We Distribute, Yet Things Multiply Class 8 Worksheet with Answers Maths Chapter 6

Each of our Ganita Prakash Class 8 Worksheet and Class 8 Maths Chapter 6 We Distribute, Yet Things Multiply Worksheet with Answers Pdf focuses on conceptual clarity.

Class 8 Maths Chapter 6 We Distribute, Yet Things Multiply Worksheet with Answers Pdf

We Distribute, Yet Things Multiply Class 8 Maths Worksheet

Class 8 Maths Chapter 6 Worksheet with Answers – Class 8 We Distribute, Yet Things Multiply Worksheet

Tick (✓) for the correct option

Question 1.
Which property of multiplication is shown by the following example?
5 × (3 + 7) = (5 × 3) + (5 × 7)
(a) Commutative property
(b) Associative property
(c) Distributive property
(d) Identity property

Question 2.
Which of the following is equivalent to 12 × 99?
(a) 12 × (100 – 1)
(b) 12 × 100 – 1
(c) 12 × 90 + 9
(d) 12 × (90 + 10)

Question 3.
If the product of two numbers, a and b, remains unchanged when a is increased by 3 and b is decreased by 5, what is the relationship between a and b?
(a) 3a – 56 + 15 = 0
(b) 5a – 36 + 15 = 0
(c) 3a – 56 = 15
(d) 5a – 3b = 15

Question 4.
The product of a number and 1 is the number itself. This is known as the
(a) Zero property of multiplication
(6) Commutative property of multiplication
(c) Identity property of multiplication
(d) Associative property of multiplication

Question 5.
What is the expanded form of (2x + 5)²?
(a) 4x² + 25
(6) 4x² + 10x + 25
(c) 4x² + 20x + 25
(d) 2x² + 20x + 25

Question 6.
Using a suitable identity, the value of 1032 is
(a) 10009
(b) 10609
(c) 10900
(d) 10600

Question 7.
The difference between the squares of two consecutive odd numbers is always a multiple of:
(a) 2
(b) 4
(c) 6
(d) 8

Question 8.
The pattern 1, 4, 9, 16, 25,… can be represented by the formula:
(a) n + 3
(b) n²
(c) 2n
(d) n³

Question 9.
Which of the following is the correct expansion of (3x – 2y)²?
(a) 9x² – 4y²
(b) 9x² + 4y²
(c) 9x² – 12xy + 4y²
(d) 9x² – 6xy + 4y²

Question 10.
The value of – 5(p – 4) is incorrectly written as – 5p – 20. What is the mistake?
(а) The number 5 should not be multiplied by 4.
(b) The sign of the second term inside the bracket was not changed.
(c) The multiplication should be 5(p – 4), not – 5(p – 4).
(d) The result should be 5p + 20.

Question 11.
A student says that (x – y) (y – x) – x² – y². This is incorrect because:
(a) The identity (a – b) (a + b) does not apply.
(b) The order of the terms in the brackets is different.
(c) The correct answer is (x – y)².
(d) The correct answer is – (x – y)².

Question 12.
When simplifying (2x + y) (2x – y), a student wrote the answer as 2x² – y². What mistake student make?
(a) The student did not use the correct identity.
(b) The student forgot to square the coefficient 2.
(c) The student incorrectly squared the second term.
(d) The student should have multiplied 2x by – y as well.

Question 13.
Which of the following methods will give the product of (2x + 5) (2x – 5) as 4x² – 25?
(a) Using the distributive property (FOIL method).
(b) Using the identity (a + b) (a – b) = a² – b².
(c) Both a and b.
(d) Neither a nor b.

Question 14.
The expression (x + a) (x + b) expands to x² + (a + b)x + ab. If a student uses the distributive property instead, the result would be:
(а) Different, as the methods are not equivalent.
(б) The same, as the identity is derived from the distributive property.
(c) Incorrect, because the identity is the only correct way.
(d) Impossible to determine without specific values for x, a and b.

Question 15.
To calculate 102² efficiently, which of the following expressions is the best to use?
(a) (100 + 2)²
(b) (10²) x (10²)
(c) (102)² + 2²
(d) (100)² + 2²

Question 16.
To expand (x + 2y)², which of the following are valid methods?
I. (x + 2y) (x + 2y) using the distributive property.
II. Using the identity (a + b)² – a² + 2ab + b².
III. Squaring each term to get x² + 4y²
(a) I only
(b) II only
(c) I and II only
(d) I, II and III

Assertion Reason Questions

In questions 1-4, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Question 1.
Assertion (A): The product of 5a²b and – 3ab² is -15a³b³.
Reason (R): While multiplying monomials, the coefficients are multiplied, and the powers of the same variables are added.
(а) 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 but (R) is false.
(d) (A) is false but (R) is true.

Question 2.
Assertion (A): (2x + 3)² = 4x² + 9.
Reason (R): The formula (a + b)² = a² + b² + 2ab is a special case of the distributive property.
(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 but (R) is false.
(d) (A) is false but (R) is true.

Question 3.
Assertion (A): Expanding (x + 3) (x – 5) using the distributive property gives the same result as using the identity (x + a) (x + b) – x2 + (a + b) x + ab.
Reason (R): Different algebraic approaches or paths can lead to the same final expanded form of an algebraic expression. .
(а) 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 but (R) is false.
(d) (A) is false but (R) is true.

Question 4.
Assertion (A): A common mistake when expanding (x + y) (x – y) is to get x² – xy – yx – y².
Reason (R): Combining the like terms – xy and + yx results in 0, so they are cancelled out.
(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 but (R) is false.
(d) (A) is false but (R) is true.

Fill in the blanks

1. To simplify 12 × 98, we can rewrite the problem as 12 × (100 – 2) and use the ______ property.
2. The product of (x + 5) (x + 3) can be found by distributing each term of the first binomial to every term of the second, resulting in ______.
3. When multiplying a number by 11, the first digit of the product is the first digit of the original number plus any ______.
4. The distributive property is important for simplifying ______ expressions because the variables inside the parentheses cannot always be added or subtracted directly.
5. The value of 992 can be calculated using the identity for the square of a difference by writing 99 as (______ – 1).
6. To multiply a number by 11 using the distributive property, we can rewrite 11 as (______ + 1).
7. The difference between the squares of two consecutive numbers, say (n + 1)² – n², is always equal to ______.
8. The sum of the first 5 odd numbers is ______, which is a perfect square of ______.
9. A common mistake is writing (a + b)² as a² + b². The correct identity includes the term ______
10. The expression 9x² + 6x + 4 is not a perfect square because the middle term is not equal to _________
11. After expanding the product of two binomials, one should combine like terms, not _______ terms.
12. When multiplying x² by x³, the exponents should be __________, not multiplied. result is x⁵.
13. To evaluate 103 × 97, one can use the identity a² – b² = (a – b) (a + b), where a = ______ and b = ______.
14. To check an answer obtained by applying an algebraic identity, one can use the method to see if the result is the same.
15. The phrase “This Way or That Way, All Ways Lead to the Bay” means that using different correct mathematical ______ will produce the same outcome.
16. By choosing the correct algebraic identity, complex numerical calculations can be made much ______ and faster.

Write (T) for true or (F) for false for the given statements

1. The product of (x – 3)(x- 5) is x² – 8x – 15.
2. The value of 98 x 102 can be calculated using the identity (a – b) (a + b).
3. The product of (x + 2) and (y + 3) is xy + 6.
4. To multiply a number by 101, we can use the distributive property by rewriting 107 as (100 + 7).
5. Multiplying any number by 99 is equivalent to multiplying the number by 100 and then adding the number to the result.
6. The sum of the first four consecutive odd numbers starting from 1 is a perfect square.
7. The difference between the squares of two consecutive natural numbers is always an even number.
8. The expression x² + 2x + 1 is a perfect square trinomial.
9. The value of (a + b) (a – b) is the same as (b – a) (a + b).
10. The distributive property is correctly applied in the expression 5(x – 2) = 5x – 10.
11. The value of 1012 can be calculated as (100 – 1)².
12. The final answer to an algebraic problem can be different if we use a different but valid method.
13. If we get the same answer using two different methods, it confirms that the answer is highly likely to be correct.
14. The product of a binomial and a binomial can only be found by applying an identity.
15. The expansion of (a + b)² using the identity is a longer process than using the distributive property.

Very Short Answer Type Questions

Question 1.
State the distributive property of multiplication over addition for three numbers, a, b and c.
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Question 2.
Simplify the expression 10(2a – 3b).
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Question 3.
What is the result of multiplying a number by 11 using the distributive property?
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Question 4.
Evaluate 2453 × 11 in one line.
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Question 5.
What is the expanded form of (x – 5)?
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Question 6.
Using the identity (a + b) (a – b) – a² – b², what is the value of 101 × 99?
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Question 7.
If the side of a square is (x – 1) units, what is its area?
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Question 8.
The sum of the first two consecutive odd numbers is a perfect square. To which perfect square does it correspond?
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Question 9.
A student incorrectly wrote (x – y)² =x² – y². What is the missing term in the correct expansion?
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Question 10.
What identity should be used to correct the mistake in (p + 2) (p – 2) = p² + 4?
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Question 11.
When subtracting an expression like – (2x – 3y), what happens to the signs inside the parentheses?
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Question 12.
When multiplying 2x by 3x², a student got 6x². What was the error?
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Question 13.
Show how we can use the distributive property to find the product of 104 × 103. We don’t need to perform the final calculation.
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Question 14.
Simplify the expression (x + 2)² – (x² + 4). Show that using the identity for (a + b)2 gives the same result as direct expansion.
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Question 15.
Is the product of (a -b)(a + b) the same as (b + a) (a – b)? Justify your answer.
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Question 16.
When expanding (x – y)², what is the role of the distributive property in verifying the identity x² – 2xy + y²?
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Short Answer Type Questions

Question 1.
What was Brahmagupta’s explicit statement regarding the distributive property?
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Question 2.
Simplify the expression 10(2a – 3b).
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Question 3.
A rectangular garden has a length of (x + 5) meters and a width of (x – 2) meters. Find the area of the garden in terms of x.
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Question 4.
What happens to the product of two numbers if one of them is increased by 3 and the other is decreased by 5? Find three examples where the product remains unchanged.
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Question 5.
Simplify the expression (4p + 5q)² – (4p – 5q)².
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Question 6.
A rectangular area has a length of(3x + 2) and a width of(3x – 2). Find the area. If x = 10, what is the numerical value of the area?
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Question 7.
Observe the pattern: 1 + 3 = 4 = 22, 1 + 3 + 5 = 9 = 32, 1 + 3 + 5 + 7 = 16 = 42. Explain this pattern using the concept of sums of consecutive odd numbers. What would be the sum of the first 6 odd numbers?
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Question 8.
Prove that the product of two numbers, a and b, can be expressed as the difference of the squares of their average and half their difference.
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Question 9.
A student is asked to simplify 4(x – 2) – 2(x – 3). He writes 4x – 8 – 2x – 6.
(a) Identify the mistake made by the student.
(b) Explain the correct application of the distributive property in this case.
(c) Provide the correct, simplified final answer.
Solution:
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Question 10.
A student tries to find the value of 99² as (100 – 1)² and gets the answer as 10000 – 1 = 9999.
(a) Explain the error in this calculation.
(b) Show the correct application of the identity (a – b)² = a²– 2ab + b² to find the correct value.
(c) State the correct value of 99².
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Question 11.
A student correctly expands (y – 4)² but then incorrectly combines the terms to y2 – 16.
(a) What is the correct expansion of (y – 4)²?
(b) Identify the mistake the student made in combining the terms.
(c) Explain whyy2 – 8y and 16 cannot be combined into a single term.
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Question 12.
Explain the difference between a perfect square trinomial and a difference of squares. Give an example for each.
(a) What identity governs a perfect square trinomial?
(b) What identity governs a difference of squares?
(c) Provide an example for both.
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Question 13.
A student wants to calculate 20 x 18 using the distributive property. He rewrites the expression as 20(20 – 2). ‘
(а) Is this a valid method for calculating the product?
(b) Explain the mathematical principle behind this approach.
(c) Show the complete calculation to demonstrate the correct final answer.
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Question 14.
Let a number leave a remainder of 4 when divided by 9, and another number leave a remainder of 6 when divided by 9.
(a) What is the remainder when the sum of these two numbers is divided by 9?
(b) What is the remainder when the difference of these two numbers is divided by 9?
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Question 15.
A field’s area is given by the expression 4x² – 9y².
(a) What is the identity that can be used to find its length and width?
(b) Factorize the expression 4x² – 9y².
(c) What are the algebraic expressions for the length and width of the field?
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Case Based Questions

A gardener has a square garden with a side length of x meters. He decides to increase its area to create a new rectangular garden with dimensions of (x + y) meters and (x – y) meters.

Based on the above information answer the following questions:

Question 1.
What is the area of the original square garden?
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Question 2.
What is the area of the new rectangular garden?
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Question 3.
How can we use a special algebraic identity to relate the areas of the two gardens?
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