Inequalities Involving Mathematical Means

Q.

If a, b, x, y are positive, then (ab + xy) (ax + by) is

1. > 2abxy

2. > 4abxy

3. 2abxy

4. < 4abxy

Q.

If $a,b,c,d$ are positive numbers, then $ab+cd<2\sqrt{abcd}$

1. False

2. True

Q.

If $a^2+b^2+c^2=1,$ then $bc+ca+ab$ lies in the interval

1. [-1, 2]

2. $[\frac{-1}{2},1]$

3. $[-1,\frac{1}{2}]$

4. $[\frac{-1}{2},3]$

Q.

If roots of the equation
$x^4-8x^3+b{x}^2+cx+16=0$
are positive, then

1. b=8

2. c=-32

3. b=24

4. c=32

Q.

If a, b and c are three positive real numbers, which one of the following are true?

1. $a^2+b^2+c^2\ge bc+ca+ab$
   

2. $a^3+b^3+c^3\ge 3abc$
   

3. $(b+c)(c+a)(a+b)\ge 8abc$
   

4. $\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\ge a+b+c$

Q.

Let a, b, c be the sides of a triangle. The least value of expression$E=\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}$ is ___

Q. If roots of the equation $f\left(x\right)=x^6-12x^5+{\mathrm{bx}}^4+{\mathrm{cx}}^3+{\mathrm{dx}}^2+\mathrm{ex}+64=0$ are positive, then the remainder when f(x) is divided by x-1 is

1. 2
2. 1
3. 3
4. 10

Q. If x and y are positive real numbers and m, n are positive integers, then $\frac{x^n{y}^m}{\left(1+x^{2n}\right)\left(1+y^{2m}\right)}>\frac{1}{4}$

1. False
2. True

Q. If $a_1,a_2,.......,a_n$are positive real numbers whose product is a fixed number c, then the minimum value of $a_1+a_2+.......+a_{n-1}+2a_n$ is

1. $n(2c{)}^{\frac{1}{n}}$
2. $(n+1)c{}^{\frac{1}{n}}$
3. $2n{c}^{\frac{1}{n}}$
4. $(n+1)(2c{)}^{\frac{1}{n}}$

Q. The product of n positive numbers is unity. Then their sum is

1. a positive integer
2. divisible by n
3. equals to $n+\frac{1}{n}$
4. never less than n