JEE Advanced · Mathematics · 28. Area Under Curves

A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P (0, 0), Q (1, 1)$ and $R (2, 0) .$ From this land, a neighbouring farmer $F_{2}$ takes away the region which lies between the side $P Q$ and a curve of the form $y = x^{n}, n > 1$ . If the area of the region taken away by the farmer $F_{2}$ is exactly $30%$ of the area of $∆ P Q R,$ then the value of $n$ is

Verified Solution

Answer & Solution

Correct Answer

(B) 4

Step-by-step Solution

Detailed explanation

Area

= \int_{0}^{1} (x - x^{n}) d x = \frac{3}{10}

{[\frac{x^{2}}{2} - \frac{x^{n + 1}}{n + 1}]}_{0}^{1} = \frac{3}{10}

\frac{1}{2} - \frac{1}{n + 1} = \frac{3}{10}

∴ n + 1 = 5

\Rightarrow n = 4

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

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty) .

•

Column 1 contains information about zeros of

f (x), f^{'} (x)

and

f^{''} (x) .

•

Column 2 contains information about the limiting behaviour of

f (x), f^{'} (x)

and

f^{''} (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f^{'} (x) .

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f^{'} (x) = 0$ for some $x i n (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in (0, 1)
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only Incorrect combination?

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