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
For a∈R (the set of all real numbers), a=−1),(lim)n→∞((n+1)a−1[(na+1)+(na+2)+……(na+n)]1a+2a++na=60.1Then a=(a)5 (b) 7 (c) 2−15 (d) 2−17
Let f:[21,1]→R (the set of all real numbers) be a positive, non-constant, and differentiable function such that fprime(x)<2f(x)andf(21)=1 . Then the value of ∫211f(x)dx lies in the interval (a)(2e−1,2e) (b) (3−1,2e−1)(c)(2e−1,e−1) (d) (0,2e−1)
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 ?
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)
Let y(x) be a solution of the differential equation (1+ex)yprime+yex=1. If y(0)=2 , then which of the following statements is (are) true? (a)y(−4)=0 (b)y(−2)=0 (c)y(x) has a critical point in the interval (−1,0) (d)y(x) has no critical point in the interval(−1,0)