Relation between Roots and Coefficients for Quadratic
If tan A and ...
Question
If tanA and tanB are the roots of the quadratic equation, 3x2−10x−25=0, then the value of 3sin2(A+B)−10sin(A+B)⋅cos(A+B)−25cos2(A+B) is :
A
−10
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B
10
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C
−25
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D
25
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Solution
The correct option is C−25 3x2−10x−25=0⇒3x2−15x+5x−25=0⇒(3x+5)(x−5)=0⇒x=−53,5
Now, tan(A+B)=tanA+tanB1−tanAtanB=5−531+253=514cos(A+B)=14√221
Now, S=3sin2(A+B)−10sin(A+B)⋅cos(A+B)−25cos2(A+B)⇒S=cos2(A+B)[3tan2(A+B)−10tan(A+B)−25]⇒S=196221[3×25196−10×514−25]⇒S=25221[3−28−196]⇒S=−25