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
$$-{1 \over 2}$$
Explanation
Equation of a line passing through $$\left( {{x_1},{y_1}} \right)$$ having
slope $$m$$ is given by $$y - {y_1} = m\left( {x - {x_1}} \right)$$
Since the line $$PQ$$ is passing through $$(1,2)$$ therefore its
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 $$x=2$$.
Hence both the statements are true and statement $$2$$ is a correct explanation for $$1.$$
3
AIEEE 2012
MCQ (Single Correct Answer)
A spherical balloon is filled with $$4500\pi $$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $$72\pi $$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $$49$$ minutes after the leakage began is :
A
$${{9 \over 7}}$$
B
$${{7 \over 9}}$$
C
$${{2 \over 9}}$$
D
$${{9 \over 2}}$$
Explanation
Volume of spherical balloon $$ = V = {4 \over 3}\pi {r^3}$$
$$ \Rightarrow 4500\pi = {{4\pi {r^3}} \over 3}$$
( as Given, volume $$ = 4500\pi {m^3}$$ )
Differentiating both the sides, $$w.r.t't'$$ we get,