Answer Key with Short Reasoning (for Chapter 3 – Number Play)
1. MCQs (20)
(a) 121 – Reads same forward and backward (palindrome).
(a) 6174 – Kaprekar’s constant.
(b) 324 – Sum of digits = 9, divisible by 9.
(a) 18 – 1+0+8+9 = 18.
(c) 625 – 25×25 = 625.
(d) Both (a) & (c) – 121 and 484 are palindrome squares.
(b) 6174 – Result of Kaprekar’s process.
(d) 50 – Not a cube number.
(b) 1 – Collatz Conjecture ends at 1.
(c) 366 days – Leap year.
(a) 18 – 1+8=9, 18 divisible by 9.
(d) 1234 – Not a palindrome.
(a) 8 – 8³ = 512.
(c) 24 – LCM(8,12)=24.
(b) 97 – Only prime among options.
(c) Friday – 100÷7=14 remainder 2 → Monday+2=Wednesday. Oops correction: let’s recalc → If today Monday, after 100 days remainder 2 → Wednesday. Correct answer = Wednesday (not in options, revise Q).
(a) 143 – Divisible by 11.
(a) 30 – 2×3×5.
(d) All of these – All palindromes.
(c) 1729 – Ramanujan number.
2. Assertion–Reasoning (20)
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(c) A true, R false (4 not prime).
(a) Both true, R explains A.
(a) Both true, R explains A.
(c) A false, R false (not all palindromes divisible by 11).
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(b) Both true, but R doesn’t fully explain.
(c) A false, R true.
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
(a) Both true, R explains A.
3. True or False (10)
False – Not all palindromes divisible by 11.
True – Digit reversal trick → 1089.
False – Kaprekar’s constant = 6174.
True – 6174 is Kaprekar’s number.
True – Collatz ends at 1.
True – Leap year has 366 days.
False – Not every palindrome is square.
True – 121 is palindrome & square.
True – 1331 divisible by 11.
False – 1001 = 7×11×13.
4. Short Answer I (15)
111, 121, 131 – All palindromes.
6174 – Kaprekar’s constant.
1+2+3+4 = 10.
18, 20 – Both divisible by digit sum.
324 ÷ 9 = 36 → Yes divisible.
∛216 = 6.
a² + 2ab + b².
LCM(6,8) = 24.
Collatz: Any number → eventually 1.
Monday+30 = Wednesday.
2,3,5,7,11.
11,22,33 etc.
15²=225.
7³=343.
1000 = 10³.
5. Short Answer II (10)
121 is 11² and reads same → palindrome & square.
HCF(18,24)=6, LCM=72.
Example: 3524 → 5432–2345 = 3087 … → 6174.
1331 ÷ 11=121 → divisible.
6→3→10→5→16→8→4→2→1.
12³=1728 using (a+b)³ expansion.
1+3+5+7+9=25=5².
498≈500, 52≈50 → 500×50=25000.
200÷7=28 r4 → Friday+4=Tuesday.
Eg: 12 & 60 → HCF=12, LCM=60.
6. Long Answer (10)
3524 process → converges to 6174 (Kaprekar constant).
11 Collatz → 11→34→17→52→26→13→40→20→10→5→16→…→1.
(n+1)² – n² = 2n+1 = sum of consecutive numbers.
20×15=300 tiles needed.
Odd square remains odd, ex: 7²=49.
1729=10³+9³=12³+1³.
500÷7=71 r3 → Wednesday+3=Saturday.
Any 2-digit number ab=10a+b → (10a+b)–(a+b)=9a → divisible by 9.
9261=21³ → cube root=21.
Estimation saves time, e.g. 198≈200, quick calculations.
7. Case-Based Qs (5 Sets)
Case 1 (Kaprekar)
a) 5432 b) 2345 c) 3087 d) 6174.
Case 2 (Collatz)
a) 5 b) 16 c) Yes → 1 d) Conjecture states all numbers reach 1.
Case 3 (Calendar)
a) Tuesday b) Friday c) 11 years later d) 52 Sundays.
Case 4 (Estimation)
a) 200+100+300=600 b) 198+102+298=598 c) Difference=2 d) Saves time.
Case 5 (Number Games)
a) 132 b) 231–132=99 c) 9+9=18 d) Digits add to multiples of 9.
