The real value of ‘a’ for which 3i^3 – 2ai^2 + (1 – a)i + 5 is real is ________.

Question

(i) The real value of ‘a’ for which 3i³ – 2ai² + (1 – a)i + 5 is real is ________.

(ii) If |z| = 2 and arg (z) = π/4, then z = ________.

(iii) The locus of z satisfying arg (z) = π/3 is _______.

(iv) The value of (− √−1)^4n–3 , where n ∈ N, is ______.

(v) The conjugate of the complex number (1-i)/(1+i) is _____.

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

(vii) If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n²) = ______.

Answer 1

(i) 3i³ – 2ai² + (1 – a)i + 5 = –3i + 2a + 5 + (1 – a)i

= 2a + 5 + (– a – 2) i, which is real if – a – 2 = 0 i.e. a = – 2.

(iii) Let z = x + iy. Then its polar form is z = r (cos θ + i sin θ), where tan θ = y/x and θ is arg (z). 

Given that θ =π/3 . Thus.

Hence, locus of z is the part of y = √3x in the first quadrant except origin.

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

(vii) Given that

(2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy ... (1)

⇒ Bar (2 + i) Bar(2 + 2i) Bar(2 + 3i) ... Bar(2 + ni) = Bar(x + iy)=(x - iy)

i.e., (2 – i) (2 – 2i) (2 – 3i) ... (2 – ni) = x – iy ... (2)

Multiplying (1) and (2), we get 5.8.13 ... (4 + n²) = x² + y².