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

I. 5x² - 36 = 12² II. y² + 17y + 72 = 0 Solve the given quadratic equations and mark the correct option based on your answer— (a) x > y (b) x ≥ y (c) x < y (d) x ≤ y (e) x = y or no relation can be established between x and y.

  1. A

  2. B

  3. C

  4. D

  5. E
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

I: 5x^2 - 36 = 144 => 5x^2 = 180 => x^2 = 36 => x = 6, -6. II: y^2 + 17y + 72 = 0 => (y+8)(y+9) = 0 => y = -8, -9. Comparing x and y, 6 > -8, 6 > -9, -6 > -8, -6 > -9. Thus x > y.

Multiple choice

The equation x-2/(x-1)=1-2/(x-1) has

  1. no root

  2. one root

  3. two equal root

  4. infinitely many roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation is x - 2/(x-1) = 1 - 2/(x-1). Adding 2/(x-1) to both sides gives x = 1. However, substituting x = 1 into the original equation results in division by zero, which is undefined. Thus, there is no valid root.

Multiple choice

Directions: In the following question, two equations (I) and (II) are given. You have to solve both the equations and give answer as per the codes given below: I. x2 – 5x + 6 = 0 II. 2y2 – y – 3 = 0

  1. if x > y

  2. if x ≥ y

  3. if x < y

  4. if x ≤ y

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

I: x^2 - 5x + 6 = 0 => (x-2)(x-3) = 0, x = 2, 3. II: 2y^2 - y - 3 = 0 => (2y-3)(y+1) = 0, y = 1.5, -1. Comparing x values (2, 3) and y values (1.5, -1), x is always greater than y.

Multiple choice

If α and β are the roots of the equation ax2 + bx + c = 0, is α equal to β? (1) α + β, α2 + β2, α3 + β3 are in G.P. (2) α ≠ 0, β ≠ 0.

  1. Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.

  5. Statements (1) and (2) TOGETHER are NOT sufficient.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Statement 1 implies a relationship between roots that can lead to equality, but doesn't guarantee it without knowing the coefficients. Statement 2 ensures roots are non-zero. Together, they allow solving for the condition where the roots are equal.

Multiple choice

Two persons A and B solved a quadratic equation of the form x2 + bx + c = 0. A made a mistake in noting down the coefficient of x and obtained the roots as 18 and 2, where as B obtained the roots as – 9 and – 3 by misreading the constant term, find the correct roots of the equation.

  1. – 6 , – 3

  2. – 6 , 6

  3. – 6 , – 5

  4. – 6 , – 6
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A misread b, so c is correct: roots 18, 2 => c = 18*2 = 36. B misread c, so b is correct: roots -9, -3 => -b = -9 + -3 = -12 => b = 12. Equation is x^2 + 12x + 36 = 0. Roots are (x+6)^2 = 0 => x = -6, -6.

Multiple choice

Ram and Sham solved a quadratic equation. In solving it, Ram made a mistake in the constant term only and got the roots as 8 and 2, while Sham made a mistake in the coefficient of x only and obtained - 9 and - 1 as roots. The correct roots of the equation are

  1. 8 and -1

  2. -9 and 2

  3. -8 and -2

  4. 9 and 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Ram's roots 8, 2 imply x^2 - 10x + 16 = 0. Since he missed the constant, the coefficient of x is -10. Sham's roots -9, -1 imply x^2 + 10x + 9 = 0. Since he missed the coefficient of x, the constant is 9. The correct equation is x^2 - 10x + 9 = 0. Roots are 9 and 1.