Practice Questions
Concept Explanation
Algebraic identities are equations true for all values of the variables. They speed up expansion and factorization.
Key Identities
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Example
99² = (100 − 1)² = 10000 − 200 + 1 = 9801.
Common Trap
(a + b)² ≠ a² + b². The middle term 2ab is often forgotten.
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Q1 Easy
What is the expanded form of (x + 3)²?
Explanation(x + 3)² = x² + 2(x)(3) + 9 = x² + 6x + 9. -
Q2 Easy
What is the expanded form of (2a − 5)²?
Explanation(2a − 5)² = 4a² − 20a + 25. -
Q3 Easy
Which of the following is the factorized form of x² − 49?
Explanationx² − 49 = x² − 7² = (x + 7)(x − 7). -
Q4 Medium
Using the identity (100 + 3)², what is the value of 103²?
Explanation10000 + 600 + 9 = 10609. -
Q5 Medium
Using the identity (100 − 3)², what is the value of 97²?
Explanation10000 − 600 + 9 = 9409. -
Q6 Hard
If x + 1/x = 5, find the value of x² + 1/x².
ExplanationSquare: x² + 2 + 1/x² = 25, so x² + 1/x² = 23. -
Q7 Medium
What is the value of (a + b + c)² when a = 1, b = 2, and c = 3?
Explanation(1 + 2 + 3)² = 6² = 36.
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Q8 Easy
Which of the following is the factorized form of 4x² − 9?
Explanation4x² − 9 = (2x)² − 3² = (2x + 3)(2x − 3). -
Q9 Medium
If x + y = 8 and xy = 15, what is the value of x² + y²?
Explanationx² + y² = (x + y)² − 2xy = 64 − 30 = 34. -
Q10 Easy
Using the difference of squares identity, what is the value of (15)² − (5)²?
Explanationa² − b² = (15 + 5)(15 − 5) = 20 × 10 = 200. -
Q11 Medium
What is the expanded form of (3x + 2y)²?
Explanation(3x + 2y)² = 9x² + 12xy + 4y². -
Q12 Hard
If a − b = 3 and a² + b² = 29, find the value of ab.
Explanation2ab = (a² + b²) − (a − b)² = 29 − 9 = 20, so ab = 10. -
Q13 Hard
Using the difference of squares identity, what is the value of (1.1)² − (0.9)²?
Explanation(1.1 + 0.9)(1.1 − 0.9) = 2 × 0.2 = 0.4. -
Q14 Hard
What is the simplified form of (x + 5)² − (x − 5)²?
ExplanationDifference: 4 × x × 5 = 20x.
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DS1 Easy
What is the value of (a + b)²?
Statement Ia = 6 and b = 4.
Statement IIa + b = 10.
ExplanationStatement I alone gives (6 + 4)² = 100. Statement II alone only gives the sum, not individual values. -
DS2 Medium
What is the value of x² + y²?
Statement Ix + y = 10.
Statement IIxy = 24.
ExplanationNeed both: x² + y² = 100 − 48 = 52.
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DS3 Medium
What is the value of a² − b²?
Statement Ia + b = 12.
Statement IIa − b = 3.
ExplanationNeed both: a² − b² = (a + b)(a − b) = 36. -
DS4 Easy
What is the value of (x − 3)²?
Statement Ix = 8.
Statement IIx² = 64.
ExplanationStatement I alone: (8 − 3)² = 25. Statement II gives x = ±8, not unique.