Find the number of pairs \( (x,y)\) of positive real numbers that satisfy \( x^3+y^3=3xy-1 \) .

Let A be the number of ordered triples of sets \( (A,B,C) \) such that \( A\cup B\cup C=\{1,2,3,\dots, 2003\} \) and \( A\cap B\cap C=\phi \) .Then find the value of \( \frac{A}{6^{2000}} \).

If the polynomial p (x) of degree 9 has the following property \( p (k) = \frac{1}{k(1 + k)} \), for k=1,2,3,4….,10

Calculate \( 66 \times P(11) \) .

\( a,b>1 \) – are naturals, and \( a^2+b,a+b^2\) are primes. Find the value of \( gcd(ab+1,a+b)\) .

ABC is a triangle where AB=10 and AC=12.CE and BD are the bisector of angle ABC and angle ACB. AM and AN are perpendicular to CE and BD. If MN=4, find BC.

Find the remainder, when \( {72}^{1001} \) is divided by 31 .

Let \( a_{1}, a_{2}, \cdots, a_{n}\) be positive real numbers.

Find the value of k

\( \frac{{a_{1}}^{2}}{a_{1}+a_{2}}+ \frac{{a_{2}}^{2}}{a_{2}+a_{3}}+\cdots + \frac{{a_{n}}^{2}}{a_{n}+a_{1}} \ge k (a_{1}+a_{2}+ \cdots +a_{n} ) \)

Find the rank of the word M A K E when its letters are arranged as in a dictionary.

The co-ordinates of the points A and B are \( (3,\sqrt{3}) \) and \( (0,2\sqrt{3}) \) respectively; if \( A B C \) be an equilateral triangle, find the co-ordinates of C.After finding the possible coordinates find their sum and just write the value of x coordinate as answer.

To test the quality of electric bulbs produced in a factory, two bulbs are randomly selected from a large sample without replacement. If either bulb is defective, the entire lot is rejected. Suppose a sample of 300 bulbs contains 5 defective bulbs. Find the probability that the sample will be rejected. ( Round your answer to 2 decimal places )