If the line x=α divides the area of the region R={x,y∈R2:x3≤y≤x, 0≤x≤1} into two equal parts, then

JEE Advanced · Mathematics · 28. Area Under Curves

If the line $x = α$ divides the area of the region $R = {(x, y) \in R^{2} : x^{3} \leq y \leq x, 0 \leq x \leq 1}$ into two equal parts, then

A $\frac{1}{2} < α < 1$
B $α^{4} + 4 α^{2} - 1 = 0$
C $0 < α \leq \frac{1}{2}$
D $2 α^{4} - 4 α^{2} + 1 = 0$

Verified Solution

Answer & Solution

Correct Answer

(D) $2 α^{4} - 4 α^{2} + 1 = 0$

Step-by-step Solution

Detailed explanation

A = \int_{0}^{1} (x - x^{3}) d x = \frac{1}{4}

Area of curvilinear triangle

O P Q = \frac{A}{2} = \frac{1}{8}

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 $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	(1)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$,is an increasing function on $\mathbb{R}$,is	(2)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$,is	(3)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}$,is NOT differentiable at $x_0$,is	(4)	6
		(5)	10

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Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the

n^{t h}

orbit of the atom and List-II contains options showing how they depend on

n

List - I	List - II
(I) Radius of the $n^{t h}$ orbit	$(P)$ $\propto n^{- 2}$
(II) Angular momentum of the electron in the $n^{t h}$ orbit	$(Q)$ $\propto n^{- 1}$
(III) Kinetic energy of the electron in the $n^{t h}$ orbit	$(R)$ $\propto n^{0}$
(IV) Potential energy of the electron in the $n^{t h}$ orbit	$(S)$ $\propto n^{1}$
	$(T)$ $\propto n^{2}$
	$(U)$ $\propto n^{1 / 2}$

Which of the following options has the correct combination considering List-I and List-II?

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	LIST - I		LIST - II
(P)	The minimum value of \(n\) for which the function \(f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]\) is continuous on the interval \([1,2]\),is	(1)	8
(Q)	The minimum value of \(n\) for which \(g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}\),is an increasing function on \(\mathbb{R}\),is	(2)	9
(R)	The smallest natural number \(n\) which is greater than 5,such that \(x=3\) is a point of local minima of \(h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)\),is	(3)	5
(S)	Number of \(x_0 \in \mathbb{R}\) such that \(l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}\),is NOT differentiable at \(x_0\),is	(4)	6
		(5)	10