Let P be a point in the first octant , whose image Q in the plane x+y= 3 (that is, the line segment PQ is perpendicular to the plane x+y=3 and the mid- point of PQ lies in the plane x+y=3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P  in the XY-plane, then the length of PR is .....

A) 4

B) 6

C) 12

D) 8


Let  $f:R\rightarrow R$  be a differentiable function with f(0) =1 and satisfying the equation f(x+y) =f(x)  f ' (y)+ f ' (x) f(y)  for 

all x , $y \in R $, Then, the value of loge (f(4)) is..............

A) 2

B) 6

C) 8

D) 4


Let $f:R\rightarrow R$ be a differentiable function with f(0)=0,  y=f(x) satisfies the differential equation

 $\frac{\text{d}y}{\text{d}x}= (2+5y)(5y-2)$  , then the value of  $\lim_{x \rightarrow \infty} f(x) $  is .............

A) 0.6

B) 0.4

C) 0.8

D) 0.1


Let X be a set with exactly 5 elements and Y be a set  exactly  7 elements. If α is the numbers of one - one functions from X to Y  and β is the number of onto functions from Y to X  , then the value of $\frac{1}{5!}(\beta -\alpha) $ is ............

A) 118

B) 119

C) 121

D) 115


The value of integral $\int_{0}^{1/2} \frac{1+\sqrt{3}}{((x+1)^{2}(1-x)^{6})^{1/4}}dx$ is ............

A) 3

B) 4

C) 1

D) 2