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JEE MainMathematicsModerate

The sides of a triangle are 3x +2, 2x +1 and 2x. If the cosine
of greatest angle of the triangle is −37- \frac{3}{7}, then x is equal to

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• 3x+2>0⇒x>−233x + 2 > 0 \Rightarrow x > - \frac{2}{3}...............(i)
• 2x + 1 > 0 ⇒\Rightarrow x >- 12\frac{1}{2}...............(ii)
• 2x > 0 ⇒\Rightarrow x > 0...............(iii)
Also sum of two sides is greater than third side
• 3x + 2 + 2x + 1 > 2x ⇒ \Rightarrow \ x > - 1...............(iv)
• 2x + 1 + 2x > 3x + 2 ⇒\Rightarrow x > 1...............(v)
• 3x + 2 + 2x > 2x + 1 ⇒\Rightarrow x > -1/3...............(vi)
from (i), (ii), (iii), (iv), (v) and (vi)
⇒\Rightarrow x > 1
for x > 1, 3x + 2 is greatest side ⇒\Rightarrow by cosine rule
⇒−37=−x2−8x−38x2+4x\Rightarrow - \frac{3}{7} = \frac{- x^{2} - 8x - 3}{8x^{2} + 4x}
⇒17x2−44x−21=0\Rightarrow 17x^{2} - 44x - 21 = 0
(x -3) (17x + 7) - 0
x = 3

Source: Grass-G / M-24 Mathematics Paper 31 Dec.docx

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