Happy e day!

This Quiz was uploaded on 7 February celebrating e day.

Only 77 responses were recorded (Unfortunately). Question in order were:

Q1. The number e is also called as __(Euclid's number/ Euler's number/ Euler-Mascheroni number/ Archimedes number/ None of the above)

Q2. e is __ (an Irrational Number/ a Rational number/ a non real-complex number/ None of the above)

Q3. There exists non zero polynomial p(x)p(x) with rational coefficients such that p(e)=0 (True/ False)

Q4. Note : ⌊ ⋅ ⌋ denotes the floor function. e=?\lfloor e\rfloor=?

Q5.limx1e×(1+x1)x=?Q5. \lim_{x\to\infty}\frac{1}{e}\times(1+x^{-1})^x=?

Q6.n=0(1)nn!=?Q6. \sum_{n=0}^\infty\frac{(-1)^n}{n!}=?

Q7. If ex=1e^x=-1, then x=?x=?

Q8.limh0ex+hexh=?Q8.\lim_{h\to 0}\frac{e^{x+h}-e^x}{h}=?

Q9.n=0(1)nx2n(2n)!+in=0(1)nx2n+1(2n+1)!, i2=1Q9. \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}+ i\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!},\space i^2=-1

Q10. f(x)=yf(x)=y is a straight line which is tangent to y=exy=e^x at x=0.5x=0.5, find f(x)f(x)

Q11.nWbn+1bn=1an,b0=1Q11. \forall n\in\mathbb{W}\frac{b_{n+1}}{b_n}=\frac{1}{a}-n,b_0=1 lima0n=0bnann!=?\lim_{a\to 0}\sum_{n=0}^\infty\frac{b_na^n}{n!}=?

Q12. Note : x(t):RRx(t):\mathbb{R}\to\mathbb{R} ad2xdt2+cx+d=0a\frac{d^2x}{dt^2}+cx+d=0 Find x(t)x(t) if it satisfies above differential equation.

Solutions by Jeff Giff

Attempts

Note by Zakir Husain
2 months, 4 weeks ago

No vote yet
1 vote

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I did it:

827\frac{8}{27}

Yajat Shamji - 2 months, 4 weeks ago

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I don’t quite understand Q12 though... :(

Jeff Giff - 2 months, 3 weeks ago

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Substitute z:CCz:\mathbb{C}\to\mathbb{C} and z(t)=Aei(αt+β)+γz(t)=Ae^{i(\alpha t +\beta)}+\gamma d2zdz2=α2Aei(αt+β)\therefore \frac{d^2z}{dz^2}=-\alpha^2 Ae^{i(\alpha t +\beta)} Putting this in the equation aα2Aei(αt+β)+cAei(αt+β)+cγ=d-a\alpha^2 Ae^{i(\alpha t +\beta)}+cAe^{i(\alpha t +\beta)}+c\gamma=-d Aei(αt+β)(aα2+c)=(d+cγ)\Rightarrow Ae^{i(\alpha t +\beta)}(-a\alpha^2+c)=-(d+c\gamma) Because this must be true for all tR(d+cγ)=0t\in\mathbb{R}\therefore -(d+c\gamma)=0 γ=dc\Rightarrow \boxed{\gamma=\frac{-d}{c}} Aei(αt+β)(aα2+c)=0\Rightarrow Ae^{i(\alpha t +\beta)}(-a\alpha^2+c)=0 Now either A=0A=0 or aα2+c=0-a\alpha^2+c=0

Because zz is a non zero function. A=0\therefore A\cancel{=}0 aα2+c=0\Rightarrow -a\alpha^2+c=0 α=ca\Rightarrow \boxed{\alpha=\sqrt{\frac{c}{a}}}

Now let x(t)=x(t)= real part of z(t)z(t) x(t)=Acos(αt+β)+γ\Rightarrow x(t)=A\cos(\alpha t +\beta)+\gamma

Zakir Husain - 2 months, 3 weeks ago

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@Zakir Husain Oh I understand now! Thanks :)

Jeff Giff - 2 months, 3 weeks ago

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I shall make a note with all the solutions :)

Jeff Giff - 2 months, 4 weeks ago

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You may post your answers without the solutions below this comment. I will mark it.

Jeff Giff - 2 months, 4 weeks ago

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@Valentin Duringer

Jeff Giff - 2 months, 3 weeks ago

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Interesting !

Valentin Duringer - 2 months, 3 weeks ago

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solutions page

Jeff Giff - 2 months, 3 weeks ago

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