# Search results

1. ### Bulbs and Buttons

It can be proved by induction. Consider any vertex i. Let it be connected to k other vertices in the bulbs section. If you switch vertex i, we switch on k bulbs. Consider the subgraph H formed by the other n-k switches and n-k bulbs. Since this subgraph also satisfies the condition that...
2. ### Probability related to an Appointment Scheduling Simulation

Since the distribution is not given,a fair assumption is to take exactly 6 bookings that day Let us say he books 1,2,3 as his time slot(of 1,2,3,4,5,6,7,8) We first compute the probability that all the 3 time-slots are filled with all the dentists Pr(of the 6 appointments,3 appointments fall...
3. ### Sum of dice Problem

The answer is P(\text{getting n})=\frac{r_1^n+r_2^n+r_3^n+r_4^n+r_5^n}{7} +\frac{2}{7} where r_1,r_2,r_3,r_4,r_5 are the roots of the equation 6x^5+5x^4+4x^3+3x^2+2x+1=0 The roots are: r_1 = 0.29419455636014125+0.66836709744330092i r_2=0.29419455636014125000+-0.66836709744330092000i...
4. ### Limit of an integral

If f converges to a point say f_{\infty} then the answer would be \frac{f_{\infty}}{(b-y)^2}
5. ### Radius probability of random cut hemisphere.

Just do some integration . You will get \frac{\pi R}{4} where R is the radius of the sphere
6. ### Newtonian mechanics of a car

The net acceleration is the coriolis acceleration and the centrepetal acceleration. So So the radial part is -rw^2 angular part is 2wv0 So the net acceleration is sqrt(r^2.w^4+4w^2.v0^2)=mu.g which gives r=v0.t=sqrt(mu^2g^2-4w^2v0^2)/w^2 So t=sqrt(mu^2.g^2-4w^2v0^2)/(v0.w^2) It is clear...
7. ### How to find pdf given moment generating function

Any bonafide moment generating function always maps to a unique distribution. The random variable for the moment generating functi`on that you have given takes values at 1,2,3...infinity with probability 1/2,1/4,1/8... ie p(k)=1/2^k for k in natural numbers and zero otherwise.
8. ### Card question

I did not know that.I just looked up the net and somewhere it was written cards have 4 colors. In any case for a 2 color suite,I guess the answer would be 3/4
9. ### Card question

you have 2 cards left. Since you already know the colors of all other cards,you know the colors that the last 2 cards can take each of the card can take one of the 4 possible colors and you know the color pair C1 C1 C1 C2 . . .16 pairs for C1,C1 C2C2 C3C3 C4C4 you will judge it...
10. ### Frisbee Dog Pursuit Problem

@Leach I think you misread the question.The problem is not to find the time of catch when the dog is always heading towards the frisbee. The dog is chasing the frisbee such that the angle formed by the line joining frisbee and the dog with the east west line is a constant as the equation...
11. ### Frisbee Dog Pursuit Problem

It will follow the equation y=\frac{16+\sqrt{91}}{9} x ie y=x*{16+sqrt(9)}/9 which is a straight line Since the equation is a straight line,it is the optimal way of catching the Frisbee.So the time is minimum.
12. ### Infinite Product

You are almost done.Note that n^2-1 is (n-1)(n+1) So product(n^2-1) for n=2 to n is 1.2.3^2.4^2.....=product(n^2)/2 Product(n^2+1) for n=1 to n =2.Product(1+n^2) for n=2 to n So from the above expression pi*cosech(pi)=2*product(n^2-1)/2*product(n^2+1)=product(n^2-1/n^2+1) for n=2 to infty
13. ### Average Number of tracks

Let us find the fraction of the number line(from 0 to 1) that will be filled: Now we will find E(1) which is nothing but the expected fraction of the line(x=0 to x=1) given that each filling width is a small h (h=1/N) E(1)=h+E(1-h) E(1-h)=(1-h){h+E(1-2h)} and so on hence we get...
14. ### No. of ways

~^{n-r+1} C_{r}
15. ### Boundaries of Coulumb's law

Now turn the plane such that the 2 electrons are symmetrically moving. ie e1 starts from -infty with speed v_0 at angle a/2 and e2 start at +infty at angle a/2 The vertical component does not change as there is no force in that direction Energy(total) at infinity=mv^2/2+mv^2/2=mv^2 AlsoAt...
16. ### Boundaries of Coulumb's law

The minimum distance happens to be kq^2/(mv^2 Cos^2(a/2)) ie if they are head on it is kq^2/mv^2 where k is the 1/(4*pi*epsilon_0) in vaccum
17. ### Kinematics of particles

Derive that x(t)=64/t+t+6/5
18. ### 6 generals propose locking a safe with a number of different locks

There are 6 people and you need that only if 4 combines all the locks will be opened ie if you take any 3 people there will be 1 lock which needs to be opened and the key to that is available only with any of the other 3 people Hence for any 3 people you need to distribute one lock among the...
19. ### Probability problem

Pr(5% broken is declared ok/1/2 are checked)=Pr(you get either 0 broken or 1 broken in the pile of 50) =Pr(you get 0 or 1 faulty in 50/5 are faulty in 100)=50C5+50.50C4/{2*50C5+2*50C4*50+2*50C3*50C2}=0.1810892429
20. ### Please tell me how to solve these

For problem 2,the position vector is given by p(t)=a \left(Cos(\frac{kt^2}{2a}) \vec{i}+Sin(\frac{kt^2}{2a}) \vec{j}\right) if the center is the origin and the line joining the center and the starting point is the x-axis Now \frac{dp(t)}{dt}=kt \left(-Sin(\frac{kt^2}{2a})...
21. ### Finding distribution by using mgf(moment generating function)

E(e^{t\sum X_i})=\prod_{i}E(e^{t X_i})=e^{(e^t-1)(\lambda_1+..\lambda_n)} Hence the resultant distribution is poisson with poisson parameter \alpha_r which is given by \alpha_r=\sum_{i=1}^{n}\alpha_i
22. ### Question of counting of coins

Using 2 dimes and 3 nickels,all the amonts within 1 dollar thar can be paid are 0c,5c,10c,15c,20c,25c,30c,35c which is 8 So when combined with $it is 8*3-1(exclude 0$)=23
23. ### Leg of Tripod

If m is the mass of the object kept, ]each has compressional force= \frac{mg}{\sqrt{3}}
24. ### Vector question-

A.C=D AxC=B So we have (AXC)XA=|A|^2 C-(C.A)A=BXA So C=\frac{BXA+dA}{|A|^2}
25. ### What is the total energy of the point charge?

Energy of point charge is given by E where E is given by the expression that is written below E=-\frac{Qq}{4 \pi \epsilon_0 R}
26. ### What is the total energy of the point charge?

If the charge is free to move through the sphere,the motion will be a simple harmonic one with time period=T where T is given by T=2 \pi \sqrt{\frac{4 \pi \epsilon_0 R^3 m}{Qq}}
27. ### The missile and the aircraft

Consider any instant at which the missile is making an angle A with the horizontal Consider along the direction in which the plane is moving We have the relative velocity= V-Ucos(A) this is the rel vel along the line joining missile and plane so we have I(V-Ucos(A))dt,t:0->T)=d..(I is...
28. ### A.P. French, Mechanics Problem

The force exerted by the ground is 3mg and the downward gravity is mg Hence the deceleration is 3g-g=2g hence we have v^2=2(2g)(h) or h=v^2/(4g) and v^2=2g.1.5 So h=2*1.5/4=3/4=0.75 metres or 75 centimetres
29. ### The missile and the aircraft

Dont integrate at all. You will get \frac{Hv}{v^2-u^2}
30. ### What is the size of angle ABE?

ABE=40 degress. I think I have already posted it