1
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
2
AIEEE 2009
+4
-1
Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only
real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( { - 1} \right) < P\left( 1 \right),$$ then in the interval $$\left[ { - 1,1} \right]:$$
A
$$P(-1)$$ is not minimum but $$P(1)$$ is the maximum of $$P$$
B
$$P(-1)$$ is the minimum but $$P(1)$$ is not the maximum of $$P$$
C
Neither $$P(-1)$$ is the minimum nor $$P(1)$$ is the maximum of $$P$$
D
$$P(-1)$$ is the minimum and $$P(1)$$ is the maximum of $$P$$
3
AIEEE 2008
+4
-1
Suppose the cubic $${x^3} - px + q$$ has three distinct real roots
where $$p>0$$ and $$q>0$$. Then which one of the following holds?
A
The cubic has minima at $$\sqrt {{p \over 3}}$$ and maxima at $$-\sqrt {{p \over 3}}$$
B
The cubic has minima at $$-\sqrt {{p \over 3}}$$ and maxima at $$\sqrt {{p \over 3}}$$
C
The cubic has minima at both $$\sqrt {{p \over 3}}$$ and $$-\sqrt {{p \over 3}}$$
D
The cubic has maxima at both $$\sqrt {{p \over 3}}$$ and $$-\sqrt {{p \over 3}}$$
4
AIEEE 2008
+4
-1
How many real solutions does the equation
$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?
A
$$7$$
B
$$1$$
C
$$3$$
D
$$5$$
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