We Distribute, Yet Things Multiply — Mastery

Q1. By the distributive property of multiplication over addition, (a + b) × c is equal to

• A. a × b + c
• B. abc
• C. a + bc
• D. ac + bc

Q2. In this chapter, an 'identity' is best described as

• A. An equation that is true for some particular value of the variable
• B. A number that does not change in multiplication
• C. A formula that gives the area of a rectangle
• D. An equation that is true for every value of the variables involved

Q3. Using Identity 1A, the expansion of (m + 3)^2 is

• A. m^2 + 9
• B. m^2 + 3m + 9
• C. m^2 + 6m + 9
• D. m^2 + 9m + 9

Q4. Statement P: Identity 1B can be derived from Identity 1A by replacing b with (−b). Statement Q: When we replace b with (−b) in a^2 + 2ab + b^2, the term b^2 also changes sign. Which is correct?

• A. Both P and Q are true
• B. Both P and Q are false
• C. P is false; Q is true
• D. P is true; Q is false

Q5. Aarti notices the pattern 9 × 9 − 1 = 10 × 8, 8 × 8 − 1 = 9 × 7, 7 × 7 − 1 = 8 × 6. Which identity explains this pattern?

• A. (a + b)^2 = a^2 + 2ab + b^2
• B. (a − b)^2 = a^2 − 2ab + b^2
• C. a^2 − 1 = (a + 1)(a − 1)
• D. a(b + 1) = ab + a

Q6. Brahmagupta named the distributive-property method khanda-gunanam in his Brāhmasphuṭasiddhānta. The literal meaning of khanda-gunanam is

• A. Multiplication by parts
• B. Multiplication by sums
• C. Squaring of a number
• D. Cross-multiplication of two fractions

Q7. A rectangular field has length 27 m and breadth 23 m. Its length is increased by 3 m and breadth by 2 m. By how many square metres does the area increase?

• A. 5
• B. 6
• C. 121
• D. 127

Q8. For which pairs of real numbers (a, b) is the statement (a + b)^2 > a^2 + b^2 TRUE?

• A. For every pair (a, b)
• B. Only when a = b
• C. Only when a and b have opposite signs
• D. Only when both a and b are positive, OR both are negative

Q9. To picture (a − b)^2 a teacher draws a square of side a, cuts off an L-shaped strip of width b along two adjacent sides, and computes the remaining area. Why must b^2 be ADDED back in the final formula a^2 − 2ab + b^2?

• A. Because the L-strip area is exactly 2ab + b^2
• B. Because Identity 1A also has +b^2
• C. Because every difference of squares has a +b^2 at the end
• D. Because the small b × b corner square is counted twice when we subtract two ab strips, so we add it once to correct

Q10. A Class 8 student is asked to state Identity 1 in plain words. Which sentence is the BEST plain-English statement?

• A. Add the first terms and add the second terms
• B. Multiply each term of the first bracket with each term of the second bracket and add the four products
• C. Multiply the first terms together and the second terms together
• D. Add all four letters and double the result

Q11. Which is the BEST first activity for a Class 8 teacher to introduce the concept of 'like terms' before any symbolic rule?

• A. Write the rule 'same letter and same power → like terms' on the board and ask students to copy it
• B. Give a bag with paper cut-outs labelled x, x, x^2, x, x^2 and ask students to sort them into piles that 'go together'
• C. Solve 10 expansion problems from Identity 1 on the board
• D. Read aloud the definition from the textbook

Q12. Tenzin computes 53 × 47 in one step. Writing 53 = 50 + 3 and 47 = 50 − 3, what value does he get?

• A. 2491
• B. 2500
• C. 2509
• D. 2350

Q13. A student simplifies 4 × (5 × 6) as (4 × 5) × (4 × 6) = 480, instead of 4 × 30 = 120. Which mistake has she made?

• A. She applied distributivity correctly; both 480 and 120 are right
• B. She wrongly distributed multiplication over multiplication, but distributivity is over ADDITION only
• C. She forgot to use commutativity
• D. She should have written 4 + 5 × 6

Q14. On the area of a square of side 65 is shown as a single big square split into four pieces. Which four pieces correctly add to 65^2?

• A. 60 × 60, 60 × 5, 5 × 60, 5 × 5
• B. 60 × 60 and 5 × 5 only
• C. 65 × 60 and 65 × 5
• D. 60 × 60, 60 × 5, 60 × 5, 60 × 5

Q15. The picture for Identity 1 splits a rectangle of (a + m) rows and (b + n) columns into four sub-rectangles. Which sub-rectangle area corresponds to the cross-term 'mb'?

• A. Top-left: a rows × b columns
• B. Top-right: a rows × n columns
• C. Bottom-left: m rows × b columns
• D. Bottom-right: m rows × n columns

Q16. A Class 8 teacher wants students to discover (a + b)^2 = a^2 + 2ab + b^2 with deep understanding. Which sequence of teaching steps is the BEST?

• A. State the formula → solve 10 examples on the board → give homework
• B. Draw a square of side (a + b) on graph paper → ask students to count the four sub-regions → name each area → write the sum
• C. Give a worksheet of (a + b)^2 problems → mark right and wrong → review errors
• D. Read aloud Identity 1A from the textbook → ask students to memorise it overnight

Q17. Identity 1 says (a + m)(b + n) = ab + mb + an + mn. By choosing which substitution do we obtain Identity 1A, (a + b)^2 = a^2 + 2ab + b^2?

• A. Put m = b and n = a in Identity 1
• B. Put m = a and n = b in Identity 1, treating the second bracket as (a + b)
• C. Put b = a and n = m in Identity 1
• D. Put a = m and b = n in Identity 1

Q18. Using Identities 1A and 1B, the value of (a + b)^2 − (a − b)^2 equals

• A. 2a^2 + 2b^2
• B. 4ab
• C. 2ab
• D. 0

Q19. Reading Identity 1 backwards, the expression x^2 + 7x + 12 can be written as (x + m)(x + n) where m + n = 7 and mn = 12. The correct pair (m, n) is

• A. (2, 6)
• B. (3, 4)
• C. (1, 12)
• D. (−3, −4)

Q20. Using Identity 1C in reverse, the expression x^2 − 49 factorises as

• A. (x − 7)^2
• B. (x + 7)(x − 7)
• C. (x + 7)^2
• D. (x − 49)(x + 1)

Q21. Imran writes (x − 4)^2 = x^2 − 8x − 16. A teacher diagnosing the error should say

• A. He should have written x^2 − 16 only
• B. The last term must be +16 because (−4)^2 = +16, not −16
• C. The middle term must be +8x, not −8x
• D. Both middle and last terms have the wrong sign

Q22. Using the distributive property, the fastest one-line computation of 487 × 9 is

• A. 487 × 10 + 9 = 4879
• B. 487 × 10 − 487 = 4383
• C. 487 × 9 done by column multiplication only
• D. 487 + 9 = 496

Q23. Using Identity 1A, the value of 27^2 written as (25 + 2)^2 is

• A. 625 + 4 = 629
• B. 625 + 100 + 4 = 729
• C. 625 + 50 + 4 = 679
• D. 625 + 54 = 679

Q24. By extending the distributive property, a(b + c + d) is equal to

• A. abcd
• B. ab + cd
• C. ab + ac + ad
• D. a + b + c + d

Q25. Expanding (x + 3)(x + 5) by Identity 1 and then substituting x = 2 gives the value

• A. 35
• B. 30
• C. 15
• D. 25

Q26. In 'Mind the Mistake' a Class 8 student writes 3a + 3a = 9a. A teacher should explain that the student has confused

• A. Adding like terms with multiplying like terms
• B. The distributive property with associativity
• C. Identity 1A with Identity 1B
• D. Variables with constants

Q27. Aarti notices that for any whole number n ≥ 2, the product (n − 1)(n + 1) is always 1 LESS than n^2. The reason — by Identity 1C — is

• A. (n − 1)(n + 1) = n^2 − 1, so it falls short of n^2 by exactly 1
• B. (n − 1)(n + 1) = n^2 + 1, so it exceeds n^2 by 1
• C. (n − 1)(n + 1) = 2n − 1, which is less than n^2 for n ≥ 2
• D. Because consecutive numbers always differ by 1

Q28. A teacher claims: 'For any whole number ending in 9, its square always ends in 1.' Using Identity 1B with the form (10k − 1)^2, which option BEST justifies this?

• A. (10k − 1)^2 = 100k^2 − 20k + 1, and only the +1 contributes to the units place
• B. Numbers ending in 9 are odd, so their squares are odd
• C. 9 × 9 = 81 alone proves the claim
• D. The claim is false; 19^2 = 361 ends in 1 by coincidence

Q29. In algebra the expression 2(x + 3) is read as

• A. 2 added to (x + 3)
• B. 2 multiplied by (x + 3)
• C. The number formed by writing 2 next to (x + 3)
• D. 2 raised to the power (x + 3)

Q30. Match the three special identities of this chapter with their names. (i) (a + b)^2 = a^2 + 2ab + b^2 (ii) (a − b)^2 = a^2 − 2ab + b^2 (iii) (a + b)(a − b) = a^2 − b^2

• A. (i) Identity 1A, (ii) Identity 1B, (iii) Identity 1C
• B. (i) Identity 1B, (ii) Identity 1A, (iii) Identity 1C
• C. (i) Identity 1C, (ii) Identity 1B, (iii) Identity 1A
• D. All three are Identity 1