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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`

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 $0<x<1?$

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 ?

A pack contains $n$cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of het numbers on the removed cards is $k,$then $k−20=$____________.

Late $a∈R$and let $f:R$be given by $f(x)=x_{5}−5x+a,$then$f(x)$has three real roots if $a>4$$f(x)$has only one real roots if $a>4$$f(x)$has three real roots if $a<−4$$f(x)$has three real roots if $−4<a<4$

Let $S$be the circle in the $xy$-plane defined by the equation $x_{2}+y_{2}=4.$(For Ques. No 15 and 16)Let $P$be a point on the circle $S$with both coordinates being positive. Let the tangent to $S$at $P$intersect the coordinate axes at the points $M$and $N$. Then, the mid-point of the line segment $MN$must lie on the curve$(x+y)_{2}=3xy$(b) $x_{2/3}+y_{2/3}=2_{4/3}$(c) $x_{2}+y_{2}=2xy$(d) $x_{2}+y_{2}=x_{2}y_{2}$

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

Word of length 10 are formed using the letters A,B,C,D,E,F,G,H,I,J. Let $x$be the number of such words where no letter is repeated; and let $y$be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $9xy =$