1
AIEEE 2010
+4
-1
Let $$f:R \to R$$ be defined by $$f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.$$$If $$f$$has a local minimum at $$x=-1$$, then a possible value of $$k$$ is A $$0$$ B $$- {1 \over 2}$$ C $$-1$$ D $$1$$ 2 AIEEE 2010 MCQ (Single Correct Answer) +4 -1 The equation of the tangent to the curve $$y = x + {4 \over {{x^2}}}$$, that is parallel to the $$x$$-axis, is A $$y=1$$ B $$y=2$$ C $$y=3$$ D $$y=0$$ 3 AIEEE 2010 MCQ (Single Correct Answer) +4 -1 Let $$f:R \to R$$ be a continuous function defined by $$f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$$$

Statement - 1 : $$f\left( c \right) = {1 \over 3},$$ for some $$c \in R$$.

Statement - 2 : $$0 < f\left( x \right) \le {1 \over {2\sqrt 2 }},$$ for all $$x \in R$$

A
Statement - 1 is true, Statement -2 is true; Statement - 2 is not a correct explanation for Statement - 1.
B
Statement - 1 is true, Statement - 2 is false.
C
Statement - 1 is false, Statement - 2 is true.
D
Statement - 1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement - 1.
4
AIEEE 2009
+4
-1
Let $$f\left( x \right) = x\left| x \right|$$ and $$g\left( x \right) = \sin x.$$
Statement-1: gof is differentiable at $$x=0$$ and its derivative is continuous at that point.
Statement-2: gof is twice differentiable at $$x=0$$.
A
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B
Statement-1 is true, Statement-2 is false
C
Statement-1 is false, Statement-2 is true
D
Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-1
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