User:Blueeighthnote/Answers

Answers
For number 1:

A. While pi (π) is irrational, 3.1416 is not (3.1416 = $$\tfrac{31416}{10000}$$). It is rational. Sorry about this trick question!

B. Square root of 3 is irrational (1.73205080757……)

C. log10√5 + log10√2 = log10√10 = 1/2 log1010 = 1/2

D. $$\dfrac{\sin15^\circ}{\cos15^\circ} + \dfrac{\cos15^\circ}{\sin15^\circ} = \dfrac{sin^215^\circ + cos^215^\circ}{cos15^\circ \cdot sin15^\circ} = \dfrac{2}{2sin15^\circ \cdot cos15^\circ} = \dfrac{2}{sin30^\circ} = 4$$

E. With Linear factor test and determination of rational root, the possible real roots that are rational can only be positive or negative 1, and they both failed. Therefore, x must be irrational.

For number 2: Dalek-James got it correct! Originally it was going to be harder and harder to prove when more people are answering.

For number 3: $$(\tfrac{1}{5})^{17}$$ is actually 1/762939453125. Each question answered is completely independent to the other, and the chances of answering each question correct was 1/5. There are 17 questions, therefore $$(\tfrac{1}{5})^{17}$$ = 1/762939453125 is the answer ($$(\tfrac{1}{5})^{17}$$ was also accepted).

Answers
For number 1: Simply divide 19 by 27. It is equal to 0.703703703...... The 122th digit is equal to the 2nd. Therefore, the answer was 0.

For number 2: Using the binomial theorem,

$$\begin{align} 40^{255} &= (39 + 1)^{255} \\ &= {255 \choose 0} 39^{255} + {255 \choose 1} 39^{254} + \ldots + {255 \choose 254} 39 +{255 \choose 255} \\ &= 39k + 1(k \in N)\end{align}$$

Therefore the remainder of $$40^{255}$$ is 1.

For number 3:

$$\frac{79 * 13 - 97 - 41 - 59}{10} = 83$$

Answers
For number 1: The answer was actually "E". However, I just noticed that the display of these numbers made them look confusing, as if they were regular numbers. The smaller numbers on the right were all exponents. The way the number is displayed makes it look like $$(\dfrac{1}{8}) - 2$$, and that affected the answers so much that I have to award points for free, despite that the answer B alone is certainly very large, due to $$(\dfrac{1}{8})^{-2} = (2^{(-3)})^{(-2)} = 2^6 (= 64)$$, making it the least likely answer at the start.

To sort out the numbers' order, the solution is to convert all numbers into $$2^n$$, and then compare it from there.

The order was: $$(\dfrac{1}{8})^{-2} (B) > 2^{\tfrac{1}{3}} (A) > 2^{-\tfrac{1}{4}} (C) > (\dfrac{1}{2})^{\tfrac{1}{2}} (D) > 8^{-\tfrac{1}{3}} (E) $$

For number 2: Correct! ALPHA is less than DELTA.

For number 3: When x equals 0, the value is 9.

For number 4: Correct! The answer was indeed 150, because as it is a 3-digit number, x cannot be 0. As y can be any digit from 0~9, there are 10 numbers each from 2y0, 1y3, 3y1, 2y4, 4y2, ...... 7y9, 9y7.

Answers
For number 1: 43679 = 34 x 72 x 11. Therefore, the answer should be 3.

For number 2: The answer is 4.

For number 3: I'm going to extend this one to be revealed in September 4. Until then, this question can still be answered!

a, b, c are all real numbers. For the following system of linear equations, which of the following are correct? There are multiple correct choices. [60 points if all correct; 36 points if 1 is incorrect, and 12 points if 2]

$$\begin{cases} x + 2y + az = 1 \\ 3x + 4y + bz = -1 \\ 2x + 10y + 7z = c \end{cases}$$

A) (If there is a solution) Exactly one set of answers is available.

B) (If there is a solution) 11a - 3b &ne; 7

C) (If there is a solution) c = 14

D) (If there are no solutions) 11a - 3b = 7

E) (If there are no solutions) c &ne; 14

Sep 5 ~ 8
For number 2: The answer is 6. (The last digit runs in a cycle with each 2 multiplied: 2, 4, 8, 6, 2, 4, 8, 6......)

For number 3: I'm going to extend this one to be revealed in September 8. Until then, this question can still be answered!

a, b, c are all real numbers. For the following system of linear equations, which of the following are correct? There are multiple correct choices. [80 points]

$$\begin{cases} x + 2y + az = 1 \\ 3x + 4y + bz = -1 \\ 2x + 10y + 7z = c \end{cases}$$

A) (If there is a solution) Exactly one set of answers is available.

B) (If there is a solution) 11a - 3b &ne; 7

C) (If there is a solution) c = 14

D) (If there are no solutions) 11a - 3b = 7

E) (If there are no solutions) c &ne; 14