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)≤ex,x∈[0,1].Which of the following is true for 0<x<1?
In R', consider the planes P1,y=0 and P2:x+z=1. Let P3, be a plane, different from P1, and P2, which passes through the intersection of P1, and P2. If the distance of the point (0,1,0) from P3, is 1 and the distance of a point (α,β,γ) from P3 is 2, then which of the following relation is (are) true ?
A pack contains ncards 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∈Rand let f:Rbe given by f(x)=x5−5x+a,thenf(x)has three real roots if a>4f(x)has only one real roots if a>4f(x)has three real roots if a<−4f(x)has three real roots if −4<a<4
Let Sbe the circle in the xy-plane defined by the equation x2+y2=4.(For Ques. No 15 and 16)Let Pbe a point on the circle Swith both coordinates being positive. Let the tangent to Sat Pintersect the coordinate axes at the points Mand N. Then, the mid-point of the line segment MNmust lie on the curve(x+y)2=3xy(b) x2/3+y2/3=24/3(c) x2+y2=2xy(d) x2+y2=x2y2