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

Directions : Select the correct alternative from the given choices. If the difference between the roots of the equation x2 - px + 2p = 0 is 3, where both the roots are positive, find the equation whose roots are 2 more than the roots of the given equation.

  1. x2 - 5x + 4 = 0

  2. x2 - 9x + 18 = 0

  3. x2 + 5x + 18 = 0

  4. x2 - 13x + 40 = 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For x^2 - px + 2p = 0, let roots be a and b. a+b=p, ab=2p. Given |a-b|=3, so (a-b)^2 = 9, which is (a+b)^2 - 4ab = 9. Thus p^2 - 8p - 9 = 0. Solving gives p=9 or p=-1. Since roots are positive, p=9. Roots are 3 and 6. New roots are 3+2=5 and 6+2=8. The equation is (x-5)(x-8) = x^2 - 13x + 40 = 0.

Multiple choice

Directions: Select the correct alternative from the given choices. The graph of a quadratic expression, ax2 + bx + c, attains a maximum value of 10 at x = 2. If the graph intersects the x-axis at two points, one positive and the other negative, then which of a, b and c is/are positive?

  1. Only c.

  2. Only b.

  3. Both b and c.

  4. Both a and c.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A quadratic with a maximum value must have a < 0 (downward parabola). The vertex is at x = 2, so -b/(2a) = 2, meaning b = -4a. Since a < 0, b must be positive. The y-intercept is c. Since the parabola has a maximum of 10 at x=2 and crosses the x-axis at positive and negative values, the y-intercept (c) must be positive to allow the curve to pass through the positive y-axis region.

Multiple choice

If the equation x2 - (k + 4)x + (4k + 1) = 0 has equal roots,

  1. k = 6

  2. k = 2

  3. k = 6, 2

  4. k = ±6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For equal roots, the discriminant b^2 - 4ac must be zero. Here, (-(k+4))^2 - 4(1)(4k+1) = 0. Expanding gives k^2 + 8k + 16 - 16k - 4 = 0, which simplifies to k^2 - 8k + 12 = 0. Factoring gives (k-6)(k-2) = 0, so k = 6 or k = 2.

Multiple choice

If the equations K(6x2 + 3) + rx + 2x2 - 1 = 0 and 6K(2x2 + 1) + Px + 4x2 -2 = 0 have both the roots common, then (2r - P) is

  1. K

  2. 1

  3. 0

  4. 3K
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For two quadratic equations to have both roots common, the ratios of their coefficients must be equal. Dividing the second equation by 2 gives 3K(x^2 + 0.5) + (P/2)x + 2x^2 - 1 = 0, which simplifies to (3K+2)x^2 + (P/2)x + (1.5K-1) = 0. Comparing this to the first equation K(6x^2+3) + rx - 1 = 0, or (6K+2)x^2 + rx + (3K-1) = 0, we find the ratios match when 2r = P, making 2r - P = 0.

Multiple choice

If the equation ax2 + bx + c = 0 has no real roots and a + b + c < 0,

  1. c = 0

  2. c > 0

  3. c < 0

  4. c = ab
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If a quadratic has no real roots, the parabola never crosses the x-axis, meaning it is always above or below the axis. Since a+b+c < 0 (the value of the function at x=1 is negative), the parabola must be entirely below the x-axis, implying a < 0. For a parabola to be entirely below the x-axis, its y-intercept c must also be negative.

Multiple choice

The equation x4 - 5x3 + 3x2 + 35x - 70 = 0 has a root between

  1. 2 and 3

  2. 3 and 4

  3. 5 and 6

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

Let f(x) = x^4 - 5x^3 + 3x^2 + 35x - 70. Evaluating at x = 2 gives 16 - 40 + 12 + 70 - 70 = -12. Evaluating at x = 3 gives 81 - 135 + 27 + 105 - 70 = 8. Since the sign changes between 2 and 3, a root exists in that interval.

Multiple choice

x f(x) -1 18 0 24 1 32 For the quadratic function f, the table shows three values of x and their corresponding values of f(x). Which equation defines f?

  1. f(x) = 3x2 + 5x + 24

  2. f(x) = x2 + 7x + 24

  3. f(x) = 5x2 – x + 24

  4. f(x) = 9x2 + x + 24
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Substitute the given x-values into the options to check for consistency. For x=0, all options yield f(0)=24, which matches the table. For x=1, f(1) must be 32; option B gives 1^2 + 7(1) + 24 = 32, which is correct.