Quadratic Equations Level-3 Quiz-2

1. If the equation 3x^4 – 13x^3 + 7x^2 + 17x + a – 10 = 0 has exactly three positive roots, then a can be?

2. If x+1/x = -sqrt(3), the value of x^42 + x^48 + x^54 + x^60 + x^66 +x^72 is?

3. What should be added to 2x^3 + 4x^2 – 7x + 4 to make it exactly divisible by x – 1?

4. If px^3 + qx^2 + rx + s is exactly divisible by x^2 – 1, then which of the following is / are necessarily true?

(a) p = r
(b) q = s
(c) p = – r                                                                                                                                                                                                                                               (d) q = – s

5. The remainders of a polynomial f(x) in x are 10 and 15, respectively, when f(x) is divided by (x – 3) and (x– 4). Find the remainder when f(x) is divided by (x – 3)(x – 4).

6. Let f(x) = a12x^12 + a10x^10 + —— + a2x^2 + a0 The coefficients a 12 , a 10 , a 2 , a 0 are real. There are three sign changes of f(x) and f(x) = 0 has four non-real roots. Which of the following is true?

7. x⁴ – ax³ + bx² – cx + 8 = 0 divided by x – 1 leaves a remainder of 4, divided by x + 1 leaves remainder 3, find b.

8. All the roots of x^4 – 12x^3 + ax^2 + bx + 81 = 0 are non-negative. The ordered pair (a, b) can be?

9. The roots of the equation x^3 + lx^2 + mx + n = 0 are the real numbers α , β and γ , where α < β < γ . Also, one of these roots is 2. If l – 3 = m, m + 8 = n, then the ratio β : γ is

10. The denominator of a fraction is one more than the
square of the numerator. When the numerator is in-
creased by 2 and the denominator is increased by 9, the
new fraction formed is equal to 1/5. If the numerator is +ve, find the original fraction

positive, fi nd the original fraction.