We Distribute, Yet Things Multiply — Hard

Q1. Consider the following two statements about Identity 1A. Statement A: (a + b)^2 = a^2 + b^2 holds for every pair of real numbers a, b. Statement B: (a + b)^2 = a^2 + b^2 holds only when 2ab = 0. Which is correct?

• A. A is true, B is false
• B. Both A and B are true
• C. A is false, B is true
• D. Both A and B are false

Q2. Aarti is asked to compute 397 × 403 in one step using an algebraic identity. Writing 397 = 400 − 3 and 403 = 400 + 3, which identity should she apply, and what value does she get?

• A. Identity 1A; 160009
• B. Identity 1B; 159991
• C. Identity 1C; 159991
• D. Identity 1C; 160009

Q3. A Class 8 student writes (y + 2)(y + 2) = y^2 + 4y + 4 and another writes (y + 2)(y + 2) = y^2 + 4. Which evaluation is correct?

• A. Only the first; the second forgets the middle term 2 × y × 2 = 4y
• B. Only the second; (y + 2)(y + 2) means y^2 + 2^2
• C. Both are correct; they are just two forms
• D. Neither; the answer should be 2y + 4

Q4. Sridharacharya's ishta-gunana writes a^2 = (a + b)(a − b) + b^2 to compute squares quickly. Using b = 3, what is 97^2 by this method?

• A. 9400 + 9 = 9409
• B. 100 × 94 + 9 = 9409
• C. 100 × 94 − 9 = 9391
• D. 94 × 100 + 3 = 9403

Q5. Priya computes 91^2 in two ways. Way 1: as (90 + 1)^2 using Identity 1A. Way 2: as (100 − 9)^2 using Identity 1B. Which statement is correct?

• A. Way 1 gives 8281; Way 2 gives a different value
• B. Way 2 gives 8281; Way 1 gives a different value
• C. Both ways give 8281
• D. Both ways give 9181

Q6. Yusuf expands (2/3)(15 + 6a) using the distributive property. Which expression is the correctly simplified result?

• A. 10 + 6a
• B. 10 + 4a
• C. 30 + 4a
• D. 10 + 12a/3

Q7. Assertion (A): The distributive property a(b + c) = ab + ac holds even when a, b or c is a negative integer. Reason (R): Multiplying a sum by (−1) reverses the sign of each term, so (−1)(b + c) = −b − c.

• A. Both A and R are true, and R is the correct explanation of A
• B. Both A and R are true, but R is NOT the correct explanation of A
• C. A is true, R is false
• D. A is false, R is true

Q8. To compute 9734 × 99 mentally, the chapter splits 99 as (100 − 1). Which expression equals 9734 × 99, and what is its value?

• A. 9734 × 100 + 9734 = 983134
• B. 9734 × 100 − 9734 = 963666
• C. 9734 × 100 − 1 = 973399
• D. 9734 + 100 − 1 = 9833

Q9. Manju expands (3j + 2k)^2 using Identity 1A. Which expansion is correct?

• A. 9j^2 + 4k^2
• B. 3j^2 + 12jk + 2k^2
• C. 9j^2 + 6jk + 4k^2
• D. 9j^2 + 12jk + 4k^2

Q10. A teacher asks students to compute 58^2 in one step using Identity 1B. Writing 58 = 60 − 2, the correct sequence is

• A. 3600 − 240 + 4 = 3364
• B. 3600 − 120 + 4 = 3484
• C. 3600 − 240 − 4 = 3356
• D. 3600 + 240 + 4 = 3844

Q11. A Class 8 teacher uses an area model — a rectangle of length (a + m) and breadth (b + n) split into 4 sub-rectangles — to teach Identity 1. Which learning outcome does this representation BEST support?

• A. Memorising the formula (a + m)(b + n) = ab + mb + an + mn by repetition
• B. Connecting each of the four algebraic terms to a visible rectangular area, making the cross-terms unmissable
• C. Computing the value of (a + m)(b + n) faster than direct multiplication
• D. Replacing algebra with arithmetic for low achievers

Q12. Tenzin computes 197^2 using ishta-gunana with b = 3. The expression becomes (197 + 3)(197 − 3) + 3^2. What is the final value?

• A. 38800
• B. 38791
• C. 38809
• D. 38818

Q13. Identity 1 says (a + m)(b + n) = ab + mb + an + mn. Putting m = 1, n = −1, the 'increase' over ab simplifies to mb + an + mn = b − a − 1. Which scenario does this correspond to?

• A. Both numbers increased by 1
• B. Both numbers decreased by 1
• C. First number increased by 1, second decreased by 1
• D. Numbers unchanged

Q14. A teacher gives the task 'Find (5 + 7)^2'. Three students answer: Rohan writes 74; Sara writes 144; Imran writes 24. Which student is correct, and what error did each of the others make?

• A. Rohan is correct; Sara forgot the middle term; Imran added instead of squaring
• B. Sara is correct; Rohan used (a + b)^2 = a^2 + b^2; Imran computed 5 + 7 + 12
• C. Imran is correct; Rohan and Sara both miscomputed squares
• D. Sara is correct; Rohan used (a + b)^2 = a^2 + b^2; Imran computed 2(5) + 2(7)

Q15. Match each algebraic notation with what it stands for. (i) a(b + c) (ii) ab + c (iii) (a + b)c (iv) a + bc

• A. (i)–(iii) mean the same thing; (ii)–(iv) mean the same thing
• B. (i) and (iii) both mean 'a times the sum (b + c)'; (ii) and (iv) are different from each other
• C. All four mean the same thing
• D. Only (i) is correctly written; the rest are wrong