Quantitative Aptitude

Algebra

3,715 Questions
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Algebra Questions

Multiple choice

Directions: The following question has two statements, (1) and (2). Answer the question using the following options: What is the value of k, if y 2 - 12y + k = 0? (1) One root of the equation is 3. (2) One root of the equation is 6 more than the other.

  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
D Correct answer
Explanation

For y^2 - 12y + k = 0, sum of roots = 12. (1) One root is 3, so other is 9. k = 3*9 = 27. Sufficient. (2) Roots are r and r+6. 2r+6 = 12, so r=3. Roots are 3 and 9. k = 27. Sufficient.

Multiple choice

Directions: The following question has two statements, (1) and (2). Answer the question using the following options: Is n in the function f(x) = xn + 2x2 + 4 an even? (1) f(- x) = f(x). (2) The equation f(x) = 0 has more than 4 roots.

  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
A Correct answer
Explanation

f(x) = x^n + 2x^2 + 4. (1) f(-x) = f(x) implies (-x)^n + 2(-x)^2 + 4 = x^n + 2x^2 + 4. This simplifies to (-x)^n = x^n, which is true if n is even. (2) f(x)=0 having more than 4 roots is impossible for a polynomial of degree n unless it is the zero polynomial, which it is not. Statement 1 is sufficient.

Multiple choice

Which of the following equations has real and distinct roots?

  1. x2 - 5x + 5 = 0

  2. 2x2 - 12x + 2 = 0

  3. 2x2 + 5x + 1 = 0

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A quadratic equation has real and distinct roots if the discriminant D = b^2 - 4ac > 0. For A: 25 - 20 = 5 > 0. For B: 144 - 16 = 128 > 0. For C: 25 - 8 = 17 > 0. All satisfy the condition.

Multiple choice

Find the quadratic equation with roots -1 and +3.

  1. x2 + 2x - 3 = 0

  2. x2 - 2x - 3 = 0

  3. x2 - 3x + 5 = 0

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation with roots r1, r2 is x^2 - (r1+r2)x + (r1*r2) = 0. Roots are -1 and 3. Sum = 2, Product = -3. Equation: x^2 - 2x - 3 = 0.

Multiple choice

Find the factors of quadratic polynomial x2 - 9x + 20.

  1. (x + 4)(x + 5)

  2. (x + 4)(x - 5)

  3. (x - 4)(x + 5)

  4. (x - 4)(x - 5)

  5. (x + 2)(x + 10)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To factor x^2 - 9x + 20, find two numbers that multiply to 20 and add to -9. These numbers are -4 and -5, so the factors are (x - 4)(x - 5).

Multiple choice

Find all the roots of the equation 4x4 - 144x2 = 0

  1. 0, 0, 2 and 3

  2. 0, 4, 2 and - 3

  3. 0, 0, 6 and - 6

  4. 2, 4, - 3 and 6

  5. 3, - 5, 4 and 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Factor the equation: 4x^2(x^2 - 36) = 0. This gives 4x^2(x - 6)(x + 6) = 0. The roots are x = 0, 0, 6, -6.