## Nature of roots - Advance

In algebra, a quadratic equation is any equation that can be rearranged in standard form as where x represents an unknown, and a, b, and c represent known numbers, where a is not equal to 0. If a = 0, then the equation is linear, not quadratic, as there is no term.

Q1.  If f(x) is a continuous function and attains only rational values and f(0)=3 , then roots of equation f(1)x2+f(3)x+f(5)=0 are

•   Imaginary
•   Rational
•  Irrational
•   Real and equal

Imaginary

Q2.  The roots of ax2+bx+c=0 , where a is not equal to 0 and coefficients are real, are non-real complex and a+c

•  4a+c<2b
•  4a+c>2b
•  4a+c=2b
•  None of these

4a+c=2b

Q3.  If a, b, c, d are four consecutive terms of an increasing AP then the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are

•   Real and distinct
•  Nonreal complex
•  Real and equal
•  Integers

Real and distinct

Q4.  If a, b, c are three distinct positive real numbers then the number of real roots of ax2+2b|x|-c=0 is

•   4
•   2
•  0
•  1

2

Q5.  The number of integral values of a for which x2-(a-1)x+3=0 has both roots positive and x2+3x+6=0 has both roots negative is

•  0
•  1
•  2
•  Infinite

Infinite

Q6.  If f(x)= (x2-1)/(x2+1) for every real number x then the minimum value of f

•  Does not exist because f is unbounded
•   Is not attained even though f is bounded
•  Is equal to 1
•  Is equal to -1

Is equal to -1

Q7.  If a, b, c are in H.P. then the expression a(b-c)x2+b(c-a)x+c(a-b)

•   Has real and distinct factors
•  Is a perfect square
•  Has no real factor
•  None of these

Is a perfect square

Q8.  If a, b, c are in G.P., where a, c are positive, then the equation ax2+bx+c=0 has

•  Real roots
•  Imaginary roots
•  Ratio of roots = 1 : w where w is a nonreal cube root of unity
•   Ratio of roots = b : ac

Imaginary roots

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Nature of roots - Advance
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