JEE 2023 Mathematics PYQ — For a triangle , the value of is least. If its inradius is and in… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
For a triangle ABC, the value of cos2A+cos2B+cos2C is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?
Choose the correct answer:
A.
perimeter of ΔABC is 183
B.
sin2A+sin2B+sin2C=sinA+sinB+sinC
C.
MA⋅MB=−18
D.
area of ΔABC is 2273
(Correct Answer)
Correct Answer:
area of ΔABC is 2273
Explanation
1. Minimizing the Expression
In any triangle ABC, the sum of the cosines of the double angles is given by:
cos2A+cos2B+cos2C=−1−4cosAcosBcosC
To make this value the least (minimum), the term 4cosAcosBcosC must be at its maximum value. In a triangle, the product cosAcosBcosC is maximized when the triangle is equilateral.
Therefore:
A=B=C=60∘
2. Finding the Side Length
We are given the inradius r=3. For an equilateral triangle with side length a, the formula for the inradius is:
r=23a
Substituting the given value:
3=23a
a=63
3. Calculating the Area of the Triangle
The area (Δ) of an equilateral triangle is given by:
Δ=43a2
Substituting the value of a:
Δ=43(63)2
Δ=43(36×3)
Δ=43(108)
Δ=273
4. Conclusion
The actual area of the triangle is 273 sq. units.
The option you mentioned states that the "Area of ΔABC is 2273". Since our calculated area is 273, this statement is mathematically false.
Thus, the statement "Area of ΔABC=2273
Explanation
1. Minimizing the Expression
In any triangle ABC, the sum of the cosines of the double angles is given by:
cos2A+cos2B+cos2C=−1−4cosAcosBcosC
To make this value the least (minimum), the term 4cosAcosBcosC must be at its maximum value. In a triangle, the product cosAcosBcosC is maximized when the triangle is equilateral.
Therefore:
A=B=C=60∘
2. Finding the Side Length
We are given the inradius r=3. For an equilateral triangle with side length a, the formula for the inradius is:
r=23a
Substituting the given value:
3=23a
a=63
3. Calculating the Area of the Triangle
The area (Δ) of an equilateral triangle is given by:
Δ=43a2
Substituting the value of a:
Δ=43(63)2
Δ=43(36×3)
Δ=43(108)
Δ=273
4. Conclusion
The actual area of the triangle is 273 sq. units.
The option you mentioned states that the "Area of ΔABC is 2273". Since our calculated area is 273, this statement is mathematically false.
