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Write the expanded form of (3a + 6b + 5c) 2 .
9a 2 + 36b 2 + 25c 2 + 40ab + 60bc + 30ac
9a 2 + 36b 2 + 25c 2 + 36ab + 60bc + 30ac
9a 2 + 36b 2 + 25c 2 + 36ab + 60bc + 40ac
9a 2 + 36b 2 + 25c 2 + 36ab + 55bc + 30ac
9a 2 + 36b 2 + 25c 2 + 36ab + 60bc + 30ac
The expansion follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x=3a, y=6b, and z=5c yields (3a)² + (6b)² + (5c)² + 2(3a)(6b) + 2(6b)(5c) + 2(5c)(3a), which simplifies to 9a² + 36b² + 25c² + 36ab + 60bc + 30ac.
The expansion follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x=3a, y=6b, and z=5c yields (3a)² + (6b)² + (5c)² + 2(3a)(6b) + 2(6b)(5c) + 2(5c)(3a), which simplifies to 9a² + 36b² + 25c² + 36ab + 60bc + 30ac.
Algebraic identities form the foundation of Quantitative Aptitude, often linked to the curriculum mandated by the Staff Selection Commission (SSC) for competitive exams in India.
The identity (x + y + z)² includes three squares and three distinct cross-product terms.
Coefficient multiplication is crucial; 2 * 3 * 6 = 36, 2 * 6 * 5 = 60, and 2 * 3 * 5 = 30.
Order of terms does not change the algebraic sum due to the commutative property.
The square of a trinomial always results in six terms: three squares and three pairs of double products.
Option A is incorrect because it calculates 40ab instead of 36ab.
Option C is incorrect because it calculates 40ac instead of 30ac.
Option D is incorrect because it miscalculates the 55bc coefficient.
B is correct — The expansion follows the identity resulting in 9a² + 36b² + 25c² + 36ab + 60bc + 30ac.
Algebraic identities like these are frequently used in SSC and Banking exams to solve complex simplification questions; remember to always check the signs of the coefficients.
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