NCERT Solutions: Exercise: Miscellaneous - Complex Numbers

# NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Question 1: Evaluate:

Question 2: For any two complex numbers z1 and z2, prove that

Re (z1z2) = Re zRe z2 – Im z1 Im z2

Question 3: Reduce  to the standard form.

[ On Multiplying numerator and denomunator by (14 + 5i)

Question 4: If xiy =  prove that .

[ On Multiplying numerator and denomunator by (c + id)]

Question 5: Convert the following in the polar form:

(i) ,

(ii)

(i) Here,

Let cos θ = –1 and r sin θ = 1

On squaring and adding, we obtain

r2 (cos2 θ + sin2 θ) = 1+ 1

⇒ r2 (cos2 θ + sin2 θ) = 2
⇒ r2 = 2     [cos2 θ + sin2 θ = 1]

zr cos θ +  i r sin θ

This is the required polar form.

(ii) Here,

Let cos θ = –1 and r sin θ = 1

On squaring and adding, we obtain

r2 (cos2 θ + sin2 θ) = 1+ 1
r2 (cos2 θ  +sin2 θ) = 2

⇒ r2 = 2                        [cos2 θ + sin2 θ = 1]

zr cos θ  + i r sin θ

This is the required polar form.

Question 6: Solve the equation

This equation can also be written as

On comparing this equation with ax2 +  bx +  c = 0, we obtain

a = 9, b = –12, and c = 20

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–12)2 – 4 × 9 × 20 = 144 – 720 = –576

Therefore, the required solutions are

Question 7: Solve the equation

This equation can also be written as

On comparing this equation with ax2 +  bx +  c = 0, we obtain

a = 2, b = –4, and c = 3

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–4)2 – 4 × 2 × 3 = 16 – 24 = –8

Therefore, the required solutions are

Question 8: Solve the equation 27x2 – 10+ 1 = 0

The given quadratic equation is 27x2 – 10x + 1 = 0

On comparing the given equation with ax2 +  bx +  c = 0, we obtain

a = 27, b = –10, and c = 1

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–10)2 – 4 × 27 × 1 = 100 – 108 = –8

Therefore, the required solutions are

Question 9: Solve the equation 21x2 – 28+ 10 = 0

The given quadratic equation is 21x2 – 28x + 10 = 0

On comparing the given equation with ax2 +  bx +  = 0, we obtain

a = 21, b = –28, and c = 10

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–28)2 – 4 × 21 × 10 = 784 – 840 = –56

Therefore, the required solutions are

Question 9: If   find .

Question 10: If a +  ib = , prove that a2 +  b2 =

On comparing real and imaginary parts, we obtain

Hence, proved.

Question 10: Let  . Find

(i) ,

(ii)

(i)

On multiplying numerator and denominator by (2 – i), we obtain

On comparing real parts, we obtain

(ii)

On comparing imaginary parts, we obtain

Question 11: Find the modulus and argument of the complex number .

Let , then

On squaring and adding, we obtain

Therefore, the modulus and argument of the given complex number are   respectively.

Question 12: Find the real numbers x and y if (xiy) (3+5i) is the conjugate of –6 – 24i.

Let

It is given that,

Equating real and imaginary parts, we obtain

Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them, we obtain

Putting the value of x in equation (i), we obtain

Thus, the values of and y are 3 and –3 respectively.

Question 13: Find the modulus of  .

Question 14: If (x  + iy)3 = u  + iv, then show that .

On equating real and imaginary parts, we obtain

Hence, proved.

Question 15: If α and β are different complex numbers with  = 1, then find .

Let α = a  + ib and β = x +  iy

It is given that,

Question 16: Find the number of non-zero integral solutions of the equation .

Thus, 0 is the only integral solution of the given equation. Therefore, the number of non-zero integral solutions of the given equation is 0.

Question 17: If (a  + ib) (c +  id) (e +  if) (g  + ih) = A + iB, then show that

(a2  + b2) (c2+   d2) (e2 +  f2) (g2 +  h2) = A2  +B2.

On squaring both sides, we obtain

(a2  + b2) (c2 +  d2) (e2 +  f2) (g2  + h2) = A2 + B2

Hence, proved.

Question 18: If , then find the least positive integral value of m.

Therefore, the least positive integer is 1.

Thus, the least positive integral value of m is 4 (= 4 × 1).

The document NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations is a part of the Commerce Course Mathematics (Maths) Class 11.
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## Mathematics (Maths) Class 11

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## Mathematics (Maths) Class 11

75 videos|238 docs|91 tests

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