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

Rajasthan Board RBSE Class 11 Maths Important Questions Chapter 5 Complex Numbers and Quadratic Equations Questions and Answers.

Rajasthan Board RBSE Solutions for Class 11 Maths in Hindi Medium & English Medium are part of RBSE Solutions for Class 11. Students can also read RBSE Class 11 Maths Important Questions for exam preparation. Students can also go through RBSE Class 11 Maths Notes to understand and remember the concepts easily.

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

Question 1.
Find conjugate of \(\frac{(4-3 i)(3+4 i)}{(1+3 i)(3-i)}\).
Answer:
Here,
RBSE Class 11 Maths Important Questions Chapter 5 Complex Numbers and Quadratic Equations 1

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

Question 2.
Find conjugate of \(\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}\).
Answer:
Here,
RBSE Class 11 Maths Important Questions Chapter 5 Complex Numbers and Quadratic Equations 2

Question 3.
Find modulus and argument of the following complex numbers:
(i) i
(ii) 1 - i
Answer:
(i) Le z = i = r(cos θ + i sin θ)
Comparing real and imaginary parts on both sides
or 0 + i = r(cos θ + i sin θ)
r cos θ = 0
and r sin θ = 1
Squaring and adding,
r2 cos2θ + r2 sin2 θ = 1
or r2 = 1 (∵ sin2 θ + cos2 θ = 1)
or r = 1
Now, cos θ = 0 and sin θ = \(\frac{1}{1}\) = 1
or θ = \(\frac{\pi}{2}\) or θ = \(\frac{\pi}{2}\)
Thus, θ = \(\frac{\pi}{2} \)Thus, modulus of i, is 1 and argument \(\frac{\pi}{2}\).

(ii) Let z = 1 - i = r (cos θ + i sin θ)
Comparing real and imaginary values of on both sides
r cos θ = 1, r sin θ = - 1
Squaring and adding,
r2 cos2 θ + r2 sin2 θ = 1 + 1 = 2
or r2 (cos2 θ + sin2 θ) = 2
or r2 = 2 [∵ cos2 θ + sin2 θ = 1]
or r = √2
∵ cos θ = \(\frac{1}{\sqrt{2}}\) and sin θ = - \(\frac{1}{\sqrt{2}}\)
or θ = - \(\frac{\pi}{4}\)
Thus, modulus of 1 - i is √2 and argument -\(\frac{\pi}{4}\).

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

Question 4.
If \(\frac{2+2 i \cos \theta}{1-2 i \cos \theta}\) is real then find real value of θ.
Answer:
RBSE Class 11 Maths Important Questions Chapter 5 Complex Numbers and Quadratic Equations 3

Question 5.
If x + iy = \(\frac{a+i b}{a-i b}\) then prove that x2 + y2 = 1.
Answer:
RBSE Class 11 Maths Important Questions Chapter 5 Complex Numbers and Quadratic Equations 4

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

Question 6.
Write polar form of complex number
z = \(\frac{i+1}{\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}}\).
Answer:
RBSE Class 11 Maths Important Questions Chapter 5 Complex Numbers and Quadratic Equations 5

Question 7.
If z1 and z2 are two complex numbers that |z1 + z2| = |z1 - z2|, then prove that:
Argument (z1) - Argument (z2) = \(\frac{\pi}{2}\)
Answer:
Let z1 = r1 (cos θ1 + i sin θ1)
and z2 = r2 (cos θ2 + i sin θ2)
then z1 + z2 = (r1 cos θ1 + r2 cos θ2) + i(r1 sin θ1 + r2 sin θ2)
and z1 - z2 = (r1 cos θ1 - r2 cos θ2) + i(r1 sin θ1 - r2 sin θ2)
|z1 + z2|2 = [(r1 cos θ1 + r2 cos θ2)2 + (r1 sin θ1 + r2 sin θ2)2]
= r12 cos2 θ1 + r22 cos2 θ2 + 2 r1r2 cos θ1 cos θ2 + r12 sin2 θ1 + r22 sin2 θ2 + 2r1r2 sin θ1 sin θ2
= r12 + r22 + 2r1 r2 cos (θ1 - θ2) (∵ sin2θ + cos2θ = 1)
and |z1 - z2|2 = ((r1 cos θ1 - r2 cos θ2)2 + (r1 sin θ1 - r2 sin θ2)2]
= r12 cos2 θ1 + r22 cos2 θ2 - 2r1r2 cos θ1 cos θ2 + r12 sin2 θ1 + r22 sin2 θ2 - 2r1r2 sin θ1 sin θ2
= r12 + r2 - 2r1r2 cos (θ1 - θ2)2 cos (θ1 - θ2)
∵ |z1 + z2|2 = |z1 - z2|2
⇒ r12 + r22 + 2r1r2 cos(θ1 - θ2)
⇒ r12 + r22 - 2r1r2 cos (θ1 - θ2)
⇒ 4r1r2 cos(θ1 - θ2) = 0
⇒ cos(θ1 - θ2) = 0 (∵r1r2 = 0)
⇒ θ1 - θ2 = \(\frac{\pi}{2}\)
⇒ Argumentz1 - Argumentz2 = \(\frac{\pi}{2}\) Hence Proved.

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

Question 8.
Find multiplicative inverse of complex number 3 + 2i.
Answer:
Let 3 + 2i is multiplicative inverse of x + iy
then (3 + 2i)(x + iy) = 1 = 1 + oi
or (3x - 2y) + i(2x + 3y) = 1 + 0.i
On comparing on both sides,
3x - 2y = 1
and 2x + 3y = 0
∴ x = \(\frac{3}{13}\), y = \(\frac{2}{13}\)
∴ Multiplicative inverse of 3 + 2i = \(\frac{3}{13}-\frac{2}{13}\) i

Multiple choice Questions

Question 1.
i + \(\frac{1}{i}\) equals:
(a) 0
(b) - 1
(c) 1
(d) None of these
Answer:
(a) 0

Question 2.
i-101 equals:
(a) - i
(b) 4
(c) - 1
(d) None of these
Answer:
(a) - i

Question 3.
\(\frac{1}{(1-i)^2}-\frac{1}{(1+i)^2}\) equals:
(a) - i
(b) i
(c) 1
(d) None of these
Answer:
(b) i

Question 4.
(1 + i) (1 + i2) + (1 + i3) (1 + i4) (1 + i5)equals:
(a) 1
(b) - 1
(c) 4
(d) None of these
Answer:
(c) 4

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

Question 5.
If \(\frac{2+3 i \sin \theta}{1-3 i \sin \theta}\) is a real number and 0 < θ < 2π, then θ =
(a) π
(b) \(\frac{\pi}{2}\)
(c) \(\frac{3 \pi}{2}\)
(d) None of these
Answer:
(a) π

Question 6.
If x is real then satisfy \(\frac{1-i x}{1+i x}\) = a - ib, then a2 + b2 equals:
(a) - 1
(b) 1
(c) 0
(d) None of these
Answer:
(b) 1

Question 7.
If x + iy = \(\sqrt{\frac{a+i b}{c+i d}}\), then value of (x2 + y2)2 is:
(a) \(\frac{c^2+d^2}{a^2+b^2}\)
(b) \(\frac{a^2+b^2}{c^2+d^2}\)
(c) \(\frac{a^2-b^2}{a^2-d^2}\)
(d) None of these
Answer:
(b) \(\frac{a^2+b^2}{c^2+d^2}\)

Question 8.
Modulus of complex number i is:
(a) 1
(b) 0
(c) \(\frac{1}{2}\)
(d) None of these
Answer:
(a) 1

Question 9.
Argument of - 1 + i is:
(a) 135°
(b) 150°
(c) 270°
(d) None of these
Answer:
(a) 135°

Question 10.
Modulus of (2 + 3i)2 is:
(a) 13
(b) 15
(c) 12
(d) None of these
Answer:
(a) 13

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

Question 11.
If x + iy = (1 + i) (1 + 2i) (1 + 3i) then x2 + y2 equals:
(a) 0
(b) 1
(c) 100
(d) None of these
Answer:
(c) 100

Question 12.
If z = \(\frac{1}{1-\cos \theta-i \sin \theta}\) then Re (z)equals:
(a) 0
(b) \(\frac{1}{2}\)
(c) tan \(\frac{\theta}{2}\)
(d) None of these
Answer:
(b) \(\frac{1}{2}\)

Question 13.
If θ is argument of \(\frac{3+2 i}{3-2 i}\), then tan θ equals:
(a) \(\frac{6}{13}\)
(b) \(\frac{12}{5}\)
(c) \(\frac{5}{13}\)
(d) None of these
Answer:
(b) \(\frac{12}{5}\)

Question 14.
If a = 1 + i, then a2 equals:
(a) (1 - i)
(b) 2i
(c) (1 + i) (1 - i)
(d) None of these
Answer:
(b) 2i

Question 15
Principal argument of (1 + i) is:
(a) \(\frac{\pi}{4}\)
(b) \(\frac{\pi}{6}\)
(c) \(\frac{3 \pi}{4}\)
(d) None of these
Answer:
(a) \(\frac{\pi}{4}\)

Question 16.
If z = x + iy, then real part of \(\frac{1}{z-1}\) is:
(a) \(\frac{x-1}{x^2+y^2-2 x+1}\)
(b) \(\frac{x+1}{x^2+y^2-2 x+1}\)
(c) \(\frac{x-1}{x^2-y^2+2 x+1}\)
(d) None of these
Answer:
(a) \(\frac{x-1}{x^2+y^2-2 x+1}\)

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

Question 17.
1 + i2 + i4 + i6 equals:
(a) 0
(b) 1
(c) - 1
(d) None of these
Answer:
(a) 0

Question 18.
i108 + i117 + i110 + i115 equals:
(a) - 1
(b) 0
(c) 1
(d) None of these
Answer:
(b) 0

Question 19.
1 + i10 + i100 - i100 equals:
(a) 1
(b) 0
(c) - 1
(d) None of these
Answer:
(c) - 1

Question 20.
(1 + i)4 (1 + \(\frac{1}{i}\))4 equals:
(a) 16
(b) - 16
(c) 8
(d) None of these
Answer:
(a) 16

Question 21.
Multiplicative inverse of 2 + 3i is:
(a) \(\frac{2}{13}-\frac{3 i}{13}\)
(b) \(\frac{2}{13}+\frac{3 i}{13} \)
(c) -\(\frac{2}{13}-\frac{3 i}{13}\)
(d) None of these
Answer:
(a) \(\frac{2}{13}-\frac{3 i}{13}\)

Question 22.
Multiplicative inverse of \(\frac{(i+1)(i+2)}{(i-1)(i-2)}\) is:
(a) \(\frac{-4}{5}-\frac{3}{5}\) i
(b) \(\frac{4}{5} i+\frac{3}{5}\) i
(c) \(\frac{4}{5} i-\frac{3}{5} i\)
(d) None of these
Answer:
(a) \(\frac{-4}{5}-\frac{3}{5} i\)

Question 23.
For complex number z, \(\overline{(\bar{z})}\) equals:
(a) z̅
(b) z
(c) |z|2
(d) None of these
Answer:
(b) z

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

Question 24.
For complex number z, zz̅ equals:
(a) |z|2
(b) |z|
(c) \(\left|\frac{1}{z}\right|^2\)
(d) None of these
Answer:
(a) |z|2

Question 25.
Conjugate of 5 + 12i is:
(a) - 5 + 12i
(b) 5 - 12i
(c) - 5 - 12i
(d) None of these
Answer:
(b) 5 - 12i

Question 26.
If z̅ = z then z:
(a) Purely real
(b) Purely imaginary
(c) Neither real nor imaginary.
(d) None of the above
Answer:
(a) Purely real

Question 27.
If z̅ = - z, then z:
(a) Purely imaginary
(b) Purely real
(c) Both real and imaginary
(d) None of these
Answer:
(a) Purely imaginary

Question 28.
For any complex number z, z + z̅:
(a) Real
(b) imaginary
(c) Both (a) and (b)
(d) None of these
Answer:
(a) Real

Question 29.
If a + ib = \(\sqrt{\frac{1+i}{1-i}}\) then a2 + b2 equals:
(a) 1
(b) - 1
(c) 0
(d) None of these
Answer:
(a) 1

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

Question 30.
If a = 3 + 2i then a4 - 12a3 + 62a2 - 156a equals:
(a) - 169
(b) 169
(c) 0
(d) None of these
Answer:
(a) - 169

Question 31.
Principal argument of (cos \(\frac{\pi}{6}\) + i sin \(\frac{\pi}{6}\))
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) None of these
Answer:
(a) \(\frac{\pi}{6}\)

Question 32.
\(\left|\frac{1+\cos \theta+i \sin \theta}{1+\cos \theta-i \sin \theta}\right|\) equals:
(a) 1
(b) 2
(c) \(\frac{1}{2}\)
(d) None of these
Answer:
(a) 1

Question 33.
If (x + iy) = \(\frac{(a+i)^2}{2 a-i}\), then x2 + y2 equals:
(a) \(\frac{\left(a^2+1\right)^2}{4 a^2+1}\)
(b) \(\frac{\left(a^2-1\right)^2}{4 a^2+1}\)
(c) \(\frac{\left(a^2+1\right)^2}{4 a^2-1}\)
(d) None of these
Answer:
(a) \(\frac{\left(a^2+1\right)^2}{4 a^2+1}\)

Question 34.
If complex numbers z1, z2 and z3 are verticies of an equilateral triangle, then z1 + z2 + z3 equals:
(a) 0
(b) 1
(c) - 1
(d) None of these
Answer:
(a) 0

Question 35.
Polar form of (1 + i) is:
(a) \(\frac{1}{\sqrt{2}}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\)
(b) \(\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\)
(c) \(\left(\cos \frac{\pi}{4}-i \sin \frac{\pi}{4}\right)\)
(d) None of these
Answer:
(d) None of these

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

Question 36.
Polar form of 3 + 4i is:
(a) \(\cos \left(\tan ^{-1} \frac{4}{3}\right)+i \sin \left(\tan ^{-1} \frac{4}{3}\right)\)
(b) 5\(\left[\cos ^{-1}\left(\frac{4}{3}\right)+i \sin ^{-1}\left(\frac{4}{3}\right)\right]\)
(c) \(\cos \left(\tan ^{-1} \frac{4}{3}\right)-i \sin \left(\tan ^{-1} \frac{4}{3}\right)\)
(d) None of the above
Answer:
(d) None of the above

Fill in the Blanks

Question 1.
The conjugate of complex number z = 3 - i is ................................
Answer:
3 + i

Question 2.
If (2 + i) (2 + 2i) (2 + 3i) ....... (2 + ni) = x + iy, then 5.8.13 ...................... (4 + n2) = ...........................
Answer:
x2 + y2

Question 3.
If |z| = 5 and arg(z) = \(\frac{\pi}{4}\) then z = .....................................
Answer:
z = 5\(\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\)

Question 4.
The value of i111 + 2i29 - \(\frac{3}{i^{99}}\) is ..............................
Answer:
- 2i

Question 5.
For any complex number z = 3 + 4i the multiplicative inverse is ...............................
Answer:
\(\frac{3-4 i}{25}\)

Question 6.
Principal arg(z) is lie ............................
Answer:
- π < θ ≤ π.

Question 7.
The roots of equation x2 + 4 = 0 are ..................................
Answer:
± 2i

Question 8.
arg\(\left(\frac{z_1}{z_2}\right)\) = ........................................
Answer:
arg(z1) - arg(z2)

Question 9.
If arg of z - 2 - 3i is π/4 than locus of z is ........................................
Answer:
Straight line

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

True/False

Question 1.
The modulus of z = 3 - i is 10.
Answer:
True

Question 2.
The conjugate of 5 + 3i is 5 - 3i
Answer:
True

Question 3.
The polar form of 1 - i is √2 (cos \(\frac{\pi}{4}\) + i sin \(\frac{\pi}{4}\))
Answer:
False

Question 4.
For any complex number z is
z̄ = \(\frac{1}{z}\)
Answer:
False

Question 5.
If z = 4 +3i then its multiplicative inverse is 4 - 3i.
Answer:
False

Question 6.
If z + z̄ = 0 then z is purelyded.
Answer:
False

Question 7.
If a + ib = \(\sqrt{\frac{x+i}{x-i}}\) then a2 + b2 = 1.
Answer:
True

Question 8.
For an two complex number z1 and z2, z1 + z2 = z2 + z1.
Answer:
True

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

Question 9.
If z = √3 + i the arg(z) = π/6.
Answer:
True

Bhagya
Last Updated on Feb. 28, 2023, 11:32 a.m.
Published Feb. 28, 2023