Complex Numbers(11th And 12th > Mathematics ) Questions and Answers
Question 1.
For positive integers n1,n2 the value of the expression (1+i)n1+(1+i3)n1+(1+i5)n2 + (1+i7)n2 where i=2√−1 is a real number if and only if
n1=n2+1
n1=n2-1
n1=n2
n1>0,n2>0
Explanation:-
Answer: Option D. -> n1>0,n2>0
: D
Using i3=−i,i5=iandi7=−i, we can write the given expression as (1+i)n1+(1+i3)n1+(1+i5)n2+(1+i7)n2wherei=2√1=2[n1C0+n1C2(i)2+n1C4(i)4+n1C6(i)6+⋯⋯]+2[n2C0+n2C2(i)2+n2C4(i)4+n2C6(i)6+⋯⋯]=2[n1C0−n1C2+n1C4−n1C6+⋯⋯]+2[n2C0−n2C2+n2C4−n2C6+⋯⋯] This is a real number irrespective of values of n1 and n2.
Question 2.
The conjugate of (2+i)23+i , in the form of a+ib, is
132+i(152)
1310+i(−152)
1310+i(−910)
1310+i(910)
Explanation:-
Answer: Option C. -> 1310+i(−910) : C
z = (2+i)23+i=3+4i3+i×3−i3−i=1310+i910
Conjugate = 1310−i910.
Question 3.
If (1+i)(1+2i)(1+3i)......(1+ni) = a+ib, then 2.5.10.....(1+n2) is equal to