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JEE Advanced

The desired product X can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II. (given, order of migratory aptitude: aryl $>$ alkyl $>$ hydrogen)

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Four person independently solve a certain problem correctly with probabilities $21 ,43 ,41 ,81 ˙$Then the probability that he problem is solve correctly by at least one of them is$256235 $b. $25621 $c. $2563 $d. $256253 $

Let m be the smallest positive integer such that the coefficient of $x_{2}$ in the expansion of $(1+x)_{2}+(1+x)_{3}+(1+x)_{4}+……..+(1+x)_{49}+(1+mx)_{50}$ is $(3n+1)._{51}C_{3}$ for some positive integer n. Then the value of n is

The value of $(((g)_{2}9)_{2})_{(log)((log)9)1}×(7 )_{(log)71}$is ________.

Let $f:R→R$be a differentiable function with $f(0)=1$and satisfying the equation $f(x+y)=f(x)f_{prime}(y)+f_{prime}(x)f(y)$for all $x,y∈R$. Then, the value of $(g)_{e}(f(4))$is _______

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3,4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_{i}$ be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that $x_{1}+x_{2}+x_{3}$ is odd isThe probability that $x_{1},x_{2},x_{3}$ are in an aritmetic progression is

The value of $∫_{0}4x_{3}{dx_{2}d_{2} (1−x_{2})_{5}}dxis$

Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S(21 ,21 ,21 )$ be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If $p =SP,q =SQ ,r=SR$ and $t=ST$ then the value of $∣(p ×q )×(r×(t)∣is$

Let the curve C be the mirror image of the parabola $y_{2}=4x$ with respect to the line $x+y+4=0$. If A and B are the points of intersection of C with the line $y=−5$, then the distance between A and B is