class 12

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JEE Advanced

Consider the following complex ions P, Q and R. $P=[FeF_{6}]_{3−},Q=[V(H_{2}O)_{6}]_{2+}andR=[Fe(H_{2}O)_{6}]_{2+}.$ The correct order of the complex ions, according to their spin-only magnetic moment values (in B.M.) is

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Suppose that the foci of the ellipse $9x_{2} +5y_{2} =1$are $(f_{1},0)and(f_{2},0)$where $f_{1}>0andf_{2}<0.$Let $P_{1}andP_{2}$be two parabolas with a common vertex at (0, 0) and with foci at $(f_{1}.0)and$(2f_2 , 0), respectively. Let$T_{1}$be a tangent to $P_{1}$which passes through $(2f_{2},0)$and $T_{2}$be a tangents to $P_{2}$which passes through $(f_{1},0)$. If $m_{1}$is the slope of $T_{1}$and $m_{2}$is the slope of $T_{2},$then the value of $(m121 +m22)$is

Let $T$be the line passing through the points $P(−2,7)$and $Q(2,−5)$. Let $F_{1}$be the set of all pairs of circles $(S_{1},S_{2})$such that $T$is tangent to $S_{1}$at $P$and tangent to $S_{2}$at $Q$, and also such that $S_{1}$and $S_{2}$touch each other at a point, say, $M$. Let $E_{1}$be the set representing the locus of $M$as the pair $(S_{1},S_{2})$varies in $F_{1}$. Let the set of all straight lines segments joining a pair of distinct points of $E_{1}$and passing through the point $R(1,1)$be $F_{2}$. Let $E_{2}$be the set of the mid-points of the line segments in the set $F_{2}$. Then, which of the following statement(s) is (are) TRUE?The point $(−2,7)$lies in $E_{1}$(b) The point $(54 ,57 )$does NOT lie in $E_{2}$(c) The point $(21 ,1)$lies in $E_{2}$(d) The point $(0,23 )$does NOT lie in $E_{1}$

Â·If the normals of the parabola $y_{2}=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x−3)_{2}(y+2)_{2}=r_{2}$ , then the value of $r_{2}$ is

Let $S={xϵ(−π,π):x=0,+2π }$The sum of all distinct solutions of the equation $3 secx+cosecx+2(tanx−cotx)=0$ in the set S is equal to

Let $S$be the set of all non-zero real numbers such that the quadratic equation $αx_{2}−x+α=0$has two distinct real roots $x_{1}andx_{2}$satisfying the inequality $∣x_{1}−x_{2}∣<1.$Which of the following intervals is (are) a subset (s) of $S?$$(21 ,5 1 )$b. $(5 1 ,0)$c. $(0,5 1 )$d. $(5 1 ,21 )$

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 $x_{i}$ be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that $x_{1}+x_{2}+x_{3}$ is odd isThe probability that $x_{1},x_{2},x_{3}$ are in an aritmetic progression is

Let $f:(0,π)→R$be a twice differentiable function such that $(lim)_{t→x}t−xf(x)sint−f(x)sinx =sin_{2}x$for all $x∈(0,π)$. If $f(6π )=−12π $, then which of the following statement(s) is (are) TRUE?$f(4π )=42 π $(b) $f(x)<6x_{4} −x_{2}$for all $x∈(0,π)$(c) There exists $α∈(0,π)$such that $f_{prime}(α)=0$(d) $f(2π )+f(2π )=0$

Let $PR=3i^+j^ −2k^andSQ=i^−3j^ −4k^$determine diagonals of a parallelogram $PQRS,andPT=i^+2j^ +3k^$be another vector. Then the volume of the parallelepiped determine by the vectors $PT$, $PQ$and $PS$is$5$b. $20$c. $10$d. $30$