1
JEE Advanced 2015 Paper 1 Offline
Numerical
+4
-0
A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V$$ $$m{m^3}$$, has a $$2$$ mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness $$2$$ mm and is of radius equal to the outer radius of the container.

If the volume of the material used to make the container is minimum when the inner radius of the container is $$10$$ mm,
then the value of $${V \over {250\pi }}$$ is

2
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Let $$y(x)$$ be a solution of the differential equation
$$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$$
If $$y(0)=2$$, then which of the following statement is (are) true?
A
$$y(-4)=0$$
B
$$y(-2)=0$$
C
$$y(x)$$ has a critical point in the interval $$(-1, 0)$$
D
$$y(x)$$ has no critical point in the interval $$(-1,0)$$
3
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Consider the family of all circles whose centres lie on the straight line $$y=x,$$ If this family of circle is represented by the differential equation $$Py'' + Qy' + 1 = 0,$$ where $$P, Q$$ are functions of $$x,y$$ and $$y'$$ $$\left( {here\,\,\,y' = {{dy} \over {dx}},y'' = {{{d^2}y} \over {d{x^2}}}} \right)$$ then which of the following statements is (are) true?
A
$$P = y + x$$
B
$$\,P = y - x$$
C
$$\,P + Q = 1 - x + y + y' + {\left( {y'} \right)^2}$$
D
$$\,P - Q = 1 - x + y - y' - {\left( {y'} \right)^2}$$
4
JEE Advanced 2015 Paper 1 Offline
Numerical
+4
-0
Let $$f:R \to R$$ be a function defined by
$$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.$$ where $$\left[ x \right]$$ is the greatest integer less than or equal to $$x$$, if $$I = \int\limits_{ - 1}^2 {{{xf\left( {{x^2}} \right)} \over {2 + f\left( {x + 1} \right)}}dx,}$$ then the value of $$(4I-1)$$ is