જો \(\int \frac{\sin x}{\sin (x-\alpha)} d x=p x-q \log |\sin (x-\alpha)|+c\) હોય, તો \(p q\) = _________.

A \(-\frac{1}{2} \sin 2 \alpha\)
B \(\sin 2 \alpha\)
C \(\frac{1}{2} \sin 2 \alpha\)
D \(-\sin 2 \alpha\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(-\frac{1}{2} \sin 2 \alpha\)

Step-by-step Solution

Detailed explanation

Let \(x-\alpha=t \Rightarrow x=t+\alpha \Rightarrow dx=dt\) \(\int \frac{\sin (t+\alpha)}{\sin t} d t = \int \frac{\sin t \cos \alpha+\cos t \sin \alpha}{\sin t} d t\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From GUJCET

સ્તંભ I ને સ્તંભ II સાથે જોડો અને સાચો વિકલ્પ લખો.

સ્તંભ-I		સ્તંભ-II
1.	કોષીય અવરોધ	(A)	મોંમાં રહેલી લાળ
2.	દેહધાર્મિક અવરોધ	(B)	ઇન્ટરફેરોન
3.	સાયટોકાઇન અવરોધ	(C)	કુદરતી મારક કોષો (એક પ્રકારના લિમ્ફોસાઇટ્સ)
4.	ભૌતિક અવરોધ	(D)	શ્વસન માર્ગનું શ્લેષ્મસ્તર

GUJCET 2021 Easy

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