The Good, the Bad, and the Useful
Ring around the Unit Circle
The Theory of Remainders
Transformers
Next Top Sinusoidal Model
100
If JEOPARDY is the key in a Vigenère cipher, what would the plaintext “crypto” encipher as?
LVMETF
100
240° is equivalent to how many radians?
(4π)/3
100
What is 2009 mod 32?
25
100
Specifically how does the graph of
y=sin(4x)
differ from
y=sin(x)?
sin(4x)" has a period " 1/4 " as long as "
sin(x) [π/2 " compared to " 2π].
100
Radio station WGBH-FM broadcasts at 89,700,000 Hz. Find the period of the radio wave.
1/89700000 " or "1.115*10^-8
200
Jordan sent Fax a message that was enciphered with a Caesar cipher shift of 11. If his message read NZXZPDELD, what was his original plaintext?
Como estas
200
Margaret claims that cos(3.67) = –1.5. Felix says that Margaret must be wrong. Who is right, and how do you know? [no calculators, no iPads!]
Felix is right, because the range of the cosine function is –1 to 1.
200
If
k equiv 2 (mod 12),
give three different values for k.
2, 14, 26, 38, 50, 62, 74, 86, 98, 110, etc.
200
Find the period of
y=6.5cos(5(x-1))+2.
(2π)/5 " or " 0.4π.
200
X-rays from the star X0726260 reach a maximum of 12 units on day 1, and then reach a minimum of 2 units on day 18.25. Find its amplitude and period.
Amplitude = 5
period = 34.5
300
The Scottish Gaelic alphabet has 18 letters, and the Arabic alphabet has 28 letters. Give an affine cipher that would be non-invertible in Gaelic but invertible in Arabic.
Any affine with C = aP + b where a is 3, 9, or 15 works, because
28 = 2 * 2 * 7
and
18 = 2 * 3 * 3
300
1.85 radians is located in which quadrant on the unit circle?
Second.
300
Solve for x:
7x + 4 equiv-5 (mod 12)
x = 9
300
Find the amplitude, phase shift, period, and vertical shift for this graph of a cosine function:
amplitude = 3
phase shift = 0
period = 0.5
vertical shift = –2
300
Write an equation for a 320 Hz sound with amplitude of 5.
y=5sin(640πx)
400
What makes the RSA cipher different and more powerful than all the other ciphers we’ve learned? (Be specific.)
One or both of these:
- RSA is a public-key system so the private key need not be distributed or shared.
- The use of extremely large primes make it virtually impossible for anyone to break the code.
400
Find
sin((-5π)/3)
without a calculator or iPad.
sqrt(3)/2
400
Find the reciprocal of 11 in mod 18.
5
(because
11 * 5 = 55 equiv 1 mod 18)
400
Write a cosine equation with an amplitude of 7, a period of 3, and a vertical shift of –2.
y = 7cos((2π)/3x) - 2
400
The tides on the planet Krypton reach a height of 7 meters exactly 5 hours after their low of 3 meters. Write an equation for the tide there if the first high tide occurs at hour 0.
y=2cos(π/5x)+5
500
What is the inverse matrix (in mod 26) for
[[5,6],[-4,-3]]?
[[-9,-18],[12,15]]
or
[[17,8],[12,15]]
500
Between 0 and 2π, what two values satisfy the equation
cos(x)=-sqrt(3)/2
(5π)/6,(7π)/6
500
Larry was writing a QR problem and accidentally smudged the mod. This is what he had:
157 equiv 93 (mod cc "A")
What are 3 different possible mods?
Any factor of 64
[1, 2, 4, 8, 16, 32, 64],
since 157 – 93 = 64.
500
What sine function is equivalent to
y=4cos(1/2(x-π/4))-3?
y=4sin(1/2(x+(3π)/4)-3
500
Temperatures at the CSA House vary from a high of 78° to a low of 68° over a 24-hour period. If the first low is at 6 AM, write an equation for the temperature as a function of hours past midnight.
y=5cos(π/12(x-30))+73