Question

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1. Möbius strip is so-called after the German mathematician August Ferdinand Möbius. The amazing properties of the Möbius strip had also been discovered by another mathematician 2 months earlier in 1858. He was?
a) George Barker Jeffery
b) Edwin Thomson Jaynes
c) Edward Jan Habich
d) Johann Benedict Listing

2. Ernst Kummer tried in vain to prove Fermat's Last Theorem, and his method (similar to Fermat's own) broke down because not all rings have the "nice" property of unique factorization. Studying these unique factorization domains did not lead to a proof of Fermat's Last Theorem, but it did lead to the study of these:
a) Prime numbers
b) Algebraic numbers
c) Ideals
d) Ring generators

3. Angle between two lines What is the size of the angle between the two lines 3x-2y+1=0 and x-3y=0 (answer in degrees and minutes)
a) 15 degrees, 15 minutes
b) 62 degrees, 6 minutes
c) 43 degrees, 31 minutes
d) 37 degrees, 52 minutes

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Answer 1

Answer:

A. d) Johann Benedict Listing

B. b) Algebraic numbers

C. a) 15 degrees, 15 minutes

Answer 2

ᴬᴺˢᵂᴱᴿ ⃗

→ D) Johann Benedict Listing

• He independently discovered Properties of half twisted Strip at the same time (1858) as August Ferdinand Mobius.

• He first introduced the term 'Topology'

→ MOBIUS ⃗ ᴵˢ ᵃ ˢᵘʳᶠᵃᶜᵉ ʷᶦᵗʰ ᵒⁿˡʸ ᵒⁿᵉ ˢᶦᵈᵉ ᵃⁿᵈ ᵒⁿˡʸ ᵒⁿᵉ ᵇᵒᵘⁿᵈᵃʳʸ ᶜᵘʳᵛᵉ.

⇒ᵐᵒᵇᶦᵘˢ ˢᵗʳᶦᵖ ᶦˢ ᵗʰᵉ ˢᶦᵐᵖˡᵉˢᵗ ⁿᵒⁿ ᵒʳᶦᵉⁿᵗᵃᵇˡᵉ ˢᵘʳᶠᵃᶜᵉ.

♦ C) Ideals

• Ernest kummer worked on 'Ideals numbers' which are defined as a special subgroup of ring.

• attempted proof of Fermat’s last theorem, which states that the formula xn + yn = zn, where n is an integer greater than 2, has no solution for positive integral values of x, y, and z. Dirichlet found an error, and Kummer continued his search and developed the concept of ideal numbers.

• 3rd question ;

Solution →

∆ The angle between two lines can be calculated from the slopes of the lines.

⇒y = mx + C

• here m is the slope of line

Given Equations → 1) 3x - 2y + 1 = 0

2) x - 3y = 0

According to Formula y = mx + c ,

1) 3x - 2y + 1 = 0

→ 3x - 2y = -1

→ - 2y = - 3x - 1

→ y = 3/2 x - 1 ( y = mx + c ) m is the slope

i.e. m = 3/2 (slope)

2) x - 3y = 0

→ - 3y = - x

→ y = x/3 ( Slope m = 1/3 )

⇔ACUTE ANGLE BETWEEN TWO SLOPES -

$\tan \theta = \frac{m1 - m2}{1 \: + \: m1 \: \times m2}$

$tan \theta \: = \frac{ \frac{3}{2} - \frac{1}{3} }{1 + \frac{3}{2} \times \frac{1}{3} } \\ \\ \frac{ \frac{9 - 2}{6} }{1 + \frac{3}{6} } = \frac{ \frac{7}{6} }{ \frac{9}{6} } \\ \\ = \frac{7}{9} \times \frac{6}{6} = \frac{7}{9} \\ \\ tan \theta = 0.77777$

⇒π radians = 180⁰

⇒1 radian = 180/π

⇒tan θ = 0.77777 so ( in radians)

⇒tan-¹ ( 0.77777)

assuming -π /2 < θ < π/2 we get,

θ = 0.77777 radians.

♦ To convert from radians to degrees we Multiply the value in radians by 180 /π

so value in degrees,

⇒0.77777 × 180 / π

⇒0.77777 × 180 / 3.14

⇒44.58⁰

♦ In minutes,

⇔1⁰ = 60 minutes → 1/ 60⁰

♪ for minutes, we have to take decimal of degree 44.58

From 44.58⁰ we get, 0.58 × 60

i.e. ≈ 31 minutes.

Solution → C) 43 degree and 31 minutes.

ᴺᴼᵀᴱ ⃗ ᴬᴺᴳᴸᴱ ᴵᴺ ᴰᴱᴳᴿᴱᴱˢ = arctan (tan θ )

Arctan is reverse of tan function.