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JEE Advanced

The total number of diprotic acids among the following is $H_{3}PO_{4}H_{3}BO_{3} H_{2}SO_{4}H_{3}PO_{2} H_{3}PO_{3}H_{2}CrO_{4} H_{2}CO_{3}H_{2}SO_{3} H_{2}S_{2}O_{7} $

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A farmer $F_{1}$has a land in the shape of a triangle with vertices at $P(0,0),Q(1,1)$and $R(2,0)$. From this land, a neighbouring farmer $F_{2}$takes away the region which lies between the side $PQ$and a curve of the form $y=x_{n}(n>1)$. If the area of the region taken away by the farmer $F_{2}$is exactly 30% of the area of $PQR$, then the value of $n$is _______.

For every pair of continuous functions $f,g:[0,1]→R$ such that $max{f(x):x∈[0,1]}=max{g(x):x∈[0,1]}$ then which are the correct statements

For every twice differentiable function $f:R→[−2,2]$with $(f(0))_{2}+(f_{prime}(0))_{2}=85$, which of the following statement(s) is (are) TRUE?There exist $r,s∈R$where $r<s$, such that $f$is one-one on the open interval $(r,s)$(b) There exists $x_{0}∈(−4,0)$such that $∣∣ f_{prime}(x_{0})∣∣ ≤1$(c) $(lim)_{x→∞}f(x)=1$(d) There exists $α∈(−4,4)$such that $f(α)+f(α)=0$and $f_{prime}(α)=0$

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

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

Let $P$be a matrix of order $3×3$such that all the entries in $P$are from the set ${−1,0,1}$. Then, the maximum possible value of the determinant of $P$is ______.

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 $

Let $f:R→R,g:R→Randh:R→R$ be the differential functions such that $f(x)=x_{3}+3x+2,g(f(x))=xandh(g(g(x)))=x,forallx∈R.Then$ (a)g'(2)=$151 $ (b)h'(1)=666 (c)h(0)=16 (d)h(g(3))=36