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

Directions: In the following question, two equations numbered (I) and (II) are given. Student should solve both the equations and mark appropriate answer. I. 9x2 - 18x + 8 = 0 II. 7y2 - 17y + 6 = 0

  1. If x > y

  2. If x = y or no relation can be established

  3. If x < y

  4. If x ≥ y

  5. If x ≤ y
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solve 9x^2 - 18x + 8 = 0: x = (18 +/- sqrt(324 - 288))/18 = (18 +/- 6)/18. x = 4/3 or 2/3. Solve 7y^2 - 17y + 6 = 0: y = (17 +/- sqrt(289 - 168))/14 = (17 +/- 11)/14. y = 2 or 3/7. Comparing values, no consistent relation exists.

Multiple choice

Directions: In the following item, Quantity I and Quantity II are given. Determine the relationship between the quantities and choose the appropriate option. In the quadratic equation ax2 + bx + c = 0 with roots α and β, where sum of roots is 10 and α2 + β2 = 34, Quantity I: The value of c/a Quantity II: 36

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I ≤ Quantity II

  5. Quantity I = Quantity II, or No relation
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sum of roots alpha + beta = -b/a = 10. alpha^2 + beta^2 = (alpha+beta)^2 - 2*alpha*beta = 34. 100 - 2*alpha*beta = 34, so 2*alpha*beta = 66, alpha*beta = 33. Since alpha*beta = c/a, Quantity I = 33. Quantity II = 36. Thus, 33 < 36.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and choose the correct option. I: 6x2 - 13x + 6 = 0 II: 4y2 - 20y + 24 = 0

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

  5. x = y or the relationship cannot be established.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

I: 6x^2 - 13x + 6 = 0 => (2x-3)(3x-2) = 0 => x = 1.5, 0.66. II: 4y^2 - 20y + 24 = 0 => y^2 - 5y + 6 = 0 => (y-2)(y-3) = 0 => y = 2, 3. Comparing, all values of x are less than all values of y.

Multiple choice

The roots of the quadratic equation mx - x2 = 12, where m is a negative integer, are p and q. If |p - q| ≥ 12, what is the greatest possible integer value of m?

  1. -7

  2. -8

  3. -13

  4. -14

  5. -15
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

x^2 - mx + 12 = 0. Roots p, q satisfy p+q = m and pq = 12. |p-q| = sqrt((p+q)^2 - 4pq) = sqrt(m^2 - 48). Given sqrt(m^2 - 48) >= 12, m^2 - 48 >= 144, m^2 >= 192. Since m is negative, m <= -sqrt(192) approx -13.85. The greatest integer is -14.

Multiple choice

Consider non-zero c, where the roots of the equation 14y2 - 31y + 3m = 0, are a and b, and the roots of 35y2 - 53y + 4m = 0, are a and c. Which of the following is the equation whose roots are 3a/b and 4a/c?

  1. 4y2 – 245y + 250 = 0

  2. 9y2 – 45y + 220 = 0

  3. 49y2 – 245y + 250 = 0

  4. 49y2 – 425y + 520 = 0

  5. 94y2 – 425y + 520 = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the properties of roots (sum and product), we can express a, b, and c in terms of m. Substituting these into the required roots 3a/b and 4a/c allows us to form the new quadratic equation. The resulting equation is 49y^2 - 245y + 250 = 0.

Multiple choice

The quadratic function f(x) = ax2 + bx + c, where a, b, and c are constants. The graph of y = f(x) in the xy-plane passes through the points (−2, 0) and (12, 0). If a is an integer greater than 1, what is the greatest possible value of 5b − 2a?

  1. −200

  2. −156

  3. −130

  4. −104
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The roots are -2 and 12, so f(x) = a(x + 2)(x - 12) = a(x^2 - 10x - 24) = ax^2 - 10ax - 24a. Here b = -10a. We want to maximize 5b - 2a = 5(-10a) - 2a = -52a. Since a is an integer greater than 1, the maximum value occurs at the smallest possible a, which is a=2. Thus, -52 * 2 = -104.

Multiple choice

If f(x) = x2 + bx + c be a quadratic function with real coefficients, and the roots of the function are α and β such that (α + β = 9) and α2 + β2 = 41, find the quadratic function f(x).

  1. x2 - 9x + 20

  2. x2 + 9x - 20

  3. x2 - 9x - 20

  4. x2 + 9x + 20

  5. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the sum of roots alpha + beta = 9 and the sum of squares alpha^2 + beta^2 = 41, we use the identity (alpha + beta)^2 = alpha^2 + beta^2 + 2*alpha*beta. Substituting the values, 81 = 41 + 2*alpha*beta, which gives alpha*beta = 20. The quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0, resulting in x^2 - 9x + 20.

Multiple choice

Let k be an integer such that the equation 2x2 + (k - 4)x + 14 = 0 has no real roots, and the equation x2 + kx + 4 = 0 has two distinct real roots for x. If kmin and kmax denote the least and greatest integer values of k satisfying both the equatons, find the value of kmax - kmin.

  1. 9

  2. 12

  3. 14

  4. 20

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

For the first equation to have no real roots, (k - 4)^2 - 112 < 0, giving integer values from -6 through 14. The second equation has two distinct real roots when k^2 - 16 > 0, so the valid integers are -6, -5 and 5 through 14. Hence kmax - kmin = 14 - (-6) = 20.