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, there are two quadratic equations. Study both the equations carefully and answer accordingly. I. 9x2 - 31x + 26 = 0 II. 5y2 - 12y + 7 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or no relation can be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For 9x^2 - 31x + 26 = 0, roots are x = (31 +/- sqrt(961 - 936)) / 18 = (31 +/- 5) / 18, so x = 2 or 13/9 (approx 1.44). For 5y^2 - 12y + 7 = 0, roots are y = (12 +/- sqrt(144 - 140)) / 10 = (12 +/- 2) / 10, so y = 1.4 or 1. Comparing values, x > y.

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. 4x2 - 16x + 7 = 0 II. 5y2 - 14y + 8 = 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

I: 4x^2 - 16x + 7 = 0. Roots: (16 +/- sqrt(256 - 112))/8 = (16 +/- 12)/8. x = 28/8 = 3.5 or x = 4/8 = 0.5. II: 5y^2 - 14y + 8 = 0. Roots: (14 +/- sqrt(196 - 160))/10 = (14 +/- 6)/10. y = 20/10 = 2 or y = 8/10 = 0.8. Comparing: x=3.5 > y=2, but x=0.5 < y=0.8. No relation.

Multiple choice

Directions: In the question, two equations numbered (I) and (II) are given. You have to solve both the equations and then mark the appropriate answer. (I) x2 = (8)2 + (6)2 (II) 35y2 + 11y – 6 = 0

  1. x = y or no relationship can be established between them.

  2. x > y

  3. x < y

  4. x ≥ y

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

I: x^2 = 64 + 36 = 100, so x = 10 or -10. II: 35y^2 + 11y - 6 = 0. Using quadratic formula: y = (-11 +/- sqrt(121 - 4*35*-6)) / 70 = (-11 +/- sqrt(121 + 840)) / 70 = (-11 +/- 31) / 70. y = 20/70 = 2/7 or y = -42/70 = -0.6. Comparing x and y, no consistent relationship exists.

Multiple choice

Directions: In the following question, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 3x2 + 13x + 14 = 0 (ii) 5y2 + 18y + 9 = 0

  1. x ≥ y

  2. x < y

  3. x ≤ y

  4. x = y or no relation can be established between x & y.

  5. x > y
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(i) 3x^2 + 13x + 14 = 0 -> (3x + 7)(x + 2) = 0 -> x = -7/3, -2. (ii) 5y^2 + 18y + 9 = 0 -> (5y + 3)(y + 3) = 0 -> y = -3/5, -3. Comparing values: -2.33, -2 vs -0.6, -3. Since -2 > -3 and -2 < -0.6, no relation can be established.

Multiple choice

Directions: In the following question, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 6x2 – 17x + 7 = 0 (ii) 2y2 – 18y + 28 = 0

  1. x = y or no relation can be established between x & y.

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x > y
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

i) 6x^2 - 17x + 7 = 0 -> (2x-1)(3x-7) = 0 -> x = 0.5, 2.33. ii) 2y^2 - 18y + 28 = 0 -> y^2 - 9y + 14 = 0 -> (y-2)(y-7) = 0 -> y = 2, 7. Comparing: x=0.5 < y=2, x=0.5 < y=7, x=2.33 > y=2, x=2.33 < y=7. Since x can be greater or less than y, no relation can be established.

Multiple choice

Directions: In the following question, two equations are given. You have to solve both the equations and choose the correct relationship accordingly. I. 5x2 - 32x + 51 = 0 II. y2 - 9 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or no relationship can be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

I: 5x^2 - 32x + 51 = 0. Roots: x = (32 +/- sqrt(1024 - 1020))/10 = (32 +/- 2)/10. x = 3.4 or 3. II: y^2 = 9, so y = 3 or -3. Comparing: x=3.4, y=3 (x>y); x=3.4, y=-3 (x>y); x=3, y=3 (x=y); x=3, y=-3 (x>y). Thus x >= y.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and give answer. I. 15x2 + 26x + 8 = 0 II. 25y2 + 15y + 2 = 0

  1. x < y

  2. x > y

  3. x ≤ y

  4. x ≥ y

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

I: 15x^2 + 26x + 8 = 0. Roots: (-26 +/- sqrt(676 - 480))/30 = (-26 +/- 14)/30. x = -40/30 = -4/3 or -12/30 = -0.4. II: 25y^2 + 15y + 2 = 0. Roots: (-15 +/- sqrt(225 - 200))/50 = (-15 +/- 5)/50. y = -20/50 = -0.4 or -10/50 = -0.2. Comparing: -1.33 <= -0.4 and -0.4 <= -0.2. Thus x <= y.

Multiple choice

𝑥³﹣𝑝𝑥² + 63𝑥﹣90 = 0 is a polynomial equation, all of whose roots are positive integers. Which of the following gives the value of 𝑝?

  1. 12

  2. 13

  3. 14

  4. 15
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let roots be r1, r2, r3. r1*r2*r3 = 90. r1+r2+r3 = p. r1r2 + r2r3 + r3r1 = 63. Possible factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15... Testing sets: 2, 5, 9 (sum 16, prod 90, sum of products 10+45+18=73 no). Testing 3, 5, 6 (sum 14, prod 90, sum of products 15+30+18=63 yes). So p = 14.