CBSE Class 9 Maths Ganita Manjari Part 1 Ch 04 Exploring Algebraic Identities MCQs with Answers Set 01

Answer: (a) The term \( 6x \) is missing

Explanation:
1. Area = \( (x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9 \).
2. The student forgot the middle term \( 2ab = 6x \).

Answer: (b) \( x^2 - 25 \)

Explanation:
1. Use \( (a + b)(a - b) = a^2 - b^2 \).
2. \( (x + 5)(x - 5) = x^2 - 25 \) cm².

Answer: (a) \( (a + b)^2 \)

Explanation:
1. Write 102 = 100 + 2.
2. \( (100 + 2)^2 = 10000 + 400 + 4 = 10404 \).

Answer: (b) \( x^3 + 6x^2 + 12x + 8 \)

Explanation:
1. Volume = \( (x + 2)^3 \).
2. Use \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \).
3. \( = x^3 + 3x^2(2) + 3x(4) + 8 = x^3 + 6x^2 + 12x + 8 \) cm³.

Answer: (b) \( 4p^2 - 4p + 1 \)

Explanation:
1. Area = \( (2p - 1)^2 \).
2. Use \( (a - b)^2 = a^2 - 2ab + b^2 \): \( 4p^2 - 2(2p)(1) + 1 = 4p^2 - 4p + 1 \).

Answer: (c) \( (7m - 8n)(7m + 8n) \)

Explanation:
1. \( 49m^2 = (7m)^2 \) and \( 64n^2 = (8n)^2 \), so this is a difference of two squares.
2. \( a^2 - b^2 = (a - b)(a + b) \) gives \( (7m - 8n)(7m + 8n) \).

Answer: (c) \( (a + b)^2 = a^2 + 2ab + b^2 \)

Explanation:
1. \( (a + b)^2 = 1^2 = 1 \) and \( a^2 + 2ab + b^2 = 9 - 12 + 4 = 1 \), so (c) holds.
2. (a) gives 1 ≠ 13 and (b) gives 25 ≠ 5, so both are false.
3. Identities are true for all real numbers, positive or negative, so (d) is wrong.

Answer: (b) \( 11y \)

Explanation:
1. \( (y + 4)(y + 7) = y^2 + (4 + 7)y + 4 \times 7 = y^2 + 11y + 28 \).
2. The student has \( y^2 \) and 28 but has missed \( 11y \).

Answer: (a) 23

Explanation:
1. Square both sides: \( \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 25 \).
2. \( x^2 + \frac{1}{x^2} = 25 - 2 = 23 \).

Answer: (b) \( 24x \)

Explanation:
1. Use \( (a + b)^2 - (a - b)^2 = 4ab \) with \( a = 3x \), \( b = 2 \).
2. \( 4 \times 3x \times 2 = 24x \).

Answer: (b) 196 cm²

Explanation:
1. Original side = \( \sqrt{144} = 12 \) cm.
2. New side = 14 cm, so new area = \( 14^2 = 196 \) cm².

Answer: (b) \( x^2 - 10x + 25 \)

Explanation:
1. \( x^2 - 10x + 25 = x^2 - 2(x)(5) + 5^2 = (x - 5)^2 \).
2. In the other options, the constant is not the square of half the x-coefficient.

Answer: (b) p = 2, q = 1

Explanation:
1. For p = 2, q = 1: \( (p - q)^2 = 1 \) but \( p^2 - q^2 = 4 - 1 = 3 \). These differ, so the claim is false.
2. With q = 0 or with both zero, both sides happen to be equal, so those checks cannot disprove it.

Answer: (c) 100

Explanation:
1. \( (x + 7)(x - 7) = x^2 - 49 \).
2. \( x^2 - 49 = 51 \Rightarrow x^2 = 100 \).

Answer: (a) \( (100 - 2)(100 + 2) \)

Explanation:
1. 98 = 100 − 2 and 102 = 100 + 2.
2. \( (100 - 2)(100 + 2) = 100^2 - 2^2 = 10000 - 4 = 9996 \).

Answer: (a) \( 5x - 24 \)

Explanation:
1. \( (x + 8)(x - 3) = x^2 + (8 - 3)x + (8)(-3) = x^2 + 5x - 24 \).
2. So the area differs from \( x^2 \) by \( 5x - 24 \).

Answer: (b) 40

Explanation:
1. \( (m + n)^2 = m^2 + n^2 + 2mn \).
2. \( 169 = 89 + 2mn \Rightarrow 2mn = 80 \Rightarrow mn = 40 \).

Answer: (a) \( x^2 + 2(x)(5) + 5^2 \)

Explanation:
1. \( 10x = 2(x)(5) \) and \( 25 = 5^2 \).
2. So \( x^2 + 10x + 25 = x^2 + 2(x)(5) + 5^2 = (x + 5)^2 \).

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Using R with a = x and b = 4: \( 4 \times x \times 4 = 16x \). The Assertion is true.
2. R is a standard identity, so it is true, and it is exactly what gives the Assertion.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. \( (a - b)^2 = 50 - 2(7) = 50 - 14 = 36 \). The Assertion is true.
2. R is the identity used in this calculation, so it is true and correctly explains A.