Embibe Experts Solutions for Chapter: Fundamentals of Mathematics, Exercise 1: EXERCISE-1

The inequality $n!>2^{n-1}$ is true

Consider a number $N=21P53Q4$, then the number of ordered pairs $\left(P, Q\right)$ so that the number ${}^{\text{'}}N^{\text{'}}$ is divisible by $44$ is equal to :-

$\frac{\sqrt{15}+\sqrt{35}+\sqrt{21}+5}{\sqrt{3}+2\sqrt{5}+\sqrt{7}}=?$ :-

If $m$ and $n$ are positive integers such that $m^2+2n=n^2+2m+5$, then the value of $n$ is :-

If $c>1$ and $x=\frac{\sqrt{c+2}-\sqrt{c+1}}{\sqrt{c}-\sqrt{c-1}},y=\frac{\sqrt{c+2}-\sqrt{c+1}}{\sqrt{c+1}-\sqrt{c}},z=\frac{\sqrt{c}-\sqrt{c+1}}{\sqrt{c+2}-\sqrt{c+1}}$ then arrange, $x, y, z$ in ascending order

Find the value of ${\left[\sqrt[3]{3}{\left(\sqrt[3]{\frac{4}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{1}{9}}\right)}^{-1}-1\right]}^3=$ ?

The value of expression ${\left(\sqrt{{\left(\sqrt{5}-\frac{5}{2}\right)}^2}-\sqrt[3]{{\left(\frac{3}{2}-\sqrt{5}\right)}^3}\right)}^{1/2}+\sqrt{2}\sin \frac{\pi }{4}$