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 the following questions, there are two equations. Solve the equations and answer accordingly. (I. p^2 – 5.8p + 8.41 = 0 ) (II. q^2 - 3.8q + 3.12 = 0)

  1. p < q

  2. p > q

  3. p = q or relationship cannot be established

  4. p ≤ q

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

The first equation factors to (p-2.9)² = 0, so p = 2.9. The second equation factors to (q-1.2)(q-2.6) = 0, giving q = 1.2 or q = 2.6. In both cases, p > q, so p > q is the correct relationship.

Multiple choice

In the following questions, there are two equations. Solve the equations and answer accordingly. (I. x^{3/2} = 25^2 + 104 ) (II. y^{1/4} = 4^2 – 13)
  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
C Correct answer
Explanation

The first equation gives x^(3/2) = 625 + 104 = 729, so x = 729^(2/3) = 81. The second gives y^(1/4) = 16 - 13 = 3, so y = 3^4 = 81. Since x = y = 81, the relationship is that they are equal.

Multiple choice

In the given question two equations numbered I and II are given. You have to solve both the equations and give answer. (I. 4x^2 + x - 3 = 0 )(II. 6y^2 - 5y + 1 = 0)

  1. If x > y

  2. If x < y

  3. If x ≥ y

  4. If x ≤ y

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

Equation I: 4x² + x - 3 = 0 gives x = 3/4 or x = -1. Equation II: 6y² - 5y + 1 = 0 gives y = 1/2 or y = 1/3. Comparing all pairs: when x = -1, x < both y values. When x = 3/4, x > both y values. No single relationship holds consistently.

Multiple choice

In the given question two equations numbered I and II are given. You have to solve both the equations and give answer $(I. 12x^2 + 29x + 14 = 0 )(II. 12y^2 - 22y + 8 = 0)$

  1. If x > y

  2. If x < y

  3. If x ≥ y

  4. If x ≤ y

  5. If x = y or relation between x and y can't be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation I: 12x² + 29x + 14 = 0. Splitting 29x as 8x + 21x and factoring by grouping gives x = -2/3 or x = -7/4. Equation II: 12y² - 22y + 8 = 0. Simplifying: 6y² - 11y + 4 = 0 gives y = 1/3 or y = 2/3. For all x values, x < all y values, so x < y.

Multiple choice

In each of the following questions two equations are given. You have to solve both the equation and find out the values of P and Q and $( (I) \quad 20P^2 + 31P + 12 = 0 \ (II) \quad 21Q^2 - 23Q + 6 = 0 )$

  1. If P > Q

  2. If P < Q

  3. If P ≥ Q

  4. If P ≤ Q

  5. P = Q or the relationship can not be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving equation (I): 20P² + 31P + 12 = 0. By factorization: (5P + 4)(4P + 3) = 0, so P = -4/5 = -0.8 or P = -3/4 = -0.75. Solving equation (II): 21Q² - 23Q + 6 = 0. By factorization: (7Q - 3)(3Q - 2) = 0, so Q = 3/7 ≈ 0.429 or Q = 2/3 ≈ 0.667. Comparing all values: Both P values (-0.8, -0.75) are less than both Q values (0.429, 0.667). Therefore P < Q is correct.

Multiple choice

If α and β are the roots of the equation x2 – 6x + 6 = 0, what is α3 + β3 + α2 + β2 + α + β equal to? यदि समीकरण x2 – 6x + 6 = 0 के मूल α और β है, तो α3 + β3 + α2 + β2 + α + β किसके बराबर है?

  1. 150

  2. 138

  3. 128

  4. 124
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

From x²-6x+6=0: α+β=6, αβ=6. The expression = (α³+β³)+(α²+β²)+(α+β). Using identities: α³+β³ = (α+β)³-3αβ(α+β) = 216-108 = 108. α²+β² = (α+β)²-2αβ = 36-12 = 24. Total = 108+24+6 = 138. This tests Vieta's formulas and algebraic identities.

Multiple choice

In each of the following question, two equations are given. You have to solve them and give answer- I. (18x^2 + 18x + 4 = 0) II. (12y^2 + 29y + 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
B Correct answer
Explanation

Solving Equation I: 18x² + 18x + 4 = 0. Using quadratic formula: x = (-18 ± √(324-288)) / 36 = (-18 ± √36) / 36 = (-18 ± 6) / 36. So x = -12/36 = -1/3 or x = -24/36 = -2/3. Solving Equation II: 12y² + 29y + 14 = 0. Factoring: 12y² + 8y + 21y + 14 = 0, so 4y(3y+2) + 7(3y+2) = 0, giving (4y+7)(3y+2) = 0. Thus y = -7/4 = -1.75 or y = -2/3 ≈ -0.67. Comparing: x values are -0.33 and -0.67, y values are -1.75 and -0.67. Since x = -2/3 = y, but x also has -1/3 which is greater than both y values, we have x ≥ y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. $( (I) \quad 8x – 3y = 85 )$ $( (II) \quad 4x – 5y = 67 )$

  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

Solve the system: 8x-3y=85 and 4x-5y=67. Multiply second equation by 2: 8x-10y=134. Subtract from first: (8x-3y)-(8x-10y)=85-134, giving 7y=-49, so y=-7. Substitute: 8x-3(-7)=85 gives 8x+21=85, so 8x=64 and x=8. Since 8 > -7, x > y is correct.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. (I) (2x^2 + 13x + 21 = 0) (II) (3y^2 + 34y + 63 = 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 2x²+13x+21=0: factorizing gives (2x+7)(x+3)=0, so x = -7/2 or -3. For 3y²+34y+63=0: factorizing gives (3y+7)(y+9)=0, so y = -7/3 or -9. Comparing: x=-3.5 or -3; y≈-2.33 or -9. We have x=-3.5 < y=-2.33, x=-3.5 > y=-9, x=-3 < y=-2.33, x=-3 > y=-9. Since x is sometimes less than y and sometimes greater, the 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.(\text{(I)}) (12x^2 + 22x + 8 = 0)(\text{(II)}) (4y^2 - y - 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

For 12x²+22x+8=0: factorizing gives 2(6x²+11x+4)=0, then (2x+1)(3x+4)=0, so x = -1/2 or -4/3. For 4y²-y-3=0: factorizing gives (4y+3)(y-1)=0, so y = -3/4 or 1. Comparing: x=-0.5 or -1.33; y=-0.75 or 1. x=-0.5 > y=-0.75, x=-0.5 < y=1, x=-1.33 < y=-0.75, x=-1.33 < y=1. Since x is sometimes less than y and sometimes greater, the 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) ) ( x^2 + 4.9x + 5.44 = 0 )( (II) ) ( y^2 + 2.8y + 1.95 =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

Solve x²+4.9x+5.44=0: discriminant = 24.01-21.76=2.25, roots = (-4.9±1.5)/2 = -1.7, -3.2. Solve y²+2.8y+1.95=0: discriminant = 7.84-7.8=0.04, roots = (-2.8±0.2)/2 = -1.3, -1.5. Both x values (-1.7, -3.2) are less than both y values (-1.3, -1.5), so x < y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. $( (I) 5x + 7y = 10 \ (II) 6x – 2y =12 )$

  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

Solve 5x+7y=10 and 6x-2y=12. From second: 3x-y=6, y=3x-6. Substitute: 5x+7(3x-6)=10, 5x+21x-42=10, 26x=52, x=2. Then y=3(2)-6=0. Since x=2 > y=0, the relationship is x > y.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. $( (I) x^{2} + 18x – 943 = 0 )( (II) y^{2} + 40y + 391 = 0 )$

  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 x²+18x-943=0: discriminant = 324+3772=4096, roots = (-18±64)/2 = 23, -41. For y²+40y+391=0: discriminant = 1600-1564=36, roots = (-40±6)/2 = -17, -23. Comparing: 23 > -17, -23 but -41 < -17, -23. Since the relationship varies, it cannot be established.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and answer the question. $(I.) (35x^{2} - 53x + 20 = 0) (II.) (56y^{2} - 97y + 42 = 0)$

  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

For 35x²-53x+20=0: factors as (5x-4)(7x-5)=0, so x = 4/5 = 0.8 or x = 5/7 ≈ 0.714. For 56y²-97y+42=0: factors as (7y-6)(8y-7)=0, so y = 6/7 ≈ 0.857 or y = 7/8 = 0.875. In all cases, x values are less than y values, so x < y.

Multiple choice

In the given question two equations numbered (I) and (II) are given. You have to solve both the equations and give answer.\newline I. (4p^2 - 8p + 3 = 0)\newline II. (2q^2 - 13q + 15 = 0)

  1. If p > q

  2. If p < q

  3. If p ≥ q

  4. If p ≤ q

  5. If p = q or no relation between p and q can be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solve equation (I): 4p² - 8p + 3 = 0 factors to (2p-1)(2p-3) = 0, giving p = 1/2 or p = 3/2. Solve equation (II): 2q² - 13q + 15 = 0 factors to (2q-3)(q-5) = 0, giving q = 3/2 or q = 5. Since p can be 0.5 or 1.5, and q can be 1.5 or 5, we have p ≤ q in all cases (0.5 ≤ 1.5, 0.5 ≤ 5, 1.5 ≤ 1.5, 1.5 ≤ 5).