Mathematics Textbook for Class 11>Chapter -1 - Complex Numbers and Quadratic Equations>EXERCISE 5.1>Q 13

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11th CBSE

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Find the multiplicative inverse of the complex number $- i$ .

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Important Points to Remember in Chapter -1 - Complex Numbers and Quadratic Equations from NCERT Mathematics Textbook for Class 11 Solutions

1. Complex Numbers:

(i) $\sqrt{- 1}$ is an imaginary quantity and is denoted by $i$ .

(ii) $i^{2} = - 1, i^{3} = - i, i^{4} = 1$ and, $i^{\pm n} = i^{\pm k}, n \in N$ where $k$ is the remainder when $n$ is divided by $4 .$

(iii) For any positive real number $a,$ $\sqrt{- a} = i \sqrt{a} .$

(iv) For any two real numbers $a$ and $b,$ $\sqrt{a} \sqrt{b} = \{\begin{cases} \sqrt{a b}, if at least one of a and b is positive \\ - \sqrt{a b}, if a < 0, b < 0 . \end{cases}$

(v) If $a, b$ are real numbers, then a number $z = a + i b$ is called a complex number.

(vi) Real number $a$ is known as the real part of $z$ and $b$ is known as its imaginary part. We write $a = Re (z), b = lm (z)$ .

(vii) A complex number $z$ is purely real if $lm (z) = 0$ and $z$ is purely imaginary if $Re (z) = 0$

2. Algebra of complex numbers:

For any two complex numbers, $z_{1} = a_{1} + i b_{1}$ and $z_{2} = a_{2} + {i b}_{2}$ .

(i) Addition: $z_{1} + z_{2} = (a_{1} + a_{2}) + i (b_{1} + b_{2})$

(ii) Subtraction: $z_{1} - z_{2} = (a_{1} - a_{2}) + i (b_{1} - b_{2})$

(iii) Multiplication: $z_{1} z_{2} = (a_{1} a_{2} - b_{1} b_{2}) + i (a_{1} b_{2} + a_{2} b_{1})$

(iv) Reciprocal: $\frac{1}{z_{1}} = \frac{a_{1}}{{a_{1}}^{2} + {b_{1}}^{2}} - i \frac{b_{1}}{{a_{1}}^{2} + {b_{1}}^{2}}$

(v) Division: $\frac{z_{1}}{z_{2}} = z_{1} (\frac{1}{z_{2}}) = (a_{1} + i b_{1}) (\frac{a_{2}}{{a_{2}}^{2} + {b_{2}}^{2}} - i \frac{b_{2}}{{a_{2}}^{2} + {b_{2}}^{2}}) = \frac{a_{1} a_{2} + b_{1} b_{2}}{{a_{2}}^{2} + {b_{2}}^{2}} + i \frac{a_{2} b_{1} - a_{1} b_{2}}{{a_{2}}^{2} + {b_{2}}^{2}}$

(vi) Every non-zero complex number $z = a + i b$ has its multiplicative inverse $\frac{1}{z}$ (also known as reciprocal of $z$ ) such that $\frac{1}{z} = \frac{a - i b}{a^{2} + b^{2}} = \frac{\bar{z}}{{|z|}^{2}} .$

3. Conjugate of a complex number and its properties:

(i) The conjugate of a complex number $z = a + i b$ is denoted by $\bar{z}$ and is equal to $a - i b$

(ii) For any three complex numbers $z, z_{1}, z_{2}$ we have

(a) $(\bar{z}) = z$

(b) $z + \bar{z} = 2 Re (z)$

(c) $z - \bar{z} = 2 i Im (z)$

(d) $z = \bar{z} \Leftrightarrow z$ is purely real.

(e) $z + \bar{z} = 0 \Leftrightarrow z$ is purely imaginary.

(f) $z z = {\{Re (z)\}}^{2} + {\{Im (z)\}}^{2} = {|z|}^{2}$

(g) $\bar{z_{1} \pm z_{2}} = \bar{z_{1}} \pm \bar{z_{2}}$

(h) $\bar{z_{1} z_{2}} = z_{1} z_{2}$

(i) $\bar{(\frac{z_{1}}{z_{2}})} = \frac{\bar{z_{1}}}{\bar{z_{2}}}, z_{2} \neq 0$

4. Modulus of a complex number and its properties:

(i) The modulus of a complex number $z = a + i b$ is denoted by $|z|$ and is defined as $|z| = \sqrt{a^{2} + b^{2}} = \sqrt{{\{R e (z)\}}^{2} + {\{I m (z)\}}^{2}}$

(ii) If $z, z_{1}, z_{2}$ are three complex numbers, then

(a) $|z| = 0 \Leftrightarrow z = 0$ i.e. $Re (z) = Im (z) = 0$

(b) $|z| = |\bar{z}| = |- z|$

(c) $- |z| \leq Re (z) \leq |z|; - |z| \leq lm (z) \leq |z|$

(d) $z \bar{z} = {|z|}^{2} = {|z|}^{2}$

(e) $|lm (z^{n})| \leq n |lm (z)| {|z|}^{n - 1}, n \in N$

(f) $|R e (z)| + |I m (z)| \leq \sqrt{2} |z|$

5. Argument or Amplitude of a complex number:

(i) The angle $θ$ which $O P$ makes with the positive direction of $x$ -axis in anti-clockwise sense is called the argument or amplitude of $z$ and is denoted by arg $(z)$ or amp $(z)$ .

(ii) $\tan θ = \frac{I m (z)}{R e (z)} \Rightarrow θ = \tan^{- 1} (\frac{Im (z)}{Re (z)})$

6. Representation of a complex number:

(i) $z = x + i y = r (\cos θ + i \sin θ)$ is called as polar form of a complex number.

(ii) The Euler's notations is $e^{i θ} = \cos θ + i \sin θ$ . $z = r e^{i θ}$ is known as the Euler's form of $z .$

7. Quadratic equations:

(i) A quadratic equation cannot have more than two roots.

(ii) If $a x^{2} + b x + c = 0, a \neq 0$ is a quadratic equation with real coefficients, then its roots $α$ and $β$ given by $α = \frac{- b + \sqrt{D}}{2 a}$ and $β = \frac{- b - \sqrt{D}}{2 a}$ , where $D = b^{2} - 4 a c$ is the discriminant of the quadratic equation.

8. Nature of roots of a quadratic equation:

(i) If $D = 0,$ then roots are real and equal.

(ii) If $a, b, c \in Q$ and $D$ is positive and a perfect square, then roots are rational and unequal.

(iii) If $a, b, c \in R$ and $D$ is positive and a perfect square, then the roots are real and distinct.

(iv) If $D > 0$ but it is not a perfect square, then roots are irrational and unequal.

(v) If $D < 0,$ then the roots are imaginary and are given by $α = \frac{- b + i \sqrt{4 a c - b^{2}}}{2 a}$ and $β = \frac{- b - i \sqrt{4 a c - b^{2}}}{2 a}$ .