Quadratic Equations Level-3 Quiz-3

1. If x and y are real and x^2 – 8x + y^2 – 4y + 20 = 0, find the relation between x and y.

2.

If 1 is a root of the quadratic equation 3x^2 + ax – 2 = 0 and the quadratic equation a(x2 + 6x) – b = 0 has equal roots, The value of b is ___ ?

3.

Two quadratic equations with positive roots have one common root. The sum of the product of all four roots taken two at a time is 192, the equation whose roots are the two different roots is x^2 − 15x + 56 = 0. The sum of all the four roots is ___ ?

4. The equation x^6 + 3x^5 – x^4 – 7x^3 – 14x^2 – 10x – 12 = 0 has exactly one negative root. How many imaginary roots does it have?

5. Find the least positive value of k for which the equation x^2 + kx + 4 = 0 has real roots.

6. The expression ax^2 + bx + c takes a maximum value of 5 at x = 1 and takes the value of −1 at x = 0. Find the value of the expression at x = – 5

7. When P(z) is divided by z+1 the remainder is -8, when divided by z-3 the remainder is 4. What is the remainder of P(z) when divided by (z-3) (z+1)?

8. If one root of x^2 + px + 12 = 0 is 4, while the equation x^2 + px + q = 0 has equal roots, then the value of q is?

9. What is the value of m which satisfies 3m^2 – 21m + 30 < 0 ?

10. A, B and C are 3 cities that form a triangle and where every city is connected to every other one by at least one direct roots. There are 33 routes direct & indirect from A to C and there are 23 direct routes from B to A. How many direct routes are there from A to C ?