Quantitative Aptitude

Algebra

3,715 Questions
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Algebra Questions

Multiple choice

Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the roots as (3, 2). What will be the exact roots of the original quadratic equation?

  1. (6, 1)

  2. (-3, -4)

  3. (4, 3)

  4. (-4, -3)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Ujakar's roots (4, 3) imply the equation x^2 - 7x + C = 0. Keshab's roots (3, 2) imply the equation x^2 + Bx + 6 = 0. Combining these, the original equation is x^2 - 7x + 6 = 0. Factoring this gives (x - 6)(x - 1) = 0, so the roots are 6 and 1.

Multiple choice

Match the following columns: Quadratic Equation Relation Between Roots 1. m2 - 9m + 20 = 0 A. The difference between the two roots is 26. 2. n2 + 7n + 10 = 0 B. If we add both the roots, the sum is 9. 3. p2 + 6p - 160 = 0 C. The product of both the roots is 10.

  1. 1 - A, 2 - B, 3 - C

  2. 1 - B, 2 - C, 3 - A

  3. 1 - A, 2 - C, 3 - B

  4. 1 - B, 2 - A, 3 - C

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

1: m^2-9m+20=0 -> roots 4, 5. Sum=9 (B). 2: n^2+7n+10=0 -> roots -2, -5. Product=10 (C). 3: p^2+6p-160=0 -> roots 10, -16. Difference=26 (A). Matches 1-B, 2-C, 3-A.

Multiple choice

Given below are three statements. A. Statement 1: Solution for x2 + 9x – 36 = 0 is (x = 3 & x = -12) B. Statement 2: x2 + 5x + 50 = 0 has no real roots. C. Statement 3: In equation, x2 + 7x + 10 = 0, one root is double of other. Which of the following is correct?

  1. (A), (B) and (C)

  2. (A) and (C) only

  3. (A) and (B) only

  4. Only (A)

  5. NA
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Statement 1: x^2 + 9x - 36 = (x+12)(x-3) = 0, roots are -12 and 3. Correct. Statement 2: Discriminant D = 5^2 - 4(1)(50) = 25 - 200 = -175. Since D < 0, there are no real roots. Correct. Statement 3: x^2 + 7x + 10 = (x+5)(x+2) = 0, roots are -5 and -2. -5 is not double -2. Incorrect.

Multiple choice

Directions: In the given question, two equations are given. You have to solve both the equations and mark the correct option. I. 5x2 - 19x + 12 = 0 II. 2y2 - 13y = -21

  1. x < y

  2. x ≤ y

  3. x > y

  4. x ≥ y

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

Eq I: 5x^2 - 19x + 12 = 0 -> (5x - 4)(x - 3) = 0 -> x = 0.8, 3. Eq II: 2y^2 - 13y + 21 = 0 -> (2y - 7)(y - 3) = 0 -> y = 3.5, 3. Comparing roots: x=0.8 < y=3.5, x=0.8 < y=3, x=3 < y=3.5, x=3 = y=3. Thus x <= y.

Multiple choice

Directions: In the given question, two equations are given. You have to solve both the equations and mark the correct option. I. 15x2 - 22x + 8 = 0 II. 21y2 - 17y + 2 = 0

  1. x < y

  2. x ≤ y

  3. x > y

  4. x ≥ y

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

Equation I: 15x^2 - 22x + 8 = 0. Roots: x = (22 +/- sqrt(484 - 480))/30 = (22 +/- 2)/30. x = 24/30 = 0.8 or 20/30 = 0.66. Equation II: 21y^2 - 17y + 2 = 0. Roots: y = (17 +/- sqrt(289 - 168))/42 = (17 +/- 11)/42. y = 28/42 = 0.66 or 6/42 = 0.14. Comparing roots: 0.8 >= 0.66 and 0.8 >= 0.14 and 0.66 >= 0.66 and 0.66 >= 0.14. Thus x >= y.

Multiple choice

Consider the following statements in respect of the roots of the equation x3 – 8 = 0: 1. The roots are non-collinear. 2. The roots lie on a circle of unit radius. Which of the above statements is/are correct?

  1. Only 1

  2. Only 2

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The roots of x^3 - 8 = 0 are 2, 2w, and 2w^2, where w is a complex cube root of unity. These roots are 2, -1 + i*sqrt(3), and -1 - i*sqrt(3), which are non-collinear in the complex plane, and they lie on a circle of radius 2, not unit radius.