The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic KOH is
Let f:R→Rbe a differentiable function with f(0)=0. If y=f(x)satisfies the differential equation dxdy=(2+5x)(5x−2)1, then the value of (lim)x→∞f(x)is ______
Let P1:2x+y−z=3and P2:x+2y+z=2be two planes. Then, which of the following statement(s) is (are) TRUE?The line of intersection of P1and P2has direction ratios 1, 2, −1(b) The line 93x−4=91−3y=3zis perpendicular to the line of intersection of P1and P2(c) The acute angle between P1and P2is 60o(d) If P3is the plane passing through the point (4, 2, −2)and perpendicular to the line of intersection of P1and P2, then the distance of the point (2, 1, 1)from the plane P3is 32
Let f:[0,2]→R be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1Let:F(x)=∫0x2f(t)dtforx∈[0,2]I˙fFprime(x)=fprime(x) . for all x∈(0,2), then F(2) equals (a)e2−1 (b) e4−1(c)e−1 (d) e4
Let P and Q be distinct points on the parabola y2=2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle ΔOPQ is 32 , then which of the following is (are) the coordinates of P?
A box B1, contains 1 white ball, 3 red balls and 2 black balls. Another box B2, contains 2 white balls, 3 red balls and 4 black balls. A third box B3, contains 3 white balls, 4 red balls and 5 black balls.
Suppose that the foci of the ellipse 9x2+5y2=1are (f1,0)and(f2,0)where f1>0andf2<0.Let P1andP2be two parabolas with a common vertex at (0, 0) and with foci at (f1.0)and(2f_2 , 0), respectively. LetT1be a tangent to P1which passes through (2f2,0)and T2be a tangents to P2which passes through (f1,0). If m1is the slope of T1and m2is the slope of T2,then the value of (m121+m22)is
Let fprime(x)=2+sin4πx192x3forallx∈Rwithf(21)=0.Ifm≤∫211f(x)dx≤M, then the possible values of mandM are (a)m=13,M=24 (b) m=41,M=21(c)m=−11,M=0 (d) m=1,M=12