If in a triangle ABC a cos^2 (C/2 )+c^2 cos^2 (A/2 )=(3 ~b)/2 , then the sides a, b and c

Correct answer: (B) are in A.P.. If a cos^2 (C/2 )+cos^2 (A/2 )=(3 ~b)/2 a[cos C+1]+c[cos A+1]=3 ~b (a+c)+(a cos C+c cos B)=3 ~b a+c+b=3 ~b or a+c=2 ~b or a, b, c are in A.P.

Q119 / 225 · JEE Main 2003 · Mathematics

If in a triangle $\mathrm{ABC} \mathrm{a} \cos ^2\left(\frac{\mathrm{C}}{2}\right)+\mathrm{c}^2 \cos ^2\left(\frac{\mathrm{A}}{2}\right)=\frac{3 \mathrm{~b}}{2}$, then the sides $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$

A
satisfy $a+b=c$
Correct answer
B
are in A.P.
C
are in G.P.
D
are in H.P.

Solution

If $a \cos ^2\left(\frac{\mathrm{C}}{2}\right)+\cos ^2\left(\frac{\mathrm{A}}{2}\right)=\frac{3 \mathrm{~b}}{2}$ $\mathrm{a}[\cos \mathrm{C}+1]+\mathrm{c}[\cos \mathrm{A}+1]=3 \mathrm{~b}$ $(\mathrm{a}+\mathrm{c})+(\mathrm{a} \cos \mathrm{C}+\mathrm{c} \cos \mathrm{B})=3 \mathrm{~b}$ $\mathrm{a}+\mathrm{c}+\mathrm{b}=3 \mathrm{~b}$ or $\mathrm{a}+\mathrm{c}=2 \mathrm{~b}$ or $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P.

Hence, the correct answer is option: 'B'.

If in a triangle $\mathrm{ABC} \mathrm{a} \cos ^2\left(\frac{\mathrm{C}}{2}\right)+\mathrm{c}^2 \cos ^2\left(\frac{\mathrm{A}}{2}\right)=\frac{3 \mathrm{~b}}{2}$, then the sides $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$

Solution

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