A Square and A Cube — Hard

Q1. Consider the two statements about Queen Ratnamanjuri's lockers. Statement I: For every factor d of a number n, the partner n/d is also a factor. Statement II: A number has an odd number of factors only when some factor pairs with itself, i.e., when n is a perfect square. Which of the following is correct?

• A. Only Statement I is correct.
• B. Only Statement II is correct.
• C. Both statements are correct, and II is the correct explanation of I.
• D. Both statements are correct, but II is not the correct explanation of I.

Q2. Which of the following numbers will have 1 as the units digit of its square? I. 161 II. 249 III. 358 IV. 471

• A. I and II only
• B. I, II and IV only
• C. II and IV only
• D. I and IV only

Q3. A student writes that the number 4000 is a perfect square because 4 is a perfect square. Using rule about trailing zeros in squares, the student's claim is

• A. Correct, because the square of 20 ends in 00.
• B. Correct, because 4000 = 4 × 1000 and both have square roots.
• C. Incorrect, because a perfect square must have an even number of trailing zeros and 4000 has three.
• D. Incorrect, because 4000 ends in 0 and squares cannot end in 0.

Q4. Using the prime factorisation method which of the following is NOT a perfect square?

• A. 324
• B. 441
• C. 156
• D. 576

Q5. Find the smallest natural number by which 9408 must be multiplied so that the product is a perfect square. (Hint: 9408 = 2⁶ × 3 × 7²)

• A. 2
• B. 3
• C. 6
• D. 7

Q6. Without computing exactly, the square root of 1936 lies between which two consecutive natural numbers?

• A. 40 and 41
• B. 43 and 44
• C. 44 and 45
• D. 45 and 46

Q7. A square handkerchief has area 125 sq cm. What is the largest whole number length, in centimetres, that its side cannot exceed?

• A. 10 cm
• B. 11 cm
• C. 12 cm
• D. 13 cm

Q8. Which of the following statements about trailing zeros of perfect cubes is correct?

• A. A perfect cube can end in exactly one zero.
• B. A perfect cube can end in exactly two zeros.
• C. A perfect cube can end in exactly three zeros.
• D. A perfect cube cannot end in any zero.

Q9. Extends cubes to negative integers. The value of (−6)³ is

• A. −216
• B. −18
• C. 36
• D. 216

Q10. Using prime factorisation in triplets find the cube root of 3375.

• A. 5
• B. 15
• C. 25
• D. 45

Q11. Find the smallest natural number by which 1323 must be multiplied so that the product is a perfect cube. (Hint: 1323 = 3³ × 7²)

• A. 3
• B. 7
• C. 9
• D. 21

Q12. A pattern: 1 = 1³; 3 + 5 = 8 = 2³; 7 + 9 + 11 = 27 = 3³; 13 + 15 + 17 + 19 = 64 = 4³. The first odd number used to express 6³ as a sum of consecutive odd numbers is

• A. 27
• B. 29
• C. 31
• D. 33

Q13. Which of the following is NOT a perfect cube?

• A. 216
• B. 343
• C. 500
• D. 729

Q14. A Class 8 student in Rampur writes 12² = 24 in his notebook. The teacher's best diagnosis of the student's error is that the student is

• A. treating squaring as multiplying the number by 2 instead of multiplying it by itself
• B. confusing perimeter with area of a square
• C. ignoring the units digit pattern of squares
• D. using the formula for the nth odd number instead of the square

Q15. A Class 8 teacher wants her students to discover that 1 + 3 + 5 + … + (2n − 1) = n² without telling them the formula. The pedagogically richest classroom task for this discovery is

• A. ask students to memorise the formula and write it five times
• B. give each pair an 8 × 8 dot grid and let them build successive L-shaped 'gnomon' layers of odd dots around a corner
• C. compute 1 + 3 + 5 + … + 19 with calculators and compare to 100
• D. show a PowerPoint slide with the proof and quiz students afterwards