1
IIT-JEE 2012 Paper 2 Offline
+4
-1
Let $$PQR$$ be a triangle of area $$\Delta$$ with $$a=2$$, $$b = {7 \over 2}$$ and $$c = {5 \over 2}$$; where $$a, b,$$ and $$c$$ are the lengths of the sides of the triangle opposite to the angles at $$P.Q$$ and $$R$$ respectively. Then $${{2\sin P - \sin 2P} \over {2\sin P + \sin 2P}}$$ equals.
A
$${3 \over {4\Delta }}$$
B
$${45 \over {4\Delta }}$$
C
$${\left( {{3 \over {4\Delta }}} \right)^2}$$
D
$${\left( {{45 \over {4\Delta }}} \right)^2}$$
2
IIT-JEE 2012 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
If $$f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt$$ for all $$x \in \left( {0,\infty } \right),$$ then
A
$$f$$ has a local maximum at $$x=2$$
B
$$f$$ is decreasing on $$(2, 3)$$
C
there exists some $$c \in \left( {0,\infty } \right),$$ such that $$f'(c)=0$$
D
$$f$$ has a local minimum at $$x=3$$
3
IIT-JEE 2012 Paper 2 Offline
+4
-1
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}$$ for all $$x \in \left( {1,\,\infty } \right)$$.

Which of the following is true?

A
$$g$$ is increasing on $$\left( {1,\infty } \right)$$
B
$$g$$ is decreasing on $$\left( {1,\infty } \right)$$
C
$$g$$ is increasing on $$(1, 2)$$ and decreasing on $$\left( {2,\infty } \right)$$
D
$$g$$ is decreasing on $$(1, 2)$$ and increasing on $$\left( {2,\infty } \right)$$
4
IIT-JEE 2012 Paper 2 Offline
+4
-1
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}$$ for all $$x \in \left( {1,\,\infty } \right)$$.

Consider the statements:
$$P:$$ There exists some $$x \in R$$ such that $$f\left( x \right) + 2x = 2\left( {1 + {x^2}} \right)$$
$$Q:\,\,$$ There exists some $$x \in R$$ such that $$2\,f\left( x \right) + 1 = 2x\left( {1 + x} \right)$$
Then

A
both $$P$$ and $$Q$$ are true
B
$$P$$ is true and $$Q$$ is false
C
$$P$$ is false and $$Q$$ is true
D
both $$P$$ and $$Q$$ are false
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