Match the following List-I with List-II P)sin[π900{∑r=110(r−ω)(r−ω2)}] is equal to1)0Q)If the roots of t2+t+1=0 be α,β, then α4+β4+α−1β−1 is equal to2)4R)If [1+cosθ+isinθsinθ+i(1+cosθ)]4=cosnθ+isinnθ, then n is equal to3)iS)If zr=cosπ3r+isinπ3r (where r=1,2,3,....), then value of z1z2z3...∞ is equal to4)1

(P) ∑e=110(r−ω)(r−ω2)=∑r=110{r2−r(−1)+1}(Q) α4+β4+α−1β−1=ω4+ω8+ω−1ω−2=ω+ω2+1=0(R) NT=1+cosθ+isinθ=2cosθ2[cosθ2+isinθ2]=2cosθ2⋅eiθ/2 Dr=−i2sinθ+i(1+cosθ)=i[conjugate of Nr]=i2cosθ2⋅e−iθ/2(S) π3+π32+π33+......∞=π2

Match the following List-I with List-II P)sin[π900{∑r=110(r−ω)(r−ω2)}] is equal to1)0Q)If the roots of t2+t+1=0 be α,β, then α4+β4+α−1β−1 is equal to2)4R)If [1+cosθ+isinθsinθ+i(1+cosθ)]4=cosnθ+isinnθ, then n is equal to3)iS)If zr=cosπ3r+isinπ3r (where r=1,2,3,....), then value of z1z2z3...∞ is equal to4)1

(P) ∑e=110(r−ω)(r−ω2)=∑r=110{r2−r(−1)+1}(Q) α4+β4+α−1β−1=ω4+ω8+ω−1ω−2=ω+ω2+1=0(R) NT=1+cosθ+isinθ=2cosθ2[cosθ2+isinθ2]=2cosθ2⋅eiθ/2 Dr=−i2sinθ+i(1+cosθ)=i[conjugate of Nr]=i2cosθ2⋅e−iθ/2(S) π3+π32+π33+......∞=π2

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Match the following List-I with List-II

P) $\sin [\frac{π}{900} {\sum_{r = 1}^{10} (r - ω) (r - ω^{2})}]$ is equal to 1) 0
Q) If the roots of $t^{2} + t + 1 = 0$ be $α, β,$ then $α^{4} + β^{4} + α^{- 1} β^{- 1}$ is equal to 2) 4
R) If ${[\frac{1 + \cos θ + i \sin θ}{\sin θ + i (1 + \cos θ)}]}^{4} = \cos n θ + i \sin n θ,$ then n is equal to 3) i
S) If $z_{r} = \cos \frac{π}{3^{r}} + i \sin \frac{π}{3^{r}}$ (where r=1,2,3,....), then value of $z_{1} z_{2} z_{3} ... \infty$ is equal to 4) 1

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P-4; Q-1; R-3; S-2

P-1; Q-4; R-3; S-2

P-2; Q-3; R-4; S-1

P-4; Q-1; R-2; S-3

answer is D.

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Detailed Solution

(P) $\sum_{e = 1}^{10} (r - ω) (r - ω^{2}) = \sum_{r = 1}^{10} {r^{2} - r (- 1) + 1}$
(Q) $α^{4} + β^{4} + α^{- 1} β^{- 1} = ω^{4} + ω^{8} + ω^{- 1} ω^{- 2} = ω + ω^{2} + 1 = 0$
(R) $N^{T} = 1 + \cos θ + i \sin θ = 2 \cos \frac{θ}{2} [\cos \frac{θ}{2} + i \sin \frac{θ}{2}] = 2 \cos \frac{θ}{2} \cdot e^{i θ / 2}$
$D^{r} = - i^{2} \sin θ + i (1 + \cos θ) = i [conjugate of N^{r}] = i 2 \cos \frac{θ}{2} \cdot e^{- i θ / 2}$
(S) $\frac{π}{3} + \frac{π}{3^{2}} + \frac{π}{3^{3}} + ...... \infty = \frac{π}{2}$

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P)	$\sin [\frac{π}{900} {\sum_{r = 1}^{10} (r - ω) (r - ω^{2})}]$ is equal to	1)	0
Q)	If the roots of $t^{2} + t + 1 = 0$ be $α, β,$ then $α^{4} + β^{4} + α^{- 1} β^{- 1}$ is equal to	2)	4
R)	If ${[\frac{1 + \cos θ + i \sin θ}{\sin θ + i (1 + \cos θ)}]}^{4} = \cos n θ + i \sin n θ,$ then n is equal to	3)	i
S)	If $z_{r} = \cos \frac{π}{3^{r}} + i \sin \frac{π}{3^{r}}$ (where r=1,2,3,....), then value of $z_{1} z_{2} z_{3} ... \infty$ is equal to	4)	1