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JEE Advanced

Geometrical shapes of the complexes formed by the reaction of $Ni_{2+}$ with $Cl_{−}$, $CN_{−}$ and $H_{2}O$, respectively, are

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For a point $P$in the plane, let $d_{1}(P)andd_{2}(P)$be the distances of the point $P$from the lines $x−y=0andx+y=0$respectively. The area of the region $R$consisting of all points $P$lying in the first quadrant of the plane and satisfying $2≤d_{1}(P)+d_{2}(P)≤4,$is

Let $XandY$be two events that $P(X)=31 ,P(X|Y)=21 andP(Y|X)=52 $then:$P(Y)=154 $ (b) $P(X∪Y)=52 $$P(X_{prime}|Y)=21 $ (d) $P(X∩Y)=51 $

A line L : y = mx + 3 meets y-axis at E (0, 3) and the arc of the parabola $y_{2}=16x$ $0≤y≤6$ at the point art $F(x_{0},y_{0})$. The tangent to the parabola at $F(X_{0},Y_{0})$ intersects the y-axis at $G(0,y)$. The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum P) m= Q) = Maximum area of $△EFG$ is (R) $y_{0}=$ (S) $y_{1}=$

For $x∈(0,π),$ the equation $sinx+2$sin$x−sin3x=3$ has (A)infinitely many solutions (B)three solutions (C)one solution (D)no solution

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 ______.

Let the curve C be the mirror image of the parabola $y_{2}=4x$ with respect to the line $x+y+4=0$. If A and B are the points of intersection of C with the line $y=−5$, then the distance between A and B is

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 _______.

Let $f:(0,∞)→R$ be a differentiable function such that $f_{′}(x)=2−xf(x) $ for all $x∈(0,∞)$ and $f(1)=1$, then