**NCERT Exemplar Problems Class 11 Mathematics Chapter 5 Complex Numbers and Quadratic Equations**

**Short Answer Type Questions
**

**Q6. If a = cos θ + i sin θ, then find the value of (1+a/1-a) **

** Sol:** a = cos θ + i sin θ

**Q10. Show that the complex number z, satisfying the condition arg lies on arg (z-1/z+1) = π/4 lies on a circle.
**

**Sol:** Let z = x + iy

**Q11. Solve the equation |z| = z + 1 + 2i.**

**Sol:** We have |z| = z + 1 + 2i

Putting z = x + iy, we get

|x + iy| = x + iy + 1+2i

**Long Answer Type Questions**

**Q12. If |z + 1| ****= z + 2( 1 + i), then find the value of z. **

**Sol:** We have |z + 11 = z + 2(1+ i)

Putting z = x + iy, we get

Then, |x +iy + 11 = x + iy + 2(1 + i)

⟹|x + iy + l|=x + iy + 2(1 +i)

**Q13. If arg (z – 1) = arg (z + 3i), then find (x – 1) : y, where z = x + iy. **

**Sol:** We have arg (z – 1) = arg (z + 3i), where z = x + iy

=> arg (x + iy – 1) = arg (x + iy + 3i)

=> arg (x – 1 + iy) = arg [x + i(y + 3)]

**Q14. Show that | z-2/z-3| = 2 represents a circle . Find its center and radius .
Sol:**We have | z-2/z-3| = 2

Puttingz=x + iy, we get

**Q15. If z-1/z+1 is a purely imaginary number (z ≠1), then find the value of |z|.**

**Sol:** Let z = x + iy

**Q17. If |z _{1} | = 1 (z_{1}≠ -1) and z_{2 }= z_{1} – 1/ z_{1} + 1 , then show that real part of z_{2} is zero** .

**Q18. If Z _{1}, Z_{2} and Z_{3}, Z_{4} are two pairs of conjugate complex numbers, then find arg (Z_{1/} Z_{4}) + arg (Z_{2/} Z_{3})**

Sol. It is given that z_{1} and z_{2} are conjugate complex numbers.

**Q20. If for complex number z _{1}**

**and z**

_{2}, arg (z_{1}) – arg (z_{2}) = 0, then show that |z_{1}– z_{2}| = | z_{1}|- |z_{2}|

Q21. Solve the system of equations Re (z^{2}) = 0, |z| = 2.

**Sol:** Given that, Re(z^{2}) = 0, |z| = 2

**Q22. Find the complex number satisfying the equation z + √2 |(z + 1)| + i = 0.**

**Fill in the blanks **

True/False Type Questions

**Q26. State true or false for the following.**

**(i) The order relation is defined on the set of complex numbers.**

**(ii) Multiplication of a non-zero complex number by -i rotates the point about origin through a right angle in the anti-clockwise direction.**

**(iii) For any complex number z, the minimum value of |z| + |z – 11 is 1.**

**(iv) The locus represented by |z — 11= |z — i| is a line perpendicular to the join of the points (1,0) and (0, 1).**

**(v) If z is a complex number such that z ≠ 0 and Re(z) = 0, then Im (z ^{2}) = 0.**

**(vi) The inequality |z – 4| < |z – 2| represents the region given by x > 3.**

**(vii) Let Z**

_{1}and Z_{2}be two complex numbers such that |z, + z_{2}| = |z_{1}j + |z_{2}|, then arg (z_{1}– z_{2}) = 0.**(viii) 2 is not a complex number.**

**Sol:**(i) False

We can compare two complex numbers when they are purely real. Otherwise comparison of complex numbers is not possible or has no meaning.

(ii) False

Let z = x + iy, where x, y > 0

i.e., z or point A(x, y) lies in first quadrant. Now, —iz = -i(x + iy)

= -ix – i^{2}y = y – ix

Now, point B(y, – x) lies in fourth quadrant. Also, ∠AOB = 90°

Thus, B is obtained by rotating A in clockwise direction about origin.

**Matching Column Type Questions**

**Q24. Match the statements of Column A and Column B.
**

Column A | Column B | ||

(a) | The polar form of i + √3 is | (i) | Perpendicular bisector of segment joining (-2, 0) and (2,0) |

(b) | The amplitude of- 1 + √-3 is | (ii) | On or outside the circle having centre at (0, -4) and radius 3. |

(c) | It |z + 2| = |z – 2|, then locus of z is | (iii) | 2/3 |

(d) | It |z + 2i| = |z – 2i|, then locus of z is | (iv) | Perpendicular bisector of segment joining (0, -2) and (0,2) |

(e) | Region represented by |z + 4i| ≥ 3 is | (v) | 2(cos /6 +I sin /6) |

(0 | Region represented by |z + 4| ≤ 3 is | (Vi) | On or inside the circle having centre (-4,0) and radius 3 units. |

(g) | Conjugate of 1+2i/1-I lies in | (vii) | First quadrant |

(h) | Reciprocal of 1 – i lies in | (viii) | Third quadrant |

**Q28. What is the conjugate of 2-i / (1 – 2i) ^{2
}**

**Q29. If |Z _{1}| = |Z_{2}|, is it necessary that Z_{1} = Z_{2}?**

**Sol:**If |Z

_{1}| = |Z

_{2}| then z

_{1}and z

_{2}are at the same distance from origin.

But if arg(Z

_{1}) ≠arg(z

_{2}), then z

_{1}and z

_{2}are different.

So, if (z

_{1}| = |z

_{2}|, then it is not necessary that z

_{1}= z

_{2}.

Consider Z

_{1}= 3 + 4i and Z

_{2}= 4 + 3i

**Q30.If (a ^{2}+1)^{2 }/ 2a –i = x + iy, then what is the value of x^{2} + y^{2}?**(a

Sol:

^{2}+1)

^{2 }/ 2a –i = x + iy

**Q31. Find the value of z, if |z| = 4 and arg (z) = 5π/6
**

**Q34. Where does z lies, if | z – 5i / z + 5i | = 1?
Sol:**We have | z – 5i / z + 5i |

**Instruction for Exercises 35-40: Choose the correct answer from the given four options indicated against each of the Exercises.**

**Q35. sin x + i cos 2x and cos x – i sin 2x are conjugate to each other for
**

**Q41. Which of the following is correct for any two complex numbers z _{1} and z_{2}?**