The number of points in -∞,∞, for which x2-xsinx-cosx=0, is:

JEE Advanced · Mathematics · 25. AOD

The number of points in $(- \infty, \infty)$ , for which $x^{2} - x \sin x - \cos x = 0$ , is:

A $6$
B $4$
C $2$
D $0$

Verified Solution

Answer & Solution

Correct Answer

(C) $2$

Step-by-step Solution

Detailed explanation

f (x) = x^{2} - x \sin x - \cos x

f^{'} (x) = 2 x - x \cos x - \sin x + \sin 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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For a polynomial $g (x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g (x)$ . Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{{(x^{2} - 1)}^{2} (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}) : a_{0}, a_{1}, a_{2}, a_{3} \in R\}$ . For a polynomial $f$ , let $f'$ and $f^{″}$ denote its first and second order derivatives respectively. Then the minimum possible value of $(m_{f^{'}} + m_{f^{″}})$ , where $f \in S$ , is ____

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Match the following.

	List – I		List – II
(A)	Let $y (x) = \cos (3 \cos^{- 1} x), x ϵ [- 1, 1], x \neq \pm \frac{\sqrt{3}}{2} .$ Then $\frac{1}{y (x)} \{(x^{2} - 1) \frac{d^{2} y (x)}{d x^{2}} + x \frac{d y (x)}{d x}\}$ equals	(P)	1
(B)	Let $A_{1}, A_{2}, \dots, A_{n} (n > 2)$ be the vertices of a regular polygon of n sides with its centre at the origin. Let $\vec{a_{k}}$ be the position vector of the point $A_{k}, k = 1, 2, \dots n .$ If $\|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \times \vec{a_{k + 1}})\| = \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \cdot \vec{a_{k + 1}})\|$ , then the minimum value of n is	(Q)	2
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(D)	Number of positive solutions satisfying the equation $\tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}})$ is	(S)	9

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List-I shows various functional dependencies of energy $(E)$ on the atomic number $(Z)$. Energies associated with certain phenomena are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

List - I	List - II
(P) $E\text { propto } Z^2$	(1) energy of characteristic x-rays
(Q) $E\text { propto } (Z-1)^2$	(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
(R) $E \text { propto } Z(Z-1)$	(3) energy of continuous x-rays
(S) E is practically independent of Z	(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
	(5) energy of radiation due to electronic transitions from hydrogen-like atoms

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List - I	List - II
(P) \(E\text { propto } Z^2\)	(1) energy of characteristic x-rays
(Q) \(E\text { propto } (Z-1)^2\)	(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
(R) \(E \text { propto } Z(Z-1)\)	(3) energy of continuous x-rays
(S) E is practically independent of Z	(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
	(5) energy of radiation due to electronic transitions from hydrogen-like atoms