Properties of Inequalities

Q. If n is a positive integer, then $2^n>1+n\sqrt{\left(2n-1\right)}$.

Q. a, b, c are in H.P. and n does not lie between 0 and 1 then $a^n+c^n>2b^n$

Q. The area of an arbitrary triangle is less than one-fourth the square of its semi-perimeter.

Q. If $lo{g}_{\frac{1}{\sqrt{2}}}sinx>0,xϵ[0,4\pi ]$ then the number of values of x which are integral multiples of $\frac{\pi }{4}$ is

1. 4
2. 12
3. 3
4. None of these

Q. The sum of the hypotenuse and the altitude of a right angled triangle dropped on the hypotenuse exceeds the half-perimeter of the triangle.

Q. If a, b, c be three real numbers of the same sign then the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ lies in the interval

1. $[2,\infty )$
2. $[3,\infty )$
3. $(3,\infty )$
4. $(-\infty ,3)$

Q. If a and b are two positive quantities whose sum is $\lambda$, then the minimum value of $\sqrt{(1+\frac{1}{a})(1+\frac{1}{b})}$ is

1. $\lambda -\frac{1}{\lambda }$
2. $\lambda -\frac{2}{\lambda }$
3. $1+\frac{1}{\lambda }$
4. $1+\frac{2}{\lambda }$

Q.

All possible values of expression $x^2-4x+9$ is

1. ($-\infty$, 9]

2. ($-\infty$, -9]

3. (5, $\infty$)

4. [5, $\infty$)

Q.

Given that $y=-6x+1$ and $-3<x<\frac{2}{3}$. Then $y$ lies in which of the following intervals.

1. $\left(18,-4\right)$

2. $\left(3,19\right)$
3. $\left(-19,3\right)$
4. $\left(-3,19\right)$

Q.

If p, q, n are three positive real numbers and p > q then which of the following is correct.

1. $p^{\frac{-1}{n}}<q^{\frac{-1}{n}}$

2. $p^{\frac{1}{n}}<q^{\frac{1}{n}}$

3. $p^{\frac{-1}{n}}>q^{\frac{-1}{n}}$

4. $p^n<q^n$