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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 i^{2} = -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 z_{1} = a + ib and z_{2} = 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 z_{1}=a + ib and z_{2}=c + id be any two complex numbers. Then addition of z_{1 }and z_{2 }defined as z_{1} + z_{2} = (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 z_{1} and z_{2}

we have z_{1} + z_{2} = z_{2} + z_{1}

(iii) Associative properly: For any complex numbers z_{1},z_{2 }and Z_{3}, 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 z_{1} and z_{2} be any two complex numbers. The different z_{1} – z_{2} is defined as follows:

z_{1 – }z_{2 }=z_{1 }+(-z_{2})

Question 7.

Define multiplication numbers.

Answer :

Let z_{1}=a + ib and z_{2} = c + id be any two complex numbers. Then, the product z_{1}z_{2} is defined as follows:

z_{1}z_{2} = (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 z
_{1 }and z_{2 }we have z_{1}z_{2 }= z_{2}z_{1} - Associative law: For any three complex numbers z
_{1}z_{2}and z_{3}we have (z_{1}z_{2})z_{3}=z_{1}(z_{2}z_{3}). - 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 z_{1 }and z_{2 }be two complex numbers and z_{2 }≠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, (z_{1} + z_{2})^{2} = (z, + z_{2})(z_{1} + z_{2})

= (z_{1} + z_{2})z_{1} + (z_{1} + z_{2})z_{2} (by distributive law)

= z^{2}_{1} + z_{2}z_{1} +z_{1}z_{2} + z_{2} (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 Z_{1} and Z_{2}, prove that

Re(z_{1}z_{2}) = Re(z_{2}) . Re(z_{2}) – Im(z_{1}) . Im(z_{2})

Answer :

z_{1}=a + ib and z_{2}=c + id

z_{1}z_{2} = {a + ib)(c + id) = (ac-bd) + i(ad + bc)

∴ Re(z_{1}z_{2}) = ac-bd

= Re(z_{1}).Re(z_{2})-1m(z_{1})\m{z_{2})

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 z_{1} = 2 – i, z_{2} = -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 (a^{2}+b^{2})(c^{2}+d^{2})(e^{2}+f^{2}) (g^{2}+h^{2}) = A^{2}+B^{2}

Answer:

Given: A + iB = (a + ib)(c + id)(e + if)(g + ih)

∴ A-iB = (a- ib)(c – id)(e – if)(g – ih)

But A^{2} + B^{2} = (A + iB)(A – iB)

= (a^{2} +b^{2})(c^{2}+d^{2})(c^{2}+f^{2})(g^{2}+h^{2})

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) x^{2} + 2 = 0

(ii) x^{2} + 3 = 0

(iii) x^{2} + x +1 = 0

(iv) 2x^{2} + x +1 = 0

(v) x^{2} + 3x + 9 = 0

(vi) -x^{2} + x – 2 = 0

(vii) x^{2} + 3x + 5 = 0

(viii) x^{2} – x + 2 = 0

(ix)\(3 x^{2}-4 x+\frac{20}{3}=0\)

(x) \(x^{2}-2 x+\frac{3}{2}=0\)

(xi) 27x^{2} -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) x^{2} -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)