Important Questions for Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations

Get Important questions for class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations here. Students can get different types of questions covered in this chapter. This chapter provides detailed information about the complex numbers and how to represent complex numbers in the Arg and plane. These important questions are taken from the previous year question papers. They also cover the NCERT problems, as the majority of the questions are asked from the NCERT textbook. Practice and solve all the important Maths questions from class 11 chapters at BYJU’S.

Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations cover the following important concepts, such as:

• Quadratic Equations
• Algebra of Complex Numbers
• Modulus and Conjugate of Complex Numbers
• Argand Plane and Polar Representation
• Euler’s Formula
• De Moivre’s Theorem

Also, Check:

• Important 1 Mark Questions for CBSE Class 11 Maths
• Important 4 Marks Questions for CBSE Class 11 Maths
• Important 6 Marks Questions for CBSE Class 11 Maths

Class 11 Chapter 5 – Complex Numbers and Quadratic Equations Important Questions with Solutions

Practice the important problems from class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations here.

Question 1:

Write the given complex number (1 – i) – ( –1 + i6) in the form a + ib

Solution:

Given Complex number: (1 – i) – ( –1 + i6)

Multiply (-) by the term inside the second bracket ( –1 + i6)

= 1 – i +1 – i6

= 2 – 7i, which is of the form a + ib.

Question 2:

Express the given complex number (-3) in the polar form.

Solution:

Given, complex number is -3.

Let r cos θ = -3 …(1)

and r sin θ = 0 …(2)

Squaring and adding (1) and (2), we get

r²cos²θ + r²sin²θ = (-3)²

Take r² outside from L.H.S, we get

r²(cos²θ + sin²θ) = 9

We know that, cos²θ + sin²θ = 1, then the above equation becomes,

r² = 9

r = 3 (Conventionally, r > 0)

Now, subsbtitute the value of r in (1) and (2)

3 cos θ = -3 and 3 sin θ = 0

cos θ = -1 and sin θ = 0

Therefore, θ = π

Hence, the polar representation is,

-3 = r cos θ + i r sin θ

3 cos π + 3 sin π = 3(cos π + i sin π)

Thus, the required polar form is 3 cos π+ 3i sin π = 3(cos π+i sin π)

Question 3:

Solve the given quadratic equation 2x² + x + 1 = 0.

Solution:

Given quadratic equation: 2x² + x + 1 = 0

Now, compare the given quadratic equation with the general form ax² + bx + c = 0

On comparing, we get

a = 2, b = 1 and c = 1

Therefore, the discriminant of the equation is:

D = b²– 4ac

Now, substitute the values in the above formula

D = (1)² – 4(2)(1)

D = 1- 8

D = -7

Therefore, the required solution for the given quadratic equation is

x =[-b ± √D]/2a

x = [-1 ± √-7]/2(2)

We know that, √-1 = i

x = [-1 ± √7i] / 4

Hence, the solution for the given quadratic equation is (-1 ± √7i) / 4.

Question 4:

For any two complex numbers z₁ and z₂, show that Re(z₁z₂) = Rez₁ Rez₂– Imz₁Imz₂

Solution:

Given: z₁ and z₂ are the two complex numbers

To prove: Re(z₁z₂) = Rez₁ Rez₂– Imz₁Imz₂

Let z₁ = x₁+iy₁ and z₂ = x₂+iy₂

Now, z₁ z₂ =(x₁+iy₁)(x₂+iy₂)

Now, split the real part and the imaginary part from the above equation:

⇒ x₁(x₂+iy₂) +iy₁(x₂+iy₂)

Now, multiply the terms:

= x₁x₂+ix₁y₂+ix₂y₁+i²y₁y₂

We know that, i² = -1, then we get

= x₁x₂+ix₁y₂+ix₂y₁-y₁y₂

Now, again seperate the real and the imaginary part:

= (x₁x₂ -y₁y₂) +i (x₁y₂+x₂y₁)

From the above equation, take only the real part:

⇒ Re (z₁z₂) =(x₁x₂ -y₁y₂)

It means that,

⇒ Re(z₁z₂) = Rez₁ Rez₂– Imz₁Imz₂

Hence, the given statement is proved.

Question 5:

Find the modulus of [(1+i)/(1-i)] – [(1-i)/(1+i)]

Solution:

Given: [(1+i)/(1-i)] – [(1-i)/(1+i)]

Simplify the given expression, we get:

= (1+i²+2i-1-i²+2i)) / (1²+1²)

Now, cancel out the terms,

= 4i/2

= 2i

Now, take the modulus,

| [(1+i)/(1-i)] – [(1-i)/(1+i)]| =|2i| = √2² = 2

Therefore, the modulus of [(1+i)/(1-i)] – [(1-i)/(1+i)] is 2.

Practice Problems for Class 11 Maths Chapter 5

Solve the below-given problems from class 11 Maths Chapter 5:

1. If |z²-1|= |z|²+1, prove that z lies on the imaginary axis.
2. Compute the value of p, such that the difference of the roots of the equation is x²+px+8=0 is 2.
3. Express each of the following complex numbers in the form a+ib
   (i) 3(7 + i7) + i (7 + i7)
   (ii) i⁹ +i¹⁹
   (iii) [(⅓)+3i]³
4. Solve the following quadratic equations:
   (a) x²+3x+5 = 0
   (b) x²+x+(1/√2)= 0
5. Determine the real numbers x and y if (x-iy)(3+5i) is the conjugate of -6-24i.
6. If arg (z – 1) = arg (z + 3i), then find (x – 1) : y, where z = x + iy.
7. Find the complex number satisfying the equation z + √2 |(z + 1)| + i = 0.
8. Solve the system of equations Re (z²) = 0, |z| = 2.

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