Quantitative Aptitude

Algebra

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Algebra Questions

Multiple choice

What is the condition that the equation $ax^{2} + bx + c = 0$, where $a\neq 0$, has both the roots positive?
  1. $a, b $ and $c$ are of same sign
  2. $a$ and $b$ are of same sign
  3. $b$ and $c$ have the same sign opposite to that of $a$
  4. $a$ and $c$ have the same sign opposite to that of $b$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For roots to be positive, the product of roots (c/a) must be positive, and the sum of roots (-b/a) must be positive. This implies a and c have the same sign, and a and b have opposite signs.

Multiple choice

(a) Show that if x is real, the expression $\frac { { x }^{ 2 }-bc }{ 2x-b-c } $ has no real values between b and c. (b) For real x, the function $\frac { (x-a)(x-b) }{ x-c } $ will assume all real values provided (i) $a>b>c$ (ii) $ac>b$ (iv) $a

  1.  Positive

  2. Negative

  3. Neal and of opposite sign

  4. Imaginary
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If alpha and beta are the roots of the first quadratic, then alpha beta < 0 because the roots have opposite signs. Expanding the second equation shows that its root product is also alpha beta, which is negative, while its discriminant is positive. Therefore, its roots are real and of opposite signs.

Multiple choice

If $\displaystyle a, : b, : c : \in : R$ and $\displaystyle 3b^{2} - 8ac < 0$, then the equation $\displaystyle ax^{4} + bx^{3} + cx^{2} + 5x + 7 = 0$ has

  1. all real roots

  2. all are imaginary roots

  3. can not have all real roots

  4. exactly two real and two imaginary roots
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The condition 3b^2 - 8ac < 0 relates to the discriminant of the derivative or properties of the polynomial. For a quartic to have all real roots, the discriminant must be non-negative. This condition implies it cannot have all real roots.