1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations

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Karnataka 1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations

Question 1.
Define a complex number.
Answer :
A number of the form a + ib, where a and b are real numbers, is defined to be a complex number, where i² = -1.
Here ‘a’ is called the real part, denoted by Re (z) and ‘b’ is called the imaginary part, denoted by Im (z), of the complex number z = a + ib.

Question 2.
When you say that two complex numbers are equal?
Answer :
Two complex numbers z₁ = a + ib and z₂ = c + id are equal if a = c and b – d.

Question 3.
Define purely real and purely imaginary numbers.
Answer :
A complex number z is said to be:

• Purely real, if Im(z) = 0,
• Purely imaginary, if Re(z) = 0

Question 4.
If 4x + i(3x – y) = 3 +i (-6), where x and y are real numbers, then find the values of
x and y.
Answer :
Given 4x + i(3x – y) = 3 + i(-6), Equating the real and the imaginary parts,we get
4x = 3 and 3x – y = -6
\(\Rightarrow x=\frac{3}{4} \text { and } y=3\left(\frac{3}{4}\right)+6=\frac{33}{4}\)

Question 5.
Define addition of two complex numbers.
Answer :
Let z₁=a + ib and z₂=c + id be any two complex numbers. Then addition of z₁ and z₂ defined as z₁ + z₂ = (a + c) + i(b + d) which is again a complex number.
Properties of addition of complex numbers:

(i) Closure property: The sum of two complex numbers is always a complex number, i.e., addition is closed in C set of complex numbers.

(ii) Commutative property: For any two complex numbers z₁ and z₂
we have z₁ + z₂ = z₂ + z₁

(iii) Associative properly: For any complex numbers z₁,z₂ and Z₃, we have
\( z_{1}+\left(z_{2}+z_{3}\right)=\left(z_{1}+z_{2}\right)+z_{3}\)

(iv) Existence of additive identity: For any complex number z, we have z + 0 = 0 +-z = z
Thus, 0 is the additive identity for complex numbers.

(v) Existence of additive inverse: Every complex number z-a + ib has -z = (-a) + i(-b) as its additive inverse, as z + (-z) = (-z) + z = 0.

Question 6.
Define difference of two complex numbers.
Answer :
Let z₁ and z₂ be any two complex numbers. The different z₁ – z₂ is defined as follows:
z_{1 –} z₂ =z₁ +(-z₂)

Question 7.
Define multiplication numbers.
Answer :
Let z₁=a + ib and z₂ = c + id be any two complex numbers. Then, the product z₁z₂ is defined as follows:
z₁z₂ = (ac – bd) + i(ad + bc)

Properties of product of complex numbers:

• Closure property: The product of two complex numbers is always a complex number.
• Commutative law: For any two complex numbers z₁ and z₂ we have z₁ z₂ = z₂z₁
• Associative law: For any three complex numbers z₁ z₂ and z₃ we have (z₁z₂)z₃=z₁(z₂z₃).
• Existence of multiplicative identity: For every complex number z, we have z . 1 = 1. z = z. Thus, 1 is the multiplicative identity.
• Existence of multiplicative inverse: Let z = a + ib. Then
  \(z^{-1}=\frac{1}{z}=\frac{1}{a+i b}=\frac{1}{a+i b} \cdot \frac{a-i b}{a-i b}=\frac{a-i b}{a^{2}+b^{2}} \)
  Clearly
  \(z \cdot \frac{1}{z}=\frac{1}{z} \cdot z=1\)
  Thus, every z = a + ib has its multiplicative inverse, given
  \( z^{-1}=\frac{1}{z}=\frac{a}{a^{2}+b^{2}}-\frac{a}{a^{2}+b^{2}} i \quad(z \neq 0)\)

Question 8.
Define division of two complex numbers.
Answer :
Let z₁ and z₂ be two complex numbers and z₂ ≠0.Then \( \frac{z_{1}}{z_{2}} \) is defined by  \( \frac{z_{1}}{z_{2}}=z_{1}\left(\frac{1}{z_{2}}\right) \)

Question 9.
Prove the following
(i) \(\left(z_{1}+z_{2}\right)^{2}=z_{1}^{2}+2 z_{1} z_{2}+z_{2}^{2}\)
(ii) \(\left(z_{1}-z_{2}\right)^{2}=z_{1}^{2}-2 z_{1} z_{2}+z_{2}^{2}\)
(iii) \(\left(z_{1}+z_{2}\right)^{3}=z_{1}^{3}+3 z_{1}^{2} z_{2}+3 z_{1} z_{2}^{2}+z_{2}^{3}\)
(iv) \(\left(z_{1}-z_{2}\right)^{3}=z_{1}^{3}-3 z_{1}^{2} z_{2}+3 z_{1} z_{2}^{2}-z_{2}^{3}\)
(v) \(z_{1}^{2}-z_{2}^{2}=\left(z_{1}+z_{2}\right)\left(z_{1}-z_{2}\right) \)
Answer:
(i) We have, (z₁ + z₂)² = (z, + z₂)(z₁ + z₂)
= (z₁ + z₂)z₁ + (z₁ + z₂)z₂ (by distributive law)
= z²₁ + z₂z₁ +z₁z₂ + z₂ (by distributive law)
\( =z_{1}^{2}+2 z_{1} z_{2}+z_{2}^{2} \quad \ z_{1} z_{2}=z_{2} z_{1}\)
Remaining try yourself.

Question 10.
Express the following in the form of a+ib
(i) \( (-5 i)\left(\frac{1}{8} i\right)\)
(ii) \((5 i)\left(-\frac{3}{8} i\right)\)
(iii) \((2+i 3)+(-6+i 5)\)
(iv) \((6+3 i)-(2-i)\)
(v) \((2-i)-(6+3 i)\)
(vi) \((3+5 i)(2+6 i)\)
(vii) \(3(7+7 i)+i(7+7 i)\)
(viii) \((1-i)-(-1+6 i)\)
(ix)\(\left(\frac{1}{5}+\frac{2}{5} i\right)-\left(4+\frac{5}{2} i\right)\)
(x)\(\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(-\frac{4}{3}+i\right) \)
(xi) \(i^{9}+i^{19}\)
(xii)\( i^{-39}\)
(xiii)\((-i)(2 i)\left(-\frac{1}{8} i\right)^{3}\)
(xiv) \((5-3 i)^{3}\)
(xv) \(\left(\frac{1}{3}+3 i\right)^{3}\)
(xvi) \(\left(-2-\frac{1}{3} i\right)^{3}\)
(xvii) \((\mathbf{1}-i)^{4}\)
(xviii) \(\frac{5+\sqrt{2} i}{1-\sqrt{2 i}}\)
(xix) \( \frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-\sqrt{2} i)}\)
Answer:

Question 11.
Define modulus of a complex number.
Answer :
Let. z = a + ib be a complex number. Then, the modulus of z, denoted by |z| to be the non-negative real number \(\sqrt{a^{2}+b^{2}}\)

Question 12.
Define conjugate of a complex number.
Answer :
Let z = a + ib be a complex number. Then, the conjugate of complex number z, denoted \(\bar{z} \) as , is the complex number a – ib, i.e., i= a – ib

Question 13.
Write down the modulus of
(i) i
(ii) \( 3+\sqrt{-3}\)
(iii)\((-1-i)^{3}\)
Answer:

Question 14.
Write down the conjugate of
(i) \((2+\sqrt{-2})\)
(ii) \( i^{3}\)
(iii) \((3-4 i)^{2} \)
Answer:

Question 15.
Find the multiplicative inverse of the following complex numbers:
(i) 2-3i
(ii) 4-3i
(iii) \(\sqrt{5}+3i\)
(iv) \(\sqrt{5}+3i \)
(v) \(-i\)
Answer:

Question 16.
Find the conjugate of  \(\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}\)
Answer:

Question 17.
Evaluate:
\( \left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}\)
Answer:

Question 18.
Reduce \( \left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)\) to the standard form.
Answer:

Question 19.
Find the modulus of
\(\frac{1+i}{1-i}-\frac{1-i}{1+i}\)
Answer:

Question 20.
Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of -6 – 24i
Answer:

Question 21.
For any two complex numbers Z₁ and Z₂, prove that
Re(z₁z₂) = Re(z₂) . Re(z₂) – Im(z₁) . Im(z₂)
Answer :
z₁=a + ib and z₂=c + id
z₁z₂ = {a + ib)(c + id) = (ac-bd) + i(ad + bc)
∴ Re(z₁z₂) = ac-bd
= Re(z₁).Re(z₂)-1m(z₁)\m{z₂)

Question 22.
\(\text { If } z_{1}=2-i, z_{2}=1+i, \text { find }\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\right| \)
Answer:

Question 23.
Let z₁ = 2 – i, z₂ = -2 + i .Find

Answer:

Question 24.
If \(x+i y=\frac{a+i b}{a-i b}, \text { prove that } x^{2}+y^{2}=1\)
Answer:

Question 25.
If \(x-i y=\sqrt{\frac{a-i b}{c-i d}}\) Prove that
\(\left(x^{2}+y^{2}\right)^{2}=\frac{a^{2}+b^{2}}{c^{2}+d^{2}}\)
Answer:

Question 26.
If \(a+i b=\frac{(x+i)^{2}}{2 x^{2}+1}\), Prove that \(a^{2}+b^{2}=\frac{\left(x^{2}+1\right)^{2}}{\left(2 x^{2}+1\right)^{2}}\)
Answer:

Question 27.
If (x + iy)³ =u + iv, then show that
\(\frac{u}{x}+\frac{v}{y}=4\left(x^{2}-y^{2}\right)\)
Answer:

Question 28.
If (a + ib)(c + id){e + if) (g + ih) = A + iB then show that (a²+b²)(c²+d²)(e²+f²) (g²+h²) = A²+B²
Answer:
Given: A + iB = (a + ib)(c + id)(e + if)(g + ih)
∴  A-iB = (a- ib)(c – id)(e – if)(g – ih)
But A² + B² = (A + iB)(A – iB)
= (a² +b²)(c²+d²)(c²+f²)(g²+h²)

Question 29.
If α and β are different complex numbers with |β|=1 then find
\(\left|\frac{\beta-\alpha}{1-\bar{\alpha} \beta}\right|\)
Answer:

Question 30.
If \(\left(\frac{1+i}{1-i}\right)^{m}=1 \)then find the least positive integral value of m.
Answer:

Question 31.
Find the number of non-zero integral solutions of the equation \(|1-i|^{x}=2^{x}\)
Answer:

Question 32.
Find real θ such that \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is purely real
Answer:

Question 33.
Define the polar form of complex number.
Answer:

Let the complex number z = x + iy be represented by the point P(x,y) in the complex plane.
Let \(| X O P=\theta \text { and }|O P|=r>0\)
Then
P(r, θ) are called the polar coordinates of P.
where x = rcosθ
y = rsinθ
∴ z = r(cosθ+ i sin θ)
This is called the polar form or trigonometric form or modulus – amplitude form of z.

Keen Eye;

• \( r=\sqrt{x^{2}+y^{2}}=1 z 1 \) is called the modulus of z and 0 is called the argument (or amplitude) of z, written as arg (z) or amp (z).
• The value of such that -π<θ≤n is called principal argument of z.
• To find θ

Case (i):
When z is purely real. Then, it lies on the x-axis.
(i) If x > 0, then θ=0
(ii) If x < 0, then θ = π

Case (ii):
When z is purely imaginary. Then, it lies on the y-axis
(i) \(\text { If } y>0, \text { then } \theta=\frac{\pi}{2}\)
(ii) \(\text { If } y<0, \text { then } \theta=-\frac{\pi}{2}\)

Case(iii):
Let tan α =|tan θ|
\(\text { where } 0<\alpha<\frac{\pi}{2}\)

(i) θ = α, when z lies in I quadrant
(ii) θ – π – α, when z lies in II quadrant
(iii) θ = α – π, when z lies in III quadrant
(iv) θ = -α, when z lies in IV quadrant

Question 34.
Find the modulus and the arguments of the following:
(i) 1
(ii) -3
(iii) i
(iv) -8i
(v) 1+i
(vi) \(\sqrt{3}-i\)
(vii)\(-\sqrt{3}+i\)
(viii)\(-1-i \sqrt{3}\)
(ix)\(\frac{1+2 i}{1-3 i}\)
(x)\(\frac{1+i}{1-i}\)
(xi) \(\frac{1}{1+i}\)
Answer:

Question 35.
Convert the Following complex numbers in the polar form:
(i) 1+i
(ii) 1-i
(iii) -1+i
(iv) -1-i
(v) \(\sqrt{3}+i\)
(vi) \(\sqrt{3}-i\)
(vii) \(-\sqrt{3}-i\)
(viii) \(-\sqrt{3}+i\)
(ix) \(1+i \sqrt{3}\)
(x) \(1-i \sqrt{3}\)
(xi) \(-1+i \sqrt{3}\)
(xii) \(-1-i \sqrt{3}\)
(xiii) -3
(xiv) i
(xv) \(\frac{-16}{1+i \sqrt{3}}\)
(xvi) \(\frac{1+7 i}{(2-i)^{2}}\)
(xvii) \(\frac{1+3 i}{1-2 i}\)
(xviii)
\(\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}\)
Answer:

Question 36.
Keen Eye:

• A polynomial equation has at least one root.
• A polynomial equation of degree V has n roots. Solve each of the following equations:

(i) x² + 2 = 0
(ii) x² + 3 = 0
(iii) x² + x +1 = 0
(iv) 2x² + x +1 = 0
(v) x² + 3x + 9 = 0
(vi) -x² + x – 2 = 0
(vii) x² + 3x + 5 = 0
(viii) x² – x + 2 = 0
(ix)\(3 x^{2}-4 x+\frac{20}{3}=0\)
(x) \(x^{2}-2 x+\frac{3}{2}=0\)
(xi) 27x² -10x + 1 = 0
(xii) \(\sqrt{2} x^{2}+x+\sqrt{2}=0\)
(xiii) \(\sqrt{3} x^{2}-\sqrt{2} x+3 \sqrt{3}=0\)
(xiv)\( x^{2}+x+\frac{1}{\sqrt{2}}=0\)
(xv) \(x^{2}+\frac{x}{\sqrt{2}}+1=0\)
Answer:



(viii) x² -x+2=0

Question 37.
Find the equation roots of the following:
(i) i
(ii) i
(iii) 1+i
(iv) 1-i
(v) -8-6i
(vi)-15-8i
(vii) -7-24i
Answer:
(i)

(ii)


(iii)

(iv)