## Todai Entrance Exam: Subject 2014 – Problem 3

(1) func f(n): if n == 0: return 0 if n == 1: return 1 return f(n – 1) + f(n – 2) (2) func f(n): left = 1 right = 0 d = 1 if d > n: return right while d < n: temp = left left = left + right right = [...]

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## Todai Entrance Exam: Subject 2014 – Problem 3

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(1) func f(n): if n == 0: return 0 if n == 1: return 1 return f(n – 1) + f(n – 2) (2) func f(n): left = 1 right = 0 d = 1 if d > n: return right while d < n: temp = left left = left + right right = [...]

(1) (2) If the input impedance is not infinity then the input voltage will be decreased by a factor before getting amplified. If the input offset voltage is non-zero, then the input-output characteristics curve will also be offsetted. I don’t know about frequency response. (3) Check the last slide, this and this. Note: [...]

(1) (2) The solution to this question is highly similar to Characteristic Function of Normal Distribution (without the imaginary part). I give up. And the solution to the probability distribution of can be found in Wikipedia. Well, I give up this problem.

(1) Proof for foci of a ellipse Proof for foci of a hyperbola is similar, with few notes: Let : (2) It’s worth noting that: So: So the statement follows. (3) Similar to question (2) (4) Let [...]

(1) (2) Thanks to Cayley–Hamilton theorem: Therefore: (3) (4) So: (4) (5)

(1) (2) (3) We see that and with and . After each iteration, decreases by as: So the algorithm in (2) will eventually terminate as is still the same while keep decreasing after each iteration. (4) If we continue that loop, eventually one of the function parameters [...]

(1) Slide 4: (2) (3) Reference. I’m not sure about this question but let me guess… Let with . We get . So: (4) I will think about that later. (5) Similar to slide 7: (6) According to this and make [...]

(1) Distributing equivalent balls to distinguishable boxes is equivalent to choose balls from balls (first ball always go to first box). These chosen balls take the role of bars splitting remaining balls into boxes. So, the number of possible ways: Reference for validation: Wikipedia (2) This is the same as choosing balls from [...]

(1) The surface area of around is determined by the product of the circumference and the curve length AB , so: Remember that depends on . So the statement follows. (2) (3) (4) This is a separable non-linear first-order differential equation. So the [...]

(1) (2) It is easy to compute that . The characteristic equation: (3) I don’t understand this question. (4) Let , we get and the solutions to are and . We examines the changes in sign of to have a picture of changes in values of . – + – – [...]