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JEE Advanced

The packing efficiency of the two -dimensional square unit cell shown belwo is

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$L_{1}=x+3y−5=0,L_{2}=3x−ky−1=0,L_{3}=5x+2y−12=0$ are concurrent if k=

A curve passes through the point $(1,6π )$ . Let the slope of the curve at each point $(x,y)$ be $xy +sec(xy ),x>0.$ Then the equation of the curve is

Let $XandY$be two events that $P(X)=31 ,P(X|Y)=21 andP(Y|X)=52 $then:$P(Y)=154 $ (b) $P(X∪Y)=52 $$P(X_{prime}|Y)=21 $ (d) $P(X∩Y)=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 in which 5 boys and 5 girls stand in such a way that exactly four girls stand consecutively in the queue. Then the value of $nm $ 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

Three randomly chosen nonnegative integers $x,yandz$are found to satisfy the equation $x+y+z=10.$Then the probability that $z$is even, is:$125 $ (b) $21 $ (c) $116 $ (d) $5536 $

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) $f(2π )+f(2π )=0$

For $3×3$matrices $MandN,$which of the following statement (s) is (are) NOT correct ?$N_{T}MN$is symmetricor skew-symmetric, according as $m$is symmetric or skew-symmetric.$MN−NM$is skew-symmetric for all symmetric matrices $MandN˙$$MN$is symmetric for all symmetric matrices $MandN$$(adjM)(adjN)=adj(MN)$for all invertible matrices $MandN˙$