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JEE Advanced

Let $f(x)=x_{2}sinπx ,x>0$ The $x_{1}<x_{2}<x_{3}...Ltx_{n}<...$ be all points of local maximum of f(x) and $y_{1}<y_{2}<y_{3}...<y_{n}<...$ be all the points of lcal minimum of f(x) then correct options is/zer

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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 $

Let $M$be a $2×2$symmetric matrix with integer entries. Then $M$is invertible ifThe first column of $M$is the transpose of the second row of $M$The second row of $M$is the transpose of the first column of $M$$M$is a diagonal matrix with non-zero entries in the main diagonalThe product of entries in the main diagonal of $M$is not the square of an integer

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

Let $f:R→R$be a differentiable function with $f(0)=0$. If $y=f(x)$satisfies the differential equation $dxdy =(2+5x)(5x−2)1 $, then the value of $(lim)_{x→∞}f(x)$is ______

Let `f(x)``=x+log_ex-xlog_ex ,x(0,oo)dot`Column 1 contains information about zeros of `f^(prime)(x)f^(prime)(x)a n df^(x)dot`Column 2 contains information about the limiting behaviour of `f^(prime)(x)f^(prime)(x)a n df^(x)`at infinity.Column 2 contains information about the increasing/decreasing nature of `f(x)a n df^(prime)(x)dot`Column I, Column 2, Column 3I, `f(x)=0forsom ex(l , e^2)`, (i), `("lim")_("x"vecoo"")f^(prime)(x)=0`, (P), `f`is increasing in (0,1)II, `f'(x)=0forsom ex(l , e)`, (ii), `("lim")_("x"vecoo"")f^(x)=-oo`, (Q), `f`is decreasing in `(e ,e^2)`III, `f'(x)=0forsom ex(0,1)`, , `("lim")_("x"vecoo"")f^(prime)(x)=-oo`, (R), `f`is increasing in (0,1)IV, `f^(' '(x))=0forsom ex(1, e)`, , `("lim")_("x"vecoo"")f^prime^'(x)=0`, (S), `f`is decreasing in (`e , e^2`)Which of the following options is the only CORRECT combination?(I) (ii) (P) (b) (IV) (iv) (S) (III) (iii) (R) (d) (II) (ii) (Q)Which of the following option is the only incorrect combination?(III) (i) (R) (b) (I) (iii) (P)(II) (iii) (P) (d) (II) (iv) (Q)Which of the following options is the only CORRECT combination?(I) (ii) (R) (b) (II) (iii) (S)(III) (iv) (P) (d) (IV) (i) (S)

The circle $C_{1}:x_{2}+y_{2}=3,$ with centre at O, intersects the parabola $x_{2}=2y$ at the point P in the first quadrant. Let the tangent to the circle $C_{1}$ at P touches other two circles $C_{2}andC_{3}atR_{2}andR_{3},$ respectively. Suppose $C_{2}andC_{3}$ have equal radii $23 $ and centres at $Q_{2}$ and $Q_{3}$ respectively. If $Q_{2}$ and $Q_{3}$ lie on the y-axis, then (a)$Q2Q3=12$(b)$R2R3=46 $(c)area of triangle $OR2R3$ is $62 $(d)area of triangle $PQ2Q3is=42 $

Â·If the normals of the parabola $y_{2}=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x−3)_{2}(y+2)_{2}=r_{2}$ , then the value of $r_{2}$ is

Consider the set of eight vector $V={ai^+bj^ +ck^;a,bc∈{−1,1}}˙$Three non-coplanar vectors can be chosen from $V$is $2_{p}$ways. Then $p$is_______.