⇒ a=-1,2. Also, a²–7a+10=0 ⇒ a=2,5. Hence, a=2.
Hence required area=π(4a²–a)=14π.

14. Find the area of the region common to the sets S₁={z∈C: |z|<4}, S₂={z∈C:Im[(z-1+√3i)/(1-√3i)]>0} and S₃={z∈C: Re z>0}.
a) 10π/3
b) 20π/3
c) 16π/3
d) 32π/3
View Answer

Answer: b
Explanation: S₁: |z|₂: √3x+y>0, z lies above the line √3x+y=0; S₃: Re(z)>0, z lies to the right of the imaginary circle. Therefore, required area=π×4²/4+π×4²/6=20π/3.
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15. In the triangle shown, if the angle corresponding to z₃ is said to be π/2,
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Find a possible value of z₃ in terms of z₁, z₂ and z₄.
a) z₄+3/5(z₂– z₁)e^iπ/2
b) z₄-3/5(z₂– z₁)e^iπ/2
c) z₄+3/5(z₂– z₁)e^-iπ/2
d) no such z₃ is possible
View Answer

Answer: d
Explanation: If we draw a circle with z₂-z₁ as a diameter, we see that z₃ would be outside the circle, since the radius of the circle is 5(₃ is possible.
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