Inequalities Involving Mathematical Means

Q. Let $f:R\to R$ be such that for all $x\in R,(2^{1+x}+2^{1-x}),f\left(x\right)$ and $(3^x+3^{-x})$ are in A.P., then the minimum value of $f\left(x\right)$ is:

1. $0$
2. $4$
3. $3$
4. $2$

Q. $\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{If the roots of the equation}& \text{(P)}& 7\\ & x^3-9x^2+26x-k=0\text{are positive}& & \\ & \text{and in A.P., then}k\text{is equal to}& & \\ \text{(B)}& \text{If the roots of the equation}& \text{(Q)}& 11\\ & x^3-14x^2+kx-64=0\text{are positive}& & \\ & \text{and in G.P., then}k\text{is equal to}& & \\ \text{(C)}& \text{If the roots of the equation}& \text{(R)}& 24\\ & 6x^3-k{x}^2+6x-1=0\text{are positive}& & \\ & \text{and in H.P., then}k\text{is equal to}& & \\ \text{(D)}& \text{The harmonic mean for the roots of}& \text{(S)}& 26\\ & \text{equation}{x}^3-11x^2+3x-26=0\text{is}& & \\ & & \text{(T)}& 56\end{array}$
Which of the following is the only CORRECT combination?

1. $\left(A\right)\to \left(Q\right),\left(B\right)\to \left(P\right),\left(C\right)\to \left(R\right),\left(D\right)\to \left(S\right)$
2. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(S\right),\left(C\right)\to \left(Q\right),\left(D\right)\to \left(Q\right)$
3. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(T\right),\left(C\right)\to \left(Q\right),\left(D\right)\to \left(S\right)$
4. $\left(A\right)\to \left(Q\right),\left(B\right)\to \left(T\right),\left(C\right)\to \left(S\right),\left(D\right)\to \left(R\right)$

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

1. > 2abxy

2. $\le$ 2abxy

3. > 4abxy

4. < 4abxy

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.

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 $a^2+b^2+c^2=1,$ 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. 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. 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, 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. The number of real solutions of $|x+3|-|4-x|=|8+x|$ is

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 $\sum _{i=1}^{10}(x_i-5)=20$ and $\sum _{i=1}^{10}(x_i-5{)}^2=660$ $y=S.D.$ of $10$ items $x_1,x_2,x_3.....,x_{10}$, then $\left[y\right]$ is equal to (where $[.]$ represents the greatest integer function)

1. $8$
2. $9$
3. $7$
4. $6$

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.

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

1. True

2. False

Q.

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

1. > 2abxy

2. $\le$ 2abxy

3. > 4abxy

4. < 4abxy

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. The sum of real roots of the equation $|x-2{|}^2+|x-2|-2=0$ is

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