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 the equation $|x|^2-5|x|+6=0$

  1. there is only one root

  2. there are only two distinct roots

  3. there are only three distinct roots

  4. there are four distinct roots
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let y = |x|. The equation becomes y^2 - 5y + 6 = 0, which factors to (y - 2)(y - 3) = 0. Thus, |x| = 2 or |x| = 3. This gives x = 2, -2, 3, -3, resulting in four distinct roots.

Multiple choice

If $2,2$ are the roots of $x^2+px+b=0$ and $1,4$ are the roots of $x^2+ax+q=0$, then roots of the equation $x^2+ax+b=0$ are

  1. real and distinct

  2. imaginary

  3. real and equal

  4. irrational
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Roots 2, 2 for x^2+px+b=0 implies sum = -p = 4 (p=-4), product = b = 4. Roots 1, 4 for x^2+ax+q=0 implies sum = -a = 5 (a=-5), product = q = 4. The equation x^2+ax+b=0 becomes x^2 - 5x + 4 = 0. Roots are (x-1)(x-4)=0, so roots are 1 and 4. These are real and distinct.

Multiple choice

State the following statement is True or False If both the roots of the equation $x^2-6ax+2-2a+9a^2=0$ exceed $3$, then $a<\dfrac {11}9$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots > 3. Let f(x) = x^2 - 6ax + 9a^2 - 2a + 2 = 0. (x-3a)^2 = 2a - 2. x = 3a +/- sqrt(2a-2). Roots are 3a + sqrt(2a-2) and 3a - sqrt(2a-2). Both > 3: 3a - sqrt(2a-2) > 3 => 3a - 3 > sqrt(2a-2). Square both sides: 9(a-1)^2 > 2a-2. 9(a-1)^2 - 2(a-1) > 0. (a-1)(9a-9-2) > 0 => (a-1)(9a-11) > 0. So a > 11/9 or a < 1. The statement a < 11/9 is false.