class 12

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JEE Advanced

Match the crystal system/unit cells mentioned in column I with their characteristic features mentioned in Column II. Indicate your answer by darkening the appropriate bubbles of the $4×4$ matrix given in the ORS.

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Word of length 10 are formed using the letters A,B,C,D,E,F,G,H,I,J. Let $x$be the number of such words where no letter is repeated; and let $y$be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $9xy =$

For $a>b>c>0$, if the distance between $(1,1)$ and the point of intersection of the line $ax+by−c=0$ is less than $22 $ then,

Let $ω$be a complex cube root of unity with $ω=1andP=[p_{ij}]$be a $n×n$matrix withe $p_{ij}=ω_{i+j}˙$Then $p_{2}=O,whe∩=$a.$57$b. $55$c. $58$d. $56$

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 $f:RR$be a differentiable function such that $f(0),f(2π )=3andf_{prime}(0)=1.$If $g(x)=∫_{x}[f_{prime}(t)cosect−cottcosectf(t)]dtforx(0,2π ],$then $(lim)_{x0}g(x)=$

There 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.The probability that, on the examination day, the student $S_{1}$gets the previously allotted seat $R_{1}$, and NONE of the remaining students gets the seat previously allotted to him/her is$403 $(b) $81 $(c) $407 $(d) $51 $

The function $y=f(x)$ is the solution of the differential equation $dxdy +x_{2}−1xy =1−x_{2} x_{4}+2x $ in $(−1,1)$ satisfying $f(0)=0.$ Then $∫_{23}f(x)dx$ is

If $α=3sin_{−1}(116 )$and $β=3cos_{−1}(94 )$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are)