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JEE Advanced

All the compounds listed in column I react with water. Match the result of the respective reactions with the appropriate options listed in column II

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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)

Let $a,b,andc$ be three non coplanar unit vectors such that the angle between every pair of them is $3π $. If $a×b+b×x=pa+qb+rc$ where p,q,r are scalars then the value of $q_{2}p_{2}+2q_{2}+r_{2} $ is

The option(s) with the values of a and L that satisfy the following equation is (are) $∫_{0}e_{t}(sin_{6}at+cos_{4}at)dt∫_{0}e_{t}(sin_{6}at+cos_{4}at)dt =L$

Let $X=(_{10}C_{1})_{2}+2(_{10}C_{2})_{2}+3(_{10}C_{3})_{2}+¨+10(_{10}C_{10})_{2}$, where $_{10}C_{r}$, $r∈{1,2,,¨ 10}$denote binomial coefficients. Then, the value of $14301 X$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

How many $3×3$ matrices $M$ with entries from ${0,1,2}$ are there, for which the sum of the diagonal entries of $M_{T}M$ is 5? (A) 126 (B)198 (C) 162 (D) 135

Let RS be the diameter of the circle $x_{2}+y_{2}=1,$ where S is the point $(1,0)$ Let P be a variable apoint (other than $RandS$) on the circle and tangents to the circle at $SandP$ meet at the point Q.The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. then the locus of E passes through the point(s)- (A) $(31 ,3 1 )$ (B) $(41 ,21 )$ (C) $(31 ,−3 1 )$ (D) $(41 ,−21 )$

For each positive integer $n$, let $y_{n}=n1 ((n+1)(n+2)n+n˙ )_{n1}$For $x∈R$let $[x]$be the greatest integer less than or equal to $x$. If $(lim)_{n→∞}y_{n}=L$, then the value of $[L]$is ______.