Let p and q be real number such that `x^(2)+px+q ne0` for every real number x. Prove that if n is an odd positive integer, then `X^(2)+pX+qI_(n)ne 0_(n)` for all real matrices X of order of `n xx n`.

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4

If f(x) =(p-x^n)^(1/n) , p >0 and n is a positive integer then f[f(x)] is equal to

If x,y are positive real numbers and m, n are positive integers, then prove that (x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)

Suppose x is a positive real number such that {x}, [x] and x are in the geometric progression. Find the least positive integer n such that x ^(n) gt 100. (Here [x] denotes the integer part of x and {x} = x- [x])

Let nge1, n in Z . The real number a in 0,1 that minimizes the integral int_(0)^(1)|x^(n)-a^(n)|dx is

If a is defined by f (x)=a_(0)x^(n)+a_(1)x^(n-2)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n) where n is a non negative integer and a_(0),a_(1),a_(2),…….,a_(n) are real numbers and a_(0) ne 0, then f is called a polynomial function of degree n. For polynomials we can define the following theorem (i) Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and 'a' be a real number. if p(x) is divided by (x-a), then the remainder is equal to p(a). (ii) Factor theorem : Let p(x) be a polynomial of degree greater than or equal to 1 and 'a' be a real number such that p(a) = 0, then (x-a) is a factor of p(x). Conversely, if (x-a) is a factor of p(x). then p(a)=0. The number of real roots of the equation, (x-1)^(2)+(x-2)^(2)+(x-3)^(2)=0 is

p : For every integer x, x^2 is positive integer, (x ne 0) statement p can be interpreted as

Show that if p,q,r and s are real numbers and pr=2(q+s) , then atleast one of the equations x^(2)+px+q=0 and x^(2)+rx+s=0 has real roots.

Let f(x)=A x^2+B x+c ,w h e r eA ,B ,C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the numbers 2A ,A+B ,a n dC are all integer. Conversely, prove that if the number 2A ,A+B ,a n dC are all integers, then f(x) is an integer whenever x is integer.

For a real number x let [x] denote the largest integer less than or equal to x and {x}=x-[x] Let n be a positive integer. Then int_0^n cos(2pi)[x]{x})dx is equal to