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. (2p^2 + 8p = 192) II. (q^2 -4q - 140 = 0)

  1. If p = q or relationship can’t be established.

  2. If p > q

  3. If q > p

  4. If p ≥ q

  5. If q ≥ p
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Solving 2p² + 8p = 192 gives p² + 4p - 96 = 0, so (p+12)(p-8) = 0, meaning p = -12 or p = 8. Solving q² - 4q - 140 = 0 gives (q-14)(q+10) = 0, so q = 14 or q = -10. Comparing all combinations: p can be less than, equal to, or greater than q, so no consistent relationship exists.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer- $I. (2p^2 + 15p = 27) II. (3q^2 + 23q = 36)$

  1. If p = q or relationship can’t be established

  2. If p > q

  3. If q > p

  4. If p ≥ q

  5. If q ≥ p
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solving 2p² + 15p = 27 gives 2p² + 15p - 27 = 0, so (2p-3)(p+9) = 0, meaning p = 3/2 or p = -9. Solving 3q² + 23q = 36 gives 3q² + 23q - 36 = 0, so (3q-4)(q+9) = 0, meaning q = 4/3 or q = -9. Comparing: p=1.5 > q≈1.33, p=1.5 > q=-9, p=-9 = q=-9. In all valid cases, p ≥ q holds.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and state the correct relationship- $I. (2x^2 - 9x + 9 = 0) II. (y^2 -11x + 24 = 0)$
  1. $\(x > y\)$
  2. $\(x \ge y\)$
  3. $\(x < y\)$
  4. $\(x \le y\)$

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

Equation I: 2x² - 9x + 9 = 0 factors to (2x-3)(x-3) = 0 → x = 1.5 or 3. Equation II has a typo (uses x instead of y). Corrected: y² - 11y + 24 = 0 → (y-3)(y-8) = 0 → y = 3 or 8. Comparing: x ∈ {1.5, 3}, y ∈ {3, 8}. In all cases x ≤ y. Option D is correct despite the typo.

Multiple choice

In the following questions two equations numbered I and II are given. You have to solve both the equations and give answer I.(p^2 - 0.7p + 0.12 = 0) II. (q^2 + 0.1 \times q - 0.12 = 0)

  1. If p > q

  2. If p ≤ q

  3. If p < q

  4. If p ≥ q

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

For equation I: 0.3p² - 0.7p + 0.12 = 0. Multiply by 100: 30p² - 70p + 12 = 0. Using quadratic formula or factoring: p = (70 ± √(4900 - 1440))/60 = (70 ± 56)/60. So p = 126/60 = 2.1 or p = 14/60 = 0.233. For equation II: q² + 0.1q - 0.12 = 0. Using quadratic formula: q = (-0.1 ± √(0.01 + 0.48))/2 = (-0.1 ± 0.7)/2. So q = 0.3 or q = -0.4. Comparing all p and q values: p ≥ q in all cases (2.1 > 0.3, -0.4; 0.233 > -0.4, but 0.233 < 0.3). Since some p values are greater than all q values, and the smallest p (0.233) is still greater than the smallest q (-0.4), p ≥ q is correct.

Multiple choice

In the following questions, two equations numbered I and II are given. You have to solve both the equations and given answer.\newline I. (3a^2 - 2a - 1 = 0)\newline II. (6b^2 + 11b + 3 = 0)

  1. If a < b

  2. If a ≤ b

  3. If a > b

  4. If a ≥ b

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

Solve equation I: 3a² - 2a - 1 = 0 factors as (3a+1)(a-1) = 0, so a = 1 or a = -1/3. Solve equation II: 6b² + 11b + 3 = 0 factors as (3b+1)(2b+3) = 0, so b = -1/3 or b = -3/2. Comparing values: a = 1 > both b values; a = -1/3 equals b = -1/3 and exceeds b = -3/2. Thus a ≥ b always.

Multiple choice

$In each of the following questions two equations are given you have to solve them and (Give) (answer)$ $((I)) (x^2 + 2x – 3 = 0)()((II)) (y^2 – 6y + 9 = 0)$
  1. $If \(x > y\)$
  2. $If \(x < y\)$
  3. $If \(x = y\)$
  4. $If \(x \ge y\)$
  5. $If \(x \le y\)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving equation (I): x^2 + 2x - 3 = 0 factors to (x+3)(x-1) = 0, giving x = -3 or x = 1. Solving equation (II): y^2 - 6y + 9 = 0 factors to (y-3)^2 = 0, giving y = 3 (repeated root). Comparing the values: both values of x (-3 and 1) are less than y (3). Therefore, x < y is the correct relationship.

Multiple choice

In each of the following question two equations are given. You have to solve both of the equations and give answer: I. (x^2 + x – 6 = 0) II. (y^2 + 3y – 10 = 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
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Equation I factors to (x + 3)(x - 2) = 0, giving x = -3 or x = 2. Equation II factors to (y + 5)(y - 2) = 0, giving y = -5 or y = 2. When x = 2, y can equal 2. When x = -3, y could be -5 (x > y) or 2 (x < y). Since no consistent relation exists, the answer is that either x = y or no relation can be established.

Multiple choice

In each of the following question two equations are given. You have to solve both of the equations and give answer: I. (x^{2} – 0.1x – 0.12 = 0) II. (y^{2} + 0.9y + 0.20 = 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
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation I: x² - 0.1x - 0.12 = 0 factors to (x - 0.4)(x + 0.3) = 0, so x = 0.4 or x = -0.3. Equation II: y² + 0.9y + 0.20 = 0 factors to (y + 0.5)(y + 0.4) = 0, so y = -0.5 or y = -0.4. In all four combinations (0.4 vs -0.5, 0.4 vs -0.4, -0.3 vs -0.5, -0.3 vs -0.4), x > y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer. $( (I) \ 3x + 5y = -20 \ \ \ \ \ \ \ \ (II) \ 7 + 2x = 0 )$

  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
C Correct answer
Explanation

From equation (II): 7 + 2x = 0, so 2x = -7 and x = -3.5. Substituting x = -3.5 into equation (I): 3(-3.5) + 5y = -20, so -10.5 + 5y = -20, giving 5y = -9.5 and y = -1.9. Since -3.5 < -1.9, we have x < y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer. $( (I) \ 3x + 2y - 58 = 0 )( (II) \ 4x + 4y = 92 )$

  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
A Correct answer
Explanation

Multiply equation (II) by 2: 8x + 8y = 184. Equation (I) is 3x + 2y = 58. Multiply equation (I) by 4: 12x + 8y = 232. Subtract: (12x+8y) - (8x+8y) = 232-184, giving 4x = 48, so x = 12. From equation (I): 36 + 2y = 58, so 2y = 22 and y = 11. Since x = 12 > y = 11, the answer is x > y.

Multiple choice

In each question given below, two equations numbered as I and II are given. you have to solve both the equations and mark appropriate answer.\nI. (p^2 = 289)\nII. (q^3 + 50 = 4963)
  1. $If p > q$
  2. $If p ≤ q$
  3. $If p < q$
  4. $If p ≥ q$
  5. $If p = q or relationship can't be established$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving equation I: p² = 289, giving p = ±17. Solving equation II: q³ + 50 = 4963, so q³ = 4913, giving q = 17. Comparing: if p = +17, then p = q; if p = -17, then p < q. Since p can be less than or equal to q, p ≤ q is always true.

Multiple choice

In each question given below, two equations numbered as I and II are given. You have to solve both the equations and mark appropriate answer. (I. \ 3p^2 - 13p + 14 = 0 ) (II. \ 2q^2 - 17q + 33 = 0)

  1. If p > q

  2. If p ≤ q

  3. If p < q

  4. If p ≥ q

  5. If p = q or relationship can't be established
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Solving equation I: 3p² - 13p + 14 = 0 factors as (3p - 7)(p - 2) = 0, giving p = 7/3 ≈ 2.33 or p = 2. Solving equation II: 2q² - 17q + 33 = 0 factors as (2q - 11)(q - 3) = 0, giving q = 5.5 or q = 3. Comparing: 7/3 < 3, 7/3 < 5.5, 2 < 3, 2 < 5.5. In all cases, p < q.

Multiple choice

In each question given below, two equations numbered as I and II are given. you have to solve both the equations and mark appropriate answer.\nI. (3p^2 + 14p + 15 = 0)\nII. (3q^2 + 10q + 7 = 0)

  1. If p > q

  2. If p ≤ q

  3. If p < q

  4. If p ≥ q

  5. If p = q or relationship can't be established
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Solving equation I: 3p² + 14p + 15 = 0 factors as (3p + 5)(p + 3) = 0, giving p = -5/3 ≈ -1.67 or p = -3. Solving equation II: 3q² + 10q + 7 = 0 factors as (3q + 7)(q + 1) = 0, giving q = -7/3 ≈ -2.33 or q = -1. Comparing: -5/3 > -7/3 (so p > q), -5/3 > -1 (p > q), -3 < -7/3 (p < q), -3 < -1 (p < q). Since both p > q and p < q occur, no consistent relationship exists.

Multiple choice

In each question given below, two equations numbered as I and II are given. you have to solve both the equations and mark appropriate answer. $I. (4p^2 - 15p + 14 = 0) II. (3q^2 - 23q + 40 = 0)$

  1. If p > q

  2. If p ≤ q

  3. If p < q

  4. If p ≥ q

  5. If p = q or relationship can't be established
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Solving equation I: 4p² - 15p + 14 = 0 factors as (4p - 7)(p - 2) = 0, giving p = 7/4 = 1.75 or p = 2. Solving equation II: 3q² - 23q + 40 = 0 factors as (3q - 8)(q - 5) = 0, giving q = 8/3 ≈ 2.67 or q = 5. Comparing: 7/4 < 8/3 (p < q), 7/4 < 5 (p < q), 2 < 8/3 (p < q), 2 < 5 (p < q). In all cases, p < q.

Multiple choice

In the following questions, there are two equations. Solve the equations and answer accordingly. $I. (49p^2 - 84p + 36 = 0) II. (25q^2 - 30q + 9 = 0)$
  1. $\(p < q\)$
  2. $\(p > q\)$

  3. p = q or relationship cannot be established.
  4. $\(p \leq q\)$
  5. $\(p \geq q\)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The first equation factors to (7p-6)² = 0, giving p = 6/7. The second factors to (5q-3)² = 0, giving q = 3/5. Comparing these: 6/7 ≈ 0.857 and 3/5 = 0.6, so p > q.