Question

One of these six people has murdered one of the others. Each has made 3 statements. The murderer has made 3 errors. The victim has made zero errors. The other suspects have made one or two errors. Find out who the murderer is and who the victim is.
Catwoman says.
1) If $y=x^{5}then\frac{dy}{dx}=5x^4$
2) If $y=2x^3+3x^5,then\frac{dy}{dx}=6x+15x^4$
3) $y=2x^3,$ then gradient is 6 at (2, 16)

Bane says
1) If $A=4\pi r^2,then\frac{dA}{dr}=8\pi r$
2) If $A=4x^{10}then\frac{dy}{dx}=40x^{10}$
3) If $A=7x,then\frac{dy}{dx}=7$

Joker says
1) If $y=x^5,then\frac{dy}{dx}=5x^5$
2) If $y=\sqrt{x},then\frac{dy}{dx}=\frac{1}{2}x$
3) If $y=2x^2,$ then the gradient is 6 at (3, 18)

Poison Ivy says
1) If $A=4\pi r^2,then\frac{dA}{dr}=8r$
2) If $A=4x^{10}then\frac{dy}{dx}=40x^{11}$
3) If $y=7x+2,then\frac{dy}{dx}=7$

The Scare crow says
If $y=3x^{11},then\frac{dy}{dx}=33x^{10}$
If $y=3x^2+6x+6,then\frac{dy}{dx}=6x+6$
The gradient of $x^2+3x+3is9whenx=3$

Harley Quinn say
The gradient of $y=x^3+x^2+3xis10whenx=2$
If $y=2x^3,$ then gradient is 24 at (2, 16)
If $y=2x^2,$ then gradient is 12 at (3, 18)

• A. Murderer – Joker
  Victim – Harley Quinn
• B. Murderer – Poison Ivy
  Victim – Joker
• C. Murderer – Joker
  Victim – Scarecrow
• D. Murderer – Scarecrow
  Victim – Joker

Answer 1

The correct option is C Murderer – Joker

Victim – Scarecrow
Check the correctness of everybody’s statements.
1) Catwoman
Derivative of $x^5=5x^4,$ - correct
$\frac{d}{dx}(2x^3+3x^5)=6x+15x^4$ – Incorrect
Rather derivative is $6x^2+15x^4$ (Using Addition rule of differentiation and $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$)
So Catwoman can neither be the victim nor be the murderer.

2) Bane
$\frac{d}{dr}\left(4\pi r^2\right)=8\pi r$ Correct
$\frac{d}{dx}\left(4x^{10}\right)=4x^{10}$ Incorrect
Rather it should be $40x^9.$
So bane can neither be the victim nor be the murderer.

3) Joker
$\frac{d}{dx}\left(x^5\right)=5x^5$ Incorrect
Because $\frac{d}{dx}\left(x^5\right)=5x^4$
$\frac{d}{dx}\left(\sqrt{x}\right)=\frac{1}{2\sqrt{x}}andnot\frac{1}{2}x.$
So this statement is also incorrect.
$y=2x^2,$ for gradient at a point, we calculate the derivative at that point.
So $\frac{d}{dx}\left(2x^2\right)=4x$
At (3, 10), $\frac{dx}{dy}willbe4\times 3=12$ and not 6.
So this is also incorrect.
Since joker made all the incorrect statements so he is the murderer.

4) For poison Ivy
You can see that second statement is incorrect and rest two are correct.
In second statement, she says that $\frac{d}{dx}\left(4x^{10}\right)=40x^{11}$
Which is incorrect. Rather derivative is $40x^9.$

5) Scare crow
$\frac{d}{dx}\left(3x^{11}\right)=33x^{10}\left(Using\frac{d}{dx}\right(x^n)=n{x}^{n-1})$
So the statement is correct
$\frac{d}{dx}(3x^2+6x+6)=6x+6$
(Using addition rule of differentiation and $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$)
So this is also correct statement.
For gradient of $x^2+3x+3,$ we have to calculate it’s derivative.
$\frac{d}{dx}(x^2+3x+3)=2x+3$
At x = 3
$2x+3=(2\times 3)+3$
= 6 + 3
= 9
So this is also the correct statement. Since he has made all the correct statements. So he is the victim. Hence C is the correct answer.