NIMCET 2019 Mathematics PYQ — If a, b, c are in GP and and are in AP, then a, b, c are the leng… | Mathem Solvex | Mathem Solvex
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NIMCET 2019 — Mathematics PYQ
NIMCET | Mathematics | 2019
If a, b, c are in GP and loga−log2b,log2b−log3c and log3c−loga are in AP, then a, b, c are the lengths of the sides of a triangle which is
Choose the correct answer:
A.
Acute angle
B.
Obtuse angled
(Correct Answer)
C.
Right angles
D.
Equilateral
Correct Answer:
Obtuse angled
Explanation
Concept: The sides of an obtuse triangle should satisfy the condition that the sum of the square of two smaller sides should be less than the square of the largest side. If a, b, c are in AP than b=2a+c If a, b, c are in GP than b=ac Calculation: loga−log2b,log2b−log3c and log3c−loga are in AP log2b−log3c=2log3c−loga+loga−log2b log3c2b=2log2b3c 9c24b2=2b3c 2b=3c−−−−−−−−−−−(1) ∵a,b,c are in GP ⇒b2=ac−−−−−−−−−−−(2) From equation 1 and 2 we get 9c=4a a:b:c=9:2:4 As we know that, for obtuse triangle the sum of the square of two smaller sides should be less than the square of the largest side. \Rightarrow a^{2}>b^{2}+c^{2} So, a, b and c are the sides of an obtuse angle triangle.
Explanation
Concept: The sides of an obtuse triangle should satisfy the condition that the sum of the square of two smaller sides should be less than the square of the largest side. If a, b, c are in AP than b=2a+c If a, b, c are in GP than b=ac Calculation: loga−log2b,log2b−log3c and log3c−loga are in AP log2b−log3c=2log3c−loga+loga−log2b log3c2b=2log2b3c 9c24b2=2b3c 2b=3c−−−−−−−−−−−(1) ∵a,b,c are in GP ⇒b2=ac−−−−−−−−−−−(2) From equation 1 and 2 we get 9c=4a a:b:c=9:2:4 As we know that, for obtuse triangle the sum of the square of two smaller sides should be less than the square of the largest side. \Rightarrow a^{2}>b^{2}+c^{2} So, a, b and c are the sides of an obtuse angle triangle.
