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JEE Advanced

The ratio $x_{1}/x_{2}$ is

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Let a,b ,c be positive integers such that $ab $ is an integer. If a,b,c are in GP and the arithmetic mean of a,b,c, is b+2 then the value of $a+1a_{2}+a−14 $ is

In the following reactions, the structure of the major product 'X' is

Let $S_{n}=k=1∑4n (−1)2k(k+1) k_{2}˙$Then $S_{n}$can take value (s)$1056$b. $1088$c. $1120$d. $1332$

Consider two straight lines, each of which is tangent to both the circle $x_{2}+y_{2}=21 $and the parabola $y_{2}=4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0,0)$and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $2 $, then which of the following statement(s) is (are) TRUE?For the ellipse, the eccentricity is $2 1 $and the length of the latus rectum is 1(b) For the ellipse, the eccentricity is $21 $and the length of the latus rectum is $21 $(c) The area of the region bounded by the ellipse between the lines $x=2 1 $and $x=1$is $42 1 (π−2)$(d) The area of the region bounded by the ellipse between the lines $x=2 1 $and $x=1$is $161 (π−2)$

Let $a,b,xandy$ be real numbers such that $a−b=1andy=0.$ If the complex number $z=x+iy$ satisfies $Im(z+1az+b )=y$ , then which of the following is (are) possible value9s) of x? (a)$−1−1−y_{2} $ (b) $1+1+y_{2} $(c)$−1+1−y_{2} $ (d) $−1−1+y_{2} $

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_______.

Let m be the smallest positive integer such that the coefficient of $x_{2}$ in the expansion of $(1+x)_{2}+(1+x)_{3}+(1+x)_{4}+……..+(1+x)_{49}+(1+mx)_{50}$ is $(3n+1)._{51}C_{3}$ for some positive integer n. Then the value of n is

Consider the family of all circles whose centers lie on the straight line `y=x`. If this family of circles is represented by the differential equation `P y^(primeprime)+Q y^(prime)+1=0,`where `P ,Q`are functions of `x , y`and `y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)),`then which of the following statements is (are) true?(a)`P=y+x`(b)`P=y-x`(c)`P+Q=1-x+y+y^(prime)+(y^(prime))^2`(d)`P-Q=x+y-y^(prime)-(y^(prime))^2`