CLASS XI COMPLEX NUMBER DPP 2
COMPLEX NUMBER
Answer each of the following in one word or one sentence or as per exact requirement of the question :
1. Write the values of the square root of i.
2. Write the values of the square root of -i.
3. If x + iy = √(a+ib)/(c+id), then write the value of (x² + y²)².
4. If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of |z|.
5. If n is any positive integer, write the value of (i(4n+1) - i(4n-1))/2.
6. Write the value of
(i592 + i590 + i588 + i586 + i584) / (i582 + i580 + i578 + i576 + i574).
7. Write 1 - i in polar form.
8. Write -1 + i√3 in polar form.
9. Write the argument of -i. [NCERT]
10. Write the least positive integral value of n for which
((1+i)/(1-i))n is real.
11. Find the principal argument of (1+i√3)².
12. Find z, if |z| = 4 and arg (z) = 5π/6.
13. If |z-5i| = |z+5i|, then find the locus of z.
14. If (a²+1)²/(2a-i) = x+iy, find the value of x² + y².
15. Write the value of √-25 x √-9.
16. Write the sum of the series i + i² + i³ + ...upto 1000 terms.
17. Write the value of arg (z) + arg (z̄).
18. If |z + 4| ≤ 3, then find the greatest and least values of |z + 1|.
19. For any two complex numbers z₁ and z₂ and any two real
numbers a, b, find the value of |az₁-bz₂|² + |az₂ + bz₁|².
20. Write the conjugate of (2-i)/(1-2i)².
21. If n ∈ N, then find the value of iⁿ + iⁿ⁺¹ + iⁿ⁺² + iⁿ⁺³.
22. Find the real value of a for which 3i³ - 2ai² + (1-a)i + 5 is real.
23. If |z| = 2 and arg (z) = π/4, find z.
24. Write the argument of (1+√3) (1+i) (cosθ+isinθ).
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