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

In each of the following questions, two equations are given. You have to solve them and give answer- I. (y^2 + 7y = -12) II. (3x^2 + 18 = -15x)

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Rearranging: y²+7y+12=0 gives (y+3)(y+4)=0, so y=-3 or -4. Rearranging: 3x²+15x+18=0 gives x²+5x+6=0, so (x+2)(x+3)=0, meaning x=-2 or -3. When x=-3, y could be -3 or -4. When x=-2, y could be -3 or -4. In all cases, x ≥ y (equal when both are -3, greater otherwise).

Multiple choice

In each of the following questions two equations are given. Solve these equations and give answer: (I) (x^2 - x - 42 = 0) (II) (y^2 - 7y + 12 = 0)

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

For equation (I): x² - x - 42 = 0 factors to (x - 7)(x + 6) = 0, so x = 7 or x = -6. For equation (II): y² - 7y + 12 = 0 factors to (y - 3)(y - 4) = 0, so y = 3 or y = 4. Comparing all pairs: (7,3) gives x > y; (7,4) gives x > y; (-6,3) gives x < y; (-6,4) gives x < y. Since we get both x > y and x < y depending on which roots we compare, no consistent relationship can be established. Option E is correct.

Multiple choice

In each of the following questions two equations are given. Solve these equations and give answer:( (I) x^{2} - 5x - 84 =0 ) ( (II) 19y^{2}-57y+38=0 )

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

Solving equation (I): x² - 5x - 84 = 0 factors to (x - 12)(x + 7) = 0, so x = 12 or x = -7. Solving equation (II): 19y² - 57y + 38 = 0. The discriminant is 57² - 4(19)(38) = 3249 - 2888 = 361 = 19². Using quadratic formula: y = (57 ± 19)/38, giving y = 76/38 = 2 or y = 38/38 = 1. Comparing values: 12 > 2 and 12 > 1, but -7 < 2 and -7 < 1. Since x can be greater or less than y depending on which root you pick, no consistent relationship exists.

Multiple choice

In the following question two equation numbered I and II are given you have to solve both the equations and give answer. (I.) (x^4 – 227 = 398) (II.) (y^2 + 321 = 346)

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solving equation I: x^4 - 227 = 398 gives x^4 = 625, so x = ±5. Solving equation II: y^2 + 321 = 346 gives y^2 = 25, so y = ±5. Since x can be 5 or -5, and y can be 5 or -5, we have multiple possibilities: x = y when both are 5 or both are -5, but x > y when x = 5 and y = -5, while x < y when x = -5 and y = 5. Therefore, the relationship cannot be established.

Multiple choice

In the following question two equation numbered I and II are given you have to solve both the equations and give answer. (I. ) (x^{2} – 4 = 0) (II. ) (y^{2} + 6y + 9 = 0)

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solving equation I: x^2 - 4 = 0 gives x = ±2. Solving equation II: y^2 + 6y + 9 = 0 gives (y + 3)^2 = 0, so y = -3 only. When x = 2, we have x > y (2 > -3). When x = -2, we also have x > y (-2 > -3). Therefore, x is always greater than y.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both equations to give the answer :( (I) \quad 5x^2 – 9x + 4 =0 ) ( (II) \quad y^2 – 11y - 42 =0 )

  1. If x < y

  2. If x > y

  3. If x ≥ y

  4. If x ≤ y

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

Equation (I): 5x² - 9x + 4 = 0 factors to (5x-4)(x-1)=0, so x = 1 or 0.8. Equation (II): y² - 11y - 42 = 0 factors to (y-14)(y+3)=0, so y = 14 or -3. Comparing values: when x=1, it can be less than y=-3 or greater than y=-3 depending on y's value. No consistent relationship exists.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both equations to give the answer: ( (I) x^2 + 15x + 56 = 0 ) ( (II) y^2 - 256 = 0 )

  1. If x < y

  2. If x > y

  3. If x ≥ y

  4. If x ≤ y

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

Equation (I): x² + 15x + 56 = 0 factors to (x+7)(x+8)=0, so x = -7 or -8. Equation (II): y² - 256 = 0 gives y = +16 or -16. For x = -7 or -8 and y = -16, we have x > y. For x = -7 or -8 and y = 16, we have x < y. Since x can be greater or less than y depending on values, no definite relationship exists.

Multiple choice

In each of the following questions two equations are given. Solve these equations and give answer : (I) (x^2 - 25 + 16 = 0) (II) (5y^2 - 40y + 60 = 0)

  1. If x < y

  2. If x > y

  3. If x ≥ y

  4. If x ≤ y

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

Solving equation (I): x² - 9 = 0 gives x = ±3. Solving equation (II): 5y² - 40y + 60 = 0, dividing by 5 gives y² - 8y + 12 = 0, so y = (8 ± √(64-48))/2 = (8 ± 4)/2, giving y = 6 or y = 2. Comparing: when x = 3, it can be less than, equal to, or greater than y depending on which y value is chosen. When x = -3, it's always less than y (which are positive). No single relationship (x < y, x > y, x ≤ y, or x ≥ y) holds for all combinations.

Multiple choice

In each of the following questions two equations are given. Solve these equations and give answer : ( (I) \quad x^{2} – 19x +34 = 0 ) ( (II) \quad y^{2} – 23y +120 = 0 )

  1. If x < y

  2. If x > y

  3. If x ≥ y

  4. If x ≤ y

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

Solving equation (I) using quadratic formula: x = [19 ± √(361-136)]/2 = [19 ± √225]/2 = [19 ± 15]/2, giving x = 17 or x = 2. Solving equation (II): y = [23 ± √(529-480)]/2 = [23 ± √49]/2 = [23 ± 7]/2, giving y = 15 or y = 8. When x = 17, x > y (for both y = 15 and y = 8). When x = 2, x < y (for both y = 15 and y = 8). Since x can be greater or less than y depending on which root we select, no consistent relationship can be established.

Multiple choice

In each of the following questions two equations (I) and (II) are given. Solve these equations and give the answer: (I) (-3x^2 + 5x + 42 = 0) (II) (-2y^2 + 9y + 35 = 0)
  1. $\(x > y\)$
  2. $\(x \ge y\)$
  3. $\(x < y\)$
  4. $\(x \le y\)$

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

Rewrite equation (I) as 3x^2 - 5x - 42 = 0, which factors as (x - 7)(3x + 6) = 0, giving x = 7 or x = -2. Rewrite equation (II) as 2y^2 - 9y - 35 = 0, which factors as (y - 7)(2y + 5) = 0, giving y = 7 or y = -2.5. Comparing possible pairs: if x = -2 and y = 7, then x < y; if x = -2 and y = -2.5, then x > y; if x = 7 and y = 7, then x = y. Since the relationship changes based on which values you compare, no consistent relationship exists.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer I. 49x2 + 84x – 64 = 0 II. 25y2 – 100y + 64 = 0

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solve 49x² + 84x - 64 = 0 using the quadratic formula: x = (-84 ± √(7056 + 12544))/(98) = (-84 ± √19600)/98 = (-84 ± 140)/98. This gives x = -8/7 ≈ -1.14 or x = 16/7 ≈ 2.29. For 25y² - 100y + 64 = 0: y = (100 ± √(10000 - 6400))/50 = (100 ± √3600)/50 = (100 ± 60)/50. This gives y = 4/5 = 0.8 or y = 16/5 = 3.2. Since both x values are less than both y values, x < y is correct.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. (2x^2 + 24x = -72) II. (y^2 = 36)

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Rearrange 2x² + 24x = -72 to 2x² + 24x + 72 = 0, then divide by 2: x² + 12x + 36 = 0. Factor as (x + 6)² = 0, giving x = -6 (repeated root). For y² = 36, we get y = ±6, so y = 6 or y = -6. Since x = -6 and y can be 6 or -6, we have x = y when y = -6, and x < y when y = 6. Therefore x ≤ y holds in all cases.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. (x = \sqrt{256}) II. (y^2 - 32y + 97 = -159)

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

For x = √256, we have x = 16. For y² - 32y + 97 = -159, rearrrange to y² - 32y + 256 = 0, which factors as (y - 16)² = 0, giving y = 16 (repeated root). Since x = 16 and y = 16, we have x = y. The relationship is x = y, not x > y or x < y, so the correct answer is that x = y (no relation needed).

Multiple choice

In each of the following questions two equations (I) and (II) are given. Solve these equations and give the answer:( (I) ) (39x^2 - 61x + 16 = 0) ( (II) ) (15y^2 - 237y + 180 = 0)

  1. If x < y

  2. If x > y

  3. If x ≥ y

  4. If x ≤ y

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

Solving equation (I): 39x² - 61x + 16 = 0 gives x = 16/13 ≈ 1.23 or x = 1/3 ≈ 0.33. Solving equation (II): 15y² - 237y + 180 = 0 gives y = 15 or y = 4/5 = 0.8. Comparing the values: when x = 1/3 and y = 15, x < y; when x = 16/13 and y = 0.8, x > y. Since different root pairs give different relationships, no consistent relationship can be established.

Multiple choice

In each of the following questions two equations (I) and (II) are given. Solve these equations and give the answer: (I) (x^2 - 8.8x + 12.6 = 0) (II) (y^2 - 4.4y + 4.2 = 0)

  1. If x < y

  2. If x > y

  3. If x ≥ y

  4. If x ≤ y

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

Solving equation (I): x² - 8.8x + 12.6 = 0 gives x = 6.3 or x = 2. Solving equation (II): y² - 4.4y + 4.2 = 0 gives y = 3 or y = 1.4. Comparing values: when x = 2 and y = 1.4, x > y; when x = 2 and y = 3, x < y. Since the relationship varies depending on which roots we compare, no consistent relationship exists between x and y.