Algebraic Identity II | Algebraic Expressions and Identities | Assessment | English | Grade 8

Which of the following is not equal to (a - b)²?

• (a-b)(a+b)

• (b-a)(b-a)

• a² - 2ab + b²

• a² - ab - ba + b²

Answer explanation

(a-b)² = (a-b)(a-b) = a(a-b) -b (a-b) = a² - ab - ba + b² On further simplification, we get (a-b)² = a² - 2ab + b² Also, (a-b)² = (a-b)(a-b) = (b-a)(b-a) Now, consider (a-b)(a+b) (a-b)(a+b) = a x (a + b) - b (a + b) = (a x a) + (a x b) - (b x a) - (b x b) = a² + ab - ba - b² = a² + ab - ab - b² = a² - b² But, (a-b)² = a² - 2ab + b² Therefore, a² - b² ≠ (a-b)² Since both expressions are not equal, the answer for given question is option 1.

(2 - p)² is equal to __________.

• p² - 2p + 4

• 4 - p² - 9p

• p² -4p + 4

• p² - 4p + 16

Answer explanation

(2 - p)² = (2)² - 2 (2) (p) + p² ...[Using (a-b)² = a² - 2ab + b²] = 4 - 4p + p² = p² - 4p + 4 Hence, option 3 is correct

Which of the following is not equal to 97²?

• 100² + 3² -2(3)(100)

• 100² - 3²

• (100 - 3)(100 - 3)

• 9409

Answer explanation

We can write 97² as (100 - 3)² Since, (a - b)² = a² - 2ab + b² (100 - 3)² = 100² - 2 x 100 x 3 + 3² = (10000 - 600) + 9 = 9400 + 9 = 9409 Therefore, value of option 1, 3 and 4 is equal to 97² In option 2, 100² -3² = 10000 - 9 = 9991 Therefore, 100² - 3² ≠ 97² Hence, option 2 is correct.

(2m² - 3n)² is equal to _________.

• 8m⁴ - 12m²n + 9n²

• 8m⁴ - 6m²n + 9n²

• 4m⁴ - 6m²n + 9n²

• 4m⁴ - 12m²n + 9n²

Answer explanation

We know that, (a - b)² = a² - 2ab + b² So, (2m² -3n)² = (2m²)² - 2 x (2m²) x (3n) + (3n)² = 2² x (m²)² - (2 x 2 x 3) x m² x n + 3² x n² ..... [since (mn)ⁿ = mⁿ x nⁿ] = 4m⁴ - 12m²n + 9n² Hence, option 4 is correct.

Using a suitable identity, find the value of 8.9².

• 64.81

• 79.21

• 64.21

• 69.81

Answer explanation

8.9 = 9 - 0.1 So, 8.9² = (9 - 0.1)² Since, (a-b)² = a² - 2ab + b² therefore, 8.9² = 9² - 2 x 9 x 0.1 + (0.1)² = 81 - 1.8 + 0.01 = 79.21 Therefore, option 2 is correct.

The length of base of a triangle is given by the expression (3c - 4). What is the area of the triangle if the height of the triangle is equal to length of the base?

• 9c²/2 + 12c + 8

• 9c²/2 + 24c + 16

• 9c²/2 - 12c + 8

• 9c²/2 - 24c + 16

Answer explanation

Area of triangle = ½ × (length of the base) × (height of the triangle) Here, length of the base = height of the triangle = (3c - 4) So, Area of triangle = ½ × (3c - 4) × (3c - 4) = ½ × (3c - 4)² Since, (a - b)² = a² - 2ab + b² Therefore, Area of triangle = ½ × [(3c)² - 2 (3c) (4) + (4)²] = ½ × (9c² - 24c + 16) = ½ × 9c² - ½ × 24c + ½ × 16 = 9c²/2 - 12c + 8 Therefore, option 3 is correct.

The value of (m - 2n)² - (2m - n)² is ________.

• 3(n² - m²)

• 4mn

• 0

• 5m² - 8mn + 5n²

Answer explanation

(m - 2n)² - (2m - n)² = [m² - 2 × m × 2n + (2n)²] - [(2m)² - 2 × (2m) × n + n²] (Using (a - b)² = a² - 2ab + b² ) = [m² -4mn + 2²n²] - [2²m² - 4mn + n²] [Since (mn)ⁿ = mⁿ x nⁿ] = m² - 4mn + 4n² - 4m² + 4mn - n² = m² - 4m² - 4mn + 4mn + 4n² - n² (On rearranging terms) = -3m² + 3n² = 3(n² - m²) Hence, (m - 2n)² - (2m - n)² = 3(n² - m²) Therefore, option 1 is correct.