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JEE Advanced

Aquesous solution of $Na_{2}S_{2}O_{3}$ on reaction with $CI_{2}$ gives

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In R', consider the planes $P_{1},y=0$ and $P_{2}:x+z=1$. Let $P_{3}$, be a plane, different from $P_{1}$, and $P_{2}$, which passes through the intersection of $P_{1}$, and $P_{2}$. If the distance of the point $(0,1,0)$ from $P_{3}$, is $1$ and the distance of a point $(α,β,γ)$ from $P_{3}$ is $2$, then which of the following relation is (are) true ?

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

PARAGRAPH AThere are five students $S_{1},S_{2},S_{3},S_{4}$and $S_{5}$in a music class and for them there are five seats $R_{1},R_{2},R_{3},R_{4}$and $R_{5}$arranged in a row, where initially the seat $R_{i}$is allotted to the student $S_{i},i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted five seats. For $i=1,2,3,4,$let $T_{i}$denote the event that the students $S_{i}$and $S_{i+1}$do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_{1}∩T_{2}∩T_{3}∩T_{4}$is$151 $(b) $101 $(c) $607 $(d) $51 $

For any positive integer $n$, define $f_{n}:(0,∞)→R$as $f_{n}(x)=j=1∑n tan_{−1}(1+(x+j)(x+j−1)1 )$for all $x∈(0,∞)$.Here, the inverse trigonometric function $tan_{−1}x$assumes values in $(−2π ,2π )˙$Then, which of the following statement(s) is (are) TRUE?$j=1∑5 tan_{2}(f_{j}(0))=55$(b) $j=1∑10 (1+fj_{′}(0))sec_{2}(f_{j}(0))=10$(c) For any fixed positive integer $n$, $(lim)_{x→∞}tan(f_{n}(x))=n1 $(d) For any fixed positive integer $n$, $(lim)_{x→∞}sec_{2}(f_{n}(x))=1$

Let $O$be the origin and let PQR be an arbitrary triangle. The point S is such that$OPO˙Q+ORO˙S=ORO˙P+OQO˙S=OQ$.$OR+OPO˙S$Then the triangle PQ has S as its:circumcentre (b) orthocentre (c) incentre (d) centroid

The coefficients of three consecutive terms of $(1+x)_{n+5}$are in the ratio 5:10:14. Then $n=$___________.

Football teams T1 and T2 have to play two games against each other. It is assumed that theoutcomes of the two games are independent. The probabilities of T1 winning, drawing andlosing a game against T2 are1/ 2,and1/6,1/3respectively. Each team gets 3 points for a win,1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total pointsscored by teams T1 and T2, respectively, after two gamesP $(X=Y)$ is

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is