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1

IIT-JEE 2010 Paper 1 Offline

If the angles $$A, B$$ and $$C$$ of a triangle are in an arithmetic progression and if $$a, b$$ and $$c$$ denote the lengths of the sides opposite to $$A, B$$ and $$C$$ respectively, then the value of the expression $${a \over c}\sin 2C + {c \over a}\sin 2A$$ is
A
$${1 \over 2}$$
B
$${{\sqrt 3 } \over 2}$$
C
$$1$$
D
$${\sqrt 3 }$$

Explanation

Since, A, B, C are in AP

$$\Rightarrow$$ 2B = A + C i.e., $$\angle$$B = 60$$^\circ$$

$$\therefore$$ $${a \over c}$$(2 sin C cos C) + $${c \over a}$$ (2 sin A cos A)

= 2k (a cos C + c cos A)

[using, $${a \over {\sin A}} = {b \over {\sin B}} = {c \over {\sin C}} = {1 \over k}$$]

= 2k (b)

= 2 sin B

[using, b = a cos C + c cos A]

= $$\sqrt3$$

2

IIT-JEE 2007

Let $$ABCD$$ be a quadrilateral with area $$18$$, with side $$AB$$ parallel to the side $$CD$$ and $$2AB=CD$$. Let $$AD$$ be perpendicular to $$AB$$ and $$CD$$. If a circle is drawn inside the quadrilateral $$ABCD$$ touching all the sides, then its radius is
A
$$3$$
B
$$2$$
C
$${3 \over 2}$$
D
$$1$$
3

IIT-JEE 2006

One angle of an isosceles $$\Delta$$ is $${120^ \circ }$$ and radius of its incircle $$= \sqrt 3$$. Then the area of the triangle in sq. units is
A
$$7 + 12\sqrt 3$$
B
$$12 - 7\sqrt 3$$
C
$$12 + 7\sqrt 3$$
D
$$4\pi$$
4

IIT-JEE 2005

In an equilateral triangle, $$3$$ coins of radii $$1$$ unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is
A
$$4 + 2\sqrt 3$$
B
$$6 + 4\sqrt 3$$
C
$$12 + {{7\sqrt 3 } \over 4}$$
D
$$3 + {{7\sqrt 3 } \over 4}$$

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