Find the truth 8

Algebra Level 5

Read the following statements.

I) For real $x$ , $\left\{ x \right\}$ is always positive. ( $\left\{ x \right\}$ is the fractional part of $x$ ).

II) If $\frac { 2x }{ \pi }$ is real but not an integer and $\cos { x }$ is known, then $\cot { x }, \sin { x }, \tan { x }, \sec { x }, \csc { x }$ are also known.

III) If ${ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=3abc$ , then $a+b+c=0$ .

Which of the given statements are true?

Only II Only I Only II and III Only I and II Only III and I All None Only III

1 solution

Abhijeet Verma
May 15, 2015

1) $\left\{ x \right\} \ge 0$ It can be 0 as well which the statement contradicts.

2) $\frac { 2x }{ \pi } \in \Re$ But x may not lie in the domain of tangent, or cotangent which contradicts the statement. thus it is false

3) ${ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc=\frac { 1 }{ 2 } \left( a+b+c \right) \left[ { \left( a-b \right) }^{ 2 }{ +\left( b-c \right) }^{ 2 }+{ \left( c-a \right) }^{ 2 } \right]$ For the LHS to be 0 , either a+b+c=0 or a=b=c. Thus the statement is false as well.

For statement 2, it is given that 2x/pi is not an integer, which means x is not a multiple of pi/2. So it lies in the domain of tan and cot, sec and csc also.

Archit Boobna - 6 years, 1 month ago

For 2nd, We can determine sec(x) but for the rest, we can only determine its magnitude and not its sign.

Ajinkya Shivashankar - 4 years, 4 months ago

Find the truth 8

1 solution

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