Quantitative Aptitude

Algebra

3,715 Questions
All Polynomial Roots and Coefficients 2451 Zero-Product Root Finding 667 Linear System Manipulation 162 Polynomial Theorems and Reduction 85 Algebraic Identity Simplification 82 Surds and Indices 70 Modulus and Absolute Value 54 Algebraic Expression Substitution 46 Algebraic Fraction Simplification 33 Logarithmic Equations 25 Repeated Series Sum 12 Linear Equation Word Problems 9

Algebra Questions

Multiple choice

For what value of a, the roots of the equation $2x^2+ 6x + a = 0$, satisfy the conditions $\left ( \dfrac{\alpha }{\beta } \right )+\left ( \dfrac {\beta}{\alpha } \right ) < 2 $ (where $\beta,\alpha $ are the roots of equation).
  1. $a > 0$
  2. $-1 < a < 0$
  3. $-1 < a < 1$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(a/b + b/a) < 2 => (a^2+b^2)/ab < 2. Since (a-b)^2 >= 0, a^2+b^2 >= 2ab. For the sum to be less than 2, the roots must be complex or have different signs. Discriminant D = 36 - 8a. For real roots, a <= 4.5. The condition (a/b + b/a) < 2 is impossible for real roots of the same sign. Thus, no real 'a' satisfies this.

Multiple choice

Which of the following equations has two distinct real roots ?
  1. $2x^2-3\sqrt 2 x+\dfrac 94=0$
  2. $x^{2}+x-5=0$
  3. $x^{2}+3x+2\sqrt{2}=0$
  4. $5x^{2}-3x+1=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A quadratic equation has two distinct real roots if its discriminant D = b^2 - 4ac > 0. For B, x^2 + x - 5 = 0, D = 1^2 - 4(1)(-5) = 21, which is greater than 0.

Multiple choice

State the nature of the given quadratic equation $3x^2 + 4x + 1 = 0$

  1. Real and Distinct Roots

  2. Real and Equal Roots

  3. Imaginary Roots

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The discriminant D of the quadratic equation ax^2 + bx + c = 0 is b^2 - 4ac. For 3x^2 + 4x + 1 = 0, D = 4^2 - 4(3)(1) = 16 - 12 = 4. Since D > 0, the roots are real and distinct.

Multiple choice

The roots of the equation $3x^{2} - 4x + 3 = 0$ are :

  1. real and unequal

  2. real and equal

  3. imaginary

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The discriminant of the quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac. For 3x^2 - 4x + 3 = 0, D = (-4)^2 - 4(3)(3) = 16 - 36 = -20. Since the discriminant is negative, the roots are imaginary.