# Class 11 NCERT Solutions- Chapter 5 Complex Numbers And Quadratic Equations – Miscellaneous Exercise on Chapter 5 | Set 1

### Question 1. Evaluate

Solution:

= [-1 – i]3

= (-1)3 [1 + i]3

= -[13 + i3 + 3 Ã— 1 Ã— i (1 + i)]

= -[1 + i3 + 3i + 3i2]

= -[1 – i + 3i – 3]

= -[2 + 2i]

= 2 – 2i

### Question 2. For any two complex numbers z1 and z2, prove that, Re (z1z2) = Re z1 Re z2 â€“ Im z1 Im z2

Solution:

Let’s assume z1 = x1 + iy1 and z2 = x2 + iy2 as two complex numbers

Product of these complex numbers, z1z2

z1z2 = (x1 + iy1)(x2 + iy2)

= x1(x2 + iy2) + iy1(x2 + iy2)

= x1x2 + ix1y2 + iy1x2 + i2y1y2

= x1x2 + ix1y2 + iy1x2 – y1y2             [i2 = -1]

= (x1x2 – y1y2) + i(x1y2 + y1x2)

Now,

Re(z1z2) = x1x2 – y1y2

â‡’ Re(z1z2) = Rez1Rez2 – Imz1Imz2

Hence, proved.

### Question 3. Reduce to the standard form

Solution:

On multiplying numerator and denominator by (14+5i)

Hence, this is the required standard form.

### Question 4. If x – iy =  prove that (x2 + y2)2

Solution:

Given:

x – iy =

On multiplying numerator and denominator by (c+id)

So,

(x – iy)2

x2 – y2 – 2ixy

On comparing real and imaginary parts, we get

x2 – y2, -2xy =       (1)

(x2 + y2)2 = (x2 – y2)2 + 4x2y2

Hence proved

### Question 5. Convert the following in the polar form:

(i)

(ii)

Solution:

(i) Here, z =

Multiplying by its conjugate in the numerator and denominator

= -1+i

Let r cos Î¸ = -1 and r sin Î¸ = 1

On squaring and adding, we get

r2 (cos2Î¸ + sin2Î¸) = 1 + 1 = 2

r2 = 2           [cos2Î¸ + sin2Î¸ = 1]

r = âˆš2

So,

âˆš2 cosÎ¸ = -1 and âˆš2 sinÎ¸ = 1

â‡’ cosÎ¸ =  and sinÎ¸  =

Therefore,

Î¸ =            [As Î¸ lies in II quadrant]

Expressing as, z = r cos Î¸ + i r sin Î¸

Therefore, this is the required polar form.

(ii) Let, z =

= -1 + i

Now, Let r cosÎ¸ = -1 and r sin Î¸ = 1

On squaring and adding, we get

r2(cos2Î¸ + sin2Î¸) = 1 + 1

r(cos2Î¸ + sin2Î¸) = 2

r2 = 2          [cos2Î¸ + sin2Î¸ = 1]

= r = âˆš2     [Conventionally, r > 0]

Therefore,

âˆš2 cosÎ¸ = -1 and âˆš2 sinÎ¸ = 1

cosÎ¸ =  and sinÎ¸ =

Therefore,

Î¸ =       [As Î¸ lies in II quadrant]

Expressing as, z = r cosÎ¸ + i r sinÎ¸

z =

Therefore, this is the required polar form.

### Question 6. 3x2 â€“ 4x + 20/3 = 0

Solution:

Given quadratic equation, 3x2 â€“ 4x + 20/3 = 0

It can be re-written as: 9x2 â€“ 12x + 20 = 0

On comparing it with ax2 + bx + c = 0, we get

a = 9, b = â€“12, and c = 20

So, the discriminant of the given equation will be

D = b2 â€“ 4ac = (â€“12)2 â€“ 4 Ã— 9 Ã— 20 = 144 â€“ 720 = â€“576

Hence, the required solutions are

X =

### Question 7. x2 â€“ 2x + 3/2 = 0

Solution:

Given:

Quadratic equation, x2 â€“ 2x + = 0

After re-written 2x2 â€“ 4x + 3 = 0

On comparing it with ax2 + bx + c = 0,

We get

a = 2, b = â€“4, and c = 3

So, the discriminant of the given equation will be

D = b2 â€“ 4ac = (â€“4)2 â€“ 4 Ã— 2 Ã— 3 = 16 â€“ 24 = â€“8

Hence, the required solutions are

x =

### Question 8. 27x2 â€“ 10x + 1 = 0

Solution:

Given:

Quadratic equation, 27x2 â€“ 10x + 1 = 0

On comparing it with ax2 + bx + c = 0,

We get

a = 27, b = â€“10, and c = 1

So, the discriminant of the given equation will be

D = b2 â€“ 4ac = (â€“10)2 â€“ 4 Ã— 27 Ã— 1 = 100 â€“ 108 = â€“8

Hence, the required solutions are

### Question 9. 21x2 â€“ 28x + 10 = 0

Solution:

Given:

Quadratic equation, 21x2 â€“ 28x + 10 = 0

On comparing it with ax2 + bx = 0,

We have

a = 21, b = â€“28, and c = 10

So, the discriminant of the given equation will be

D = b2 â€“ 4ac = (â€“28)2 â€“ 4 Ã— 21 Ã— 10 = 784 â€“ 840 = â€“56

Hence, the required solutions are

### Question 10. If z1 = 2 â€“ i, z2 = 1 + i, find

Solution:

Given, z1 = 2 â€“ i, z2 = 1 + i

Hence, the value of  is âˆš2

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