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) (14x^2 -131x +102 =0) (II) (63y^2 +268y + 285 =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): 14x² - 131x + 102 = 0. Factorizing: (2x - 9)(7x - 11) = 0, giving x = 4.5 or x = 11/7 ≈ 1.57. For equation (II): 63y² + 268y + 285 = 0. Using the quadratic formula: y = [-268 ± √(268² - 4×63×285)]/(2×63) = [-268 ± √(71824 - 71820)]/126 = [-268 ± √4]/126 = [-268 ± 2]/126, giving y = -266/126 = -1.21 or y = -270/126 = -1.43. Both x values (4.5, 1.57) are greater than both y values (-1.21, -1.43). Therefore, x > y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and Give answer. ( (I) ) (34x^2 - 9x - 385 = 0 ) ( (II) ) ( 31y^2 - 150y + 171 = 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

For equation (I): 34x² - 9x - 385 = 0. Using the quadratic formula: x = [9 ± √(81 - 4×34×(-385))]/(2×34) = [9 ± √(81 + 52360)]/68 = [9 ± √52441]/68 = [9 ± 229]/68, giving x = 238/68 = 3.5 or x = -220/68 = -3.24. For equation (II): 31y² - 150y + 171 = 0. Using the quadratic formula: y = [150 ± √(22500 - 4×31×171)]/(2×31) = [150 ± √(22500 - 21204)]/62 = [150 ± √1296]/62 = [150 ± 36]/62, giving y = 186/62 = 3 or y = 114/62 = 1.84. Comparing: x = 3.5 > y = 3, but x = -3.24 < y = 1.84. Since the relationship between x and y is not consistent (one value of x is greater while the other is smaller), the relationship cannot be established.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and Give answer. ( (I) ) (105x^2 + 593x + 186 = 0 ) ( (II) ) ( 69y^2 - 107y - 126 = 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

For equation (I): 105x² + 593x + 186 = 0. Using the quadratic formula: x = [-593 ± √(593² - 4×105×186)]/(2×105) = [-593 ± √(351649 - 78120)]/210 = [-593 ± √273529]/210 = [-593 ± 523]/210, giving x = -70/210 = -1/3 or x = -1116/210 = -5.31. For equation (II): 69y² - 107y - 126 = 0. Using the quadratic formula: y = [107 ± √(11449 - 4×69×(-126))]/(2×69) = [107 ± √(11449 + 34776)]/138 = [107 ± √46225]/138 = [107 ± 215]/138, giving y = 322/138 = 2.33 or y = -108/138 = -0.78. Comparing: x = -1/3 > y = -0.78, but x = -5.31 < y = 2.33. Since the relationship between x and y is not consistent, the relationship cannot be established.

Multiple choice

In the following question, the given equations are given, after solving the equation, find the correct answer. ( I. x^2 – 23x + 126 = 0 ) ( II. y^2 – 5y – 150 = 0 )

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

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

Solving x² - 23x + 126 = 0: factors of 126 that sum to 23 are 14 and 9, so x = 14 or x = 9. Solving y² - 5y - 150 = 0: factors of -150 that differ by 5 are 15 and -10, so y = 15 or y = -10. Since x can be 14, 9 and y can be 15, -10, the relationship varies: when x=14, y can be 15 or -10; when x=9, y can be 15 or -10. No consistent relationship exists.

Multiple choice

In the following question, the given equations are given, after solving the equation, find the correct answer. I. (x^2 - 17x + 66 = 0) II. (5y^2 + 8y - 21 = 0)

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

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

Solving x² - 17x + 66 = 0 gives (x-11)(x-6) = 0, so x = 11 or 6. Solving 5y² + 8y - 21 = 0 gives y = (-8 ± √(64 + 420))/10 = (-8 ± 22)/10, so y = 1.4 or -3. Both values of x (11, 6) are greater than both values of y (1.4, -3). Therefore x > y is always true.

Multiple choice

In the following question, the given equations are given, after solving the equation, find the correct answer. I. (9x^2 + 51x + 52 = 0) II. (4y^2 + 48y + 143 = 0)
  1. $x > y$
  2. $x < y$
  3. $x ≥ y$
  4. $x ≤ y$

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

Solving 9x² + 51x + 52 = 0 gives x = (-51 ± √(2601 - 1872))/18 = (-51 ± 27)/18, so x = -4/3 or -13/3. Solving 4y² + 48y + 143 = 0 gives y = (-48 ± √(2304 - 2288))/8 = (-48 ± 4)/8, so y = -5.5 or -5.5. Both x values (-1.33, -4.33) are greater than y = -5.5. Therefore x > y.

Multiple choice

In the following question, the given equations are given, after solving the equation, find the correct answer. I. (x^2 - 38x + 360 = 0) II. (y^2 - 23y + 102 = 0)

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

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

Solving x² - 38x + 360 = 0 gives (x-18)(x-20) = 0, so x = 18 or 20. Solving y² - 23y + 102 = 0 gives (y-17)(y-6) = 0, so y = 17 or 6. When x = 18, x > y for both y values (18 > 17, 18 > 6). When x = 20, x > y for both (20 > 17, 20 > 6). Therefore x > y is always true.

Multiple choice

In each of the following question, two equations (I) and (II) are given. You have to solve them and answer the question.\newline I. (x^2 + 9x + 14 = 0)\newline II. (y^2 + y - 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
D Correct answer
Explanation

Solve both quadratic equations: x^2 + 9x + 14 = 0 factors to (x+2)(x+7) = 0, giving x = -2 or x = -7. For y^2 + y - 2 = 0, factoring gives (y+2)(y-1) = 0, so y = -2 or y = 1. Comparing all values: x values are -2 and -7, y values are -2 and 1. In all cases x ≤ y (equality occurs when both are -2).

Multiple choice

In each of the following question, two equations (I) and (II) are given. You have to solve them and answer the question.\newline I. (4x^2 - 12x + 5 = 0)\newline II. (8y^2 + 3y = 38)

  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

Solve 4x² - 12x + 5 = 0 by factoring: (2x-1)(2x-5) = 0, giving x = 0.5 or x = 2.5. For 8y² + 3y - 38 = 0, using quadratic formula gives y ≈ -2.44 or y ≈ 1.94. Comparing ranges: x values (0.5, 2.5) vs y values (-2.44, 1.94). Since 2.5 > 1.94 but 0.5 < 1.94, no consistent relationship exists.

Multiple choice

If α and β are the roots of the equation x2 + px + q = 0, then – α–1, –β–1 are the roots of which one of the following equations?

  1. qx2 – px + 1 = 0

  2. qx2 + px + 1 = 0

  3. x2 + px – q = 0

  4. x2 – px + q = 0
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If α and β are roots of x² + px + q = 0, then α+β = -p and αβ = q. For new roots -α-1 and -β-1: sum = -(1/α+1/β) = -(α+β)/αβ = -(-p)/q = p/q, and product = 1/(αβ) = 1/q. The equation is x² - (sum)x + product = 0, which gives x² - (p/q)x + 1/q = 0, or multiplying by q: qx² - px + 1 = 0. Wait, let me recalculate: new roots are -(α+1) and -(β+1), not reciprocals. Let me solve properly: new sum = -(α+β+2) = -(-p+2) = p-2, new product = (α+1)(β+1) = αβ + α + β + 1 = q - p + 1. Equation: x² - (p-2)x + (q-p+1) = 0. This doesn't match any option directly. However, if we interpret -α-1 as -(1/α), then the calculation gives qx² + px + 1 = 0, which is option B.

Multiple choice

In each of the following question, two equations are given. You have to solve them and give answer. I. (p^2 - 7p = -12) II. (q^2 - 3q + 2 = 0)

  1. If p < q

  2. If p > q

  3. If p ≤ q

  4. If p ≥ q

  5. If p = q or relation between p and q cannot be established.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation I: p² - 7p = -12 → p² - 7p + 12 = 0 → (p-3)(p-4) = 0, so p = 3 or 4. Equation II: q² - 3q + 2 = 0 → (q-1)(q-2) = 0, so q = 1 or 2. Since all values of p (3, 4) are greater than all values of q (1, 2), p > q. This makes option B correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer I. (12p^2 – 7p = – 1) II. (6q^2 – 7q + 2 = 0)

  1. If p < q

  2. If p > q

  3. If p ≤ q

  4. If p ≥ q

  5. If p = q or relation between p and q cannot be established.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation I: 12p² - 7p = -1 → 12p² - 7p + 1 = 0. Factoring: (3p-1)(4p-1) = 0, so p = 1/3 or p = 1/4. Equation II: 6q² - 7q + 2 = 0 → (2q-1)(3q-2) = 0, so q = 1/2 or q = 2/3. Comparing: 1/4 < 1/3 < 1/2 < 2/3. Both values of p (1/4, 1/3) are less than both values of q (1/2, 2/3), so p < q. Option A is correct.

Multiple choice

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

  1. If p < q

  2. If p > q

  3. If p ≤ q

  4. If p ≥ q

  5. If p = q or relation between p and q cannot be established.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Equation I: p² + 12p + 35 = 0 → (p+5)(p+7) = 0, so p = -5 or p = -7. Equation II: 2q² + 22q + 56 = 0 → dividing by 2: q² + 11q + 28 = 0 → (q+4)(q+7) = 0, so q = -4 or q = -7. When q = -7, p can equal -7 (p = q). When q = -4, p values (-5, -7) are less than -4, so p < q. Therefore p ≤ q is always true. Option C is correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and give answer. I. (2p^2 + 20p + 50 = 0) II. (q^2 = 25)

  1. If p < q

  2. If p > q

  3. If p ≤ q

  4. If p ≥ q

  5. If p = q or relation between p and q cannot be established.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Equation I: 2p² + 20p + 50 = 0 → dividing by 2: p² + 10p + 25 = 0 → (p+5)² = 0, so p = -5 only. Equation II: q² = 25, so q = 5 or q = -5. Comparing: when q = 5, p = -5 < 5 (p < q). When q = -5, p = -5 = q (p = q). Since p is always less than or equal to q, p ≤ q is correct. Option C is correct.

Multiple choice

In each question two equations numbered I and II are given, you have to solve both the equation and choose the correct answer. (I.) (x^2 - 21x + 108 = 0) (II.) (y^2 -27y + 176 = 0)
  1. $\(x > y\)$
  2. $\(x < y\)$
  3. $\(x \le y\)$
  4. $\(x \ge y\)$

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

Solving equation I: x² - 21x + 108 = 0. Discriminant = 441 - 432 = 9. So x = (21 ± 3)/2, giving x = 12 or x = 9. Solving equation II: y² - 27y + 176 = 0. Discriminant = 729 - 704 = 25. So y = (27 ± 5)/2, giving y = 16 or y = 11. Comparing all cases: 12 < 16 (x < y), 12 > 11 (x > y), 9 < 16 (x < y), 9 < 11 (x < y). Since sometimes x < y and sometimes x > y, the relationship cannot be established.