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

For the quadratic equation ax2 + bx + c = 0, if the roots are in the ratio c : a, what can be inferred about their relationship?

  1. b2 = a(c + 1)2

  2. b2 = (c + a)2

  3. b = a2(c + 1)

  4. b = a(c + 1)2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the roots of the quadratic equation be k*c and k*a. The sum of the roots is k(c + a) = -b/a, and the product of the roots is k^2 * a * c = c/a, which simplifies to k^2 = 1/a^2. Squaring the sum equation gives k^2 * (c + a)^2 = b^2 / a^2, and substituting k^2 = 1/a^2 yields b^2 = (c + a)^2.

Multiple choice

If α, β are the roots of the equation x2 - x - 1 = 0 and Sn = 2023 αn + 2024 βn, then:

  1. S11 = S10 + S12

  2. 2S11 = S12 + S10

  3. 2S12 = S11 + S10

  4. S12 = S11 + S10
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since alpha and beta are roots of x^2 - x - 1 = 0, they satisfy x^2 = x + 1. Multiplying by x^(n-2) gives x^n = x^(n-1) + x^(n-2). This linear recurrence holds for both alpha and beta, so S_n = 2023*alpha^n + 2024*beta^n also satisfies S_n = S_(n-1) + S_(n-2).

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. 2p2 - 7p + 6 = 0 II. 4q2 = 9

  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
B Correct answer
Explanation

Eq I: 2p^2 - 7p + 6 = 0. (2p-3)(p-2)=0, p=1.5 or 2. Eq II: 4q^2 = 9, q^2 = 2.25, q = 1.5 or -1.5. Comparing: p=1.5, q=1.5 (p=q); p=2, q=1.5 (p>q); p=1.5, q=-1.5 (p>q); p=2, q=-1.5 (p>q). Thus p >= q.

Multiple choice

A quadratic equation is given by (a + b + c)x2 - (2a + 2b)x + (a + b - c) = 0; where a, b and c are real and distinct. Consider the following statements: 1. One of the roots of the equation is always less than 1 if a, b and c are all positive. 2. One of the roots of the equation is always negative if a, b and c are all negative. Which of the statements given above is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation is (a+b+c)x^2 - 2(a+b)x + (a+b-c) = 0. Using the quadratic formula or testing values, if a,b,c > 0, the product of roots is (a+b-c)/(a+b+c). Statement 1 holds true under specific conditions, but statement 2 is not universally true for all negative a,b,c.

Multiple choice

Directions: The given question is followed by two statements I and II. You have to identify which statements are sufficient to answer the question and mark the appropriate option. What is the sum of roots of equation 5cx2 + (a + b)x + 1 = 0? I. a + c = 0 II. 3a + 2b + c = 5

  1. Only statement I alone is sufficient to answer the question.

  2. Only statement II alone is sufficient to answer the question.

  3. Both the statements together are sufficient to answer the question.

  4. Both the statements together are insufficient to answer the question.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Sum of roots = -(a+b) / (5c). We have two equations with three variables (a, b, c). Statement I: a = -c. Statement II: 3a + 2b + c = 5. Even with both, we cannot determine the ratio (a+b)/c uniquely.

Multiple choice

If p and q are negative integers of the quadratic equation py2 + qy + r, having equal roots, then which of the following is definitely true?

  1. pqr > 0

  2. p + q + r < 0

  3. p + q - r < 0

  4. p + q + r > 0

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

For py^2 + qy + r = 0 to have equal roots, the discriminant D = q^2 - 4pr = 0, so q^2 = 4pr. Since p and q are negative integers, q^2 is positive, so 4pr must be positive. Since p is negative, r must also be negative. If p, q, r are all negative, then p+q+r must be less than 0.

Multiple choice

3x² + 5x - 2 = 0 ,What are the roots of the equation?

  1. x = -2, 1/3

  2. x = -5/6, 2/3

  3. x = -5/3, 2/3

  4. x = -5/3, -2/3

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

3x^2 + 5x - 2 = 0. Factoring: 3x^2 + 6x - x - 2 = 0 => 3x(x+2) - 1(x+2) = 0 => (3x-1)(x+2) = 0. Roots are x = 1/3 and x = -2.