Fusion of MnO2 along with KOH and O2 forms X. electrolytic oxidation of X yields Y. X undergoes disproportionation reaction in acidic medium to MnO2 and Y. The Manganese in X and Y is in the form W and Z respectively, then
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
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3,4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let xi be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that x1+x2+x3 is odd isThe probability that x1,x2,x3 are in an aritmetic progression is
Consider the circle x2+y2=9 and the parabola y2=8x. They intersect at P and Q in first and 4th quadrant,respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.
Let P be the point on parabola y2=4x which is at the shortest distance from the center S of the circle x2+y2−4x−16y+64=0 let Q be the point on the circle dividing the line segment SP internally. Then
Let Xbe a set with exactly 5 elements and Ybe a set with exactly 7 elements. If αis the number of one-one function from Xto Yand βis the number of onto function from Yto X, then the value of 5!1(β−α)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
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