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MHT CET202521 Apr 2025Morning ShiftMathematicsProperties of TrianglesActual

With usual notation, in triangle ABC , m A =30^ , then the value of (1+ a c + b c ) (1+ c b - a b ) is equal to

Options

  1. A2+ 3 2
  2. B2+ 3
  3. C1+ 3 2
  4. D1+ 3

Correct answer

B. 2+ 3

Step-by-step solution

Let E = (1 + a c + b c ) (1 + c b - a b ) . Combining terms gives E = ( a + b + c c ) ( b + c - a b ) = (a + b + c)(b + c - a) bc . Applying the Law of Sines with circumradius R yields a = 2R A , b = 2R B , c = 2R C . Substituting into the expression and simplifying: E = ( A + B + C)( B + C - A) B C . Applying the difference of squares formula: E = ( B + C)^2 - ^2 A B C = ^2 B + ^2 C + 2 B C - ^2 A B C . Separating terms: E = ^2 B + ^2 C - ^2 A B C + 2 . From the Law of Cosines converted via the Law of Sines, ^2 B

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