1
AIEEE 2012
+4
-1
Let $$a,b \in R$$ be such that the function $$f$$ given by $$f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0$$ has extreme values at $$x=-1$$ and $$x=2$$

Statement-1 : $$f$$ has local maximum at $$x=-1$$ and at $$x=2$$.

Statement-2 : $$a = {1 \over 2}$$ and $$b = {-1 \over 4}$$

A
Statement - 1 is false, Statement - 2 is true.
B
Statement - 1 is true , Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1.
C
Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1.
D
Statement - 1 is true, Statement - 2 is false.
2
AIEEE 2012
+4
-1
A line is drawn through the point $$(1, 2)$$ to meet the coordinate axes at $$P$$ and $$Q$$ such that it forms a triangle $$OPQ,$$ where $$O$$ is the origin. If the area of the triangle $$OPQ$$ is least, then the slope of the line $$PQ$$ is :
A
$$-{1 \over 4}$$
B
$$-4$$
C
$$-2$$
D
$$-{1 \over 2}$$
3
AIEEE 2012
+4
-1
Let $$A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right)$$. If $${u_1}$$ and $${u_2}$$ are column matrices such
that $$A{u_1} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right)$$ and $$A{u_2} = \left( {\matrix{ 0 \cr 1 \cr 0 \cr } } \right),$$ then $${u_1} + {u_2}$$ is equal to :
A
$$\left( {\matrix{ -1 \cr 1 \cr 0 \cr } } \right)$$
B
$$\left( {\matrix{ -1 \cr 1 \cr -1 \cr } } \right)$$
C
$$\left( {\matrix{ -1 \cr -1 \cr 0 \cr } } \right)$$
D
$$\left( {\matrix{ 1 \cr -1 \cr -1 \cr } } \right)$$
4
AIEEE 2012
+4
-1
Let $$P$$ and $$Q$$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If $${P^3} = {Q^3}$$ and
$${P^2}Q = {Q^2}P$$ then determinant of $$\left( {{P^2} + {Q^2}} \right)$$ is equal to :
A
$$-2$$
B
$$1$$
C
$$0$$
D
$$-1$$
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