This is a Test of Mathematics Solution Subjective 84 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Problem

Show that there is exactly one value of \(x\) that satisfies the equation:

\(2 cos^2(x^3+x)=2^x+2^{-x}\)

Solution:

We know that \(cos \;x \leq 1\) for all \(x \in I\!R\)

\(=> cos(x^3 + x)\leq 1\)

\(=> cos^2(x^3 + x)\leq 1\)

\(=> 2cos^2(x^3 + x)\leq 2\)

Now consider \(2^x\) and \(2^{-x}\). By AM-GM inequality we have,

\(2^x+2^{-x}\geq 2\)

So \(2 cos^2(x^3+x)=2^x+2^{-x}\), only when \(2 cos^2(x^3+x)=2=2^x+2^{-x}\).

That means \(2^x+2^{-x} = 2, => x= 0\). So \(x=0\) being the only solution.

and \(x=0\) also satisfies \(2 cos^2(x^3+x)=2\)

Thus there is exactly one solution.

Hence Proved.

This is a Test of Mathematics Solution Subjective 78 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

Problem

For real numbers $ {x}$, $ {y}$ and $ {\displaystyle{z}}$, show that
$ {\displaystyle{|x| + |y| + |z| {\le} |x + y - z| + |y + z - x| + |z + x - y|}}$.

Solution

Applying Ravi transformation
$ {x = a + b}$, $ {y = b + c}$ and $ {z = c + a}$.
Our inequality reduces to $ {\displaystyle{|a + b| + |b + c| + |c + a| {\le} |2b| + |2c| + |2a|}}$.
$ {\Leftrightarrow}$ $ {\displaystyle{|a + b| + |b + c| + |c + a| {\le} 2(|a| + |b| + |c|)}}$.
Now we know, $ {\displaystyle{|m + n| {\le} |m| + |n|}}$.
Applying this we get
L.H.S = $ {\displaystyle{|a + b| + |b + c| + |c + a| {\le} |a| + |b| + |b| + |c| + |c| + |a|}}$
= $ {\displaystyle{2(|a| + |b| + |c|)}}$
= R.H.S (proved)