Case-Based Study Questions (4 MCQs)
Case Study:
Khoisnam was part of a puzzle challenge involving 100 lockers. Each person toggled lockers according to their assigned number (Person 1 opened every locker, Person 2 toggled every second locker, and so on). At the end, Khoisnam noticed only certain lockers remained open. He also found that some lockers were toggled exactly twice and these helped him form a passcode.
Q1. Which locker numbers will remain open at the end of the toggling process?
(a) All even-numbered lockers
(b) All prime-numbered lockers
(c) All perfect square lockers
(d) All lockers with even number of factors
Answer: (c)
Q2. If the first five locker numbers toggled exactly twice form a code, what will be the code?
(a) 2, 3, 4, 5, 6
(b) 2, 3, 5, 7, 11
(c) 2, 4, 6, 8, 10
(d) 3, 5, 7, 11, 13
Answer: (b)
Q3. Which statement best explains why perfect square lockers remain open?
(a) Perfect squares have an even number of factors.
(b) Perfect squares have an odd number of factors due to a repeated factor.
(c) Perfect squares are always prime numbers.
(d) Perfect squares are always even numbers.
Answer: (b)
Q4. If locker number 49 remains open, which statement is correct?
(a) 49 has an even number of factors.
(b) 49 is a cube number.
(c) 49 is a perfect square.
(d) 49 is a prime number.
Answer: (c)
Case Study:
Akhil is designing square and cube-shaped tiles for a math exhibition. The side lengths of the tiles are whole numbers, and he labels each tile with its area (for squares) or volume (for cubes). While checking his labels, he notices patterns in the numbers and decides to play a guessing game with his friends.
Q1. Akhil has a square tile with area 256 cm². What is the length of its side?
(a) 12 cm
(b) 14 cm
(c) 15 cm
(d) 16 cm
Answer: (d)
Q2. One of the cube-shaped tiles has a volume of 512 cm³. What is the length of its side?
(a) 7 cm
(b) 8 cm
(c) 9 cm
(d) 6 cm
Answer: (b)
Q3. Akhil notices that some numbers cannot be perfect squares because of their unit digit. Which of the following cannot be a perfect square?
(a) 441
(b) 676
(c) 538
(d) 784
Answer: (c)
Q4. The largest square tile Akhil can make with an integer side length from a cloth of area 300 cm² has a side of:
(a) 16 cm
(b) 17 cm
(c) 18 cm
(d) 15 cm
Answer: (d)
(Because 15² = 225 < 300 and 16² = 256 < 300, but 17² = 289 is also possible — however, for maximum integer side length without exceeding the cloth area, we pick the largest perfect square ≤ 300, which is 289 → 17 cm)
Q5. Akhil labels a cube tile with volume 1331 cm³. Which of these is correct?
(a) It is not a perfect cube.
(b) Its side length is 11 cm.
(c) Its side length is 13 cm.
(d) Its side length is 15 cm.
Answer: (b)
Case Study 1 – The Locker Puzzle
Queen Ratnamanjuri’s 100-locker puzzle required each person to toggle lockers based on their number. At the end, only certain lockers stayed open, and some lockers were toggled exactly twice, forming a passcode.
Q1. Which lockers remain open at the end?
(a) All even-numbered lockers
(b) All prime-numbered lockers
(c) All perfect square lockers
(d) All lockers with even number of factors
Answer: (c)
Q2. If the first five lockers toggled exactly twice form the code, what is it?
(a) 2, 3, 4, 5, 6
(b) 2, 3, 5, 7, 11
(c) 2, 4, 6, 8, 10
(d) 3, 5, 7, 9, 11
Answer: (b)
Q3. Why do perfect square lockers remain open?
(a) They have an odd number of factors due to a repeated factor.
(b) They are all prime numbers.
(c) They are all even numbers.
(d) They have an even number of factors.
Answer: (a)
Q4. If locker #81 is open, which is true?
(a) 81 is a cube number.
(b) 81 is a perfect square.
(c) 81 is a prime number.
(d) 81 has an even number of factors.
Answer: (b)
Case Study 2 – Square Tile Designs
Akhil designs square tiles for a math exhibition, each with integer side lengths. He writes their areas and notices patterns in the last digits.
Q1. Which of these numbers cannot be a perfect square?
(a) 784
(b) 625
(c) 676
(d) 538
Answer: (d)
Q2. A tile has an area of 441 cm². What is the side length?
(a) 20 cm
(b) 21 cm
(c) 22 cm
(d) 23 cm
Answer: (b)
Q3. The largest square tile that can be cut from a cloth of area 300 cm² has side length:
(a) 15 cm
(b) 16 cm
(c) 17 cm
(d) 18 cm
Answer: (c) (Because 17² = 289 ≤ 300)
Q4. Which of these areas ends in 6 and is a perfect square?
(a) 256
(b) 216
(c) 676
(d) 196
Answer: (c) (676 = 26²)
Case Study 3 – Cube Blocks
Ravi stacks wooden cubes to make cube-shaped blocks. Each cube has equal side length in cm.
Q1. A block has volume 512 cm³. What is the side length?
(a) 6 cm
(b) 7 cm
(c) 8 cm
(d) 9 cm
Answer: (c)
Q2. Which of these is not a perfect cube?
(a) 729
(b) 125
(c) 216
(d) 500
Answer: (d)
Q3. The cube of a number ending in 5 will have a units digit:
(a) 5
(b) 0
(c) 1
(d) 6
Answer: (a)
Q4. Which cube has side length 11 cm?
(a) 1331 cm³
(b) 1728 cm³
(c) 2197 cm³
(d) 2744 cm³
Answer: (a)
Case Study 4 – The Hardy–Ramanujan Number
When Hardy visited Ramanujan, he mentioned taxi number 1729. Ramanujan said it was special — the smallest number expressible as the sum of two cubes in two different ways.
Q1. 1729 can be written as:
(a) 1³ + 12³ = 9³ + 10³
(b) 1³ + 12³ = 8³ + 11³
(c) 1³ + 11³ = 8³ + 10³
(d) 1³ + 13³ = 8³ + 10³
Answer: (a)
Q2. Numbers like 1729 are called:
(a) Square numbers
(b) Perfect cubes
(c) Taxicab numbers
(d) Prime numbers
Answer: (c)
Q3. The next taxicab number after 1729 is:
(a) 1331
(b) 4104
(c) 4913
(d) 32768
Answer: (b)
Q4. Which of these is a perfect cube from the list above?
(a) 1331
(b) 4104
(c) 1729
(d) 32768
Answer: (a) (1331 = 11³)
Case Study 5 – Odd Number Patterns
Manoj explores a pattern:
1 = 1²,
1 + 3 = 4 = 2²,
1 + 3 + 5 = 9 = 3²,
and so on.
Q1. The sum of the first 10 odd numbers is:
(a) 10² = 100
(b) 10³ = 1000
(c) 20² = 400
(d) 5² = 25
Answer: (a)
Q2. The 15th odd number is:
(a) 27
(b) 29
(c) 31
(d) 33
Answer: (c) (Formula: 2n – 1)
Q3. The sum of the first n odd numbers equals:
(a) n³
(b) 2n
(c) n²
(d) n(n + 1)/2
Answer: (c)
Q4. If 1225 is the sum of the first 35 odd numbers, then adding the 36th odd number gives:
(a) 1250
(b) 1261
(c) 1296
(d) 1369
Answer: (c)
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