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

$( \text{In each of the following question, two equation (I) and (II) is given. You have to solve them and answer the question.} )$ $( (I) 2x^2 + 11x - 63 = 0 )( \qquad (II) y^2 - 2y - 63 = 0 )$

  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

Solve equation (I): 2x² + 11x - 63 = 0 factors to (2x - 7)(x + 9) = 0. So x = 7/2 = 3.5 or x = -9. Solve equation (II): y² - 2y - 63 = 0 factors to (y - 9)(y + 7) = 0. So y = 9 or y = -7. Comparing all combinations: when x = 3.5, y could be 9 (x < y) or -7 (x > y). When x = -9, y could be 9 (x < y) or -7 (x < y). Since the relationship varies (sometimes x < y, sometimes x > y), it cannot be established, confirming option E.

Multiple choice

A and B attempted to solve a quadratic equation. A made a mistake in writing down the constant term and ended up in roots (4, 3). B made a mistake in writing down the coefficient of x and get roots (3, 2). The correct roots of the equation are- A और B एक द्विघात समीकरण को हल करने का प्रयास करते हैं। A अचर पद को त्रुटिवश गलत लिख देता है और मूल (4, 3) प्राप्त करता है। B त्रुटिवश X के गुणांक को गलत लिख देता है और मूल (3, 2) प्राप्त करता है। समीकरण के सही मूल है-

  1. -4, -3

  2. 6, 1

  3. 4, 3

  4. -6, -1
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A's wrong constant gives (x-4)(x-3) = x²-7x+12, so coefficient of x (-7) is correct. B's wrong coefficient gives (x-3)(x-2) = x²-5x+6, so constant (6) is correct. Correct equation: x²-7x+6=0. Roots: (7±√25)/2 = 6, 1. Option C (4,3) is A's wrong answer, A and D are sign errors.

Multiple choice

sin a and cosα are the roots of the equation x2-abx+c =0 then/ यदि sinα और cosα समीकरण x2-abx+c=0 के मूल हैं तो -

  1. a2 c2 -2b-1=0

  2. a2 -ac+b2 =0

  3. a2 b2 -2c-1=0

  4. a2 -ab+ac=0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If sin α and cos α are roots of x² - abx + c = 0, then by Vieta's formulas: sin α + cos α = ab and sin α × cos α = c. Using sin²α + cos²α = 1, we have (sin α + cos α)² = sin²α + cos²α + 2sin α cos α, so (ab)² = 1 + 2c, which gives a²b² - 2c - 1 = 0. Option C matches this.

Multiple choice

In the following questions two equations numbered I and II are given. You have to solve both equations and give answer. $( (I) \quad p^2 + q^2 = 12.50 ) ( (II) \quad p + q = 5 )$

  1. p = q or relation can't be established.

  2. p > q

  3. q > p

  4. p ≥ q

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

Solving the system: From p+q=5, q=5-p. Substituting in p^2+q^2=12.50 gives 2p^2-10p+12.50=0, which simplifies to (p-2.5)^2=0, so p=2.5 and q=2.5. Therefore p = q.

Multiple choice

In each of the following questions, two equations are given. You have to solve them and ( I. ) ( x^2 = (38)^2 \times 5^2 ) ( II. ) ( y^2 = (38 \times 5)^2 ) 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 not be established
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Both equations simplify to the same value: x² = (38)² × 5² = 190² and y² = (38×5)² = 190², so x² = y². This means x could equal y, or x could equal -y. Without additional information about whether x and y are positive or negative, we cannot determine the exact relationship.

Multiple choice

$I. ( (x+y)^2 = 7744 )\nII. ( (x-y)^2 = 7921 )$ Solve the equations and give the answer.

  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

From (x-y)^2 = 7921, we get x-y = ±89. From (x+y)^2 = 7744, we get x+y = ±88. Since x+y must be positive, x+y = 88. If x-y = 89, then adding: 2x = 177, x = 88.5, y = -0.5, so x > y. If x-y = -89, then x = -0.5, y = 88.5, but x+y = 88 is satisfied, yet x < y. However, taking positive roots: x+y = 88, x-y = 89 gives x > y.

Multiple choice

In each of the following question, equations are given. You have to solve them and state the correct relationship- (I. P^2 – 6P – 16 = 0 ) (II. Q^2 + 6Q – 16 = 0)

  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

Solve equation I: P^2 - 6P - 16 = 0 gives P = 8 or P = -2. Solve equation II: Q^2 + 6Q - 16 = 0 gives Q = 2 or Q = -8. Comparing all possible pairs: if P=8 and Q=2, then P>Q; if P=-2 and Q=2, then PQ. Since the relationship changes based on which values we pick, it cannot be established.

Multiple choice

In each of the following question, equations are given. You have to solve them and state the correct relationship- I. (P^2 + 2P + 1 = 0) II. (Q^2 = 4)

  1. If P ≥ Q

  2. If P > Q

  3. If P < Q

  4. If P ≤ Q

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

From equation I: P² + 2P + 1 = 0 means (P + 1)² = 0, so P = -1. From equation II: Q² = 4, so Q = ±2. When Q = 2, P < Q. When Q = -2, P > Q. Since we cannot establish a consistent relationship (P could be less than or greater than Q depending on which value of Q we choose), option E is correct.

Multiple choice

In the following question two equations are given. You have to solve them and give answer - ( (I) ) ( p^2 = 36 ) ( (II) ) ( 2q^2 + 20q + 50 = 0 )

  1. If p < q

  2. If p > q

  3. If p ≤ q

  4. If p ≥ q

  5. If p = q or relation can't be established.
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Equation (I): p² = 36 gives p = ±6. Equation (II): 2q² + 20q + 50 = 0 can be simplified to q² + 10q + 25 = 0, which factors as (q+5)² = 0, giving q = -5 (repeated root). Comparing p and q: If p = 6, then p > q. If p = -6, then p < q. Since p can be greater or less than q depending on which root we take, the relation cannot be uniquely established.

Multiple choice

(\text{In the following question two equations are given. You have to solve them and give answer -}\ (\text{(I)}) \quad p^2 + p - 42 = 0\ (\text{(II)}) \quad q^2 + 11q + 30 = 0

  1. If p < q

  2. If p > q

  3. If p ≤ q

  4. If p ≥ q

  5. If p = q or relation can't be established.
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Equation (I): p² + p - 42 = 0 factors as (p+7)(p-6) = 0, giving p = -7 or p = 6. Equation (II): q² + 11q + 30 = 0 factors as (q+5)(q+6) = 0, giving q = -5 or q = -6. Comparing all values: When p = -7, p < q (for both q = -5, -6). When p = 6, p > q (for both q = -5, -6). Since p can be less or greater than q, the relation cannot be uniquely established.

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. (3x^{2} - 19x = 14) II. (2y^{2} + 5y - 3 = 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
E Correct answer
Explanation

Equation I: 3x² - 19x - 14 = 0 factors to (3x + 2)(x - 7) = 0, so x = 7 or x = -2/3. Equation II: 2y² + 5y - 3 = 0 factors to (2y - 1)(y + 3) = 0, so y = 1/2 or y = -3. Comparing values: x = 7 > y = -3, x = 7 > y = 1/2, x = -2/3 < y = 1/2, x = -2/3 > y = -3. No consistent relation exists.

Multiple choice

In the following questions two equations I and II are given. Solve both the equations and give answer $( (I) \quad 2x + 3y = 11 )$ $( (II) \quad 4x + 5y = 19 )$
  1. $\( x > y \)$
  2. $\( x \ge y \)$
  3. $\( x < y \)$
  4. $\( x \le y \)$

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

Solving the system: Multiply (I) by 2: 4x+6y=22. Subtract from (II): (4x+5y)-(4x+6y)=19-22 gives -y=-3, so y=3. Substituting in (I): 2x+9=11 gives 2x=2, x=1. Since x=1 < y=3, option C is correct.