class 12

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JEE Advanced

Hyperconugation involves overlap of the following orbitals

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Let $P=⎣⎡ 1416 014 001 ⎦⎤ $and $I$ be the identity matrix of order $3$. If $Q=[q_{()}ij]$ is a matrix, such that $P_{50}−Q=I$, then $q_{21}q_{31}+q_{32} $ equals

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $27 $ on y-axis is (are)

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is :

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

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) $P→2;Q→1;R→4;S→3$

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

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 $

For $3×3$matrices $MandN,$which of the following statement (s) is (are) NOT correct ?$N_{T}MN$is symmetricor skew-symmetric, according as $m$is symmetric or skew-symmetric.$MN−NM$is skew-symmetric for all symmetric matrices $MandN˙$$MN$is symmetric for all symmetric matrices $MandN$$(adjM)(adjN)=adj(MN)$for all invertible matrices $MandN˙$