Answer Key Chapter 3: Number Play –QUESTION BANK 2( Study Material cum) Worksheet

 Chapter 3: Number Play –QUESTION BANK 2( Study Material cum) Worksheet

Chapter 3: Number Play – QUESTION BANK – Answer Key

1. Multiple Choice Questions (MCQs)

1. a) 121 (121 reads the same forwards and backwards)

2. a) 6174

3. a) 18 (1+0+8+9=18)

4. b) 6174

5. b) 1

6. d) 1234

7. d) All of these

8. b) 176 (Digit sum of 68 is 6+8=14. 1+7+6=14)

9. b) Both neighbours are taller.

10. b) It is larger than all its adjacent cells.

11. a) 6174

12. b) 2002 and d) 1001 (Both are palindromes. The question allows for one answer, but both are correct. Typically, 2002 or 1001 would be accepted.)

13. b) Find its half.

14. a) 59 (To get the smallest number, use the fewest digits, placing the largest digit in the leftmost place. 59 has a digit sum of 14 and is smaller than 68, 77, or 149.)

15. b) 17 (The winning strategy is to leave your opponent on a multiple of 4. 17 is one less than 18, which is a multiple of 4.)

16. c) About 900 (Assuming 15 breaths per minute: 15 * 60 = 900)

17. b) 22 (5+6+8+3=22)

18. d) All of the above

19. c) Palindromic time

20. a) In the corners (Corners have fewer neighbors, making it easier for a number to be larger than all of them.)

21. a) 5085 (Largest: 7432, Smallest: 2347, Difference: 7432-2347=5085)

22. c) 90 (From 10 to 99)

23. b) 400 (The number 400 is used as a common factor in the mental math example to reach 3,400.)

24. b) 94100 (To maximize the number, put the largest possible digit (9) in the highest place value. The digit sum of 94100 is 9+4+1+0+0=14.)

25. d) It depends on the leap year cycle.

26. a) 109989 (Smallest 5-digit palindrome: 10001, Largest: 99999, Sum: 10001+99999=109989)

27. b) 10 (Similar to the Game of 21, the strategy involves controlling multiples of the step+1. Since players can add 1-9, the key multiples are of 10.)

28. d) 20 (It appears 20 times: 7,17,27,37,47,57,67,70,71,72,73,74,75,76,77,78,79,87,97)

---

2. Assertion and Reasoning Questions

1. a) Both A and R are true and R is the correct explanation of A.

2. a) Both A and R are true and R is the correct explanation of A. (The sequence 0,1,2,1,0 implies the middle child is the shortest, so both neighbors are taller, hence they say '2'.)

3. a) Both A and R are true and R is the correct explanation of A. (If it's the smallest number, at least one neighbor must be larger, so it cannot be larger than all its neighbors.)

4. d) A is false but R is true. (The Collatz Conjecture is not proven; it remains an unsolved problem. However, the reason describing the cycle it falls into is correct.)

5. d) A is false but R is true. (The assertion is false, as the reason proves with a counterexample: 99,999 + 999 = 100,998, which is a 6-digit number.)

6. a) Both A and R are true and R is the correct explanation of A.

7. a) Both A and R are true and R is the correct explanation of A.

8. a) Both A and R are true and R is the correct explanation of A.

9. a) Both A and R are true and R is the correct explanation of A. (9+9+9=27)

10. a) Both A and R are true and R is the correct explanation of A. (02022020 is a palindrome.)

---

3. True or False – Justify Your Reason

1. False. (Not all palindromes are divisible by 11. Example: 131 is a palindrome but not divisible by 11.)

2. True. (The 1089 trick involves reversing a 3-digit number and subtracting.)

3. False. (Kaprekar's constant for 4-digit numbers is 6174. 1729 is the Hardy-Ramanujan number.)

4. True.

5. True. (The conjecture states all sequences will eventually reach the cycle 4, 2, 1.)

6. False. (Many palindromes are not perfect squares, e.g., 121 is, but 131 is not.)

7. True. (121 is 11² and reads the same forwards and backwards.)

8. True. (There are 900 three-digit numbers (100-999) and only 90 two-digit numbers (10-99).)

9. False. (The smallest possible sum of two 5-digit numbers is 10,000 + 10,000 = 20,000, which is a 5-digit number.)

10. True. (495 is the Kaprekar constant for 3-digit numbers.)

---

4. Short Answer I (2 Marks)

1. 101, 111, 121, 131, 141, 151, 161, 171, 181, 191 (Any three)

2. 6174

3. 1 + 2 + 3 + 4 = 10

4. (Estimates will vary) Mug: ~0.5 L, Bucket: ~10 L, Overhead Tank: ~1000 L

5. Example: 18,666 + 4 OR 18,660 + 10 (Any combination of a 1-digit and a 5-digit number that sums to 18,670)

6. A supercell is a number larger than all its adjacent cells (up, down, left, right). The supercells in the first grid are likely 4795 and 8632. (This must be checked by comparing each number to its neighbors).

7. (Open-ended task) The student must place numbers such that the colored cells are larger than all their adjacent white cells.

8. Start with any number. If it's even, divide by 2. If it's odd, multiply by 3 and add 1. Repeat. You will always eventually reach the number 1.

9. 11, 22, 33, 44, 55, 66, 77, 88, 99 (Any one)

10. Yes. The Collatz sequence for 100 is: 100 → 50 → 25 → 76 → 38 → 19 → 58 → 29 → 88 → 44 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

11. The winning strategy is to be the one who says 19. This forces the opponent to choose a number between 20-21, allowing you to reach 22. (The key is to leave your opponent on 18, which is a multiple of (1+3)=4. 22 - 4 = 18).

12. Placed on number line from left to right: 1050, 1500, 2180, 2754, 3050, 3600, 5030, 5300, 8400, 9590, 9950

13. a. Labels: 2000, 2010, 2020, 2030, 2040. Smallest: 2000, Largest: 2040.
    b. Labels: 9995, 9996, 9997, 9998, 9999, 10000. Smallest: 9995, Largest: 10000.
    c. Labels: 15,077, 15,078, 15,079, 15,080, 15,081, 15,082, 15,083. Smallest: 15,077, Largest: 15,083.
    d. Labels: 86,705, 87,705, 88,705, 89,705. Smallest: 86,705, Largest: 89,705.

14. (Open-ended task) To maximize supercells, place the largest numbers in the corners and the smallest numbers in the center. The number of supercells will be the number of corners (4).

---

5. Short Answer II (3 Marks)

1. Palindrome: 121 reads the same forwards and backwards.
   Square: 11 × 11 = 121.

2. (Task involves completing a table with given digits) The biggest number is 96,301. The smallest even number is 10,936 (if using 0,1,3,6,9). The smallest number >5000 is 60,139.

3. Example with 3524:

• Step 1: 5432 - 2345 = 3087

• Step 2: 8730 - 0378 = 8352

• Step 3: 8532 - 2358 = 6174 (Kaprekar's constant)

4. 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

5. (Open-ended puzzle) The student must find two digits in the grid that, when swapped, create four numbers that are larger than all their neighbors.

6. | 1-digit: 9 | 2-digit: 90 | 3-digit: 900 | 4-digit: 9000 | 5-digit: 90000 |

7. Estimation: 500 × 50 = 25,000
   Collatz:
   a) 12, 6, 3
   b) 17, 52, 26
   c) 21, 64, 32
   d) 22, 11, 34

8. The number of steps varies. For 5683, the process must be followed until 6174 is reached to count the steps.

9. (Drawing task)

10. The next palindromic time after 10:01 is 11:11 (in 70 minutes). The one after that is 12:21 (in another 70 minutes).

11. Smallest 2-digit palindrome: 11, Largest: 99. Sum = 110, Difference = 88.

12. (Examples)
    a) 50,000 + 40,250 = 90,250 (digit sum 9+0+2+5+0=16 > 9)
    b) 70,000 - 14,999 = 55,001 (<56,503)
    c) 99,999 + 1 = 100,000
    d) 9,999 + 9,999 = 19,998 (Not 6-digit). 9,999 + 1 = 10,000 (5-digit). *This is not possible. The maximum sum is 9,999+9,999=19,998 (5 digits).*
    e) 10,000 - 1 = 9,999
    f) 50,000 + 50,000 = 100,000
    g) 10,000 + 8,500 = 18,500
    h) 10,000 - 1 = 9,999
    i) 10,000 - 9,901 = 99
    j) 100,000 - 8,500 = 91,500 (But 100,000 is 6-digit). 99,999 - 8,499 = 91,500

13. a) Sometimes (e.g., 10,000+10,000=20,000 is 5-digit; 50,000+50,000=100,000 is 6-digit)
    b) Sometimes (e.g., 1000+10=1010 is 4-digit; 9999+99=10098 is 5-digit)
    c) Never (Max is 9999+99=10098, which is 5 digits)
    d) Sometimes (e.g., 10,000-1=9,999 is 4-digit; 10,000-0=10,000 is 5-digit)
    e) Never (Min is 10,000-99=9,901, which is 4 digits)

---

6. Long Answer (5 Marks)

1. Kaprekar's Process with 3524:

• Largest number: 5432

• Smallest number: 2345

• Difference 1: 5432 - 2345 = 3087

• Largest: 8730, Smallest: 0378 → 378

• Difference 2: 8730 - 0378 = 8352

• Largest: 8532, Smallest: 2358

• Difference 3: 8532 - 2358 = 6174 (Kaprekar's Constant)

• Largest: 7641, Smallest: 1467

• Difference 4: 7641 - 1467 = 6174 (It repeats)

2. (Personalized task) The student must take their birth year (e.g., 2012) and apply the Kaprekar steps until they reach 6174, counting the number of iterations.

3. Collatz Conjecture with 11 (odd):
   11 → (3×11+1=34) → (34/2=17) → (3×17+1=52) → (52/2=26) → (26/2=13) → (3×13+1=40) → (40/2=20) → (20/2=10) → (10/2=5) → (3×5+1=16) → (16/2=8) → (8/2=4) → (4/2=2) → (2/2=1)

4. Using digits 4,7,3,2:
   a) To get a larger difference, choose digits with a greater spread (e.g., 9,0,1,2). Largest: 9210, Smallest: 0129 → 129, Difference: 9210-129=9081 (>5085)
   b) To get a smaller difference, choose digits close together (e.g., 3,4,5,6). Largest: 6543, Smallest: 3456, Difference: 6543-3456=3087 (<5085)
   c) To get a larger sum, choose large digits (e.g., 9,8,7,6). Largest: 9876, Smallest: 6789, Sum: 9876+6789=16,665 (>9779)
   d) To get a smaller sum, choose small digits (e.g., 1,2,3,4). Largest: 4321, Smallest: 1234, Sum: 4321+1234=5555 (<9779)

5. Importance of Estimation:

• Example 1: Shopping. Quickly adding rounded prices of items (~100 + ~250 + ~50 = ~400) to check if you have enough money before going to the counter.

• Example 2: Time Management. Estimating that homework will take ~45 minutes and dinner is in ~1 hour to decide if you have time to start it.

6. Digit Sum 14:
   a) 59, 68, 77, 86, 95, 149, 158, 167, 176, 185, 194, 239...
   b) 59
   c) There is no single largest number. You can always add another zero (e.g., 95, 950, 9500, 95000). All have a digit sum of 14.
   d) You can make a number of any size. To make a bigger number with the same digit sum, add more digits (zeros). e.g., 9,500,000,000.

---

7. Case-Based Questions

Case 1 – Kaprekar’s Process
1.1) A) 5432
1.2) B) 2345
1.3) C) 3087
1.4) B) 6174

Case 2 – Collatz Conjecture
2.1) C) 5 (10/2=5)
2.2) B) Multiply by 3 and add 1 → 16 (5 is odd)
2.3) B) Yes
2.4) B) Every number eventually reaches 1

Case 4 – Estimation in Daily Life
4.1) C) 600 (200+100+300)
4.2) A) 598 (198+102+298)
4.3) B) 2 (600-598)
4.4) B) It saves time and helps in quick decisions

Case 5 – Number Games
5.1) B) 132
5.2) C) 99 (231-132)
5.3) B) 18 (9+9)
5.4) B) Always multiples of 9 (The result of this trick is always 99 or 198, etc., whose digits sum to a multiple of 9)