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

Solve the following equations and find the values of (x) and (y): I. (17x^2 + 48x = 9) II. (13y^2 = 32y - 12)

  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

Solve equation I: 17x² + 48x - 9 = 0. Using quadratic formula, x = [-48 ± √(2304 + 612)]/34 = [-48 ± √2916]/34 = [-48 ± 54]/34. This gives x = 6/34 = 3/17 ≈ 0.176 or x = -102/34 = -3. Solve equation II: 13y² - 32y + 12 = 0. Using quadratic formula, y = [32 ± √(1024 - 624)]/26 = [32 ± √400]/26 = [32 ± 20]/26. This gives y = 52/26 = 2 or y = 12/26 = 6/13 ≈ 0.462. Comparing all values, both positive solutions (3/17 ≈ 0.176 and 6/13 ≈ 0.462) give x < y, and the negative solution x = -3 is also less than both positive y values. Therefore x < y always holds.

Multiple choice

In the following questions two equations numbered I and II are given. You have to solve both the equations and ————— I. (x^{2} - 218 = 106) II. (y^{2} - 37y + 342 = 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
D Correct answer
Explanation

Equation I: x² - 218 = 106, so x² = 324, which means x = ±18. Equation II: y² - 37y + 342 = 0. Using quadratic formula: y = [37 ± √(1369 - 1368)] ÷ 2 = [37 ± √1] ÷ 2 = [37 ± 1] ÷ 2. So y = 38/2 = 19, or y = 36/2 = 18. Comparing: x can be 18 or -18. y can be 19 or 18. If x = -18, then x < y (since -18 < 18 and -18 < 19). If x = 18, then x ≤ y (since 18 = 18 when y = 18, and 18 < 19 when y = 19). In all cases, x ≤ y is true. Option D is correct.

Multiple choice

In each of the following questions, two equations are given you have to solve them and. (i) (x^2 - 6x - 16 = 0) (ii) (y^2 + 6y - 16 = 0) Give answer.

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solve both equations: x²-6x-16=0 gives x=(6±√(36+64))/2=(6±10)/2, so x=8 or x=-2. y²+6y-16=0 gives y=(-6±√(36+64))/2=(-6±10)/2, so y=2 or y=-8. Since x can be 8 (greater than both y values) or -2 (greater than -8 but less than 2), and y can be 2 or -8, no consistent relationship exists.

Multiple choice

In each of the following questions, two equations are given you have to solve them and. (i) (x^2 + 21x - 46 = 0) (ii) (y^2 + 4y - 81 = 0) Give answer

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solve both equations: x²+21x-46=0 gives x=(-21±√(441+184))/2=(-21±√625)/2=(-21±25)/2, so x=2 or x=-23. y²+4y-81=0 gives y=(-4±√(16+324))/2=(-4±√340)/2=(-4±18.44)/2, so y≈7.22 or y≈-11.22. Since x can be 2 (less than 7.22 but greater than -11.22) or -23 (less than both y values), and y has irrational values, no clear relationship can be established.

Multiple choice

Solve the following equations and determine the relationship between (x) and (y): (i) (3x^2 - 16x + 21 = 0) (ii) (6y^2 + 5y + 21 = 0)

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x ≤ y

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

Solve both equations: 3x²-16x+21=0 gives x=(16±√(256-252))/6=(16±2)/6, so x=3 or x≈2.33. For 6y²+5y+21=0, the discriminant is 25-504=-479, which is negative, meaning y has no real solutions (it's imaginary). Therefore, all real x values are greater than imaginary y values, so x > y. The claimed answer (A) is correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and $( (i) 3x^2 - 16x + 21 = 0 )\n( (ii) 6y^2 + 5y + 21 = 0 )$

  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

Solving 3x²-16x+21=0 gives x=7/3≈2.33 and x=3. The equation 6y²+5y+21=0 has discriminant 25-504=-479 < 0, so y has no real values. Since all real x values are greater than non-existent real y values, x>y.

Multiple choice

In each of the following questions, two equations (I) and (II) are given. You have to solve them and give answer. $(I) (5x + 7y = 35) (II) (10x + 14y = 42)$

  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

Equation I: 5x + 7y = 35. If we multiply by 2, we get 10x + 14y = 70. Equation II: 10x + 14y = 42. This is impossible - the same expression cannot equal both 70 and 42. The system is inconsistent with no solution, so relationship between x and y cannot be established.

Multiple choice

In the following questions two equations I and II are given. You have to solve both the equations and give answer. $( (I) \quad x^2 + y^2 = 45 \qquad (II) \quad x + y = 9 )$

  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 determined
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

From (II), y = 9 - x. Substitute into (I): x² + (9-x)² = 45 gives x² + 81 - 18x + x² = 45, so 2x² - 18x + 36 = 0, which simplifies to x² - 9x + 18 = 0. This gives x = 3 or x = 6. When x = 3, y = 6. When x = 6, y = 3. Since x can be greater than, less than, or equal to y depending on which solution we take, the relationship cannot be uniquely determined.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer. (I) (10x^2 + 19x + 9 = 0) (II) (13y^2 - 2y - 11 = 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

Solving equation (I): 10x²+19x+9=0 factors as (2x+9)(5x+1)=0, giving x=-9/2=-4.5 or x=-1/5=-0.2. Solving equation (II): 13y²-2y-11=0 factors as (13y+11)(y-1)=0, giving y=-11/13≈-0.85 or y=1. Comparing values: when x=-4.5 or x=-0.2, and y=1, we have x

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer. (I) (2x^2 + 22x + 56 = 0) (II) (y^2 + 12y + 35 = 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

Solving equation (I): 2x²+22x+56=0 simplifies to x²+11x+28=0, factoring as (x+4)(x+7)=0, giving x=-4 or x=-7. Solving equation (II): y²+12y+35=0 factors as (y+5)(y+7)=0, giving y=-5 or y=-7. When x=-7 and y=-7, x=y. When x=-4 and y=-5, x>y. When x=-4 and y=-7, x>y. When x=-7 and y=-5, xy, sometimes x

Multiple choice

$(\text{In each of the following questions two equations are given. You have to solve both the equation and give answer -} )$ $((I) \space x^2 + 7x + 10 = 0 \qquad (II) \space y^2 + 11x + 18 = 0)$

  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
E Correct answer
Explanation

Equation (I) factors to (x+2)(x+5) = 0, so x = -2 or x = -5. Equation (II) is written as y² + 11x + 18 = 0, which contains x instead of y - this is likely a typo for y² + 11y + 18 = 0. With the typo, equation (II) has no real solutions (discriminant would involve x). The relationship cannot be established.

Multiple choice

In each of the following questions two equations are given. You have to solve both the equation and give answer - $( (I) \ 4x + 3y = 16 )( (II) \ 2x + 2y = 9 )$

  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 the system: 4x + 3y = 16 and 2x + 2y = 9. Multiply second equation by 2: 4x + 4y = 18. Subtract first equation: (4x+4y) - (4x+3y) = 18 - 16, so y = 2. Substitute: 4x + 3(2) = 16, so 4x = 10, x = 2.5. Since x = 2.5 > y = 2, the answer is x > y.

Multiple choice

In each of the following questions two equations are given. You have to solve both the equation and give answer - ( (I) 2x + y = 12 )( (II) 3y + 5y - 32 = 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
E Correct answer
Explanation

Equation (II) is '3y + 5y - 32 = 0' which simplifies to 8y = 32, giving y = 4. Substituting into equation (I): 2x + 4 = 12, so 2x = 8 and x = 4. Therefore x = y = 4, meaning x = y. However, equation (II) appears to have a typo - it should likely involve two variables.

Multiple choice

In the following questions two equations numbered (I) and (II) are given. You have to solve both equations and give answer— $(I) (16x^2 + 20x + 6 = 0) (II) (10y^2 + 38y + 24 = 0)$
  1. $\(x < y\)$
  2. $\(x > y\)$
  3. $\(x \leq y\)$
  4. $\(x \geq y\)$

  5. x=y or relation cannot be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving equation (I): 16x² + 20x + 6 = 0 gives x = -0.5 or x = -0.75. Solving equation (II): 10y² + 38y + 24 = 0 gives y = -0.8 or y = -3. Both values of x (-0.5, -0.75) are greater than both values of y (-0.8, -3), so x > y is always true. Option C (x ≤ y) is false because x is never less than or equal to y.

Multiple choice

In the following questions two equations numbered (I) and (II) are given. You have to solve both equations and give answer— $(I) (8x^2 + 6x = 5) (II) (12y^2 - 22y + 8 = 0)$

  1. if x < y

  2. if x > y

  3. if x ≤ y

  4. if x ≥ y

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

Solving equation (I): 8x² + 6x - 5 = 0 gives x = 1/2 or x = -5/4. Solving equation (II): 12y² - 22y + 8 = 0 gives y = 7/6 or y = 2/3. When x = 0.5, we have x < 1.167 but x > 0.667, so the relationship between x and y varies. No single inequality consistently holds.