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JEE Advanced

In the Newman projection for 2,2 -dimethyl butane X and Y can respectively be

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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 $f:R→Randg:R→R$ be respectively given by $f(x)=∣x∣+1andg(x)=x_{2}+1$. Define $h:R→R$ by $h(x)={max{f(x),g(x)},ifx≤0andmin{f(x),g(x)},ifx>0$.The number of points at which $h(x)$ is not differentiable is

Let $MandN$ be two $3×3$ matrices such that $MN=NM˙$ Further, if $M=N_{2}andM_{2}=N_{4},$ then Determinant of $(M_{2}+MN_{2})$ is 0 There is a $3×3$ non-zeero matrix $U$ such tht $(M_{2}+MN_{2})U$ is the zero matrix Determinant of $(M_{2}+MN_{2})≥1$For a $3×3$ matrix $U,if(M_{2}+MN_{2})U$ equal the zero mattix then $U$ is the zero matrix

If a chord, which is not a tangent of the parabola $y_{2}=16x$has the equation $2x+y=p,$and midpoint $(h,k),$then which of the following is(are) possible values (s) of $p,handk?$$p=−1,h=1,k=−3$ $p=2,h=3,k=−4$ $p=−2,h=2,k=−4$ $p=5,h=4,k=−3$

The largets value of non negative integer for which $x→1lim x+sin(x−1)−1(−ax+sin(x−1)+a]1−x }_{1−x1−x}=41 $

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,

How many $3×3$ matrices $M$ with entries from ${0,1,2}$ are there, for which the sum of the diagonal entries of $M_{T}M$ is 5? (A) 126 (B)198 (C) 162 (D) 135

Let $f:R→R$be a differentiable function with $f(0)=0$. If $y=f(x)$satisfies the differential equation $dxdy =(2+5x)(5x−2)1 $, then the value of $(lim)_{x→∞}f(x)$is ______