1
IIT-JEE 2012 Paper 1 Offline
+4
-1
The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant $$K$$)
A
$-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
B
$\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
C
$-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
D
$\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
2
IIT-JEE 2012 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then
A
$$S \ge {1 \over e}$$
B
$$S \ge 1 - {1 \over e}$$
C
$$S \le {1 \over 4}\left( {1 + {1 \over {\sqrt e }}} \right)$$
D
$$S \le {1 \over {\sqrt 2 }} + {1 \over {\sqrt e }}\left( {1 - {1 \over {\sqrt 2 }}} \right)$$
3
IIT-JEE 2012 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then
A
$$y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {8\sqrt 2 }}$$
B
$$y'\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {18}}$$
C
$$y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over 9}$$
D
$$y'\left( {{\pi \over 3}} \right) = {{4\pi } \over 3} + {{2{\pi ^2}} \over {3\sqrt 3 }}$$
4
IIT-JEE 2012 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
A ship is fitted with three engines $${E_1},{E_2}$$ and $${E_3}$$. The engines function independently of each other with respective probabilities $${1 \over 2},{1 \over 4}$$ and $${1 \over 4}$$. For the ship to be operational at least two of its engines must function. Let $$X$$ denote the event that the ship is operational and Let $${X_1},{X_2}$$ and $${X_3}$$ denote respectively the events that the engines $${E_1},{E_2}$$ and $${E_3}$$ are functioning. Which of the following is (are) true?
A
$$P\left[ {X_1^c|X} \right] = {3 \over {16}}$$
B
$$P$$ [exactly two engines of the ship are functioning $$\left. {|X} \right] = {7 \over 8}$$
C
$$P\left[ {X|{X_2}} \right] = {5 \over {16}}$$
D
$$P\left[ {X|{X_1}} \right] = {7 \over {16}}$$
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