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NCERT Solutions · Maths Ch 6
We Distribute, Yet Things Multiply
20 questions · 3 exercises
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Ex 6.1
Ex 6.2
Ex 6.3
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
5 QUESTIONS
Exercise 6.1
Q1
Observe the multiplication grid (the 10 × 10 multiplication table). Each number inside the grid is formed by multiplying two numbers. If the middle number of a 3 × 3 frame is given by the expression pq (row p, column q), write the expressions for the other numbers in the frame.
(1 mark)
Q2
Expand the following products. (i) (3 + u)(v – 3) (ii) (2/3)(15 + 6a) (iii) (10a + b)(10c + d) (iv) (3 – x)(x – 6) (v) (–5a + b)(c + d) (vi) (5 + z)(y + 9)
(2 marks)
Q3
Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.
(2 marks)
Q4
Expand (i) (a + ab – 3b²)(4 + b), and (ii) (4y + 7)(y + 11z – 3).
(3 marks)
Q5
Expand (i) (a – b)(a + b), (ii) (a – b)(a² + ab + b²) and (iii) (a – b)(a³ + a²b + ab² + b³). Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?
(2 marks)
2
4 QUESTIONS
Exercise 6.2
Q1
Which is greater: (a – b)² or (b – a)²? Justify your answer.
(2 marks)
Q2
Express 100 as the difference of two squares.
(2 marks)
Q3
Find 406², 72², 145², 1097², and 124² using the identities you have learnt so far.
(3 marks)
Q4
Do Patterns 1 and 2 [(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²] hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.
(2 marks)
3
EXERCISE 6.3 · 11 QUESTIONS
End of chapter
Q1
Compute these products using the suggested identity. (i) 46² using (a + b)² (ii) 397 × 403 using (a + b)(a – b) (iii) 91² using (a – b)² (iv) 43 × 45 using (a + b)(a – b)
(2 marks)
Q2
Use either a suitable identity or the distributive property to find each of the following products. (i) (p – 1)(p + 11) (ii) (3a – 9b)(3a + 9b) (iii) –(2y + 5)(3y + 4) (iv) (6x + 5y)² (v) (2x – 1/2)² (vi) (7p) × (3r) × (p + 2)
(2 marks)
Q3
For each statement identify the appropriate algebraic expression(s). (i) Two more than a square number: 2 + s, (s + 2)², s² + 2, s² + 4, 2s², 2²s (ii) The sum of the squares of two consecutive numbers: m² + n², (m + n)², m² + 1, m² + (m + 1)², m² + (m – 1)², (m + (m + 1))², (2m)² + (2m + 1)²
(1 mark)
Q4
Consider any 2 by 2 square of numbers in a calendar. Find products of numbers lying along each diagonal — e.g. 4 × 12 = 48, 5 × 11 = 55. Do this for the other 2 by 2 squares. What do you observe about the diagonal products? Explain why this happens.
(2 marks)
Q5
Verify which of the following statements are true. (i) (k + 1)(k + 2) – (k + 3) is always 2. (ii) (2q + 1)(2q – 3) is a multiple of 4. (iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8. (iv) (6n + 2)² – (4n + 3)² is 5 less than a square number.
(3 marks)
Q6
A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7. What is the remainder when their sum, difference, and product are divided by 7?
(3 marks)
Q7
Choose three consecutive numbers, square the middle one, and subtract the product of the other two. Repeat the same with other sets of numbers. What pattern do you notice? How do we write this as an algebraic equation? Expand both sides of the equation to check that it is a true identity.
(2 marks)
Q8
What is the algebraic expression describing the following steps — add any two numbers. Multiply this by half of the sum of the two numbers. Prove that this result will be half of the square of the sum of the two numbers.
(2 marks)
Q9
Which is larger? Find out without fully computing the product. (i) 14 × 26 or 16 × 24 (ii) 25 × 75 or 26 × 74
(2 marks)
Q10
A tiny park is coming up in Dhauli. Two square plots, each of area g² sq. ft., will have a green cover. All the remaining area is a walking path w ft. wide that needs to be tiled. Write an expression for the area that needs to be tiled.
Figure: a rectangle containing two g × g squares side by side; a path of width w along the top, bottom and both ends, and a path of width 2w between the squares.
(2 marks)
Q11
For each pattern shown below, (i) Draw the next figure in the sequence. (ii) How many basic units are there in Step 10? (iii) Write an expression to describe the number of basic units in Step y.
Pattern (a): yellow squares — 9, 16, 25 squares in Steps 1, 2, 3. Pattern (b): blue squares — a square of side (step + 1) plus one extra square per step: 5, 11, 19 in Steps 1, 2, 3.
(2 marks)