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

What is the respective ratio of negative value of sum of squares of roots to the product of the roots of the equation 7x2 + 12x + 18 = 0 ? समीकरण 7x2 + 12x + 18 = 0 के मूलों के वर्गो के योगफल के ऋणात्मक का मूलों के गुणनफल से क्रमशः क्या अनुपात है?

  1. 6 : 1

  2. 1 : 6

  3. 6 : 7

  4. 7 : 6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For ax² + bx + c = 0: sum of roots = -b/a, product = c/a. Sum of squares = (sum)² - 2(product) = (b²/a²) - (2c/a). For 7x² + 12x + 18 = 0: sum = -12/7, product = 18/7. Sum of squares = (144/49) - (36/7) = (144-252)/49 = -108/49. Negative value of sum of squares = 108/49. Ratio = (108/49) / (18/7) = (108/49) × (7/18) = 6/7. Option A incorrectly uses reciprocal. Option B inverts the correct ratio. Option D swaps numerator and denominator.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and (I) (2x^2 - 21x + 55 = 0) (II) (5y^2 - 17y + 14 = 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
A Correct answer
Explanation

For 2x² - 21x + 55 = 0: using quadratic formula, x = (21 ± 1)/4, so x = 5 or 5.5. For 5y² - 17y + 14 = 0: y = (17 ± 3)/10, so y = 1.4 or 2. In all four combinations, x values (5, 5.5) are always greater than y values (1.4, 2). Therefore x > y is consistently true, making option A correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and\newline 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 relationship can not be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation I: x⁴ - 227 = 398 → x⁴ = 625 → x = ±5. Equation II: y² + 321 = 346 → y² = 25 → y = ±5. Both equations give |x| = 5 and |y| = 5. Since x could be +5 or -5, and y could be +5 or -5, we have |x| = |y|, meaning x ≥ y (x can equal y or be greater). Option B (x ≥ y) is correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answers- $(I) P = \sqrt{x} + 90 \quad (II) Q = 12$

  1. If P < Q

  2. If P ≤ Q

  3. If P > Q

  4. If P ≥ Q

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

Equation (I) states P = √x + 90, but x is not specified. If x is negative, P is undefined. If x = 0, P = 90. If x > 0, P > 90. Equation (II) states Q = 12. Since the value of x is unknown and P can be undefined or greater than 12 depending on x, the relationship between P and Q cannot be established. Therefore, option E is correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer $( (I) \space x^{2} + 9x + 14 = 0 \space\space\space\space\space\space (II) \space y^{2} + y -2 = 0 )$
  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

Solving equation (I): x² + 9x + 14 = 0 factors as (x + 7)(x + 2) = 0, giving x = -7 or x = -2. Solving equation (II): y² + y - 2 = 0 factors as (y + 2)(y - 1) = 0, giving y = -2 or y = 1. Comparing values: when x = -7, x < both y values (-7 < -2 and -7 < 1). When x = -2, x equals y = -2 and x < y = 1. Therefore x ≤ y is always true (x can equal y when both are -2, or x is less than y in all other cases).

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer- (I) 21x2 + 5x – 4 = 0 (II) 9y2 + 18y = – 8

  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

Solving (I): 21x² + 5x - 4 = 0 gives x = -4/7 or x = 4/3. Solving (II): 9y² + 18y + 8 = 0 gives y = -4/3 (repeated root). When x = 4/3 (positive) and y = -4/3 (negative), we have x > y. The positive root of x is greater than the negative root of y.

Multiple choice

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

  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

Solving (I): x² + 3x + 2 = 0 gives x = -2 or x = -1. Solving (II): 2y² - 5y = 0 gives y(2y - 5) = 0, so y = 0 or y = 5/2. Comparing: -2 < 0, -2 < 2.5, -1 < 0, -1 < 2.5. All possible x values are less than all possible y values, so x < y.

Multiple choice

In each of the following questions two equations I and II are given. You have to solve the equations and then answer. $I. (x^2 - 14x + 48 = 0) II. (y^2 + 6 = 5y)$

  1. If x < y

  2. If x ≤ y

  3. If x > y

  4. If x ≥ y

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

Solve I: x^2 - 14x + 48 = 0. This factors as (x-6)(x-8) = 0, so x = 6 or x = 8. Solve II: y^2 + 6 = 5y, rearranging gives y^2 - 5y + 6 = 0, which factors as (y-2)(y-3) = 0, so y = 2 or y = 3. Comparing all values: x can be 6 or 8, y can be 2 or 3. In all cases, x > y (6>2, 6>3, 8>2, 8>3). Therefore option C is correct.

Multiple choice

In each of the following questions two equations I and II are given. You have to solve both the equations and then answer. $I. (3x + 5y = 28) II. (8x - 3y = 42)$

  1. If x < y

  2. If x ≤ y

  3. If x > y

  4. If x ≥ y

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

Solving the system: 3x + 5y = 28 and 8x - 3y = 42. Using elimination: Multiply first equation by 3: 9x + 15y = 84. Multiply second by 5: 40x - 15y = 210. Add: 49x = 294, so x = 6. Substitute: 3(6) + 5y = 28, so 18 + 5y = 28, giving 5y = 10 and y = 2. Since x = 6 and y = 2, we have 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 answer the question.( \text{(I)} \quad 6x^2 - 11x + 4 = 0 ) ( \text{(II)} \quad 30y^2 - 23y + 3 = 0 )

  1. If x>y

  2. If x≥y

  3. If x<y

  4. If x≤y

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

Equation (I): 6x² - 11x + 4 = 0 factors as (2x-1)(3x-4) = 0, giving x = 1/2 or 4/3. Equation (II): 30y² - 23y + 3 = 0 factors as (5y-3)(6y-1) = 0, giving y = 3/5 or 1/6. Since x values (0.5, 1.33) and y values (0.6, 0.17) overlap in range, no definite relationship exists.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. $( (I) \quad 7x^2 + 10x + 3 = 0 )$ $( (II) \quad 2y^2 + 11y + 14 = 0 )$

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Equation (I): 7x² + 10x + 3 = 0 factors as (7x+3)(x+1) = 0, giving x = -3/7 or -1. Equation (II): 2y² + 11y + 14 = 0 factors as (2y+7)(y+2) = 0, giving y = -7/2 or -2. Comparing values: x ∈ {-0.43, -1} and y ∈ {-3.5, -2}. Both x values are greater than both y values, so x > y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. $( (I) \quad 2x^2 – 15 = 7x \\ (II) \quad 17y = – 7 – 6y^2 )$

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Equation (I): 2x² - 7x - 15 = 0 factors as (2x+3)(x-5) = 0, giving x = -3/2 or 5. Equation (II): 6y² + 17y + 7 = 0 factors as (2y+1)(3y+7) = 0, giving y = -1/2 or -7/3. Comparing: x ∈ {-1.5, 5} and y ∈ {-0.5, -2.33}. The ranges overlap, so relationship cannot be established.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. ( (I) ) ( 6x^2 + 31x + 35 = 0 ) ( (II) ) ( 2y^2 + 3y + 1 = 0 )

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Equation (I): 6x² + 31x + 35 = 0 factors as (2x+5)(3x+7) = 0, giving x = -5/2 or -7/3. Equation (II): 2y² + 3y + 1 = 0 factors as (2y+1)(y+1) = 0, giving y = -1/2 or -1. Numerically: x ∈ {-2.5, -2.33} and y ∈ {-0.5, -1}. Both x values are less than both y values, so x < y.

Multiple choice

In each of the following questions, two equations (I) and (II) are given. You have to solve them and answer the questions. $I. (9x^2 - 45x + 56 = 0)$ $II. (4y^2 - 7y - 2 = 0)$

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

For equation I: 9x² - 45x + 56 = 0 factors to (3x - 8)(3x - 7) = 0, giving x = 8/3 or x = 7/3. For equation II: 4y² - 7y - 2 = 0 factors to (4y + 1)(y - 2) = 0, giving y = 2 or y = -1/4. Comparing both roots of x (2.67 and 2.33) with both roots of y, both x values are greater than both y values, so x > y.

Multiple choice

In each of the following questions, two equations (I) and (II) are given. You have to solve them and answer the question. $( (I) \ 3x^2 - 4x - 32 = 0 \\ (II) \ 2y^2 - 17y + 36 = 0 )$

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solving equation (I): 3x²-4x-32=0. Using quadratic formula: x=(4±√(16+384))/6=(4±√400)/6=(4±20)/6, giving x=24/6=4 or x=-16/6=-8/3≈-2.67. Solving equation (II): 2y²-17y+36=0 factors to (y-9)(2y-4)=0, giving y=9 or y=2. Comparing: when x=4, y=2 gives x>y; y=9 gives x2 is false, but 4<9 and -2.67<9 and -2.67<2 are all true), the correct answer is x≤y.