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JEE Advanced

The total number of lone-pairs of electrons in melamine is

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Let $f:(0,∞)R$ be given by $f(x)=∫_{x1}te_{−(t+t1)}dt ,$ then (a)$f(x)$ is monotonically increasing on $[1,∞)$(b)$f(x)$ is monotonically decreasing on $(0,1)$(c)$f(2_{x})$ is an odd function of $x$ on $R$

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover cards numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done isa.$264$ b. $265$ c. $53$ d. $67$

Let $O$be the origin, and $OX,OY,OZ$be three unit vectors in the direction of the sides $QR$, $RP$, $PQ$, respectively of a triangle PQR.$∣OX×OY∣=$$s∈(P+R)$ (b) $sin2R$$(c)sin(Q+R)$(d) $sin(P+Q)˙$

Let $f:RR$be a continuous odd function, which vanishes exactly at one point and $f(1)=21 ˙$Suppose that $F(x)=∫_{−1}f(t)dtforallx∈[−1,2]andG(x)=∫_{−1}t∣f(f(t))∣dtforallx∈[−1,2]I˙G(x)f(lim)_{x1}(F(x)) =141 ,$Then the value of $f(21 )$is

From a point $P(λ,λ,λ)$, perpendicular PQ and PR are drawn respectively on the lines $y=x,z=1$ and $y=−x,z=−1$.If P is such that $∠QPR$ is a right angle, then the possible value(s) of $λ$ is/(are)

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 $

If the line $x=α$divides the area of region $R={(x,y)R_{2}:x_{3}≤x,0≤x≤1}$into equal parts, then: $2α_{4}−4α_{2}+1=0$ $α_{4}+4α_{2}−1=0$$0<α≤21 $ (d) $21 <α<1$

Consider the circle $x_{2}+y_{2}=9$ and the parabola $y_{2}=8x$. They intersect at P and Q in first and 4th quadrant,respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.