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Question

List I has four entries and List II has five entries. Each entry of List I is to be correctly matched with one or more than one entries of List II. List IList II (A)Possible value(s) of √i+√−i is (are)(P)√2(B)If z3=¯¯¯z (z≠0),(Q)ithen possible values of z is/are(C)1+14+1⋅34⋅8+1⋅3⋅54⋅8⋅12+⋯⋯∞(R)√2i(D)132+1+142+2+152+3+⋯⋯∞(S)12(T)1336 Which of the following is CORRECT combination?

A
(C)(P) ; (D)(T)
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B
(C)(S) ; (D)(P)
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C
(C)(S) ; (D)(T)
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D
(C)(P) ; (D)(S)
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Solution

The correct option is A (C)→(P) ; (D)→(T)(C) 1+14+1⋅34⋅8+1⋅3⋅54⋅8⋅12+⋯⋯∞(1+x)n=1+nx+n(n−1)x22!+⋯ Comparing both, we get nx=14⋯(1)⇒n(n−1)x22=332 Using equation (1), we get n(n−1)32n2=332⇒n−1n=3⇒n=−12⇒x=−12 ∴ Required sum =(1−12)−1/2=√2 (C)→(P) (D) 132+1+142+2+152+3+⋯⋯∞ General term of the series is Tn=1n2+(n−2)n=3,4,5,……⇒Tn=13[1n−1−1n+2] Now, S=∞∑n=3Tn ⇒3S=∞∑n=3[1n−1−1n+2]⇒3S=[(12−15)+(13−16)+(14−17)+(15−18)+⋯]⇒3S=[12+13+14]∴S=1336 (D)→(T)

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