Class 11 – Mathematics – Chapter 5 – Complex Numbers Quadratic Equations

Q6. If a = cos Î¸ + i sin Î¸, then find the value of (1+a/1-a)
Sol:  a = cos Î¸ + i sin Î¸

Q10. Show that the complex number z, satisfying the condition arg lies on arg (z-1/z+1) = Ï€/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

Q12. If |z + 1|  = z + 2( 1 + i), then find the value of z.
Sol: We have |z + 1
1 = z + 2(1+ i)
Putting z = x + iy, we get
Then, |x +
iy + 11 = x + iy + 2(1 + i)
âŸ¹|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 â‰ 1), then find the value of |z|.

Sol: Let   z = x + iy

Q17. If |z1 | = 1 (z1â‰  -1) and z2 = z1 â€“ 1/ z1 + 1 , then show that real part of z2 is zero .

Q18. If Z1, Z2 and Z3, Z4 are two pairs of conjugate complex numbers, then find arg (Z1/ Z4) + arg (Z2/ Z3)
Sol. It is given that z1 and z2 are conjugate complex numbers.

Q20. If for complex number z1 and z2, arg (z1) – arg (z2) = 0, then show that |z1 – z2| = | z1|- |z2 |

Q21. Solve the system of equations Re (z2) = 0, |z| = 2.

Sol: Given that, Re(z2) = 0, |z| = 2

Q22. Find the complex number satisfying the equation z + âˆš2 |(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 â€” 11= |z â€” 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 â‰  0 and Re(z) = 0, then Im (z2) = 0.
(vi) The inequality |z – 4| < |z – 2| represents the region given by x > 3.
(vii) Let Z1  and Z2 be two complex numbers such that |z, + z2| = |z1 j + |z2|, then arg (z1 – z2) = 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, â€”iz = -i(x + iy)
= -ix – i2y = y – ix
Now, point B(y, – x) lies in fourth quadrant. Also, âˆ 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 + âˆš3 is (i) Perpendicular bisector of segment joining (-2, 0) and (2,0) (b) The amplitude of- 1 + âˆš-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 â€“ 2i|, then locus of z is (iv) Perpendicular bisector of segment joining (0, -2) and (0,2) (e) Region represented by |z + 4i| â‰¥ 3 is (v) 2(cos /6 +I sin /6) (0 Region represented by |z + 4| â‰¤ 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 â€“ 2i)2

Q29. If |Z1| = |Z2|, is it necessary that Z1 = Z2?
Sol: If |Z1| = |Z2| then z1 and z2 are at the same distance from origin.
But if arg(Z1) â‰ arg(z2), then z1 and z2 are different.
So, if (z1| = |z2|, then it is not necessary that z1 = z2.
Consider Z1 = 3 + 4i and Z2 = 4 + 3i

Q30.If  (a2+1)2 / 2a â€“i = x + iy, then what is the value of x2 + y2?
Sol:
(a2+1)2 / 2a â€“i = x + iy

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

Q34. Where does z lies, if | z â€“ 5i / z + 5i  |  = 1?
Sol:
We have  | z â€“ 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 z1 and z2?