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1

AIEEE 2003

If in a $$\Delta ABC$$ $$a{\cos ^2}\left( {{C \over 2}} \right) + {\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},$$ then the sides $$a, b$$ and $$c$$
A
satisfy $$a+b=c$$
B
are in A.P
C
are in G.P
D
are in H.P

Explanation

If $$a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2}$$

$$a\left[ {\cos C + 1} \right] + c\left[ {\cos A + 1} \right] = 3b$$

$$\left( {a + c} \right) + \left( {a\cos C + c\cos \,B} \right) = 3b$$

$$a + c + b = 3b$$ or $$a + c = 2b$$

or $$a,b,c$$ are in $$A.P.$$
2

AIEEE 2003

In a triangle $$ABC$$, medians $$AD$$ and $$BE$$ are drawn. If $$AD=4$$,
$$\angle DAB = {\pi \over 6}$$ and $$\angle ABE = {\pi \over 3}$$, then the area of the $$\angle \Delta ABC$$ is
A
$${{64} \over 3}$$
B
$${8 \over 3}$$
C
$${{16} \over 3}$$
D
$${{32} \over {3\sqrt 3 }}$$

Explanation

$$AP = {2 \over 3}AD = {8 \over 3};\,\,PD = {4 \over 3};\,\,$$

Let $$PB=x$$

$$\tan {60^ \circ } = {{8/3} \over x}$$

or $$x = {8 \over {3\sqrt 3 }}$$

Area of $$\Delta ABD$$

$$= {1 \over 2} \times 4 \times {8 \over {3\sqrt 3 }} = {{16} \over {3\sqrt 3 }}$$

$$\therefore$$ Area of $$\Delta ABC$$

$$= 2 \times {{16} \over {3\sqrt 3 }} = {{32} \over {3\sqrt 3 }}$$

$$\left[ \, \right.$$ As median of a $$\Delta$$ divides it into two $$\Delta 's$$ of equal area. $$\left. \, \right]$$
3

AIEEE 2003

The sum of the radii of inscribed and circumscribed circles for an $$n$$ sided regular polygon of side $$a,$$ is
A
$${a \over 4}\cot \left( {{\pi \over {2n}}} \right)$$
B
$$a\cot \left( {{\pi \over {n}}} \right)$$
C
$${a \over 2}\cot \left( {{\pi \over {2n}}} \right)$$
D
$$a\cot \left( {{\pi \over {2n}}} \right)$$

Explanation

$$\tan \left( {{\pi \over n}} \right) = {a \over {2r}};\,\,\sin \left( {{\pi \over n}} \right) = {a \over {2R}}$$

$$r + R = {a \over 2}\left[ {\cot {\pi \over n} + \cos ec{\pi \over n}} \right]$$

$$= {a \over 2}\left[ {{{\cos {\pi \over n} + 1} \over {\sin {\pi \over n}}}} \right]$$

$$= {a \over 2}\left[ {{{2{{\cos }^2}{\pi \over {2n}}} \over {2\sin {\pi \over {2n}}\cos {\pi \over {2n}}}}} \right]$$

$$= {a \over 2}\cot {\pi \over {2\pi }}$$
4

AIEEE 2002

In a triangle with sides $$a, b, c,$$ $${r_1} > {r_2} > {r_3}$$ (which are the ex-radii) then
A
$$a>b>c$$
B
$$a < b < c$$
C
$$a > b$$ and $$b < c$$
D
$$a < b$$ and $$b > c$$

Explanation

$${r_1} > {r_2} > {r_3}$$

$$\Rightarrow {\Delta \over {s - a}} > {\Delta \over {s - b}} > {\Delta \over {s - c}};$$

$$\Rightarrow s - a < s - b < s - c$$

$$\Rightarrow - a < - b < - c$$

$$\Rightarrow a > b > c$$

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