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JEE Advanced

An AC voltage source of variable angular frequency $ω$ and fixed amplitude $V_{0}$ is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When $ω$ is increased

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Let $p,q$be integers and let $α,β$be the roots of the equation, $x_{2}−x−1=0,$where $α=β$. For $n=0,1,2,,leta_{n}=pα_{n}+qβ_{n}˙$FACT : If $aandb$are rational number and $a+b5 =0,thena=0=b˙$If $a_{4}=28,thenp+2q=$7 (b) 21 (c) 14 (d) 12

PARAGRAPH AThere are five students $S_{1},S_{2},S_{3},S_{4}$and $S_{5}$in a music class and for them there are five seats $R_{1},R_{2},R_{3},R_{4}$and $R_{5}$arranged in a row, where initially the seat $R_{i}$is allotted to the student $S_{i},i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted five seats. For $i=1,2,3,4,$let $T_{i}$denote the event that the students $S_{i}$and $S_{i+1}$do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_{1}∩T_{2}∩T_{3}∩T_{4}$is$151 $(b) $101 $(c) $607 $(d) $51 $

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of m/n is

Let $a,b,c$be three non-zero real numbers such that the equation $3 acosx+2bsinx=c,x∈[−2π ,2π ]$, has two distinct real roots $α$and $β$with $α+β=3π $. Then, the value of $ab $is _______.

For a point $P$in the plane, let $d_{1}(P)andd_{2}(P)$be the distances of the point $P$from the lines $x−y=0andx+y=0$respectively. The area of the region $R$consisting of all points $P$lying in the first quadrant of the plane and satisfying $2≤d_{1}(P)+d_{2}(P)≤4,$is

Consider the set of eight vector $V={ai^+bj^ +ck^;a,bc∈{−1,1}}˙$Three non-coplanar vectors can be chosen from $V$is $2_{p}$ways. Then $p$is_______.

For $a>b>c>0$, if the distance between $(1,1)$ and the point of intersection of the line $ax+by−c=0$ is less than $22 $ then,

A circle S passes through the point (0, 1) and is orthogonal to the circles $(x−1)_{2}+y_{2}=16$ and $x_{2}+y_{2}=1$. Then (A) radius of S is 8 (B) radius of S is 7 (C) center of S is (-7,1) (D) center of S is (-8,1)