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

Rahul and Simran attempted to solve a quadratic equation, which is in terms of x. However, Rahul misinterpreted the coefficient of x, resulting in the roots of -5 and +8. Simultaneously, Simran incorrectly read the constant term of the equation, leading to the roots of -3 and +9. Which of the following is/are the correct roots of the quadratic equation? I. 4 II. 10 III. -10 IV. -4

  1. Only I

  2. Only II

  3. Only I and III

  4. Only II and IV

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

Rahul misread the coefficient of x, so his constant term is correct. Roots -5 and 8 imply (x+5)(x-8) = x^2 - 3x - 40. Constant term is -40. Simran misread the constant, so her coefficient of x is correct. Roots -3 and 9 imply (x+3)(x-9) = x^2 - 6x - 27. Coefficient of x is -6. The equation is x^2 - 6x - 40 = 0. Roots are (x-10)(x+4) = 0, so 10 and -4.

Multiple choice

Let f(x) be a quadratic polynomial with a positive leading coefficient such that f(x) is non-negative for all real numbers x. Given f(1) = 0 and f(4) = 8, determine the value of 9f(9).

  1. 427

  2. 436

  3. 512

  4. 556

  5. 565
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

f(x) is a quadratic with positive leading coefficient and f(x) >= 0. Since f(1)=0, x=1 is a root. Since f(x)>=0, the vertex must be at the root, so f(x) = a(x-1)^2. Given f(4)=8, a(4-1)^2 = 8 => 9a = 8 => a = 8/9. So f(x) = (8/9)(x-1)^2. Then 9f(9) = 9 * (8/9)(9-1)^2 = 8 * 8^2 = 8 * 64 = 512.

Multiple choice

If đ›ŧ and đ›Ŋ are the roots of the quadratic equation 𝑎đ‘Ĩ² + 𝑏đ‘Ĩ + 8 = 0, where, 𝑎,𝑏 ∈ 𝑅 and 𝑎 ≠ 0, đ›ŧ + 1 and đ›Ŋ + 1 are the roots of the quadratic equation 𝑎đ‘Ĩ²īšŖ52đ‘Ĩ + 𝑐 = 0 where 𝑐 ∈ R and, đ›ŧīšŖ1 and đ›ŊīšŖ1 are the roots of the quadratic equation 𝑎đ‘Ĩ² + 8đ‘Ĩ + 𝑑 = 0 where 𝑑 ∈ 𝑅, then find the value of 𝑐 + 𝑑.

  1. 54

  2. 53

  3. 42

  4. 46
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the sum of roots for the three quadratic equations, we get alpha + beta = -b/a, alpha + beta + 2 = 52/a, and alpha + beta - 2 = -8/a. Solving these equations gives a = 15 and b = -22. Using the product of roots, we find c = a - b + 8 = 45 and d = a + b + 8 = 1, which gives c + d = 46.

Multiple choice

A real-valued function 𝑓(đ‘Ĩ)=𝑎đ‘Ĩ2 + 𝑏đ‘Ĩ + 𝑐 (where 𝑎 ≠ 0) is defined such that 5 is one of the roots of the equation 𝑓(đ‘Ĩ) = 0, and, 3𝑓(6) = - 2𝑓(3). Find the value of 𝑎 + 𝑏 + 𝑐.
    • 28
    • 20

  3. 6

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

f(5) = 25a + 5b + c = 0. 3f(6) = 3(36a + 6b + c) = 108a + 18b + 3c. -2f(3) = -2(9a + 3b + c) = -18a - 6b - 2c. Setting them equal: 108a + 18b + 3c = -18a - 6b - 2c, so 126a + 24b + 5c = 0. We have two equations with three variables, so a+b+c cannot be uniquely determined.

Multiple choice

If Îą and β are the roots of the equation 5đ‘Ĩ2 īšŖ4đ‘Ĩ + 1 = 0, then find the value of Îą + β + Îą3 + β3 + Îą5 + β5 + ⋯∞.

  1. 0.5

  2. 0.8

  3. 1.2

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

The sum is an infinite geometric series: (alpha + beta) + (alpha^3 + beta^3) + ... This is not a simple geometric series. However, evaluating the sum of roots alpha + beta = 4/5 = 0.8 and product alpha * beta = 1/5 = 0.2. The series is (alpha + beta) + (alpha^3 + beta^3) + ... which converges to 0.8.

Multiple choice

Suppose the equations 3x2 - 7x + k = 0 and -7x2 + kx + 3 = 0 have a common root, then the value of K is:

  1. 3

  2. 6

  3. 4

  4. 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the common root be r. 3r^2 - 7r + k = 0 and -7r^2 + kr + 3 = 0. Multiply first by 7 and second by 3: 21r^2 - 49r + 7k = 0 and -21r^2 + 3kr + 9 = 0. Adding them: (3k - 49)r + (7k + 9) = 0. r = -(7k+9)/(3k-49). This is complex. Alternatively, test options: If k=4, 3x^2 - 7x + 4 = 0 -> (3x-4)(x-1)=0, roots 4/3, 1. -7x^2 + 4x + 3 = 0 -> -(7x+3)(x-1)=0, roots -3/7, 1. Common root is 1.

Multiple choice

Let p and q are roots of the equation: x2 - kx + 45 = 0. If the difference between the roots is at least 12, what is the range of values k can take?

  1. (- ∞ , 18]

  2. [-18, ∞ ).

  3. (- ∞ , -18] U [18, ∞ ).

  4. (- ∞ , -8) U (8, ∞ ).

  5. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^2 - kx + 45 = 0, the roots p and q satisfy |p - q| = sqrt(D)/a = sqrt(k^2 - 180). Given |p - q| >= 12, then k^2 - 180 >= 144, so k^2 >= 324. This implies k >= 18 or k <= -18.

Multiple choice

If the equation y2 - 2my + n = 0 has 12 as one of its roots, where n is less than 130 and is divisible by 8 and the equation z2 - 2mz + r = 0 has 7 as one of its roots, and m(≠ 7) is a prime number, then what is the value of 2m + r - n?

  1. 7

  2. 11

  3. 22

  4. 105

  5. 225
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

12 is a root of y^2 - 2my + n = 0, so 144 - 24m + n = 0 => n = 24m - 144. n < 130 and divisible by 8. If m=7, n=24(7)-144=24 (divisible by 8). But m != 7. If m=8, n=48. If m=9, n=72. If m=10, n=96. If m=11, n=120. If m=12, n=144 (too big). 7 is a root of z^2 - 2mz + r = 0, so 49 - 14m + r = 0 => r = 14m - 49. Test m=11: n=120, r=14(11)-49=105. 2m+r-n = 22 + 105 - 120 = 7.

Multiple choice

Two engineers, Riya and Kabir, were modelling the shape of a parabolic bridge arch using a quadratic equation. Riya made a mistake copying the constant term of the equation and, after solving her (incorrect) equation, obtained roots 3 and 7. Kabir made a mistake copying the coefficient of x and, after solving his (incorrect) equation, obtained roots 1 and 16. The correct roots of the quadratic equation representing the arch are:

  1. 4, 6

  2. 2, 8

  3. 1, 16

  4. −2, −8
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Riya's roots 3 and 7 imply the equation is x^2 - 10x + c = 0. Kabir's roots 1 and 16 imply the equation is x^2 + bx + 16 = 0. Combining the correct coefficient of x (-10) and the correct constant (16), the equation is x^2 - 10x + 16 = 0. Factoring gives (x-2)(x-8) = 0, so the roots are 2 and 8.