• # Todai Entrance Exam: Subject 2013 – Problem 1

(1) Reference: 1 & 2 & 3 There are 2 ways to attack this problem. For the first way, we attack the differential equation subject to . For the second way, we combine the series of capacitors and into one capacitor , and solve this circuit with the charge conservation equation . As this is […]

• # 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 = […]

• # Todai Entrance Exam: Subject 2014 – Problem 1

(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: […]

• # Todai Entrance Exam: Math 2015 – Problem 3

(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.

• # Todai Entrance Exam: Math 2015 – Problem 2

(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 […]

• # Todai Entrance Exam: Math 2015 – Problem 1

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

• # Todai Entrance Exam: Subject 2015 – Problem 3

(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 […]

• # Todai Entrance Exam: Subject 2015 – Problem 1

(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 […]

• # Todai Entrance Exam: Math 2016 – Problem 3

(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 […]

• # Todai Entrance Exam: Math 2016 – Problem 2

(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 […]