# December 2015: Mathematical Sciences, Chemical Sciences and Earth Sciences

December 2015

1. Let A, B be the ends of the longest diagonal of the unit cube. The length of the shortest path from A to B along surface is

2. “The clue is hidden in this statement”, read the note handed to Sherlock by Moriarty, who hid the stolen treasure in one of the ten pillars. Which pillar is it?

3. A circle drawn in the x–y co–ordinate plane passes through the origin and has chords of lengths 8 units and 7 units on the x and y axes, respectively. The coordinates of its centre are

4. The probability that a ticket less traveller is caught during a trip is 0.1. If the traveller makes 4 trips, the probability that he/she will be caught during at least one of the trip is:

5. Three boxes are coloured red, blue and green and so are three balls. In how many ways can one put the balls one in each box such that no ball goes into the box of its own colour?

6. The number of diagonals of a convex dodecagon (12–gon) is

7. A wheel barrow with unit spacing between its wheels is pushed along a semi–circular path of mean radius 10. The difference between distances covered by the inner and outer wheels is

8. One is required to tile a plane with congruent regular polygons. With which of the following polygon is this possible?

9. Decode the below code 10. The base diameter of a glass is 20% smaller than the diameter at the rim. The glass is filled to half the height. The ratio of empty to filled volume of the glass is

11. Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor of the group attended exactly two meetings. 21 professors attended Mumbai meeting, 27 attended Delhi meeting and 30 attended Chennai meeting. How many of them attended both the Chennai and Delhi meetings?

12. Write d = 1 degree, r = 1 radian and g = 1 grad. Then which of the following is true? (100 grad = a right angle)

13. The minimum number of straight lines required to connect the nine points without lifting the pen or retracing is 14. There is an inner circle and an outer circle around a square. What is the ratio of the area of the outer circle to that of the inner circle?

15. The missing number is 16. Most Indian tropical fruit trees produce fruits in April–May. The best possible explanation for this is

17. Three circles of equal diameters are placed such that their centres make an equilateral triangle as in the figure. Within each circle, 50 points are randomly scattered. The frequency distribution of distances between all possible pairs of points will look as 18. A vendor sells articles having a cost price of Rs.100 each. He sells these articles at a premium price during first eight months, and at a sale price, which is half of the premium price, during next four months. He makes a net profit of 20% at the end of the year. Assuming that equal numbers of articles are sold each month, what is the premium price of the article?

19. How many digits are there in 316 when it is expressed in the decimal form?

20. The statement: “The father of my son is the only child of your parents”

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