AP EAMCET · Maths · Indefinite Integration

If \(\int e^{\sin x}(1+\sec x \tan x) d x=e^{\sin x} f(x)+c\), then in \(0 \leq x \leq 2 \pi\), the number of solutions of \(f(x)=1\) is

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Answer & Solution

Correct Answer

(C) 2

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Detailed explanation

\(\frac{d}{dx}(e^{\sin x} f(x)) = e^{\sin x} f'(x) + f(x) e^{\sin x} \cos x\) \(e^{\sin x} f'(x) + f(x) e^{\sin x} \cos x = e^{\sin x}(1+\sec x \tan x)\) \(f'(x) + f(x) \cos x = 1+\sec x \tan x\) Comparing terms, let \(f(x) = \sec x\). \(f'(x) = \sec x \tan x\)…

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.

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\(P\) is a variable point on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with foci \(F_1\) and \(F_2\). If \(A\) is the area of the triangle \(P F_1 F_2\), then the maximum value of \(A\) is

AP EAMCET 2019 Medium

If \(Z_1\) and \(Z_2\) are conjugate complex numbers. Match the items under the following columns?

Column-l	Column-II
A. \(Z_1 Z_2\)	1. imaginary axis
B. \(Z_1+Z_2=0\)	2. \(I_m\left(-Z_2\right)\)
C. \(I_m\left(Z_1\right)\)	3. \(\left\|Z_1\right\|^2\)
D. \(\operatorname{Re}\left(Z_1\right)\)	4. \(\operatorname{Re} Z_2\)

AP EAMCET 2021 Easy

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