Q.30 Physics ⭐ 360° Option Elimination Active

NEET 2021 Physics Question 30 Solution

Topic: Nuclei · Official NTA Booklet Code M1

Question

The half-life of a radioactive nuclide is 100 hours. The fraction of original activity that will remain after 150 hours would be:

✓

Verified Against NTA Final Answer Key

Option (2) : 1 / (2√2)

📚 Nuclei Easy

📋 Given Parameters & Problem Setup

Half-Life (T_{1/2}) 100 hours

Elapsed Time (t) 150 hours

⚡

The 30-Second Exam Hall Shortcut: ⚡ Half-Life Count: n = 150/100 = 1.5 = 3/2

⏱️ Save ~20 seconds

Number of half-lives $n = \frac{150}{100} = 1.5$ . Remaining fraction of activity is $\left(\frac{1}{2}\right)^{3/2} = \frac{1}{2^{3/2}} = \frac{1}{2\sqrt{2}}$ ! Instant 5-second calculation matching Option (2)!

1 Compute number of half-lives:

n = \frac{t}{T_{1/2}} = \frac{150\text{ h}}{100\text{ h}} = 1.5 = \frac{3}{2}

2 Remaining activity fraction:

\frac{A}{A_0} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^{3/2}

3 Simplify:

2^{3/2} = 2^1 \times 2^{1/2} = 2\sqrt{2}

4 Thus,

\frac{A}{A_0} = \frac{1}{2\sqrt{2}}

5 Match Option (2).

🛡️

Why You Can 100% Trust This Shortcut: Canonical Rutherford-Soddy exponential decay law and fractional half-life algebra.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Formulating the Radioactive Decay Law for Activity

The activity $A(t) = -\frac{dN}{dt} = \lambda N(t)$ follows the same exponential decay law as the number of undecayed nuclei. In terms of half-life $T_{1/2}$ :

A(t) = A_0 \left(\frac{1}{2}\right)^{t / T_{1/2}}

Step 2

Determining the Number of Elapsed Half-Lives

Given half-life $T_{1/2} = 100\text{ hours}$ and elapsed time $t = 150\text{ hours}$ :

n = \frac{t}{T_{1/2}} = \frac{150}{100} = 1.5 = \frac{3}{2}

Step 3

Evaluating the Remaining Fraction

Substitute $n = 3/2$ into the decay equation:

\frac{A}{A_0} = \left(\frac{1}{2}\right)^{3/2} = \frac{1}{2^{3/2}} = \frac{1}{\sqrt{2^3}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 1

✗ Incorrect

Verdict & Scientific Reason:

This is the fraction remaining after exactly 1 half-life ( $t = 100\text{ hours}$ ), not 150 hours.

⚠️

Student Fallacy (Why candidates pick this trap): Ignoring the extra 50 hours of decay.

2 2

✓ 100% Correct

Verdict & Scientific Reason:

Exact remaining fraction calculated from exponential decay law: $(1/2)^{1.5} = 1/(2\sqrt{2})$ .

💡

Why this is airtight: None; textbook radioactive decay calculation.

3 3

✗ Incorrect

Verdict & Scientific Reason:

Result of linear decay assumption: $1 - (150/100) \times (1/2)$ , which violates exponential decay.

⚠️

Student Fallacy (Why candidates pick this trap): Assuming radioactivity decays linearly with time.

4 4

✗ Incorrect

Verdict & Scientific Reason:

Arbitrary algebraic combination.

⚠️

Student Fallacy (Why candidates pick this trap): Incorrect exponent manipulation.

⚠️

NTA Trap Alert: Linear Interpolation vs. Exponential Decay Trap

Students who treat decay linearly think: "At 100 hrs it's 1/2, so at 150 hrs it's half of that again or 2/3". Exponential decay requires multiplying by factors of $1/2$ , so for 1.5 half-lives it is $(1/2)^1 \times (1/2)^{0.5} = \frac{1}{2} \times \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}}$ .

🛡️ How to avoid the -1 mark: Fraction remaining after

n

half-lives is ALWAYS

(1/2)^n

. For

n = 1.5

(1/2)^{1.5} = \frac{1}{2\sqrt{2}}

📖

Core NEET Syllabus Standard

NCERT Physics · Nuclei Standard Benchmark

Core Scientific Principle Tested:

"Activity decays exponentially according to $A = A_0 (1/2)^{t/T_{1/2}}$ ."

Syllabus Focus: Radioactive Decay Law (Half-Life)

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2021 Physics Question 30 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Half-Life Count: n = 150/100 = 1.5 = 3/2

Step-by-Step Rigorous Workflow & Mathematical Proof

Formulating the Radioactive Decay Law for Activity

Determining the Number of Elapsed Half-Lives

Evaluating the Remaining Fraction

360° Option Elimination Engine Key Feature

NTA Trap Alert: Linear Interpolation vs. Exponential Decay Trap

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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