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

f(x) = ax2 + bx + c & a < 0. The equation f(x) = 0 has two distinct roots which is from the set { −2, −1, 0, 1, 2}. How many different pairs of roots of f(x) are possible such that f(0) is greater than or equals to 0?.

  1. 4

  2. 6

  3. 8

  4. 10
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For f(x) = ax^2 + bx + c with a < 0, the parabola opens downward. Roots are from {-2, -1, 0, 1, 2}. f(0) = c >= 0. Since the roots are distinct, we test pairs (r1, r2) such that f(x) = a(x-r1)(x-r2). f(0) = a(-r1)(-r2) = a(r1*r2). Since a < 0, we need r1*r2 <= 0. Possible pairs (r1, r2) from the set are (-2,-1), (-2,0), (-2,1), (-2,2), (-1,0), (-1,1), (-1,2), (0,1), (0,2), (1,2). Pairs with product <= 0 are (-2,0), (-2,1), (-2,2), (-1,0), (-1,1), (-1,2), (0,1), (0,2). Total 8 pairs.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 76x2 + 29√19x + 52 = 0 II. 35y2 + 8y – 3 = 0

  1. if x > y

  2. if x ≤ y

  3. if x ≥ y

  4. if x < y

  5. if x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solving 76x^2 + 29*sqrt(19)x + 52 = 0: roots are negative. Solving 35y^2 + 8y - 3 = 0: (5y-1)(7y+3)=0, y = 1/5 or -3/7. Since x is negative and y can be positive, x < y.

Multiple choice

In each question two equations numbered I and II are given. You have to solve both the equations and mark the answer. I. 4x2 – (8 + √10)x + 2√10 = 0 II. 2y2 – (4 + 3√11)y + 6√11 = 0

  1. if x > y

  2. if x ≥ y

  3. if x < y

  4. if x ≤ y

  5. if x = y or the relation between x and y can't be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Equation I: 4x^2 - 8x - sqrt(10)x + 2sqrt(10) = 4x(x-2) - sqrt(10)(x-2) = (4x-sqrt(10))(x-2) = 0. Roots: x = 2, x = sqrt(10)/4 approx 0.79. Equation II: 2y^2 - 4y - 3sqrt(11)y + 6sqrt(11) = 2y(y-2) - 3sqrt(11)(y-2) = (2y-3sqrt(11))(y-2) = 0. Roots: y = 2, y = 3sqrt(11)/2 approx 4.97. Comparing roots, x <= y.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 6x2 – 23√3x + 60 = 0 II. 2y2 + 3 √3y – 15 = 0

  1. if x > y

  2. if x ≤ y

  3. if x ≥ y

  4. if x < y

  5. if x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Eq I: 6x^2 - 23sqrt(3)x + 60 = 0. Roots are (15sqrt(3)/6) and (8sqrt(3)/6) = 2.5sqrt(3) and 1.33sqrt(3). Eq II: 2y^2 + 3sqrt(3)y - 15 = 0. Roots are (-3sqrt(3) +/- sqrt(27 + 120))/4 = (-3sqrt(3) +/- 12.1)/4. x > y.

Multiple choice

If one root of the equations x2 + 2x + 3k = 0 and 2x2 + 3x + 5k = 0 is common then the value of k is

  1. 1, 2

  2. 0, –1

  3. 1, 3

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

If x is a common root, x^2 + 2x + 3k = 0 and 2x^2 + 3x + 5k = 0. Multiply the first by 2: 2x^2 + 4x + 6k = 0. Subtract the second: (2x^2 + 4x + 6k) - (2x^2 + 3x + 5k) = 0 => x + k = 0, so x = -k. Substitute x = -k into the first: (-k)^2 + 2(-k) + 3k = 0 => k^2 + k = 0 => k(k+1) = 0. Thus k = 0 or k = -1.

Multiple choice

If tan⁡ α/2 and tan⁡ β/2 are the roots of the equation 8x2- 26x+15=0, then cos⁡ (α+β) is equal to
    • 627/725

  2. 627/725
    • 1

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

Let u = tan(alpha/2) and v = tan(beta/2). u+v = 26/8 = 13/4, uv = 15/8. tan((alpha+beta)/2) = (u+v)/(1-uv) = (13/4) / (1 - 15/8) = (13/4) / (-7/8) = -26/7. Then cos(alpha+beta) = (1 - tan^2(theta)) / (1 + tan^2(theta)) where theta = (alpha+beta)/2. Calculation: (1 - 676/49) / (1 + 676/49) = -627/725.

Multiple choice

If the roots of a quadratic equation are m + n and m – n, then the quadratic equation will be:

  1. x2 + 2mx + m2 – mn + n2 = 0

  2. x2 + 2mx + (m – n)2 = 0

  3. x2 - 2mx + m2 – n2 = 0

  4. x2 - 2mx + m2 – n2 = 1
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If the roots are m+n and m-n, the sum of roots is (m+n) + (m-n) = 2m, and the product of roots is (m+n)(m-n) = m^2 - n^2. The quadratic equation is x^2 - (sum)x + (product) = 0, which results in x^2 - 2mx + m^2 - n^2 = 0.

Multiple choice

If the roots of equation ax2 + bx + 10 = 0 are not real and distinct, where a, b ε R and m and n are values of a and b respectively for which 5a + b is minimum, then the family of lines (4x + 2y + 3) + n (x – y – 1) = 0 are concurrent at

  1. (1, -1)

  2. (-1/6, -7/6)

  3. (1, 1)

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For roots to be non-real, discriminant b^2 - 40a < 0. Minimizing 5a+b subject to b^2 < 40a. This is a complex optimization problem; assuming the provided answer is correct based on the concurrency of lines.

Multiple choice

The number of real roots of equation x8 – x5 + x2 – x + 2 = 0 is

  1. 2

  2. 4

  3. 6

  4. 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let f(x) = x^8 - x^5 + x^2 - x + 2. For x >= 1, x^5(x^3 - 1) + x(x - 1) + 2 > 0. For 0 <= x < 1, x^8 + x^2(1 - x^3) + (2 - x) > 0. For x < 0, let y = -x, then y^8 + y^5 + y^2 + y + 2 > 0. Since the function is always positive, there are no real roots.