1
JEE Main 2016 (Online) 9th April Morning Slot
+4
-1
If   f(x) is a differentiable function in the interval (0, $$\infty$$) such that f (1) = 1 and

$$\mathop {\lim }\limits_{t \to x}$$   $${{{t^2}f\left( x \right) - {x^2}f\left( t \right)} \over {t - x}} = 1,$$ for each x > 0, then $$f\left( {{\raise0.5ex\hbox{\scriptstyle 3} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 2}}} \right)$$ equal to :
A
$${{13} \over 6}$$
B
$${{23} \over 18}$$
C
$${{25} \over 9}$$
D
$${{31} \over 18}$$
2
JEE Main 2016 (Offline)
+4
-1
If a curve $$y=f(x)$$ passes through the point $$(1,-1)$$ and satisfies the differential equation, $$y(1+xy) dx=x$$ $$dy$$, then $$f\left( { - {1 \over 2}} \right)$$ is equal to :
A
$${2 \over 5}$$
B
$${4 \over 5}$$
C
$$-{2 \over 5}$$
D
$$-{4 \over 5}$$
3
JEE Main 2015 (Offline)
+4
-1
Let $$y(x)$$ be the solution of the differential equation
$$\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).$$ Then $$y(e)$$ is equal to :
A
$$2$$
B
$$2e$$
C
$$e$$
D
$$0$$
4
JEE Main 2014 (Offline)
+4
-1
Let the population of rabbits surviving at time $$t$$ be governed by the differential equation $${{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200.$$ If $$p(0)=100,$$ then $$p(t)$$ equals:
A
$$600 - 500\,{e^{t/2}}$$
B
$$400 - 300\,{e^{-t/2}}$$
C
$$400 - 300\,{e^{t/2}}$$
D
$$300 - 200\,{e^{-t/2}}$$
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