Selected Sample Problems and Solutions

 

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

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

Detailed solution:

 

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

Solution: Use QM \geq AM    inequality on   \left ( 1+\frac{1}{a} \right ),\left ( 1+\frac{1}{b} \right )and\left ( 1+\frac{1}{c} \right )  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 \rightarrow 0} \frac{\theta -\sin \theta }{\theta ^{3}}

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

 

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

Solution: 1. Detailed solution   

 

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

Solution:  1/2 . Detailed solution

 

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

Sin θ + cos θ +tan θ + cot θ + sec θ + cosec θ

Solution: 

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

Solution: