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JEE Advanced

The compounds P, Q and S were separately subjected to nitration using $HNO_{3}/H_{2}SO_{4}$ mixture. The major product formed in each case respectively. Is

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Let $X$be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11,,¨ $and $Y$be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23,¨$. Then, the number of elements in the set $X∪Y$is _____.

Let $P$be a point in the first octant, whose image $Q$in the plane $x+y=3$(that is, the line segment $PQ$is perpendicular to the plane $x+y=3$and the mid-point of $PQ$lies in the plane $x+y=3)$lies on the z-axis. Let the distance of $P$from the x-axis be 5. If $R$is the image of $P$in the xy-plane, then the length of $PR$is _______.

If $w=α+iβ,$where $β=0$and $z=1$, satisfies the condition that $(1−zw−wz )$is a purely real, then the set of values of $z$is $∣z∣=1,z=2$ (b) $∣z∣=1andz=1$$z=z$ (d) None of these

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

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60

Perpendiculars are drawn from points on the line $2x+2 =−1y+1 =3z $ to the plane $x+y+z=3$ The feet of perpendiculars lie on the line (a) $5x =8y−1 =−13z−2 $ (b) $2x =3y−1 =−5z−2 $ (c) $4x =3y−1 =−7z−2 $ (d) $2x =−7y−1 =5z−2 $

Let $f:[0,∞)→R$be a continuous function such that $f(x)=1−2x+∫_{0}e_{x−t}f(t)dt$for all $x∈[0,∞)$. Then, which of the following statement(s) is (are) TRUE?The curve $y=f(x)$passes through the point $(1,2)$(b) The curve $y=f(x)$passes through the point $(2,−1)$(c) The area of the region ${(x,y)∈[0,1]×R:f(x)≤y≤1−x_{2} }$is $4π−2 $(d) The area of the region ${(x,y)∈[0,1]×R:f(x)≤y≤1−x_{2} }$is $4π−1 $

Let $f(x)=∣1−x∣1−x(1+∣1−x∣) cos(1−x1 )$ for $x=1.$ Then: (A)$(lim)_{n→1_{−}}f(x)$ does not exist (B)$(lim)_{n→1_{+}}f(x)$ does not exist (C)$(lim)_{n→1_{−}}f(x)=0$ (D)$(lim)_{n→1_{+}}f(x)=0$