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 ratio of the roots of the equation $a{ x }^{ 2 }+bx+c=0$ is same as the ratio of the roots of the equation $p{ x }^{ 2 }+qx+r=0$. If ${ D }{ 1 }$ and ${ D }{ 2 }$ are the discriminants of $a{ x }^{ 2 }+bx+c=0$ and $p{ x }^{ 2 }+qx+r=0$ respectively, then ${ D }{ 1 }:{ D }{ 2 }$ is equal to
  1. $\displaystyle \frac { { a }^{ 2 } }{ { p }^{ 2 } } $
  2. $\displaystyle \frac { { b }^{ 2 } }{ { q }^{ 2 } } $
  3. $\displaystyle \frac { { c }^{ 2 } }{ { r }^{ 2 } } $

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

Let the roots be alpha, beta. Ratio alpha/beta = k. For ax^2+bx+c=0, roots are (-b +/- sqrt(D1))/2a. Ratio = (-b+sqrt(D1))/(-b-sqrt(D1)) = k. This implies D1/b^2 is constant. Specifically, D1/b^2 = (k-1)^2 / (k+1)^2. Since this ratio is the same for both equations, D1/b^2 = D2/q^2, so D1/D2 = b^2/q^2.

Multiple choice

The equation $\displaystyle 2x^{2}+3x+1= 0$ has

  1. two rational roots.

  2. two irrational roots.

  3. integral roots.

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

Discriminant D = b^2 - 4ac = 3^2 - 4(2)(1) = 9 - 8 = 1. Since D > 0 and is a perfect square, the roots are rational and distinct.

Multiple choice

Discuss the nature of the roots of the equation $\displaystyle 4ax^{2}+3bx+2c=0$, where $a, b, c> \displaystyle \in R$ and are connected by the relation $a+b+c=0$.

  1. All roots are imaginary.

  2. All roots are real.
  3. They can be real or imaginary depends on the value of $a,b,c$.

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

The discriminant of the quadratic equation is D = 9b^2 - 32ac. Substituting b = -(a + c) into this expression yields D = 9(a + c)^2 - 32ac = 9a^2 - 14ac + 9c^2. Since the discriminant of this quadratic form in terms of a and c is negative, the expression 9a^2 - 14ac + 9c^2 is always non-negative for all real values of a and c, ensuring the roots are always real.

Multiple choice

Solve the equation obtained $x^{2} - x - 6 = 0$ and hence find the dimensions of the verandah. Verandah is in rectangular shape having area and perimeter equal.

  1. x = 3; length = 6 m and breadth = 3 m

  2. x = 3; length = 6 m and breadth = 4 m

  3. x = 3; length = 4 m and breadth = 3 m

  4. x = 4; length = 6 m and breadth = 3 m
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

x^2 - x - 6 = 0 => (x-3)(x+2) = 0. x=3. Area = L*B, Perimeter = 2(L+B). If L=6, B=3, Area = 18, Perimeter = 2(9) = 18. Matches.

Multiple choice

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.
  1. $\displaystyle x^{2}-Gx+A^{2}=0.$
  2. $\displaystyle x^{2}-Ax+G^{2}=0.$
  3. $\displaystyle x^{2}-2Gx+A^{2}=0.$
  4. $\displaystyle x^{2}-2Ax+G^{2}=0.$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If roots are r1, r2, then AM = (r1+r2)/2 = A, so r1+r2 = 2A. GM = sqrt(r1*r2) = G, so r1*r2 = G^2. The quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0, which is x^2 - 2Ax + G^2 = 0.

Multiple choice

If $\alpha, \beta $ are the roots of the equation $ax^2+bx+c=0$, then $\dfrac{\alpha}{a\beta+b}+\dfrac{\beta}{a\alpha+b}$ is equal to
  1. $\cfrac{2}{a}$
  2. $\cfrac{2}{b}$
  3. $\cfrac{2}{c}$
  4. $-\cfrac{2}{a}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given ax^2+bx+c=0, alpha+beta = -b/a, alpha*beta = c/a. Expression: (alpha(a*alpha+b) + beta(a*beta+b)) / ((a*beta+b)(a*alpha+b)) = (a*alpha^2 + b*alpha + a*beta^2 + b*beta) / (a^2*alpha*beta + ab(alpha+beta) + b^2). Numerator: a(alpha^2+beta^2) + b(alpha+beta) = a((alpha+beta)^2 - 2*alpha*beta) + b(alpha+beta) = a(b^2/a^2 - 2c/a) - b^2/a = b^2/a - 2c - b^2/a = -2c. Denominator: a^2(c/a) + ab(-b/a) + b^2 = ac - b^2 + b^2 = ac. Result: -2c / ac = -2/a.

Multiple choice

The roots of the equation $x^2 - 3x - 10 = 0$ are
  1. Roots are $1$ and $3$.
  2. Roots are $-2$ and $5$.
  3. Roots are $-8$ and $2$.
  4. Roots are $4$ and $3$.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Factor x^2 - 3x - 10 = 0 into (x - 5)(x + 2) = 0. The roots are x = 5 and x = -2.