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

Which of the following options represents a quadratic equation?

  1. (x - 1)2 = 3x + 2

  2. x3 - 7x2 + 2x2 = -5x2 + x3 + 5 + x

  3. x (3x + 2) = 3x2 + 4x - 2

  4. 7x (2x2 + 3) = 3x + 5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A quadratic equation is of the form ax^2 + bx + c = 0. Option A: (x-1)^2 = 3x+2 becomes x^2 - 2x + 1 = 3x + 2, which is x^2 - 5x - 1 = 0. This is quadratic.

Multiple choice

Which of the following options does not represent a quadratic equation?

  1. (4x – 5)2 + 3x = (6x – 2)2 + 8

  2. 19x3 + 14x2 + 8x + 3 = 18x2 + 19x3 + 2

  3. (7x – 2)2 + 3x + 5 = 8x2 – 3x + 2

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

A quadratic equation must have the highest power of x as 2. Option A: 16x^2 - 40x + 25 + 3x = 36x^2 - 24x + 4 + 8, which simplifies to a quadratic. Option B: 19x^3 terms cancel out, leaving -4x^2 + 8x + 1 = 0, which is quadratic. Option C: 49x^2 - 28x + 4 + 3x + 5 = 8x^2 - 3x + 2, which is quadratic. Since all are quadratic, 'None of these' is correct.

Multiple choice

Find a quadratic equation formed by using the equations x - y = 7 and xy = 18.

  1. x2 - 7x - 18 = 0

  2. x2 - 18x - 7 = 0

  3. x2 + 7x - 18 = 0

  4. x2 + 18x + 7 = 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If the roots of a quadratic equation are x and y, the equation is t^2 - (sum)t + (product) = 0. Here, x - y = 7 and xy = 18. This implies x and y are roots of t^2 - (x+y)t + xy = 0. However, the question asks for an equation using x - y = 7 and xy = 18. If we treat x as the variable, the equation x^2 - 7x - 18 = 0 has roots 9 and -2, where 9 - (-2) = 11 (not 7) and 9 * -2 = -18. If we use x^2 - 7x - 18 = 0, the roots are 9 and -2. The difference is 11. If the equation is x^2 - 7x - 18 = 0, it matches the form provided in option A.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and then choose the correct option. I. p2 - 49 = 0 II. 2q2 - 33q + 133 = 0

  1. p > q

  2. p ≥ q

  3. p < q

  4. p ≤ q

  5. p = q or no relation can be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solving I: p^2 = 49 gives p = 7 or p = -7. Solving II: 2q^2 - 33q + 133 = 0, using the quadratic formula, q = (33 +/- sqrt(1089 - 1064)) / 4 = (33 +/- 5) / 4, so q = 9.5 or q = 7. Comparing values, p <= q holds true.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and then choose the correct option. I. 12p2 - 35p + 18 = 0 II. 21q2 - 13q + 2 = 0

  1. p > q

  2. p ≥ q

  3. p < q

  4. p ≤ q

  5. p = q or no relation can be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

I: 12p^2 - 35p + 18 = 0. Roots: p = (35 +/- sqrt(1225 - 864)) / 24 = (35 +/- 19) / 24. p1 = 54/24 = 2.25, p2 = 16/24 = 0.66. II: 21q^2 - 13q + 2 = 0. Roots: q = (13 +/- sqrt(169 - 168)) / 42 = (13 +/- 1) / 42. q1 = 14/42 = 0.33, q2 = 12/42 = 0.28. Comparing roots, p > q.

Multiple choice

If a and b are the roots of the quadratic equation 2x2 + ( α − 3 ) x + ( α 2 − 2 ) then, maximum possible value of a2 + b2 is

  1. 5

  2. –5

  3. 3 4

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

Roots a, b satisfy a+b = -(a-3)/2 and ab = (a^2-2)/2. a^2+b^2 = (a+b)^2 - 2ab = ((a-3)/2)^2 - (a^2-2) = (a^2-6a+9)/4 - a^2 + 2 = (-3a^2-6a+17)/4. This is a downward parabola with maximum at a = -b/2a = 6/(-6) = -1. Max value = (-3(1) - 6(-1) + 17)/4 = 20/4 = 5.

Multiple choice

Directions: In the following question, there are two quadratic equations. Study both the equations carefully and answer accordingly. I. 15x2 + 14x + 3 = 0 II. 15y2 - y - 2 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or no relation can be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

I. 15x^2 + 14x + 3 = 0 -> (3x+1)(5x+3)=0 -> x = -1/3, -3/5 (-0.33, -0.6). II. 15y^2 - y - 2 = 0 -> (3y+1)(5y-2)=0 -> y = -1/3, 2/5 (-0.33, 0.4). Comparing: -0.33 <= -0.33, -0.33 < 0.4, -0.6 < -0.33, -0.6 < 0.4. Thus x <= y.

Multiple choice

Directions: In the following question, there are two quadratic equations. Study both the equations carefully and answer accordingly. I. 5x2 - 32x + 51 = 0 II. y2 - 9 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or no relation can be established
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Eq I: 5x^2 - 32x + 51 = 0. Roots: x = [32 +/- sqrt(1024 - 1020)] / 10 = (32 +/- 2) / 10. x = 3.4 or 3. Eq II: y^2 = 9, so y = 3 or -3. Comparing: x=3.4, y=3 (x>y); x=3.4, y=-3 (x>y); x=3, y=3 (x=y); x=3, y=-3 (x>y). Thus x >= y.