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 EasyWhat is the expanded form of (x + 3)²?Explanation(x + 3)² = x² + 2(x)(3) + 9 = x² + 6x + 9.
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Q2 EasyWhat is the expanded form of (2a − 5)²?Explanation(2a − 5)² = 4a² − 20a + 25.
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Q3 EasyWhich of the following is the factorized form of x² − 49?Explanationx² − 49 = x² − 7² = (x + 7)(x − 7).
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Q4 EasyWhich of the following is the factorized form of 4x² − 9?Explanation4x² − 9 = (2x)² − 3² = (2x + 3)(2x − 3).
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Q5 EasyUsing the difference of squares identity, what is the value of (15)² − (5)²?Explanationa² − b² = (15 + 5)(15 − 5) = 20 × 10 = 200.
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Q6 MediumUsing the identity (100 + 3)², what is the value of 103²?Explanation10000 + 600 + 9 = 10609.
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Q7 MediumUsing the identity (100 − 3)², what is the value of 97²?Explanation10000 − 600 + 9 = 9409.
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Q8 MediumWhat 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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Q9 MediumIf x + y = 8 and xy = 15, what is the value of x² + y²?Explanationx² + y² = (x + y)² − 2xy = 64 − 30 = 34.
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Q10 MediumWhat is the expanded form of (3x + 2y)²?Explanation(3x + 2y)² = 9x² + 12xy + 4y².
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Q11 HardIf x + 1/x = 5, find the value of x² + 1/x².ExplanationSquare: x² + 2 + 1/x² = 25, so x² + 1/x² = 23.
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Q12 HardIf a − b = 3 and a² + b² = 29, find the value of ab.Explanation2ab = (a² + b²) − (a − b)² = 29 − 9 = 20, so ab = 10.
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Q13 HardUsing 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.
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Q14 HardWhat 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 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.
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DS3 Medium
What is the value of x² + y²?
Statement Ix + y = 10.
Statement IIxy = 24.
ExplanationNeed both: x² + y² = 100 − 48 = 52. -
DS4 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.