class 12

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JEE Advanced

Match the compounds/ions in Column I with their properties/reactions in Column II. Indicate your answer by darkening the appropriate bubbles of the $4×4$ matrix in the ORS.

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Let $f_{prime}(x)=2+sin_{4}πx192x_{3} forallx∈Rwithf(21 )=0.Ifm≤∫_{21}f(x)dx≤M,$ then the possible values of $mandM$ are (a)$m=13,M=24$ (b) $m=41 ,M=21 $(c)$m=−11,M=0$ (d) $m=1,M=12$

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

Let $F_{1}(x_{1},0)$ and $F_{2}(x_{2},0)$, for $x_{1}<0$ and $x_{2}>0$, be the foci of the ellipse $9x_{2} +8y_{2} =1$ Suppose a parabola having vertex at the origin and focus at $F_{2}$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral $MF_{1}NF_{2}$ is

Let $f[0,1]→R$ (the set of all real numbers be a function.Suppose the function f is twice differentiable, $f(0)=f(1)=0$,and satisfies $f_{′}(x)–2f_{′}(x)+f(x)≤e_{x},x∈[0,1]$.Which of the following is true for $0<x<1?$

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

Let $X$be a set with exactly 5 elements and $Y$be a set with exactly 7 elements. If $α$is the number of one-one function from $X$to $Y$and $β$is the number of onto function from $Y$to $X$, then the value of $5!1 (β−α)$is _____.

Let $S={1,2,3,¨ 9}F˙ork=1,2,5,letN_{k}$be the number of subsets of S, each containing five elements out of which exactly $k$are odd. Then $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=?$210 (b) 252 (c) 125 (d) 126

Let E1 and E2, be two ellipses whose centers are at the origin.The major axes of E1 and E2, lie along the x-axis and the y-axis, respectively. Let S be the circle $x_{2}+(y−1)_{2}=2$. The straight line x+ y =3 touches the curves S, E1 and E2 at P,Q and R, respectively. Suppose that $PQ=PR=322 $.If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are):