Question

If in a $$\vartriangle ABC,$$  the values of $$\cot A,\cot B,\cot C$$    are in A.P., then

A. $$a, b, c$$  are in A.P.
B. $${a^2},{b^2},{c^2}$$  are in A.P.  
C. $$\cos A,\cos B,\cos C$$    are in A.P.
D. None of these
Answer :   $${a^2},{b^2},{c^2}$$  are in A.P.
Solution :
$$\cot A,\cot B,\cot C$$    are in A.P.
$$ \Rightarrow \,\,\frac{{{b^2} + {c^2} - {a^2}}}{{2bc \cdot \frac{a}{{2R}}}},\frac{{{c^2} + {a^2} - {b^2}}}{{2ca \cdot \frac{b}{{2R}}}},\frac{{{a^2} + {b^2} - {c^2}}}{{2ab \cdot \frac{c}{{2R}}}}$$        are in A.P.
$$ \Rightarrow \,\,{b^2} + {c^2} - {a^2},{c^2} + {a^2} - {b^2},{a^2} + {b^2} - {c^2}$$        are in A.P.
$$ \Rightarrow \,\, - 2{a^2}, - 2{b^2}, - 2{c^2}$$    are in A.P. (subtracting $${{a^2} + {b^2} + {c^2}}$$   from each).

Releted MCQ Question on
Trigonometry >> Properties and Solutons of Triangle

Releted Question 1

If the bisector of the angle $$P$$ of a triangle $$PQR$$  meets $$QR$$  in $$S,$$ then

A. $$QS = SR$$
B. $$QS : SR = PR : PQ$$
C. $$QS : SR = PQ : PR$$
D. None of these
View Answer
Releted Question 2

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is 15°. The distance of the boat from the foot of the light house is

A. $$\left( {\frac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}} \right)60\,{\text{metres}}$$
B. $$\left( {\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}} \right)60\,{\text{metres}}$$
C. $${\left( {\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}} \right)^2}{\text{metres}}$$
D. none of these
View Answer
Releted Question 3

In a triangle $$ABC,$$  angle $$A$$ is greater than angle $$B.$$ If the measures of angles $$A$$ and $$B$$ satisfy the equation $$3\sin x - 4{\sin ^3}x - k = 0, 0 < k < 1,$$       then the measure of angle $$C$$ is

A. $$\frac{\pi }{3}$$
B. $$\frac{\pi }{2}$$
C. $$\frac{2\pi }{3}$$
D. $$\frac{5\pi }{6}$$
View Answer
Releted Question 4

In a triangle $$ABC,$$  $$\angle B = \frac{\pi }{3}{\text{ and }}\angle C = \frac{\pi }{4}.$$     Let $$D$$ divide $$BC$$  internally in the ratio 1 : 3 then $$\frac{{\sin \angle BAD}}{{\sin \angle CAD}}$$   is equal to

A. $$\frac{1}{{\sqrt 6 }}$$
B. $${\frac{1}{3}}$$
C. $$\frac{1}{{\sqrt 3 }}$$
D. $$\sqrt {\frac{2}{3}} $$
View Answer

Practice More Releted MCQ Question on
Properties and Solutons of Triangle

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