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MHT CET202526 Apr 2025Evening ShiftMathematicsProperties of TrianglesActual

In a triangle ABC with usual notations if A=30^ , then the value of (1+ a c + b c ) (1+ c b - a b )=

Options

  1. A3 -2
  2. B2+ 5
  3. C3 +2
  4. D2- 5

Correct answer

C. 3 +2

Step-by-step solution

Given expression: E = (1+ a c + b c ) (1+ c b - a b ) Combine terms over common denominators: E = ( c+a+b c ) ( b+c-a b ) = (a+b+c)(b+c-a) bc Recognize the numerator as a difference of squares where X = b+c and Y = a : E = (b+c)² - a² bc Expanding gives: E = b² + c² + 2bc - a² bc Using the Law of Cosines, a² = b² + c² - 2bc A , so b² + c² - a² = 2bc A . Substituting: E = 2bc A + 2bc bc = 2( A + 1) Given A = 30^ and 30^ = 3 2 , then: E = 2 ( 3 2 + 1 ) = 3 + 2 This matches option C . Final answer: C

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