Chapter 2 – Power Play: Case-Based & Assertion–Reasoning Questions Case-Based Study Questions with Answer Keys

 

Chapter 2 – Power Play: Case-Based & Assertion–Reasoning Questions

Case-Based Study Questions with Answer Keys

Case Study 1 — Magical Pond & Doubling
A magical pond has 1 lotus flower on Day 1. Each day, the number of lotus flowers doubles. On Day 30, the pond is completely covered.
Q1.1 On which day is the pond half-covered?
Q1.2 Write the number of lotuses on Day 30 in exponential form.
Q1.3 If each lotus takes up 0.5 m², what is the total area covered by lotuses on Day 30?

Q1: On which day is the pond half-covered?
Answer: Day 29 (one day before full coverage).

Q2: Write the number of lotuses on Day 30 in exponential form.
Answer: 2²⁹ lotuses on Day 29, 2³⁰ lotuses on Day 30.

Q3: If each lotus takes up 0.5 m², what is the total area covered by lotuses on Day 30?
Answer: 2³⁰ × 0.5 m² = 536,870,912 m².

Case Study 2 — Paper Folding to the Moon

Q1: Write the thickness after 10 folds in cm using exponents.
Answer: 0.001 × 2¹⁰ = 1.024 cm.

Q2: Find the thickness after 46 folds in km.
Answer: 0.001 × 2⁴⁶ cm ≈ 7,036,874 km.

Q3: Compare exponential growth in folding to linear growth of stacking 46 sheets without folding.
Answer: Exponential growth reaches Moon in 46 folds; linear stacking = only 0.046 cm thickness.

Case Study 3 — Roxie’s Tulābhāra Donation

Q1: Find the worth of the donation.
Answer: ₹3150.

Q2: If 1-rupee coins replace the jaggery (mass = 4 g each), find the number of coins needed.
Answer: 11,250 coins.

Q3: Write your answer in scientific notation.
Answer: 1.125 × 10⁴ coins.

Case Study 4 — Locks and Combinations

Q1: How many different codes are possible?
Answer: 10⁵ = 100,000 codes.

Q2: If the lock instead uses 5 characters (A–Z letters only), how many codes are possible?
Answer: 26⁵ = 11,881,376 codes.

Q3: Which is more secure and why?
Answer: Letter-based is more secure (more possibilities).

Case Study 5 — Earth to Moon Steps vs Folding

Q1: Find number of steps needed.
Answer: 1,922,000,000 steps.

Q2: Compare with number of folds needed (paper example).
Answer: Folding paper = only 46 folds to reach Moon’s distance.

Q3: Explain the difference in terms of linear vs exponential growth.
Answer: Linear = constant addition; exponential = rapid multiplication.

Assertion and Reasoning Questions with Answer Keys

(A) Both true and R explains A
(B) Both true but R does not explain A
(C) A true but R false
(D) A false but R true

1. A: Doubling every day is an example of exponential growth.
   R: In exponential growth, the quantity increases by a fixed multiple each time.
   Answer: A
2. A: 2¹⁰ = 1024 means 2 multiplied by itself 10 times equals 1024.
   R: The base tells how many times to multiply the exponent by itself.
   Answer: C
3. A: Scientific notation expresses numbers as x × 10ⁿ with 1 ≤ x < 10.
   R: This makes it easier to write and compare very large or small numbers.
   Answer: A
4. A: In the magical pond, the pond is one-quarter full on Day 28.
   R: Each day’s lotus count is double the previous day’s.
   Answer: A
5. A: Linear growth can overtake exponential growth if given enough time.
   R: In exponential growth, increase per step decreases over time.
   Answer: D
6. A: a^m × aⁿ = a^m+n for all integers m and n.
   R: Multiplying powers with the same base adds the exponents.
   Answer: A
7. A: a^-n = 1/aⁿ for a ≠ 0.
   R: Negative exponents represent reciprocals.
   Answer: A
8. A: The number 64 is both a perfect square and a perfect cube.
   R: 64 = 8² and 64 = 4³.
   Answer: A
9. A: The cube of any even number is odd.
   R: Cube of even = even × even × even = even.
   Answer: D
10. A: The largest 3-digit power of 2 is 2⁹.
    R: 2⁹ = 512 and 2¹⁰ = 1024.
    Answer: A
11. A: Multiplying by 10³ increases a number’s value by 100.
    R: 10³ = 1000, so it multiplies the number by 1000.
    Answer: D
12. A: (a^m)ⁿ = a^mn holds for all integers m, n.
    R: Raising a power to another power multiplies exponents.
    Answer: A
13. A: The number of 6-letter passwords from A–Z is 26⁶.
    R: Each letter has 26 choices, independent of others.
    Answer: A
14. A: Linear growth means adding the same amount each step.
    R: Linear growth is faster than exponential growth for small step sizes.
    Answer: B
15. A: In scientific notation, the exponent tells the number of decimal shifts.
    R: Shifting decimal right means negative exponent.
    Answer: C