Mathematics

Q. If $a,b,c\in R$ and $a^2+b^2+c^2=1$, then $ab+bc+ca$ lies in the nterval

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

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. 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, 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.

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

1. > 2abxy

2. $\le$ 2abxy

3. > 4abxy

4. < 4abxy

Q. The least value of the expression $2{\log }_{10}x-{\log }_x\left(0.01\right)$, for $x>1$, is

1. $10$
2. $2$
3. $-0.01$
4. $4$

Q.

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

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

Q. If $x^2+y^2+z^2=1$, then $yz+zx+xy$ lies in the interval

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

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

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 ___