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. Solve these equations and give answer: ( (I) \quad 5x^2 - 20x + 15 = 0 ) ( (II) \quad 3y^2 - 20y + 17 = 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: 5x² - 20x + 15 = 0. Dividing by 5: x² - 4x + 3 = 0, giving x = 1 or x = 3. Solving equation II: 3y² - 20y + 17 = 0. Using quadratic formula: y = [20 ± √(400-204)]/6 = [20 ± √196]/6 = [20 ± 14]/6, giving y = 17/3 ≈ 5.67 or y = 1. Since x can be both less than and greater than y depending on the chosen roots, no consistent relationship can be established.

Multiple choice

In each of the following questions two equations are given. Solve these equations and give answer: (I) (2x^2 - 4.2x + 1.8 = 0) (II) (y^2 - 4.8y + 5.4 = 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
A Correct answer
Explanation

Solving equation I: 2x² - 4.2x + 1.8 = 0. Using quadratic formula: x = [4.2 ± √(17.64-14.4)]/4 = [4.2 ± √3.24]/4 = [4.2 ± 1.8]/4, giving x = 1.5 or x = 0.6. Solving equation II: y² - 4.8y + 5.4 = 0 gives y = [4.8 ± √(23.04-21.6)]/2 = [4.8 ± √1.44]/2 = [4.8 ± 1.2]/2, giving y = 3 or y = 1.8. Since all values of x (0.6, 1.5) are less than all values of y (1.8, 3), x < y.

Multiple choice

In each of the following questions two equations (I) and (II) are given. Solve these equations and give the answer: ( (I) (x – 16)^2 = 0 ) ( (II) y^2 - 256 = 0 )
  1. $If \( x < y \)$
  2. $If \( x > y \)$
  3. $If \( x \ge y \)$
  4. $If \( x \le y \)$

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

Equation I: (x-16)²=0 gives x=16 (repeated root). Equation II: y²-256=0 gives y²=256, so y=16 or y=-16. Comparing: when y=16, x=16 gives x=y; when y=-16, x=16 gives x>y. In all cases x ≥ y, so option C is correct.

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 - 10x + 7 = 0)\ (II) (5y^2 - 8y + 3 = 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
C Correct answer
Explanation

Equation I: 3x²-10x+7=0 factors as (3x-7)(x-1)=0, giving x=7/3≈2.33 or x=1. Equation II: 5y²-8y+3=0 factors as (5y-3)(y-1)=0, giving y=3/5=0.6 or y=1. Comparing all pairs: x=2.33,y=0.6 (x>y); x=2.33,y=1 (x>y); x=1,y=0.6 (x>y); x=1,y=1 (x=y). In all cases x ≥ y, so option C is correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and state the correct relationship-(i)\ (x^2 - 4x + 3.91 = 0) (ii)\ (y^2 - 2y - 4.76 = 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
E Correct answer
Explanation

For equation (i): x² - 4x + 3.91 = 0. Using quadratic formula: x = [4 ± √(16 - 15.64)]/2 = [4 ± √0.36]/2 = [4 ± 0.6]/2. So x = 2.3 or x = 1.7. For equation (ii): y² - 2y - 4.76 = 0. Using formula: y = [2 ± √(4 + 19.04)]/2 = [2 ± √23.04]/2 = [2 ± 4.8]/2. So y = 3.4 or y = -1.4. Since x values (1.7, 2.3) and y values (-1.4, 3.4) overlap without clear inequality, we cannot establish x > y, x < y, or x = y definitively.

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 - 14x + 24 = 0) (II). (y^2 - 3y + 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
C Correct answer
Explanation

Solving equation I: x² - 14x + 24 = 0 factors to (x-12)(x-2) = 0, so x = 12 or 2. Solving equation II: y² - 3y + 2 = 0 factors to (y-2)(y-1) = 0, so y = 2 or 1. Comparing values: when x=12, x > both y values; when x=2, x = y=2 or x > y=1. Therefore x ≥ y is always true.

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. (4x^2 - 33x + 63 = 0) II. (10y^2 – 113y + 318 = 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 Equation I: 4x^2 - 33x + 63 = 0. Factors: (4x-21)(x-3) = 0. Solutions: x = 21/4 = 5.25 or x = 3. Solve Equation II: 10y^2 - 113y + 318 = 0. Using quadratic formula: y = [113 ± sqrt(12769 - 12720)]/20 = [113 ± sqrt(49)]/20 = [113 ± 7]/20. Solutions: y = 120/20 = 6 or y = 106/20 = 5.3. Comparing: x values are 3 and 5.25; y values are 5.3 and 6. The maximum x (5.25) is less than the minimum y (5.3), so x < y.

Multiple choice

(\text{In each of the following questions, two equations are given. You have to solve them and give answer.}\\text{(I). } p^{2} + 16p + 63 = 0\\text{(II). } q^{2} + q = 42)

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

  2. if p < q

  3. if p > q

  4. if p ≥ q

  5. if p ≤ q
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

(I) p² + 16p + 63 = 0 → (p+7)(p+9) = 0 → p = -7, -9. (II) q² + q - 42 = 0 → (q+7)(q-6) = 0 → q = -7, 6. Comparing: p = -7 when q = -7; otherwise p < q (when q = 6). Therefore p ≤ q is correct.

Multiple choice

If the ratio of the roots of the equation x2- 2ax + b = 0 is equal to that of the roots x2 -2cx + d = 0, then find which option is correct. यदि समीकरण x2- 2ax + b = 0 के मूल और समीकरण x2 -2cx + d = 0 के मूल बराबर हो, तो कौन सा विकल्प सही होगा|

  1. a2b = c2d

  2. a2c = b2d

  3. d2b = c2a

  4. a2d = c2b
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For quadratic ax² + bx + c = 0 with roots α, β: sum = α+β = -b/a and product = αβ = c/a. Equation 1: x² - 2ax + b = 0 has sum = 2a, product = b. Equation 2: x² - 2cx + d = 0 has sum = 2c, product = d. Given ratios are equal: α/β = p/q (say). Then α²/αβ = p²/(pq). But αβ = product, so α² = (sum/product) × product²? Let's use α/β = k. Then α = kβ and αβ = kβ² = product. Also α+β = kβ+β = β(k+1) = sum. For ratio equality: α₁/β₁ = α₂/β₂ = k. Then α₁β₁ = kβ₁² = b, α₂β₂ = kβ₂² = d. Also: (α₁+β₁)² = β₁²(k+1)² = (2a)² = 4a². So β₁²(k+1)² = 4a² and kβ₁² = b. Dividing: (k+1)²/k = 4a²/b. Similarly: (k+1)²/k = 4c²/d. Equating: 4a²/b = 4c²/d, so a²/b = c²/d, giving a²d = c²b.

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 - 50x + 625 = 0) (II) (y^2 - 28y + 196 = 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
B Correct answer
Explanation

Equation (I): x² - 50x + 625 = 0 factors as (x-25)² = 0, so x = 25. Equation (II): y² - 28y + 196 = 0 factors as (y-14)² = 0, so y = 14. Since 25 > 14, we have x > y. Both are perfect square trinomials that yield single roots.

Multiple choice

In each of the following questions two equations (I) and (II) are given. Solve these equations and give the answer: (I). (2x^2 + 5x - 42 = 0) (II). (y^2 - 19y + 84 = 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
A Correct answer
Explanation

Solve equation (I): 2x² + 5x - 42 = 0 gives x = -6 or x = 3.5 using quadratic formula. Solve equation (II): y² - 19y + 84 = 0 gives y = 7 or y = 12. For all possible values of x (-6 or 3.5) and y (7 or 12), x is always less than y, so x < y is the correct relationship.

Multiple choice

In the following questions two equations (I) and (II) are given. You have to solve both the equation and give answer- ( (I) ) ( 4y^2 + 24y + 35 = 0 ) ( (II) ) ( 4x^2 – 25x + 39 = 0 )

  1. If x ≥ y

  2. If x ≤ y

  3. If x > y

  4. If x < y

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

Solving equation (I) for y: 4y² + 24y + 35 = 0. Using quadratic formula: y = (-24 ± √(576 - 560))/8 = (-24 ± √16)/8 = (-24 ± 4)/8. So y = -20/8 = -5/2 or y = -28/8 = -7/2. Solving equation (II) for x: 4x² - 25x + 39 = 0. Factorizing: (4x-13)(x-3) = 0, so x = 13/4 or x = 3. Comparing: x values are 3.25 and 3, while y values are -2.5 and -3.5. Both x values are greater than both y values, so x > y.

Multiple choice

$(\text{In the following questions two equations (I) and (II) are given. You have to solve both the equation and give answer-})$ $((I) \quad x^2 + 17x + 72 = 0\(II) \quad y^2 + 19y + 90 = 0)$

  1. If x ≥ y

  2. If x ≤ y

  3. If x > y

  4. If x < y

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

Solving both equations gives x = -8 or -9, and y = -9 or -10. In all four possible pairings, x is either equal to y or greater than y, so x ≥ y is always true. Option B (x ≤ y) is false because x is never less than y.

Multiple choice

A and B are the roots of the equation x2 – 13x + N = 0. If A – B = 3, what is the value of N? A और B समीकरण x2 - 13x + N = 0 के मूल हैं । यदि A - B = 3, N का मान क्या है?

  1. 40

  2. 27

  3. 36

  4. 45
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using Vieta's formulas: A + B = 13, AB = N. Also A - B = 3. Squaring: (A+B)² = A² + 2AB + B² = 169, (A-B)² = A² - 2AB + B² = 9. Subtracting: 4AB = 160 → AB = 40. Therefore N = 40.

Multiple choice

In each of these questions, two equations are given. You have to solve these equations and find out the values of X and Y and give answers. I. (3x^2 + 2x - 21 = 0) II. (3y^2 - 8y - 16 = 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: 3x² + 2x - 21 = (3x-7)(x+3) = 0, so x = -3 or 7/3. Equation II: 3y² - 8y - 16 = (3y+4)(y-4) = 0, so y = -4/3 or 4. Comparing values: when x=-3, xy for y=-4/3. Since the relationship varies, no definite relation exists.