NCERT Class 11-Math՚S: Chapter – 5 Complex Numbers and Quadratic Equations Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 15:

If both satisfy then find .

Answer:

Let

Then

Since both satisfy (1) , we have

Again

Therefore, , where

From (2) , we get

Objective Type Questions

Question 16:

Fill in the blanks:

(i) The real value of ‘’ for which is real is ________.

(ii) If ________.

(iii) The locus of satisfying is ________.

(iv) The value of , where , is ________.

(v) The conjugate of the complex number is ________.

(vi) If a complex number lies in the third quadrant, then its conjugate lies in the ________.

(vii) If then ________.

Answer:

(i)

which is real if

(ii)

(iii) Let . Then its polar form is , where and is . Given that Thus.

where .

Hence, locus of is the part of in the first quadrant except origin.

(iv) Here

(v)

Hence, conjugate of is .

(vi) Conjugate of a complex number is the image of the complex number about the . Therefore, if a number lies in the third quadrant, then its image lies in the second quadrant.

(vii) Given that

i.e.. ,

Multiplying (1) and (2) , we get .

Question 17:

State true or false for the following:

(i) Multiplication of a non-zero complex number by i rotates it through a right angle in the anti- clockwise direction.

(ii) The complex number can be zero for some .

(iii) If a complex number coincides with its conjugate, then the number must lie on imaginary axis.

(iv) The argument of the complex number is

(vii) If is a positive integer, then the value of is

Answer:

(i) True. Let be complex number represented by OP. Then , represented by , where if is rotated in the anticlockwise direction through a right angle, it coincides with .

(ii) False. Because and . But there is no value of for which and both are zero.

(iii) False, because number lies on .

(iv) True,

(v) False, because

which gives .

(vi) False, because if are in A. P. , then is the midpoint of and , which implies that the points are collinear.

(vii) True, because