Class 9 Exploring Algebraic Identities Mock Test (CBSE)

Topics in Exploring Algebraic Identities

• Visualising identities
• Factorisation using identities
• More identities
• Factorisation using algebra tiles
• Factorisation without algebra tiles
• Finding new identities
• Simplifying rational expressions

Exploring Algebraic Identities practice questions with answers

1. Q1Multiple choice·1 mark
   If $x + \frac{1}{x} = 5$, then the value of $x^2 + \frac{1}{x^2}$ is:

   1. (a)$25$
   2. (b)$23$
   3. (c)$27$
   4. (d)$21$
   Show answer and step marks

   1. Correct option: (b) $23$. Squaring both sides of $x + \frac{1}{x} = 5$ gives $x^2 + \frac{1}{x^2} + 2 = 25$, so $x^2 + \frac{1}{x^2} = 23$, which is option (b).+1

   Answer: (b)
2. Q2Multiple choice·1 mark
   The factorised form of $9x^2 - 16y^2$ is:

   1. (a)$(3x - 4y)(3x - 4y)$
   2. (b)$(3x + 4y)(3x - 4y)$
   3. (c)$(9x + 16y)(9x - 16y)$
   4. (d)$(3x + 4y)(3x + 4y)$
   Show answer and step marks

   1. Correct option: (b) $(3x + 4y)(3x - 4y)$. Using the difference of squares identity $a^2 - b^2 = (a + b)(a - b)$, we get $(3x + 4y)(3x - 4y)$, making option (b) correct.+1

   Answer: (b)
3. Q3Assertion and reason·1 mark
   Assertion (A): $a^2 - b^2 = (a - b)(a + b)$ is an algebraic identity. Reason (R): An algebraic identity is an algebraic equation that is true for all values of its variables.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer and step marks

   1. Correct option: (a)+1

   Answer: (a)
4. Q4Multiple choice·1 mark
   If $x + y + z = 10$, $xy + yz + zx = 25$, and $x^2 + y^2 + z^2 = 50$, then the value of $x^3 + y^3 + z^3 - 3xyz$ is:

   1. (a)$125$
   2. (b)$250$
   3. (c)$500$
   4. (d)$750$
   Show answer and step marks

   1. Correct option: (b) $250$+1

   Answer: (b)
5. Q5Short answer·2 marks
   Simplify the following expression completely using a suitable algebraic identity: $\frac{x^3 + y^3 + z^3 - 3xyz}{(x-y)^2 + (y-z)^2 + (z-x)^2}$, where $x$, $y$ and $z$ are real numbers that are not all equal.
   Show answer and step marks

   1. Use the identity for the numerator: $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$.+1
   2. Express the denominator as $2(x^2 + y^2 + z^2 - xy - yz - zx)$ and simplify the fraction to obtain $\frac{x + y + z}{2}$.+1

   Answer: $\frac{x + y + z}{2}$
6. Q6Short answer·3 marks
   Factorise $6x^2 + 7x - 20$ completely by splitting the middle term.
   Show answer and step marks

   1. Product of the coefficient of $x^2$ and the constant term: $6 \times (-20) = -120$. Two numbers with sum $7$ and product $-120$ are $15$ and $-8$.+1
   2. Split the middle term: $6x^2 + 15x - 8x - 20$.+0.5
   3. Group the terms: $3x(2x + 5) - 4(2x + 5)$.+0.5
   4. Take the common factor $(2x + 5)$: $(2x + 5)(3x - 4)$.+1

   Answer: $(2x + 5)(3x - 4)$

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