A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to maximum height of 4g3v2 with respect to the initial position. The object is
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation 3x+y−6=0 and the point D is (3 sqrt3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ. (1)The equation of circle C is (2)Points E and F are given by (3)Equation of the sides QR, RP are
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)For a=2,ifa tangent is drawn to a suitable conic (Column 1) at the point of contact (−1,1),then which of the following options is the only CORRECT combination for obtaining its equation?(I) (ii) (Q) (b) (III) (i) (P)(II) (ii) (Q) (d) (I)(i)(P)
Let Obe the origin, and OXxOY,OZbe three unit vectors in the direction of the sides QR, RP, PQ, respectively of a triangle PQR.If the triangle PQR varies, then the minimum value of cos(P+Q)+cos(Q+R)+cos(R+P)is:−23 (b) 35 (c) 23 (d) −35
In R', consider the planes P1,y=0 and P2:x+z=1. Let P3, be a plane, different from P1, and P2, which passes through the intersection of P1, and P2. If the distance of the point (0,1,0) from P3, is 1 and the distance of a point (α,β,γ) from P3 is 2, then which of the following relation is (are) true ?
If g(x)=∫sinxsin(2x)sin−1(t)dt,then: (a)gprime(2π)=−2π (b) gprime(−2π)=−2π (c)gprime(−2π)=2π (d) gprime(2π)=2π
In a triangle the sum of two sides is x and the product of the same is y. If x2−c2=y where c is the third side. Determine the ration of the in-radius and circum-radius
Consider the hyperbola H:x2−y2=1 and a circle S with centre N(x2,0) Suppose that H and S touch each other at a point (P(x1,y1) with x1>1andy1>0 The common tangent to H and S at P intersects the x-axis at point M. If (l,m) is the centroid of the triangle ΔPMN then the correct expression is (A) dx1dl=1−3x121 for x1>1 (B) dx1dm=3(x12−1)x!)forx1>1 (C) dx1dl=1+3x121forx1>1 (D) dy1dm=31fory1>0