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JEE Advanced

Aniline reacts with mixed acid conc. $HNO_{3}$ and conc. $H_{2}SO_{4}$ at 288 K to give P (51 %), Q (47%) and R (2%). The major product(s) of the following reaction sequence is (are)

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A line $l$ passing through the origin is perpendicular to the lines $l_{1}:(3+t)i^+(−1+2t)j^ +(4+2t)k^,∞<t<∞,l_{2}:(3+s)i^+(3+2s)j^ +(2+s)k^,∞<t<∞$ then the coordinates of the point on $l_{2}$ at a distance of $17 $ from the point of intersection of \displaystyle{l}&{l}_{{1}} is/are:

Let $f:[a,b]1,∞ $be a continuous function and let $g:RR$be defined as$g(x)={0ifxbThen$$g(x)$is continuous but not differentiable at a$g(x)$is differentiable on $R$$g(x)$is continuous but nut differentiable at $b$$g(x)$is continuous and differentiable at either $a$or $b$but not both.

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 $

Let $αandβ$ be nonzero real numbers such that $2(cosβ−cosα)+cosαcosβ=1$ . Then which of the following is/are true? (a) $3 tan(2α )+tan(2β )=0$ (b) $3 tan(2α )−tan(2β )=0$ (c) $tan(2α )+3 tan(2β )=0$ (d) $tan(2α )−3 tan(2β )=0$

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

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. $(5 1 ,21 )$

Let $P_{1}:2x+y−z=3$and $P_{2}:x+2y+z=2$be two planes. Then, which of the following statement(s) is (are) TRUE?The line of intersection of $P_{1}$and $P_{2}$has direction ratios $1,2,−1$(b) The line $93x−4 =91−3y =3z $is perpendicular to the line of intersection of $P_{1}$and $P_{2}$(c) The acute angle between $P_{1}$and $P_{2}$is $60o$(d) If $P_{3}$is the plane passing through the point $(4,2,−2)$and perpendicular to the line of intersection of $P_{1}$and $P_{2}$, then the distance of the point $(2,1,1)$from the plane $P_{3}$is $3 2 $

Let $X$be a set with exactly 5 elements and $Y$be a set with exactly 7 elements. If $α$is the number of one-one function from $X$to $Y$and $β$is the number of onto function from $Y$to $X$, then the value of $5!1 (β−α)$is _____.