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

The roots of the equation (V - W)X2 + (W - U)X + (U - V) = 0 are:

  1. 1 , V − U V − W

  2. 1 , U − V V − W

  3. − 1 , V − W V − U

  4. − 1 , W − U V − W

  5. − 1 , V − U U − W
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation is (V-W)X^2 + (W-U)X + (U-V) = 0. Notice the sum of coefficients: (V-W) + (W-U) + (U-V) = 0. If the sum of coefficients is 0, then X=1 is a root. The product of roots is (U-V)/(V-W). Since one root is 1, the other root is (U-V)/(V-W).

Multiple choice

Directions: In the following question, two equations (i) and (ii) are given. You have to solve both the equations and mark the correct option accordingly. (i) 4x2 + 13x - 12 = 0 (ii) 5y2 + 13y + 6 = 0

  1. x ≥ y

  2. x > y

  3. x = y, or no relation can be established between x and y.

  4. x ≤ y

  5. x < y
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Solve (i): 4x^2 + 13x - 12 = 0 -> (4x - 3)(x + 4) = 0 -> x = 3/4, -4. Solve (ii): 5y^2 + 13y + 6 = 0 -> (5y + 3)(y + 2) = 0 -> y = -3/5, -2. Comparing x and y: x can be 0.75 or -4, y can be -0.6 or -2. Since 0.75 > -0.6 and -4 < -2, no relation can be established.

Multiple choice

Directions: Equations I and II are given. You have to solve both the equations and give answer. I. 25p2 - 55p + 28 = 0 II. 25q2 + 20q - 32 = 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
A Correct answer
Explanation

I: 25p^2 - 55p + 28 = 0. Roots are p = (55 +/- sqrt(3025 - 2800)) / 50 = (55 +/- 15) / 50. p = 1.4 or 0.8. II: 25q^2 + 20q - 32 = 0. Roots are q = (-20 +/- sqrt(400 + 3200)) / 50 = (-20 +/- 60) / 50. q = 0.8 or -1.6. Comparing p and q, p >= q.

Multiple choice

Which of the following equations does not have real roots?

  1. x2 + x - 5 = 0

  2. 5x2 - 3x + 1 = 0

  3. -x2 + 2x + 3 = 0

  4. x2 + 3x - 12 = 0

  5. 7x2 - x - 12 = 0
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A quadratic equation ax^2 + bx + c = 0 has no real roots if the discriminant D = b^2 - 4ac < 0. For B: 5x^2 - 3x + 1 = 0, D = (-3)^2 - 4(5)(1) = 9 - 20 = -11. Since D < 0, it has no real roots.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and then choose the correct option. I. 6p2 – 5p + 1 = 0 II. 4q2 – 7q + 3 = 0

  1. p > q

  2. p ≥ q

  3. p < q

  4. p ≤ q

  5. p = q or no relation can be established
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

I: 6p^2 - 5p + 1 = 0 => (2p-1)(3p-1) = 0 => p = 1/2, 1/3. II: 4q^2 - 7q + 3 = 0 => (4q-3)(q-1) = 0 => q = 3/4, 1. Comparing: 1/2 < 3/4, 1/2 < 1, 1/3 < 3/4, 1/3 < 1. Thus, p < q.

Multiple choice

Directions: Equations I and II are given. You have to solve both the equations and give answer. I. 12p2 + 31p + 9 = 0 II. 6q2 - q - 1 = 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: 12p^2 + 31p + 9 = 0. Roots are p = (-31 +/- sqrt(961 - 432)) / 24 = (-31 +/- 23) / 24. p = -8/24 = -1/3 or p = -54/24 = -2.25. II: 6q^2 - q - 1 = 0. Roots are q = (1 +/- sqrt(1 + 24)) / 12 = (1 +/- 5) / 12. q = 6/12 = 0.5 or q = -4/12 = -1/3. Comparing: -2.25 <= -1/3 <= 0.5. Thus, p <= q.

Multiple choice

A and B solved a quadratic equation. While solving it, A made a mistake in the constant term and obtained the roots as 5, - 3, while B made a mistake in the coefficient of x and obtained the roots as 1, - 3. The correct roots of the equation are
    • 1, + 3
    • 1, 3
    • 1, - 3

  4. 1, - 1

  5. 1, - 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A's roots (5, -3) give the correct x-coefficient: sum = 2, so x^2 - 2x + c = 0. B's roots (1, -3) give the correct constant: product = -3, so x^2 + bx - 3 = 0. The correct equation is x^2 - 2x - 3 = 0, which factors to (x-3)(x+1) = 0, giving roots 3 and -1.

Multiple choice

Directions: In this question, two equations (I) and (II) are given. You have to solve both the equations and give the answer accordingly. (I) a2 + 6a + 8 = 0 (II) b3 + 5341 = 7538

  1. a > b

  2. a < b

  3. a ≥ b

  4. a ≤ b

  5. a = b or no relation can be established between 'a' and 'b'.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation I: a^2 + 6a + 8 = 0 factors to (a+4)(a+2) = 0, so a = -4 or -2. Equation II: b^3 + 5341 = 7538 implies b^3 = 2197, so b = 13. Comparing values, -4 < 13 and -2 < 13, so a < b.