Selected Sample Problems and Solutions

Problem 1: If a, b, c are three positive real numbers  such that  $\frac{1}{a}+&space;\frac{1}{b}+\frac{1}{c}=&space;6$ , then find the maximum value of the expression $\left&space;(&space;1+\frac{1}{a}&space;\right&space;)\left&space;(&space;1+\frac{1}{b}\right&space;)\left&space;(&space;1+\frac{1}{c}&space;\right&space;)$ ?

Solution:  Use   $AM\geq&space;GM$ inequality on $\left&space;(&space;1+\frac{1}{a}&space;\right&space;),\left&space;(&space;1+\frac{1}{b}&space;\right&space;)and\left&space;(&space;1+\frac{1}{c}&space;\right&space;)$      to get the answer as 27 .

Detailed solution:

Problem 2:  In problem 1, find the minimum value of     $\frac{1}{{a^{2}}}&space;+\frac{1}{b^{2}}+\frac{1}{c^{2}}$  ?

Solution: Use $QM&space;\geq&space;AM$    inequality on   $\left&space;(&space;1+\frac{1}{a}&space;\right&space;),\left&space;(&space;1+\frac{1}{b}&space;\right&space;)and\left&space;(&space;1+\frac{1}{c}&space;\right&space;)$  to get the answer as 12 . [QM and AM stand for quadratic and arithmetic means respectively].

Detailed solution

Problem 3:  Find the value of the limit     $\lim_{\theta&space;\rightarrow&space;0}&space;\frac{\theta&space;-\sin&space;\theta&space;}{\theta&space;^{3}}$

Solution:   $\frac{1}{6}$ . Detailed solution

Problem 4:  Find the value of the limit       $\lim_{x\rightarrow&space;0}&space;x^{x}$

Solution: 1. Detailed solution

Problem 5: If $\theta$ is a real number and $\sec&space;\theta&space;-&space;\tan&space;\theta&space;=&space;2$  , find the value of $\sec&space;\theta&space;+\tan&space;\theta$  ?

Solution:  1/2 . Detailed solution

Problem 6:  Find the minimum value of the  below expression for all real values of   $\theta$  ?

Solution:

Problem 7 : For positive real numbers a, b, x  such that $\left&space;(&space;a,b&space;\right&space;)\geq&space;0$   and  $0<&space;x>&space;\frac{\pi&space;}{2}$   show that  $\left&space;(&space;1+\frac{a}{\sin&space;x}&space;\right&space;)\left&space;(&space;1+\frac{b}{\cos&space;x}&space;\right&space;)\geq&space;\left&space;(&space;1+\sqrt{2ab}&space;\right&space;)^{2}$

Solution: