JEE-MATHS

1.Let f : N -> N be defined as f(n) =  $\left\{\frac{n+1}{2}\ if\ n\ is\ odd\ ,\frac{n}{2\ }\ if\ n\ is\ even\right\}$ Then f is 

• One to one 

• one to one and onto

• onto

• none of these 

For the sets of real numbers given by 
 $R_1=\left\{\left(X,Y\right)\ :\ X\in R,Y\in R\ X^{2\ }+\ Y^{2\ }\le\ 25\right\}$  
 $R_2=\left\{\left(X,Y\right)\ :\ X\in R,Y\in R\ ,9Y^{ }\ge4X^2\right\}$  
 $R_1\cap R_2$  is

• One - to - one

• One - one but not onto 

• onto but not one-one 

• none of  these 

 Range of  $\sin^{-1\ }\left[\frac{x^2+1}{x^2+2}\right]\ is\ :$  

•  $\left[0,\frac{\pi}{2}\right]$  

•  $\left[0,\frac{\pi}{6}\right]$  

•  $\left[\frac{\pi}{6},\frac{\pi}{2}\right]$  

• none of these

If fog =  $\left|\sin\ x\right|$  and gof = $\left|\sin\ x^2\right|\ \ \sqrt{x}$ then find f(x) and g(x) 

• f(x) =  $\sqrt{\sin\ x}$  , g(x) =  $x^2$  

• f(x) =  $\left|x\right|$ ,g(x)=  $x^2$  

• f(X) =  $\sqrt{x}$ , g(x)=  $\sin^2x$  

•  $f\left(x\right)\ =\ \sin\ \sqrt{x}\ .\ g\left(x\right)=x^2$  

If f(x) + 2f(1-x) =  $x^2\ +2\ \forall\ \ x\in R\ ,\ then\ f\left(x\right)\ is\ given\ as\$  

•  $\frac{\left(x+2\right)^2}{3}$  

•  $x^2-2$  

•  $1$  

• none of these 

 $f\left(x\right)\ =\ \frac{\cos x}{\frac{2x}{\pi}+\frac{1}{2}}\ ,$   $\ where\ x\ is\ not\ an\ integral\ multiple\ of\ \pi\$  
 $\ and\left[.\right]\ denotes\ the\ greatest\ integer\ function\ is$  

• an odd function 

• even function 

• neither odd nor even 

• none of these 

Domain of the function f(X) =  $\frac{1}{\sqrt{^{10}c_{x-1}\ -3.}^{10}c_x}$  contains the points 

• 9,10,11

• 9,10,12

• all natural numbers 

• none of these 

Domain of the function f(x) = $\frac{x}{\sqrt{\sin\left(\ln x\right)-cox\left(\ln x\right)}}$  is

•  $\left(e^{2n\pi},e^{\left(3n+\frac{1}{2}\right)\pi}\right)$  

•  $\left(e^{\left(2n+\frac{1}{4}\right)\pi},e^{\left(2n+\frac{5}{4}\right)\pi}\right)$  

•  $\left(e^{\left(2n+\frac{1}{4}\right)\pi},e^{\left(3n-\frac{3}{4}\right)\pi}\right)$  

• none of these 

The number of functions f from the set A = {0,1,2} in to the set B={0,1,2,3,4,5,6,7} such that f(i)  $\le$  f(j) for i<j and  $i,j\in A$  is   

•  $^8c_3$  

•  $^8c_3+2\left(^8c_2\right)$  

•  $^{10}c_3$  

•  $^{10}c_4$  

Let f : R  $\rightarrow\left(0,1\right]\ be\ defined\ by\$   $f\left(x\right)=\frac{1}{x^2+1}$  then is 

• many one 

• into 

• many one and into

• one - one and onto