If `x=2+a+a^2+oo,w h e r e|a|<1a n dy=1+b+b^2+oo,w h e r e|b|<1.` prove that: `1+a b+a^2b^2+oo=(x y)/(x+y-1)`

Let A and B be two sets such that n(A)=3 a n d n(B)=2. if (x ,1), (y ,2), (z ,1)a r e in AxxB , find A a n d B , w h e r e x , y , z are distinct elements.

Evaluate the following integrals: int_-5^0f(x)dx ,w h e r ef(x)=|x|+|x+2|+|x+5|

If the median of the scores 1, 2, x , 4, 5 (w h e r e 1<2

Evaluate: int(a x^2+b x+c)/((x-a)(x-b)(x-c))\ dx ,\ \ w h e r e\ a ,\ b ,\ c are distinct.

If f(x0=x+{-x}+[x],\ w h e r e\ [x] is the integral part& {x} is the fractional part of xdot Discuss the continuity of f in [-2,\ 2] .

Find the area of the region bounded by the x-axis and the curves defined by y=tanx(w h e r e-pi/3lt=xlt=pi/3) and y=cotx(w h e r epi/6lt=xlt=(3x)/2) .

Comprehension 1 Let a^((log)_b x)=c w h e r e a , b , c & x are parameters. On the basis of above information, answer the following questions: If b=(log)_(sqrt(3))3, c=2(log)_bsqrt(b) a n d s intheta=a(w h e r e x >1) t h e n theta can be a.pi/4 b. (3pi)/2 c. pi/2 d. 0

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

Comprehension 1 Let a^((log)_b x)=c w h e r e a , b , c & x are parameters. On the basis of above information, answer the following questions: If a=b=2xx3^sin^2theta+1/(1+tan^(2theta)) & c=(4 t a ntheta)/(sqrt(sec^2theta-1))(w h e r e theta lies in Ist Quadrant) , then x is- a.3 b. -3 c. 0 d. 4

Let f(x)=int_1^x(3^t)/(1+t^2)dx ,w h e r e x > 0. Then (a)for 0