Chapter 7

Find out whether the following numerical expressions are defined or not. i. \(\sqrt{\log _{2} 1.4+\log _{2} 0.7}\). \\{Ans. Not defined ii. \(\sqrt{\log 15+\log 0.07}\). \\{ns. Defined iii. logloglog11. \\{Ans. Defined\\}

Without using tables prove \(\frac{1}{\log _{3} \pi}+\frac{1}{\log _{4} \pi}>2\).

Find all pairs \((x, y)\) of real numbers such that \(16^{x^{2}+y}+16^{x+y^{2}}=1\).

Given \((2011 a+2012 b)(2012 c-2011 b)=(2011 b+2012 c)(2012 b-2011 a)\) and \(0

For each integer \(n \geq 1\), define \(a_{n}=\left[\begin{array}{l}n \\\ {[\sqrt{n}]}\end{array}\right]\), where [ ] denotes the Greatest Integer Function. Find the number of all \(n\) in the set \(\\{1,2,3, \cdots \cdots, 2010\\}\) for which $a_{n}>a_{n+1}