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

What are the roots of the quadratic equation 21x2 - 37x - 28 = 0?

  1. -7/3, 4/7

  2. 3/7, -7/4

  3. 7/3, -4/7

  4. -3/7, 7/4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

21x^2 - 37x - 28 = 0. Using the quadratic formula: x = (37 +/- sqrt(37^2 - 4*21*(-28))) / 42 = (37 +/- sqrt(1369 + 2352)) / 42 = (37 +/- sqrt(3721)) / 42 = (37 +/- 61) / 42. Roots are 98/42 = 7/3 and -24/42 = -4/7.

Multiple choice

Directions: Solve the following question and mark the best possible option. The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

  1. 1

  2. 2

  3. 3

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

Sum of roots alpha+beta = -(a+3). Product of roots alpha*beta = -(a+5). Sum of squares = (alpha+beta)^2 - 2*alpha*beta = (a+3)^2 + 2(a+5) = a^2 + 6a + 9 + 2a + 10 = a^2 + 8a + 19. To minimize, take derivative: 2a + 8 = 0 => a = -4. Minimum value = (-4)^2 + 8(-4) + 19 = 16 - 32 + 19 = 3.

Multiple choice

The coefficients b and c of the equation x2 + bx + c = 0 are determined by throwing an ordinary die. The probability that the equation has equal roots is

  1. 1/18

  2. 13/18

  3. 5/18

  4. 1/9
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For equal roots, discriminant b^2 - 4c = 0, so b^2 = 4c. Possible values for b and c (1 to 6): If b=1, c=1/4 (no). If b=2, c=1 (yes). If b=3, c=9/4 (no). If b=4, c=4 (yes). If b=5, c=25/4 (no). If b=6, c=9 (no). There are 2 successful outcomes out of 36 total combinations (6*6). Probability = 2/36 = 1/18.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and choose the correct answer accordingly. I. 4A2 - 68A + 64 = 0 II. 8B2 + 40B + 32 = 0

  1. A < B

  2. A > B

  3. A = B, or No relation can be established between A and B.

  4. A ≤ B

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

I: 4A^2 - 68A + 64 = 0 simplifies to A^2 - 17A + 16 = 0, so (A-16)(A-1) = 0, A = 16 or 1. II: 8B^2 + 40B + 32 = 0 simplifies to B^2 + 5B + 4 = 0, so (B+4)(B+1) = 0, B = -4 or -1. Since all values of A are greater than all values of B, A > B.

Multiple choice

Which of the following pairs of linear equations when solved gives the values of x and y as 1 and 4, respectively?

  1. 2x + y = 6 and 2x - y + 2 = 0

  2. 2x - y = 6 and 2x + y - 2 = 0

  3. 2x + y + 6 = 0 and 2x - y = 2

  4. 2x + y + 6 = 0 and 2x - y + 2 = 0

  5. 2x + y = 6 and 2x - y = 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Substitute x=1 and y=4 into the equations. For option A: 2(1)+4=6 (True) and 2(1)-4+2=0 (True). All other options fail to satisfy both equations simultaneously.

Multiple choice

The number of integral values of m for which the equation (1 + m2)x2 - 2(1 + 3m)x + (1 + 8m) = 0 has no real root is

  1. infinitely many

  2. 2

  3. 3

  4. 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For the quadratic equation to have no real roots, the discriminant D must be less than 0. D = [-2(1+3m)]^2 - 4(1+m^2)(1+8m) < 0. Expanding this leads to 4(1 + 6m + 9m^2) - 4(1 + 8m + m^2 + 8m^3) < 0, which simplifies to 4(1 + 6m + 9m^2 - 1 - 8m - m^2 - 8m^3) < 0, or -8m^3 + 8m^2 - 2m < 0. This inequality holds for many values of m.