Inequality

Q. For a>0, b>0, c>0, which of the following hold good?

1. $2(a^3+b^3+c^3)\ge bc(b+c)+ca(c+a)+ab(a+b)$
2. $\frac{a^3+b^3+c^3}{3}>\frac{a+b+c}{3}.\frac{a^2+b^2+c^2}{3}$
3. $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{1}{2}(a+b+c)$
4. $\frac{2}{b+c}+\frac{2}{c+a}+\frac{2}{a+b}<\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Q. If a > b > 1, then $a^n-b^n>n(a-b)$ for every + ive integer $n\ge 2$.

Q. If A, B, C be the angles of a triangle, then which of the following hold good?

1. $cotA.cotB.cotC\le \frac{1}{3\sqrt{3}}$
2. $cot\frac{A}{2}cot\frac{B}{2}cot\frac{C}{2}\ge 3\sqrt{\left(3\right)}$
3. $cosA+cosB+cosC<\frac{3}{2}$
4. $sin\left(\frac{A}{2}\right)sin\left(\frac{B}{2}\right)sin\left(\frac{C}{2}\right)\le \frac{1}{8}$

Q. Which of the following hold good? If n is a +ve integer, then

1. $n^n>\mathrm{1.3.5}\dots \dots (2n-1)$
2. $\mathrm{2.4.6}\dots \dots 2n<(n+1{)}^n$
3. $(n!{)}^3<n^n{\left(\frac{n+1}{2}\right)}^{2n}$
4. $[1^r+2^r+3^r+\dots \dots +n^r{]}^n>n^n.(n!{)}^r$

Q. Which of the following hold good?
If sum of any two of quantities x, y, z be together greater than the third, then

1. $(x+y+z{)}^3>27(y+z-x\left)\right(z+x-y\left)\right(x+y-z)$
   
2. $xyz>(y+z-x)(z+x-y)(x+y-z)$
   
3. If n is a + ive integer, then
   $(1+x^3)(1+y^3)(1+z^3)>(1+xyz{)}^3$
   
4. $2(a^2+b^2+c^2)(a+b+c)>a^3+b^3+c^3+15abc$

Q. $\frac{2}{b+c}+\frac{2}{c+a}+\frac{2}{a+b}>\frac{9}{a+b+c}$

Q. Which of the following hold good?

1. 
   $p{x}^{q-r}+q{x}^{r-p}+r{x}^{p-q}>p+q+r$, unless p = q = r, or x = 1
   
2. ${\left[\frac{x^2+y^2+z^2}{x+y+z}\right]}^{x+y+z}>x^x{y}^y{z}^z>{\left[\frac{x+y+z}{3}\right]}^{x+y+z}$
3. ${\left[\frac{a^2+b^2}{a+b}\right]}^{a+b}>a^a{b}^b$
4. $a^a{b}^b>{\left(\frac{a+b}{2}\right)}^{a+b}>a^b.b^a$

Q. If A, B, C are the angles of a triangle such that C is an obtuse angle, then

1. $tanA.tanB<1$
2. $tanA.tanB>1$
3. $tanA.tanB=1$
4. None

Q. In any triangle the semi-perimeter is greater than each of its sides.

Q. If $y=3^{x-1}+3^{-x-1}$ (x real), then the least value of y is

1. 2
2. 6
3. None of these
4. $\frac{2}{3}$