Algebraic Identities – MCQs

By: Prof. Dr. Fazal Rehman Shamil | Last updated: January 31, 2025

Question 1:

\[
\text{If } a^2 + b^2 = 29 \text{ and } ab = 10, \text{ then find } (a+b)^2.
\]
\[
\text{(a) } 39, \quad \text{(b) } 49, \quad \text{(c) } 59, \quad \text{(d) } 69
\]
Answer: B

Question 2:

\[
\text{What is the expansion of } (x + y)^3 \text{?}
\]
\[
\text{(a) } x^3 + 3x^2y + 3xy^2 + y^3, \quad \text{(b) } x^3 + y^3, \quad \text{(c) } (x + y)(x – y), \quad \text{(d) } x^3 – 3x^2y + 3xy^2 – y^3
\]
Answer: A

Question 3:

\[
\text{If } x – \frac{1}{x} = 4, \text{ then find } x^2 + \frac{1}{x^2}.
\]
\[
\text{(a) } 14, \quad \text{(b) } 15, \quad \text{(c) } 16, \quad \text{(d) } 18
\]
Answer: C

Question 4:

\[
\text{Which identity represents } a^3 – b^3?
\]
\[
\text{(a) } (a – b)(a^2 + ab + b^2), \quad \text{(b) } (a – b)(a^2 – ab + b^2), \quad \text{(c) } (a + b)(a^2 + ab + b^2), \quad \text{(d) } (a + b)(a^2 – ab + b^2)
\]
Answer: A

Question 5:

\[
\text{If } x + y = 5 \text{ and } xy = 6, \text{ then find } x^2 + y^2.
\]
\[
\text{(a) } 11, \quad \text{(b) } 13, \quad \text{(c) } 15, \quad \text{(d) } 19
\]
Answer: B

Question 6:

\[
\text{Find the value of } (a + b)^2 – (a – b)^2.
\]
\[
\text{(a) } 2ab, \quad \text{(b) } 4ab, \quad \text{(c) } a^2 – b^2, \quad \text{(d) } (a + b)(a – b)
\]
Answer: B

Question 7:

\[
\text{If } (x + 1)^2 = x^2 + 6x + 9, \text{ then find } x.
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } 3, \quad \text{(d) } 4
\]
Answer: C

Question 8:

\[
\text{Which of the following is the correct identity for } (a – b)^3?
\]
\[
\text{(a) } a^3 – 3a^2b + 3ab^2 – b^3, \quad \text{(b) } a^3 – b^3, \quad \text{(c) } (a – b)(a^2 + ab + b^2), \quad \text{(d) } a^3 + 3a^2b + 3ab^2 + b^3
\]
Answer: A

Question 9:

\[
\text{If } a^2 – b^2 = 21 \text{ and } a – b = 3, \text{ then find } a + b.
\]
\[
\text{(a) } 5, \quad \text{(b) } 6, \quad \text{(c) } 7, \quad \text{(d) } 8
\]
Answer: C

Question 10:

\[
\text{Which of the following is the identity for } (a + b + c)^2?
\]
\[
\text{(a) } a^2 + b^2 + c^2 + 2ab + 2bc + 2ca, \quad \text{(b) } a^2 + b^2 + c^2 + ab + bc + ca, \quad \text{(c) } a^2 + b^2 + c^2 – 2ab – 2bc – 2ca, \quad \text{(d) } (a+b+c)(a-b+c)
\]
Answer: A

Question 11:

\[
\text{If } x – y = 3 \text{ and } xy = 18, \text{ then find } x^2 – y^2.
\]
\[
\text{(a) } 10, \quad \text{(b) } 12, \quad \text{(c) } 15, \quad \text{(d) } 18
\]
Answer: D

Question 12:

\[
\text{If } (a – b)^2 = 81 \text{ and } ab = 20, \text{ then find } a^2 + b^2.
\]
\[
\text{(a) } 100, \quad \text{(b) } 121, \quad \text{(c) } 144, \quad \text{(d) } 169
\]
Answer: B

Question 13:

\[
\text{Which of the following represents the identity } a^4 – b^4?
\]
\[
\text{(a) } (a^2 – b^2)(a^2 + b^2), \quad \text{(b) } (a – b)(a^3 + b^3), \quad \text{(c) } (a^2 + b^2)^2, \quad \text{(d) } (a – b)^2(a + b)^2
\]
Answer: A

Question 14:

\[
\text{If } a + b = 6 \text{ and } ab = 8, \text{ then find } a^3 + b^3.
\]
\[
\text{(a) } 140, \quad \text{(b) } 144, \quad \text{(c) } 150, \quad \text{(d) } 156
\]
Answer: B

Question 15:

\[
\text{If } x + y = 10 \text{ and } xy = 21, \text{ then find } x^3 + y^3.
\]
\[
\text{(a) } 469, \quad \text{(b) } 476, \quad \text{(c) } 490, \quad \text{(d) } 500
\]
Answer: C

More MCQs on Algebra (Collected from Past Papers)

  1. Elementary Algebra – MCQs
  2. Algebraic Expressions – MCQs
  3. Algebraic Identities – MCQs
  4. Commutative Algebra – MCQs
  5. Linear Equations – MCQs
  6. Graphing Linear Equations – MCQs
  7. Inequalities in Algebra – MCQs
  8. Absolute Value Inequalities – MCQs
  9. Exponent – MCQs
  10. Exponential Functions – MCQs
  11. Logarithms – MCQs
  12. Polynomials – MCQs
  13. Factoring Methods – MCQs
  14. Polynomial Arithmetic – MCQs
  15. Quadratic Equation – MCQs
  16. Linear Algebra – MCQs
  17. Matrices in Algebra – MCQs
  18. Functions in Algebra – MCQs
  19. Sequences in Algebra – MCQs
  20. Arithmetic in Algebra – MCQs
  21. Combining Like Terms in Algebra – MCQs
  22. Abstract Algebra – MCQs
  23. Sets in Algebra – MCQs
  24. Algebra Calculator – MCQs

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