## Todai Entrance Exam: Math 2016 – Problem 1

(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 . – + – – [...]

## Todai Entrance Exam: Subject 2016 – Problem 3

(1) Let and be the minimum span tree of the graph . We find the edge which has the minimum weight among the edges connecting the vertex of and the vertex of . We prove that is the minimum span tree of the graph with . Assume that there exists a whose weight sum is [...]

## Todai Entrance Exam: Subject 2016 – Problem 1

This seems to be a Buck converter in continuous mode (in SMPS series). By “ignore the change of in the meantime”, they mean . (1)     (2)     (3)     (4) The power consumption of the resistor is:     (5) As the sum of current flow decrease in (2) equals to [...]

## Todai Entrance Exam: Subject 2017 – Problem 1

Thanks to this lecture on MIT Courseware (see the Lecture Note) and why closed loop gain formula does not work in this case. (1)     (2) As , , we get: , so:     (3) As , , we get:     So when:                 So [...]

## Todai Entrance Exam: Subject 2017 – Problem 3

(1) Let be the values of after processing .                                     (2) We use induction proof: If a list has the number of varieties of user IDs being at most one then . Base case: , so when . [...]

## Todai Entrance Exam: Math 2017 – Problem 3

(1)         (2)     (3)                 (4)     (5) From (1), we have:

## Todai Entrance Exam: Math 2017 – Problem 2

(1)             (2)     Let and . So, thanks to the given fact:     (3)         We solve the characteristic equation of the above homogenous second order linear differential equations:     So:     If :     which is the excluded solution. If [...]

## Todai Entrance Exam: Math 2017 – Problem 1

(1)         Therefore:     (2)     Then we solve the equation for each , we then arrive:     (3)     (4) As:         Therefore:     So:     (5)     Notice:     Therefore and .     (6)     Rayleigh quotient? [...]

## Todai Entrance Exam: Math 2019 – Problem 3

(1)     (2) The easy way: The projection of on is . So . The hard way: Let be the intersection between and . As the triangle ABC is symmetric with respect to the line , every point on has a corresponding point on . So:     Note that as . (3) We [...]

## Todai Entrance Exam: Math 2019 – Problem 2

(1) Let , so :     (2) (i) From Eq. (2.1), we have:     Remember that:     So:                 Notice that:     So:         Working… (ii) I speculate the general form of the solution is: . Eq. (2.3) implies that:   [...]