Which of the following is true for all possible non-zero choices of integers m, n ; m \neq n, or all possible non-zero choices of real numbers p, q ; p \neq q, as applicable?

Correct Answer is : 12π∫−ππsinpθcosqθdθ=0

\frac{1}{2 \pi} \int_{-\pi / 2}^{\pi \/ 2} \sin p \theta \sin q \theta d \theta=0

\frac{1}{\pi} \int_{0}^{\pi} \sin m \theta \sin n \theta d \theta=0

\frac{1}{2 \pi} \int_{-\pi}^{\pi} \sin p \theta \cos q \theta d \theta=0

\lim_{\alpha \to \infty} \frac{1}{2 \alpha} \int_{-\alpha}^{\alpha} \sin p \theta \sin q \theta d \theta=0

GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 21 of 65

Marks: +1, -0

Which of the following is true for all possible non-zero choices of integers

m, n ; m ≠ n

, or all possible non-zero choices of real numbers

p, q ; p ≠ q

, as applicable?

Solution:

∫↙{-1}↖{1} \cos (\;{n π x}/{1}) \cos (\;{n π x}/{1}) d x=\{\table 21 , \;\text " if "\; , n=m=0 ; 1 , \;\text " if "\; , n=m ≠ 0 ; 0 , \;\text " if "\; , m ≠ n.

∫↙{0}↖{1} \cos (\;{n π x}/{1}) \cos (\;{n π x}/{1}) d x=\{\table 1 , \;\text " if "\; , n=m=0 ; 1 / 2 , \;\text " if "\; , n=m ≠ 0 ; 0 , \;\text " if "\; , m ≠ n.

∫↙{-1}↖{1} \sin (\;{ {n} π {x}}/{ {I}}) \sin (\;{ {n} π {x}}/{ {I}}) {dx}=\{\table 1 , \;\text " if "\; , {m}= {n} ; 0 , \;\text " if "\; , {m} ≠ {n}.

∫↙{0}↖{1} \sin (\;{ {n} π {x}}/{ {I}}) \cos (\;{ {n} π {x}}/{ {I}}) {dx}=\{\table 1 / 2 , \;\text " if "\; , {n}= {m} ; 0 , \;\text " if "\; , {n} ≠ {m}.

∫↙{-1}↖{1} \sin (\;{n π x}/{1}) \cos (\;{n π x}/{1}) d x=0

\;{1}/{π} \sin m θ \cos n θ=0

Put

I=π

in rule 4

∫↙{0}↖{π} \sin (\;{n π x}/{π}) \sin (\;{m π}/{π}) d x

Given that,

m ≠ n

\;{1}/{π} ∫↙{0}↖{π} \sin x \sin m x d x=0

(C).

\;{1}/{2 π} ∫↙{-π}↖{π} \sin p θ \cos θ d θ=0

Put

1=π

in rule 5

∫↙{-π}↖{π} \sin (\;{n π x}/{π}) \cos (\;{m π x}/{π}) d x

Given that

m ≠ 0

∫↙{-π}↖{π} \sin n x \cos m x d x=0

(D).

{Lt}↙{α → ∞} \;{1}/{2 α} ∫↙{-α}↖{a} \sin p θ d θ=0

When,

a → ∞

\;{1}/{2 ∞} ∫↙{-∞}↖{∞} \sin p θ \sin θ d θ=\;{1}/{∞}(

finite

)=0

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