Assignment:
Q1. Use mathematical induction to prove that 6 is a factor of n (n + 1)(n + 2).
Q2. Use mathematical induction to prove 1^2 + 3^2 + 5^2 + . . . + (2n - 1)^2 = n (4n^2 - 1) / 3.
Q3. Expand (2x - y)^6 using binomial coefficients and then evaluate each coefficient.
Example: the first term is 6 chose 0 (2x)^6 (-y)^0 = 64x^6
Q4. Interpret what the result in problem 3 tells us about construction the next row of the Pascal triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Q5. Find the equation of the circle that has its center at the focus of the parabola X^2 - 4x + 6y - 20 = 0 and that is tangent to the parabola at its vertex. Also sketch both curves.
Q6. Find the equation of the Ellipse whose foci coincide with the foci of the hyperbola x^2 - 3y^2 - 2x - 12y + 1 = 0 and whose eccentricity is the reciprocal of that of the hyperbola.
Provide complete and step by step solution for the question and show calculations and use formulas.