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

If the equation 2x2 - 6x + m = 0 has two distinct roots a and b, and ab and (a + b) are the distinct roots of the equation y2 + ry + r = 0, then the value of 16(r - m) is ___.

  1. -16

  2. -12

  3. -8

  4. 0

  5. 8
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For 2x^2 - 6x + m = 0, a+b = 3 and ab = m/2. The roots of y^2 + ry + r = 0 are ab and a+b. Thus, m/2 + 3 = -r and (m/2)*3 = r. Solving these, 3m/2 = -m/2 - 3 gives 2m = -3, m = -1.5, r = -2.25. Then 16(r - m) = 16(-2.25 - (-1.5)) = 16(-0.75) = -12.

Multiple choice

If the equation x2 - 2x + |k| = 0, where k belongs to integers, has two real and distinct roots, then the values of k lie in the interval ____.

  1. (-2, 2)

  2. (-2, 1]

  3. (-1, 1)

  4. (-1, 2]

  5. (-2, 1)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For the quadratic equation to have two real and distinct roots, its discriminant must be strictly greater than zero. This gives 4 - 4|k| > 0, which simplifies to |k| < 1, meaning k must lie in the open interval (-1, 1). Since k is specified to be an integer, the only possible value for k is 0, which lies within this interval.

Multiple choice

Find the quadratic equation with one root 3 and other root 1/3 ?

  1. A. 3xΒ² - 10x + 3 = 0

  2. B. 3xΒ² - 8x - 3 = 0

  3. C. 3xΒ² + 10x + 3 = 0

  4. D. 3xΒ² + 8x - 3 = 0

  5. E. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Roots are 3 and 1/3. Sum = 3 + 1/3 = 10/3. Product = 3 * 1/3 = 1. Equation is x^2 - (sum)x + product = 0. x^2 - (10/3)x + 1 = 0. Multiply by 3: 3x^2 - 10x + 3 = 0.

Multiple choice

In each of the following questions two equations are given. Solve these equations and give answer: (a) if π‘₯ β‰₯ 𝑦, i.e., π‘₯ is greater than or equal to 𝑦 (b) if π‘₯ > 𝑦, i.e., π‘₯ is greater than 𝑦 (c) if π‘₯ ≀ 𝑦, i.e., π‘₯ is less than or equal to 𝑦 (d) if π‘₯ < 𝑦, i.e., π‘₯ is less than 𝑦 (e) π‘₯ = 𝑦 or no relation can be established between π‘₯ and y I. xΒ² βˆ’ 12 x + 35 = 0 II. yΒ² βˆ’ 11 y + 24 = 0

  1. e

  2. b

  3. c

  4. d

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

I: x^2 - 12x + 35 = 0 -> (x-7)(x-5) = 0, so x = 5, 7. II: y^2 - 11y + 24 = 0 -> (y-8)(y-3) = 0, so y = 3, 8. Comparing the sets {5, 7} and {3, 8}: 5 > 3, 5 < 8, 7 > 3, 7 < 8. Since x can be both greater than and less than y, no relation can be established.

Multiple choice

In a quadratic function Z(x) = px2 + mx + n, where p, m, and n are integers, if Z(5) = Z(-2) = 0 and p(β‰  1) is a positive integer, then which of the following is the maximum value of m + p?

  1. 0

  2. -2

  3. -4

  4. -8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since Z(5) = Z(-2) = 0, the roots are 5 and -2. The function is Z(x) = p(x - 5)(x + 2) = p(x^2 - 3x - 10) = px^2 - 3px - 10p. Comparing to px^2 + mx + n, we get m = -3p and n = -10p. We want to maximize m + p = -3p + p = -2p. Since p is a positive integer and p != 1, the smallest p is 2. For p = 2, m + p = -4. For larger p, the value becomes more negative.

Multiple choice

The table below lists the values of a quadratic function g(x) at three different points: x g(x) -1 10 0 14 1 20 Which equation best represents the function g(x)?

  1. g(x) = 2x2 + 4x + 14

  2. g(x) = x2 + 5x + 14

  3. g(x) = 3x2 - x + 14

  4. g(x) = 4x2 + x + 14
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Test the points: For x=0, g(0)=14. All options satisfy this. For x=1, g(1)=20. Option B: 1^2 + 5(1) + 14 = 20. Correct. For x=-1, g(-1)=(-1)^2 + 5(-1) + 14 = 1 - 5 + 14 = 10. Correct.

Multiple choice

If the roots of the quadratic equation mx2 + nx + r = 0 are sinΞΈ and cosΞΈ, which of the following is the correct relation of r in terms of m and n?

  1. (m2 - n2)/2

  2. (n2 - m2)/2

  3. (n2 - m2)/2m

  4. (m2 - n2)/2n
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots are sin(theta) and cos(theta). Sum of roots = sin(theta) + cos(theta) = -n/m. Product of roots = sin(theta) * cos(theta) = r/m. Square the sum: sin^2 + cos^2 + 2sin*cos = n^2/m^2. 1 + 2(r/m) = n^2/m^2. 2r/m = n^2/m^2 - 1 = (n^2 - m^2)/m^2. r = (n^2 - m^2) / 2m.