If at angle θthe light takes maximum time to travel in optical fiber. Then the maximum time is x×10−8, calculate x.
The function y=f(x) is the solution of the differential equation dxdy+x2−1xy=1−x2x4+2x in (−1,1) satisfying f(0)=0. Then ∫2323f(x)dx is
A farmer F1has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1)and R(2, 0). From this land, a neighbouring farmer F2takes away the region which lies between the side PQand a curve of the form y=xn (n>1). If the area of the region taken away by the farmer F2is exactly 30% of the area of PQR, then the value of nis _______.
A circle S passes through the point (0, 1) and is orthogonal to the circles (x−1)2+y2=16 and x2+y2=1. Then (A) radius of S is 8 (B) radius of S is 7 (C) center of S is (-7,1) (D) center of S is (-8,1)
Let f:RRbe a differentiable function such that f(0),f(2π)=3andfprime(0)=1.If g(x)=∫x2π[fprime(t)cosect−cottcosectf(t)]dtforx(0,2π],then (lim)x0g(x)=
Let a,b,andc be three non coplanar unit vectors such that the angle between every pair of them is 3π. If a×b+b×x=pa+qb+rc where p,q,r are scalars then the value of q2p2+2q2+r2 is
For every twice differentiable function f:R→[−2, 2]with (f(0))2+(fprime(0))2=85, which of the following statement(s) is (are) TRUE?There exist r, s∈Rwhere r<s, such that fis one-one on the open interval (r, s)(b) There exists x0∈(−4, 0)such that ∣∣fprime(x0)∣∣≤1(c) (lim)x→∞f(x)=1(d) There exists α∈(−4, 4)such that f(α)+f(α)=0and fprime(α)=0
Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, x2+y2=a, (i), my=m2x+a, (P), (m2a,m2a)II, x2+a2y2=a, (ii), y=mx+am2+1, (Q), (m2+1−ma,m2+1a)III, y2=4ax, (iii), y=mx+a2m2−1, (R), (a2m2+1−a2m,a2m2+11)IV, x2−a2y2=a2, (iv), y=mx+a2m2+1, (S), (a2m2+1−a2m,a2m2+1−1)The tangent to a suitable conic (Column 1) at (3,21)is found to be 3x+2y=4,then which of the following options is the only CORRECT combination?(IV) (iii) (S) (b) (II) (iii) (R)(II) (iv) (R) (d) (IV) (iv) (S)