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

The equation, whose roots are the squares of the roots of the equation ax2 – bx + c = 0, is

  1. a2x2 + b2x + c2 = 0

  2. a2x2 – (b2 – 4ac)x + c2 = 0

  3. a2x2 – (b2 – 2ac)x + c2 = 0

  4. a2x2 + (b2 – ac)x + c2 = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If roots of ax^2 - bx + c = 0 are alpha and beta, then alpha+beta = b/a and alpha*beta = c/a. The new equation has roots alpha^2 and beta^2. Sum = alpha^2 + beta^2 = (alpha+beta)^2 - 2*alpha*beta = (b/a)^2 - 2(c/a) = (b^2 - 2ac)/a^2. Product = (alpha*beta)^2 = c^2/a^2. Equation: x^2 - (Sum)x + Product = 0 => x^2 - ((b^2-2ac)/a^2)x + c^2/a^2 = 0 => a^2x^2 - (b^2-2ac)x + c^2 = 0.

Multiple choice

If the equations x2 – px + q = 0 and x2 – ax + b = 0 have a common root and the roots of the second equation are equal, then which of the following is correct?

  1. aq = 2(b + p)

  2. aq = b + p

  3. ap = 2(b + q)

  4. ap = b + q
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots of x^2 - ax + b = 0 are equal, so discriminant = a^2 - 4b = 0 => a^2 = 4b. Let the common root be k. k^2 - pk + q = 0 and k^2 - ak + b = 0. Subtracting: (a-p)k = b-q => k = (b-q)/(a-p). Substituting back into the first equation and using a^2=4b leads to the relation ap = 2(b+q).

Multiple choice

If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to:

  1. 2α2

  2. -2α(α + 1)

  3. 2α(α - 1)

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

The equation is 4x^2 + 2x - 1 = 0. The sum of roots α + β = -2/4 = -1/2, so β = -1/2 - α. The product of roots αβ = -1/4, so β = -1/(4α). Testing the options, if β = -2α(α + 1), then β = -2α^2 - 2α. Since 4α^2 + 2α - 1 = 0, we have 2α^2 = 0.5 - α. Substituting this: β = -(0.5 - α) - 2α = -0.5 - α, which matches the sum of roots condition.

Multiple choice

If α and β are the roots of the quadratic equation x2−10x+15=0, then find the quadratic equation whose roots are α + α/β] and [β + β/α]

  1. 15x2+71x+210= 0

  2. 5x2−22x+56= 0

  3. 3x2−44x+78= 0

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

The roots satisfy α + β = 10 and αβ = 15. The transformed roots have sum α + β + α/β + β/α = 10 + (100 - 30)/15 = 44/3, and product (α + 1)(β + 1) = 26. Hence their equation is x^2 - (44/3)x + 26 = 0, or 3x^2 - 44x + 78 = 0.

Multiple choice

In the Maths Olympiad of 2020 at Animal Planet, two representatives from the donkey's side, while solving a quadratic equation, committed the following mistakes: (i) One of them made a mistake in the constant term and got the roots as 5 and 9. (ii) Another one committed an error in the coefficient of x and he got the roots as 12 and 4. But in the meantime, they realized that they are wrong and they managed to get it right jointly. Which of the following could be the quadratic equation?

  1. x2 + 4x + 14 = 0

  2. 2x2 + 7x – 24 = 0

  3. x2 – 14x + 48 = 0

  4. 3x2 – 17x + 52 = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

First person got roots 5 and 9 (sum 14, product 45). Mistake in constant term means sum is correct: x^2 - 14x + k = 0. Second person got roots 12 and 4 (sum 16, product 48). Mistake in x-coefficient means product is correct: x^2 - kx + 48 = 0. Combining these, the equation is x^2 - 14x + 48 = 0.