એક પદાર્થ પર x-દિશામાં \(\mathrm{F}=\alpha+\beta \mathrm{x}^2\) બળ લાગે છે. પદાર્થનું 1 m જેટલું સ્થાનાંતર થાય ત્યારે બળ વડે થતું કાર્ય 5 J છે. જો અચળાંક \(\alpha=1 \mathrm{~N}\) હોય, તો \(\beta\) નું મૂલ્ય કેટલું હશે?

A \(15 \mathrm{~N} / \mathrm{m}^2\)
B \(12 \mathrm{~N} / \mathrm{m}^2\)
C \(8 \mathrm{~N} / \mathrm{m}^2\)
D \(10 \mathrm{~N} / \mathrm{m}^2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(12 \mathrm{~N} / \mathrm{m}^2\)

Step-by-step Solution

Detailed explanation

\(F=\alpha+\beta x^2\) કાર્ય \(\int d w=\int F \cdot d x\) \(\begin{aligned} & \Rightarrow \quad \Delta W=\int F \cdot d x=\int\left(\alpha+\beta x^2\right) d x \\ & \Rightarrow \quad \Delta W=\left|\alpha x+\frac{\beta x^3}{3}\right|_0^1=\alpha+\frac{\beta}{3}=5 \end{aligned}\)…

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

સૂચી \(-I\) ને સૂચી \(- II\) સાથે મેળવો.

સૂચી \(-I\)	સૂચી \(-II\)
\((a)\) \(h\) (પ્લાન્કનો અચળાંક)	\((i)\) \(\left[ M L T ^{-1}\right]\)
\((b)\) \(E\) (ગતિ ઊર્જા)	\((ii)\) \(\left[ M L ^{2} T ^{-1}\right]\)
\((c)\) \(V\) (વિદ્યુત સ્થિતિમાન)	\((iii)\) \(\left[ M L ^{2} T ^{-2}\right]\)
\((d)\) \(P\) (રેખીય વેગમાન)	\(( iv )\left[ M L ^{2} I ^{-1} T ^{-3}\right]\)

નીચે આપેલા વિકલ્પોમાંથી સાચા જવાબનું ચયન કરો.

JEE Mains 2021 Medium

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