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JEE Advanced

The compound that does NOT liberate $CO_{2}$, on treatment with aqueous sodium bicarbonate solution, is

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Let $P_{1}:2x+y−z=3$and $P_{2}:x+2y+z=2$be two planes. Then, which of the following statement(s) is (are) TRUE?The line of intersection of $P_{1}$and $P_{2}$has direction ratios $1,2,−1$(b) The line $93x−4 =91−3y =3z $is perpendicular to the line of intersection of $P_{1}$and $P_{2}$(c) The acute angle between $P_{1}$and $P_{2}$is $60o$(d) If $P_{3}$is the plane passing through the point $(4,2,−2)$and perpendicular to the line of intersection of $P_{1}$and $P_{2}$, then the distance of the point $(2,1,1)$from the plane $P_{3}$is $3 2 $

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`

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3,4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_{i}$ be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that $x_{1}+x_{2}+x_{3}$ is odd isThe probability that $x_{1},x_{2},x_{3}$ are in an aritmetic progression 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 $△PQR$ be a triangle. Let $a=Q R,b=RP$ and $c=PQ$. If $∣a∣=12,∣∣ b∣∣ =43 $ and $b.c=24$, then which of the following is (are) true ?

The coefficients of three consecutive terms of $(1+x)_{n+5}$are in the ratio 5:10:14. Then $n=$___________.

Let $a,b,c$be three non-zero real numbers such that the equation $3 acosx+2bsinx=c,x∈[−2π ,2π ]$, has two distinct real roots $α$and $β$with $α+β=3π $. Then, the value of $ab $is _______.

A cylindrica container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $Vm_{3}$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container. If the volume the material used to make the container is minimum when the inner radius of the container is $10mm$. then the value of $250πV $ is