mkerala

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1. didnt understand the question properly

2. 7, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 28, 28.

3. (Θ-1) times for the first person to know everything. as he communicates with the last person he also tells everything he knows to that person so the remaining (Θ-2) people should call once atleast to know everything which adds upto --- (Θ-1)+(Θ-2)

4. while moving around a fixed point the side will become the radius for the arc. it takes three rounds for c to come tothe original position and on calculating the whole thing it comes to 2*22/7*Θ*10/3

5. a minimum of 7 articles total. 4-periodicals, 1-novel, 1-newspaper, 1-hardcover

6. player A can win easily as Θ > 20. he has to make sure that he moves to the right end in more than 9 steps

7. 150kgs (i assumed empty ship has no wt loss)

Are these answer correct?

In the following problems, Θ = 20 + units digit of your day of birth. For example, if you

were born on April 1st, then Θ = 20 + 1 = 21. If you were born on March 30th, then Θ = 20

+ 0 = 20.

1. Alok and Bhanu play the following game on arithmetic expressions. Given the

expression

N = (Θ + A)/B + (Θ + C + D)/E

where A, B, C, D and E are variables representing digits (0 to 9), Alok would like to

maximize N while Bhanu would like to minimize it. Towards this end, they take turns in

instantiating the variables. Alok starts and, at each move, proposes a value (digit 0-9)

and Bhanu substitutes the value for a variable of her choice. Assuming both play to their

optimal strategies, what is the value of N at the end of the game? Also find a sequence

of moves (digits by Alok and variables by Bhanu) that would yield this value.

Note: Moves that lead to a divide-by-zero condition are disallowed. A non-optimal

sequence of moves is (5 → B, 6 → C , 3 → D, 2 → E, 0 → A) and the expression

evaluates to Θ/5 + (Θ+9)/2.

2. The mean, unique mode, median and range of 21 positive integers is 21. What is the

largest value that can be in this sequence? Also find such a sequence.

Note: Given a sequence of numbers a(1) ≤ a(2) ≤ ... ≤ a(n),

 The median of the sequence is the middlemost value in the sequence if n is

odd and the average of the two middle values if n is even.

 The mode is the most occurring value in the sequence

 The range is the difference between the largest and the smallest values, i.e.

a(n) - a(1).

For example, the sequence 2, 3, 4, 6, 6, 9 has mean = (2 + 3 + 4 + 6 + 6 + 9)/6 = 5,

median = (4+6)/2 =5, mode = 6, and range = 9 – 2 = 7.

3. A secret message is divided into Θ parts and each part is shared with a different

person. People communicate with each other using two-way phone calls and, in each

communication, share all the information they know until that point. What is the minimum

number of communications required for all Θ of them to know the secret? Find a

sequence of communications that achieves this minimum.

4. An equilateral triangle ABC with sides of length Θ cm is placed inside a square AXYZ

with sides of length 2*Θ cm so that side AB of triangle is along the base of the square

(as shown). The triangle is rotated clockwise about B, then C and so on along the sides

of the square until the points A, B and C return to their original positions. Find the length

of the path (in cm) traversed by point C.

5. A bag contains printed articles of 4 different kinds: periodicals, novels, newspapers

and hardcovers. When 4 articles are drawn from the bag without replacement, the

following events are equally likely:

 the selection of 4 periodicals

 the selection of 1 novel and 3 periodicals

 the selection of 1 newspaper, 1 novel and 2 periodicals and

 the selection of 1 article of each kind

What is the smallest number of articles in the bag satisfying these conditions? How

many of these are of each kind?

6. Given a 9 x Θ chessboard, a rook is placed at the lower left corner. Players A and B

take turns moving the rook. A plays first and each turn consists of moving the rook

horizontally to the right or vertically above. The last person to make a move wins the

game. At the completion of the game, the rook will be at the top right corner. For

example, the figure below shows a 3 x 4 chessboard and the sequence of moves that

leads to a win for player A.

Does player A have a winning strategy in the given 9 x Θ chessboard? If so, what is the

strategy? If not, what is player B's winning strategy?

7. A spaceship on an inter-galactic tour has to transfer some cargo from a base camp to

a station 100 light sec away through an asteroid belt. The ship can carry a maximum of

100 kgs of cargo and, as a result of colliding against the asteroids, every 2 light sec of

travel causes it to lose 1 kg of cargo. There are 300 kgs of cargo available at the base

camp. Find the maximum amount of cargo (in kg) that the ship can transfer to the

station? Assume that the spaceship can store the cargo at any intermediate point along

the way and that stored cargo is not depleted by the asteroids.

Prerequisites

Please answer as many questions as you can.

They have also given me an example

Questions on three dimensional geometry sometimes require the student to consider a

two-dimensional representation of the underlying object and use methods of plane

geometry to arrive at the solution. Here is one such example.

Example: An ant lives on the surface of a regular tetrahedron with edges of length 3cm.

It is currently at the midpoint of one of the edges and has to travel to the midpoint of the

opposite edge where a grain is located (see figure). What is the length (in cm) of the

shortest route to the destination assuming that the ant can only travel along the surface

of the tetrahedron?

Solution: The ant has several routes by which it can reach the grain. For instance, it can

travel to the vertex C and move along edge CD.

The idea behind finding the shortest route is to embed the surface of the tetrahedron on

a plane. This is done by opening the tetrahedron along some edges and spreading it out.

For example, the figure on the right is a planar representation containing the triangular

faces ABC and ACD. Notice that ABCD is a rhombus of length 3cm and the segment

joining ant and grain (which is the shortest route) is parallel to the base and thus of

length 3cm as well.

Now use the same idea to solve the problem below where the tetrahedron is replaced by

a cube.

Please use the example above to solve the problem.

An ant lives on the surface of a cube with edges of length 7cm. It is currentlylocated on an edge x cm from one of its ends. While traveling on the surface of the cube,it has to reach the grain located on the opposite edge (also at a distance xcm from oneof its ends) as shown below.

(i) What is the length of the shortest route to the grain if x = 2cm? How many routes ofthis length are there?

(ii) Find an x for which there are four distinct shortest length routes to the grain

Please tell the steps you have followed to arrive at the solution.