**MATHMAA**

# KVS Mathematical Olympiad-2015-Stage-1

KVS Mathematical Olympiad-2015-Stage-1 was held on 02-08-2015. This page contains the questions . Its solutions you can find in next page which you get by clicking the link next.

## KVS Mathematical Olympiad-2015-Stage-1

1. When the tens digit of a three digit number abc is deleted, a two digit number is formed. How many numbers abc are there such that abc=9(ac)+4c ?

2. Let P(x) = x² + bx + c, where b and c are integers. If P(x) is a factor of both x^4 + 6x^2 + 25 and 3x^4 + 4x^2 + 28x + 5 , find the value of P(1).

3. A square is inscribed in a equilateral triangle. Find the ratio of the area of the square to that of the triangle.

4. (a) Prove that \(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+......+\frac{1}{98}-\frac{1}{99}+\frac{1}{100}> \frac{1}{5}\).

(b) Find the largest prime factor of \(3^{12}+ 2^{12}-2.6^{6}\).

5. Surface area of a sphere A is 300% more than the surface area of another sphere B. If the volume of sphere B is p% less than the volume of sphere A, find the value of 'p' .

6. ABC is and isosceles triangle in which AB=AC=25 cm and BC=14 cm . Find the difference of the circum-radius and in-radius of the triangle.

7. AB and BC are two equal chords of a circle of length\(2\sqrt(5)\) cm each. If raidus of the circle is 5 cm, find the length of the chord AC.

8. Two dice are thrown simultaneously. Find the sum of the probability of "getting a prime number as a sum" and probability of "getting a doublet of prime numbers."

9. A person starts from a place P towards another place Q at a speed of 30 km/h. After every 12 minutes, he increases his speed by 5 km/h. If the distance between P and Q is 51 km., find the time taken by him to cover the whole distance.

10. Solve for 'x' ;

\(4(x-\frac{1}{x})^{2} + 8(x+\frac{1}{x})=29\).

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