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

If the roots of the equation a(b - c)x2 + b(c - a)x + c(a - b) = 0 are equal, then which of the following is true?

  1. b = (a + c)/ac

  2. 2/b = (1/a) + (1/c)

  3. 2b = (1/a) + (1/c)

  4. abc = ab + bc + ca
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac = 0. Here, the equation is Ax^2 + Bx + C = 0 where A=a(b-c), B=b(c-a), C=c(a-b). The condition D=0 leads to [b(c-a)]^2 = 4[a(b-c)][c(a-b)]. Simplifying this identity results in 2/b = 1/a + 1/c.

Multiple choice

If the difference between the roots of the equation Ax2 - Bx + C = 0 is 4, then which of the following is TRUE?

  1. B2 - 16A2 = 4AC + 4B2

  2. B2 - 10A2 = 4AC + 6A2

  3. B2 - 8A2 = 4AC + 10A2

  4. B2 - 16A2 = 4AC + 8B2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For Ax^2 - Bx + C = 0, roots are (B +/- sqrt(B^2 - 4AC)) / 2A. The difference is sqrt(B^2 - 4AC) / A = 4. Squaring both sides gives (B^2 - 4AC) / A^2 = 16, so B^2 - 4AC = 16A^2. Rearranging gives B^2 - 16A^2 = 4AC. Option B is B^2 - 10A^2 = 4AC + 6A^2, which simplifies to B^2 - 16A^2 = 4AC. Both are mathematically equivalent.

Multiple choice

If α and β are the roots of equation x2 - 2x + 4 = 0, then what is the equation whose roots are α3/β2 and β3/α2?

  1. x2 - 4x + 8 = 0

  2. x2 - 32x + 4 = 0

  3. x2 - 2x + 4 = 0

  4. x2 - 16x + 4 = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots of x^2 - 2x + 4 = 0 are alpha, beta. alpha+beta = 2, alpha*beta = 4. New roots are alpha^3/beta^2 and beta^3/alpha^2. Sum = (alpha^5 + beta^5) / (alpha*beta)^2. Product = alpha*beta = 4. Calculating alpha^5 + beta^5 using Newton's sums or expansion gives 32. Sum = 32 / 16 = 2. Equation: x^2 - (sum)x + (product) = x^2 - 2x + 4 = 0.

Multiple choice

Mani was asked to form a quadratic equation such that the square of the difference of the roots when multiplied by the sum of the roots gives an answer which is 288 times the difference of roots. If the roots are integers, how many quadratic equations are possible?

  1. 6

  2. 12

  3. 18

  4. 24

  5. Data inadequate
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let roots be alpha, beta. (alpha-beta)^2 * (alpha+beta) = 288 * (alpha-beta). (alpha-beta)(alpha+beta) = 288. alpha^2 - beta^2 = 288. Since alpha, beta are integers, we look for factors of 288. There are 24 pairs of integers satisfying this.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and give the answer. (I) 2d2 - 7d + 6 = 0 (II) 5e2 - 9e + 4 = 0

  1. d > e

  2. d ≤ e

  3. d < e

  4. d ≥ e

  5. d = e, or no relation between d and e
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Solving 2d^2 - 7d + 6 = 0: (2d-3)(d-2) = 0, so d = 1.5, 2. Solving 5e^2 - 9e + 4 = 0: (5e-4)(e-1) = 0, so e = 0.8, 1. Comparing: 1.5 > 0.8, 1.5 > 1, 2 > 0.8, 2 > 1. Thus, d > e.

Multiple choice

Directions: Equations I and II are given. You have to solve both the equations and give answer. I. 11p2 + 40p + 21 = 0 II. 11q2 - 15q - 14 = 0

  1. p ≥ q

  2. p < q

  3. p > q

  4. p ≤ q

  5. p = q or relation cannot be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

I: 11p^2 + 40p + 21 = 0. Roots: p = (-40 +/- sqrt(1600 - 924))/22 = (-40 +/- 26)/22. p = -14/22 = -7/11 or p = -66/22 = -3. II: 11q^2 - 15q - 14 = 0. Roots: q = (15 +/- sqrt(225 + 616))/22 = (15 +/- 29)/22. q = 44/22 = 2 or q = -14/22 = -7/11. Comparing: p <= q.

Multiple choice

Directions: Equations I and II are given. You have to solve both the equations and give answer. I. 21p2 - 19p + 4 = 0 II. 18q2 - 51q + 35 = 0

  1. p ≥ q

  2. p < q

  3. p > q

  4. p ≤ q

  5. p = q or relation cannot be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving 21p^2 - 19p + 4 = 0: roots are 4/7 and 1/3. Solving 18q^2 - 51q + 35 = 0: roots are 7/6 and 5/3. Comparing the values: 4/7 (approx 0.57) and 1/3 (approx 0.33) are both less than 7/6 (approx 1.16) and 5/3 (approx 1.66). Thus, p < q.