**Algebra**

**Find the minimum possible value of 2x² + 2xy + 5y² + 4y – x. **

**Find the minimum possible value of 2x² + 2xy + 5y² + 4y – x. **

**Five students – Ani, Chinu, Jigyasa, Mayank and Viru – appeared for an exam. There were total five questions – two multiple choice (a, b or c) and three true/false questions. They answered five questions each and answered as follow.
**

**Also, no two students got the same number of correct answers. **

**1. Which are the correct answers? **

**2. What are their individual score?**

**Let ABC be a triangle such that AB=3, BC=4, and AC=5. Let X be a point in the triangle. Compute the minimal possible value of AX²+BX²+CX².**

**Three mathematics classes X,Y and Z take an geometry test. The average of class X is 64, The average of class Y is 80 and the average score of class Z is 76 . The average score of class X and Y is 70 while the average score of class X and Z is 68. What is the ratio of number of students in the class X,Y and Z?**

**(A) 6:10:3
**

**In a contest, five fruits (an apple, a banana, an orange, a strawberry and a pear) have been placed in five closed boxes (one fruit per box). People may guess which fruit is in which box. 107 people participate in the contest. When the boxes are opened, it turns out that 40 people have guessed none of the fruits correctly, 30 people have guessed one fruit correctly, 20 people have guessed two fruits correctly and 10 people have guessed three fruit correctly. How many people have guessed four fruits correctly and how many people have guessed all the fruits correctly?**

**There are 100 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently switched off. The room has an entry door and an exit door. There are 100 people lined up outside the entry door. Each bulb is numbered consecutively from 1 to 100. So is each person. Person No. 1 enters the room, switches on every bulb, and exits. Person No. 2 enters and flips the switch on every second bulb (turning off bulbs 2, 4, 6, …). Person No. 3 enters and flips the switch on every third bulb (changing the state on bulbs 3, 6, 9, …). This continues until all 100 people have passed through the room.**

** **

**Point O is the center of the regular octagon ABCDEFGH, and X is the midpoint of the side AB. What fraction of the area of the octagon is shaded?
**

**(A) 7/16
(B) 11/32
(C)23/32
(D)3/8
(E)15/32**

**Let N =123456789…..4344 be the 79-digit number obtained that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when N is divided by 45 ?**

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