Find the Truth - April Fool's Edition

Algebra Level 5

So April Fool's Day has arrived and it's time for the ultimate test.

"True or False is not easy as 1-2-3"- Anonymous

Note: This is not a troll question.

(1) If the first term of a GP is 2, product of first five terms is 32, product of first 9 terms is 512, then the sum of first 3 terms is 6.

(2) When velocity and acceleration due to gravity are given, area under projectile is maximum when angle of projection is ${ 45 }^{ o }$ . Neglect all other values like air resistance

(3) In any series with only positive terms, following the rule ${ t }_{ n }={ t }_{ n-1 }+{ t }_{ n-2 }$ , where ${ t }_{ n }$ is the nth term, the ratio of a term to its previous term tends to $\frac { 1+\sqrt { 5 } }{ 2 }$ , as n approaches infinite.

(4) If f(x) is an even function, then f'(x) is always an even function.

(5) If f(x) is an even function, then f'(x) is never an even function.

(6) If f'(x) is an even function, then f(x) is always an odd function.

Add the serial numbers of all true statements in the list. If all are false, then answer is 0.

8 3 14 11 7 21 9 0

1 solution

Archit Boobna
Apr 1, 2015

(1) FALSE- The GP may be 2,-2,2,-2,2,-2,2,-2 . . . . .

(2) FALSE- In case of 45 degrees, range is highest. Area under projectile is highest when Angle of Projection=60 degrees.

(3) TRUE- You can check it by yourself. When we prove that this ratio tends to Phi, we do not consider the first 2 numbers of the series. We use the rule only to prove this.

(4) FALSE- Let f(x)= ${ x }^{ 2 }$ .

(5) FALSE- Let f(x)=0. Actually if f(x) is an even function, then f'(x) is an odd function. But this doesn't mean that it can never be an even function.

(6) FALSE- Let f(x)=1. Actually if f(x) is an even function, then f'(x) is in the form of "o(x)+c", where o(x) is an odd function and c is a constant.

Statement 3 is false as well. Let $t_n = \left( \frac{1 - \sqrt{5}}{2} \right)^n.$ Then $t_n = t_{n - 1} + t_{n - 2}$ for all $n \ge 2$ , but the ratio of any term to the previous term is $\frac{1 - \sqrt{5}}{2}.$ There is also the counter-example where $t_n = 0$ for all $n$ , which is less interesting.

Jon Haussmann - 6 years, 2 months ago

I am really sorry.. Thank you for informing me.

I should have mentioned that all terms are positive. I have changed the question

Archit Boobna - 6 years, 2 months ago

Find the Truth - April Fool's Edition

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