Consider two curves `C_1: y=1/x a n dC_2: y=1nx` on the `x y` plane. Let `D_1` denotes the region surrounded by `C_1,C_2,` and the line `x=1a n dD_2` denotes the region surrounded by `C_1,C_2` and the line `x=adot` If `D_1=D_2,` then the sum of logarithm of possible value of `a` is _____________

The area of the region bounded by the curve C :y=(x+1)/(x^(2)+1) nad the line y=1 , is

Sketch the region bounded by the curves y=x^2+2,\ y=x ,\ x=0\ a n d\ x=1. also, find the area of this region.

Consider two curves C_1: y^2=4[sqrt(y)]x a n dC_2: x^2=4[sqrt(x)]y , where [.] denotes the greatest integer function. Then the area of region enclosed by these two curves within the square formed by the lines x=1,y=1,x=4,y=4 is (a) 8/3s qdotu n i t s (b) (10)/3s qdotu n i t s (c) (11)/3s qdotu n i t s (d) (11)/4s qdotu n i t s

Let 'a' be a positive constant number. Consider two curves C_1: y=e^x, C_2:y=e^(a-x) . Let S be the area of the part surrounding by C_1, C_2 and the y axis, then Lim_(a->0) s/a^2 equals (A) 4 (B) 1/2 (C) 0 (D) 1/4

Consider two curves C1:y^2=4x ; C2=x^2+y^2-6x+1=0 . Then, a. C1 and C2 touch each other at one point b. C1 and C2 touch each other exactly at two point c. C1 and C2 intersect(but do not touch) at exactly two point d. C1 and C2 neither intersect nor touch each other

The slope of the tangent to a curve y=f(x) at (x,f(x)) is 2x+1. If the curve passes through the point (1,2) then the area of the region bounded by the curve, the x-axis and the line x=1 is (A) 5/6 (B) 6/5 (C) 1/6 (D) 1

If the line y=mx+c is a tangent to the ellipse x^(2)+2y^(2)=4 , then the minimum possible value of c is (a) -sqrt(2) (b) sqrt(2) (c) 2 (d) 1

The line y=mx+1 is a tangent to the curve y^2=4x if the value of m is(A) 1 (B) 2(C) 3(D) 1/2.

Let C : y =x^(2) -3, D : y = kx^(2) be two parabolas and L_(1) : x= a , L_(2) : x = 1 (a ne 0) be two straight lines. IF the line L_(1) meets the parabola C at a point B on the line L_(2) , other than A, then a may be equal to

The line y=m x+1 is a tangent to the curve y^2=4x , if the value of m is (a) 1 (b) 2 (c) 3 (d) 1/2