If `x in R ,a n da ,b ,c` are in ascending or descending order of magnitude, show that `(x-a)(x-c)//(x-b)(w h e r ex!=b)` can assume any real value.

A certain polynomial P(x)x in R when divided by x-a ,x-ba n dx-c leaves remainders a , b ,a n dc , resepectively. Then find remainder when P(x) is divided by (x-a)(x-b)(x-c)w h e r eab, c are distinct.

Column I, Column II If a ,b ,c ,a n dd are four zero real numbers such that (d+a-b)^2+(d+b-c)^2=0 and he root of the equation a(b-c)x^2+b(c-a)x=c(a-b)=0 are real and equal, then, p. a+b+c=0 If the roots the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 are real and equal, then, q. a ,b ,ca r einAdotPdot If the equation a x^2+b x+c=0a n dx^3-3x^2+3x-0 have a common real root, then, r. a ,b ,ca r einGdotPdot Let a ,b ,c be positive real number such that the expression b x^2+(sqrt((a+b)^2+b^2))x+(a+c) is non-negative AAx in R , then, s. a ,b ,ca r einHdotPdot

Solve: (-1)/(|x|-2)geq1,w h e r ex in R ,x!=+-2.

Solve: (-1)/(|x|-2)geq1,w h e r ex in R ,x!=+-2.

If a x^2+b x+c=0,a ,b ,c in R has no real zeros, and if c 0 (c) a >0

Prove that if x is real, the expression ((x-a)(x-c))/(x-b) is capable of assuming all values if agtbgtc or altbltc .

Evaluate int_a^b(dx)/(sqrt(x)),w h e r ea , b > 0.

Evaluate int_a^b(dx)/(sqrt(x)),w h e r ea , b > 0.

If a,b,c, in R, find the number of real root of the equation |{:(x,c,-b),(-c,x,a),(b,-a,x):}| =0

Assuming that x is a positive real number and a ,\ b ,\ c are rational numbers, show that: ((x^a)/(x^b))^(1/(a b))\ ((x^b)/(x^c))^(1/(b c))\ \ ((x^c)/(x^a))^(1/(a c))=1