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JEE Advanced

The standard enthalpies of formation of $CO_{2}(g),H_{2}O(l)and glucose(s) at25_{∘}Care−400kJ/mol,−300kJ/moland−1300kJ/mol,$ respectively. The standard enthalpy of combustion per gram of glucose at $25_{∘}C$ is

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A box $B_{1}$, contains 1 white ball, 3 red balls and 2 black balls. Another box $B_{2}$, contains 2 white balls, 3 red balls and 4 black balls. A third box $B_{3}$, contains 3 white balls, 4 red balls and 5 black balls.

Let P be the point on parabola $y_{2}=4x$ which is at the shortest distance from the center S of the circle $x_{2}+y_{2}−4x−16y+64=0$ let Q be the point on the circle dividing the line segment SP internally. Then

Let $f_{1}:R→R,f_{2}:(−2π ,2π )→Rf_{3}:(−1,e_{2π}−2)→R$and $f_{4}:R→R$be functions defined by\displaystyle{{f}_{{1}}{\left({x}\right)}}={\sin{{\left(\sqrt{{{1}-{e}^{{-{{x}}}}^{2}}}\right)}}},(ii) $f_{2}(x)={tan_{−1}x∣sinx∣ ifx=01ifx=0,$where the inverse trigonometric function $tan_{−1}x$assumes values in $(2π ,2π )$,(iii) $f_{3}(x)=[sin((g)_{e}(x+2))]$, where, for $t∈R$, $[t]$denotes the greatest integer less than or equal to $t$,(iv) $f_{4}(x)={x_{2}sin(x1 )ifx=00ifx=0$.LIST-I LIST-IIP. The function $f_{1}$is 1. NOT continuous at $x=0$Q. The function $f_{2}$is 2. continuous at $x=0$and NOTR. The function $f_{2}$is differentiable at $x=0$S. The function $f_{2}$is 3. differentiable at $x=0$and itsis NOT continuous at $x=0$4. differentiable at $x=0$and itsderivative is continuous at $x=0$The correct option is$P→2;Q→3;R→1;S→4$(b) $P→4;Q→1;R→2;S→3$(c) $P→4;Q→2;R→1;S→3$(d) $P→2;Q→1;R→4;S→3$

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 $R$

Let $S={xϵ(−π,π):x=0,+2π }$The sum of all distinct solutions of the equation $3 secx+cosecx+2(tanx−cotx)=0$ in the set S is equal to

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

Let $n_{1},andn_{2}$, be the number of red and black balls, respectively, in box I. Let $n_{3}andn_{4}$,be the number one red and b of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probablity that this red ball was drawn from box II is $31 $ then the correct option(s) with the possible values of $n_{1},n_{2},n_{3},andn_{4}$, is(are)

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