5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be T1, the work done in the process is
Let Pbe a point in the first octant, whose image Qin the plane x+y=3(that is, the line segment PQis perpendicular to the plane x+y=3and the mid-point of PQlies in the plane x+y=3)lies on the z-axis. Let the distance of Pfrom the x-axis be 5. If Ris the image of Pin the xy-plane, then the length of PRis _______.
Let a, b, cbe three non-zero real numbers such that the equation 3 acosx+2 bsinx=c, x∈[−2π,2π], has two distinct real roots αand βwith α+β=3π. Then, the value of abis _______.
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)If a tangent to a suitable conic (Column 1) is found to be y=x+8and its point of contact is (8,16), then which of the followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)
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
For x∈(0,π), the equation sinx+2sinx−sin3x=3 has (A)infinitely many solutions (B)three solutions (C)one solution (D)no solution
If a chord, which is not a tangent of the parabola y2=16xhas the equation 2x+y=p,and midpoint (h,k),then which of the following is(are) possible values (s) of p,handk?p=−1,h=1,k=−3 p=2,h=3,k=−4 p=−2,h=2,k=−4 p=5,h=4,k=−3