If `f(x)=|[a,-1, 0],[a x, a,-1],[a x^2,a x, a]|,t h e nf(2x)-f(x)` is divisible by (a) `a` (b) `b` (c) `c ,d ,e` (d) none of these

If y=tan^(-1)((2^x)/(1+2^(2x+1))),t h e n(dy)/(dx)a tx=0 is (a)1 (b) 2 (c) 1n 2 (d) none of these

If f(x)={x^2, for xgeq1x ,for x<0 ,t h e n fof(x) is given by

If f^(prime)(x)=sqrt(2x^2-1)a n dy=f(x^2),t h e n(dy)/(dx)a tx=1 is 2 (b) 1 (c) -2 (d) none of these

If f(x)=int_0^1(dt)/(1+|x-t|) ,then f^(prime)(1/2) is equal to (a)0 (b) 1/2 (c) 1 (d) none of these

The function x^x decreases in the interval (0,e) (b) (0,1) (0,1/e) (d) none of these

If f(x+1)+f(x-1)=2f(x)a n df(0),=0, then f(n),n in N , is nf(1) (b) {f(1)}^n (c) 0 (d) none of these

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

If af(x+1)+bf(1/(x+1))=x ,x!=-1,a!=b ,t h e nf(2) is equal to (a) (2a+b)/(2(a^2-b^2)) (b) a/(a^2-b^2) (c) (a+2b)/(a^2-b^2) (d) none of these

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)-|f(x)| (d) none of these

The value of int_0^1e^(x^2-x)dx is (a) 1 (c) > e^(-1/4) (d) none of these