If gx=∫sinxsin2xsin-1tdt, then

JEE Advanced · Mathematics · 27. Definite Integration

If $g (x) = \int_{\sin x}^{\sin 2 x} \sin^{- 1} t d t,$ then

A $g^{'} (- \frac{π}{2}) = 0$
B $g^{'} (- \frac{π}{2}) = 2 π$
C $g^{'} (\frac{π}{2}) = 2 π$
D $g^{'} (- \frac{π}{2}) = - 2 π$

Verified Solution

Answer & Solution

Correct Answer

(A) $g^{'} (- \frac{π}{2}) = 0$

Step-by-step Solution

Detailed explanation

Given,

g (x) = \int_{\sin x}^{\sin 2 x} \sin^{- 1} (t) d t . . . . . . . . (i)

D.w.r.to x both side

g (x) = \int_{\sin x}^{\sin 2 x} \sin^{- 1} t d t \Rightarrow g^{'} (x) = 2 \sin^{- 1} (\sin 2 x) \times \cos 2 x - \sin^{- 1} (\sin x) \cos x

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.

Same subject

Consider a triangle $P Q R$ having sides of lengths $p, q$ and $r$ opposite to the angles $P, Q$ and $R$ , respectively. Then which of the following statements is (are) TRUE?

JEE Advanced 2021 Hard

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3} x + 2 y = 4

, then which of the following options is the only Correct combination?

JEE Advanced 2017 Hard

Consider the given data with frequency distribution

\begin{array}{l} x_{i} & 3 & 8 & 11 & 10 & 5 & 4 \\ f_{i} & 5 & 2 & 3 & 2 & 4 & 4 \end{array}

Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	The mean of the above data is	$(1)$	$2.5$
$(Q)$	The median of the above data is	$(2)$	$5$
$(R)$	The mean deviation about the mean of the above data is	$(3)$	$6$
$(S)$	The mean deviation about the median of the above data is	$(4)$	$2.7$
		$(5)$	$2.4$

The correct option is

JEE Advanced 2023 Easy

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From JEE Advanced

List-I	List-II
(I)	$(P)$
(II)	$(Q)$
(III)	$(R)$
(IV)	$(S)$
	$(T)$
	$(U)$

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