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JEE Advanced

Among P,Q,R and S, the aromatic compound(s) is/are

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Suppose that the foci of the ellipse $9x_{2} +5y_{2} =1$are $(f_{1},0)and(f_{2},0)$where $f_{1}>0andf_{2}<0.$Let $P_{1}andP_{2}$be two parabolas with a common vertex at (0, 0) and with foci at $(f_{1}.0)and$(2f_2 , 0), respectively. Let$T_{1}$be a tangent to $P_{1}$which passes through $(2f_{2},0)$and $T_{2}$be a tangents to $P_{2}$which passes through $(f_{1},0)$. If $m_{1}$is the slope of $T_{1}$and $m_{2}$is the slope of $T_{2},$then the value of $(m121 +m22)$is

For $a∈R$ (the set of all real numbers), $a=−1),$$(lim)_{n→∞}((n+1)_{a−1}[(na+1)+(na+2)+……(na+n)]1_{a}+2_{a}++n_{a} =60.1 $Then $a=$(a)$5$ (b) 7 (c) $2−15 $ (d) $2−17 $

Coefficient of $x_{11}$ in the expansion of $(1+x_{2})_{4}(1+x_{3})_{7}(1+x_{4})_{12}$ is 1051 b. 1106 c. 1113 d. 1120

Let $f:[21 ,1]→R$ (the set of all real numbers) be a positive, non-constant, and differentiable function such that $f_{prime}(x)<2f(x)andf(21 )=1$ . Then the value of $∫_{21}f(x)dx$ lies in the interval (a)$(2e−1,2e)$ (b) $(3−1,2e−1)$(c)$(2e−1 ,e−1)$ (d) $(0,2e−1 )$

In R', consider the planes $P_{1},y=0$ and $P_{2}:x+z=1$. Let $P_{3}$, be a plane, different from $P_{1}$, and $P_{2}$, which passes through the intersection of $P_{1}$, and $P_{2}$. If the distance of the point $(0,1,0)$ from $P_{3}$, is $1$ and the distance of a point $(α,β,γ)$ from $P_{3}$ is $2$, then which of the following relation is (are) true ?

Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, $x_{2}+y_{2}=a$, (i), $my=m_{2}x+a$, (P), $(m_{2}a ,m2a )$II, $x_{2}+a_{2}y_{2}=a$, (ii), $y=mx+am_{2}+1 $, (Q), $(m_{2}+1 −ma ,m_{2}+1 a )$III, $y_{2}=4ax$, (iii), $y=mx+a_{2}m_{2}−1 $, (R), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 1 )$IV, $x_{2}−a_{2}y_{2}=a_{2}$, (iv), $y=mx+a_{2}m_{2}+1 $, (S), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$If a tangent to a suitable conic (Column 1) is found to be $y=x+8$and its point of contact is (8,16), then which of the followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)

Let $y(x)$ be a solution of the differential equation $(1+e_{x})y_{prime}+ye_{x}=1.$ If $y(0)=2$ , then which of the following statements is (are) true? (a)$y(−4)=0$ (b)$y(−2)=0$ (c)$y(x)$ has a critical point in the interval $(−1,0)$ (d)$y(x)$ has no critical point in the interval$(−1,0)$

Three boys and two girls stand in a queue. The probability, that the number of boys ahead is at least one more than the number of girls ahead of her, is (A) $21 $ (B) $31 $ (C) $32 $ (D) $43 $