1
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
Consider

$$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0$$

$$\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0$$

where p is a real number, and $$\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0$$

STATEMENT-1 : If line $${L_1}$$ is a chord of circle C, then line $${L_2}$$ is not always a diameter of circle C
and

STATEMENT-2 : If line $${L_1}$$ is a diameter of circle C, then line $${L_2}$$ is not a chord of circle C.

A
Statement-1 is True, Statement-2 is True; Statement-2 is a correct rexplanation for Statement-1
B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct rexplanation for Statement-1
C
Statement-1 is True, Statement-2 is False
D
Statement-1 is False, Statement-2 is True
2
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
Consider a branch of the hyperbola $$${x^2} - 2{y^2} - 2\sqrt 2 x - 4\sqrt 2 y - 6 = 0$$$

with vertex at the point $$A$$. Let $$B$$ be one of the end points of its latus rectum. If $$C$$ is the focus of the hyperbola nearest to the point $$A$$, then the area of the triangle $$ABC$$ is

A
$$1 - \sqrt {{2 \over 3}} $$
B
$$\sqrt {{3 \over 2}} - 1$$
C
$$1 + \sqrt {{2 \over 3}} $$
D
$$\sqrt {{3 \over 2}} + 1$$
3
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

Let $$g(x) = \log f(x)$$, where $$f(x)$$ is a twice differentiable positive function on (0, $$\infty$$) such that $$f(x + 1) = xf(x)$$. Then for N = 1, 2, 3, ..., $$g''\left( {N + {1 \over 2}} \right) - g''\left( {{1 \over 2}} \right) = $$

A
$$ - 4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N - 1} \right)}^2}}}} \right\}$$
B
$$4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N - 1} \right)}^2}}}} \right\}$$
C
$$ - 4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N + 1} \right)}^2}}}} \right\}$$
D
$$4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N + 1} \right)}^2}}}} \right\}$$
4
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
Let the function $$g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)$$ be given by

$$g\left( u \right) = 2{\tan ^{ - 1}}\left( {{e^u}} \right) - {\pi \over 2}.$$ Then, $$g$$ is
A
even and is strictly increasing in $$\left( {0,\infty } \right)$$
B
odd and is strictly decreasing in $$\left( { - \infty ,\infty } \right)$$
C
odd and is strictly increasing in $$\left( { - \infty ,\infty } \right)$$
D
neither even nor odd, but is strictly increasing in $$\left( { - \infty ,\infty } \right)$$
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