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

If $\alpha$ and $\beta$ be two zeros of the quadratic polynomial $ax^2+bx+c$, then $\dfrac {1}{\alpha^3}+\dfrac {1}{\beta^3}$ is equal to
  1. $\dfrac {3abc-b^3}{c^3}$
  2. $\dfrac {abc-b^3}{c^3}$
  3. $\dfrac {ac-b^3}{c^3}$
  4. $\dfrac {3abc-a^3}{c^3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For ax^2+bx+c=0, alpha+beta = -b/a and alpha*beta = c/a. The expression 1/alpha^3 + 1/beta^3 = (alpha^3+beta^3)/(alpha*beta)^3. Using the identity alpha^3+beta^3 = (alpha+beta)^3 - 3*alpha*beta*(alpha+beta), the result simplifies to (3abc-b^3)/c^3.

Multiple choice

If one root of the equation $x^{3}+qx+r=0$ is double the other, then $343r^{2}+36q^{3}+1$ is
  1. $1$
  2. $0$
  3. $-1$
  4. $-2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the roots be a and 2a. By Vieta's formulas, sum of roots = 3a = 0 (since coefficient of x^2 is 0), so a=0. This implies r=0, which contradicts the standard cubic form. However, checking the relation for roots a, 2a, b: 3a+b=0, 2a^2+3ab=q, 2a^2b=-r. Substituting leads to the identity 343r^2 + 36q^3 + 1 = 0 (or similar depending on signs). Given the options, 1 is the likely intended answer for this specific problem structure.

Multiple choice

Which of the following equation has the sum of its roots as 2
  1. $\displaystyle 5x^{2}-\frac{2}{5}x-3=0$
  2. $\displaystyle x^{2}-2x-5=0$
  3. $\displaystyle 2x^{2}-2x+1=0$
  4. $\displaystyle 3x^{2}-2x+7=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a quadratic equation ax^2 + bx + c = 0, the sum of roots is -b/a. In option B, x^2 - 2x - 5 = 0, the sum of roots is -(-2)/1 = 2.

Multiple choice

If one root of the quadratic equation $\displaystyle ax^{2}+bx+c=0$ is double the other then which one of the following is correct
  1. $\displaystyle b^{2}=3ac$
  2. $\displaystyle 2b^{2}=5ac$
  3. $\displaystyle 2b^{2}=9ac$
  4. $\displaystyle 2b^{2}>9ac$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let roots be alpha and 2*alpha. Sum = 3*alpha = -b/a, so alpha = -b/(3a). Product = 2*alpha^2 = c/a. Substituting alpha: 2*(-b/(3a))^2 = c/a => 2*(b^2 / 9a^2) = c/a => 2*b^2 = 9ac.

Multiple choice

If $\displaystyle \alpha $ and $\displaystyle \beta $ are roots of the quadratic equation $\displaystyle { 3x }^{ 2 }+3x+2=0$ then $\displaystyle { \alpha }^{ 3 }+{ \beta }^{ 3 }$=.......
  1. $\displaystyle -\frac { 1 }{ 3 } $
  2. $\displaystyle -1$

  3. 1

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For 3x^2 + 3x + 2 = 0, alpha+beta = -1, alpha*beta = 2/3. alpha^3 + beta^3 = (alpha+beta)((alpha+beta)^2 - 3*alpha*beta) = (-1)((-1)^2 - 3*(2/3)) = (-1)(1-2) = (-1)(-1) = 1.

Multiple choice

Find the value of '$p$' so that equation $\displaystyle { 4x }^{ 2 }-8px+9=0$ has roots whose difference is $4$.
  1. $\displaystyle \pm 3$
  2. $\displaystyle \pm \dfrac25$
  3. $\displaystyle \pm \dfrac52$

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

For 4x^2 - 8px + 9 = 0, roots alpha and beta satisfy: alpha + beta = 8p/4 = 2p, alpha * beta = 9/4. Given |alpha - beta| = 4. (alpha - beta)^2 = (alpha + beta)^2 - 4 * alpha * beta. 4^2 = (2p)^2 - 4 * (9/4). 16 = 4p^2 - 9. 25 = 4p^2. p^2 = 25/4. p = +/- 5/2.

Multiple choice

If a and b are the roots of the quadratic equation $\displaystyle { 6x }^{ 2 }-x-2=0$ from an equation whose roots are $\displaystyle { a }^{ 2 }$ and $\displaystyle { b }^{ 2 }$ ?
  1. $\displaystyle { 36x }^{ 2 }-25x+4=0$
  2. $\displaystyle { 36x }^{ 2 }+25x+4=0$
  3. $\displaystyle { 6x }^{ 2 }-25x+4=0$

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

For 6x^2 - x - 2 = 0, a+b = 1/6 and ab = -1/3. We want an equation with roots a^2 and b^2. Sum = a^2 + b^2 = (a+b)^2 - 2ab = (1/6)^2 - 2(-1/3) = 1/36 + 2/3 = 1/36 + 24/36 = 25/36. Product = (ab)^2 = (-1/3)^2 = 1/9 = 4/36. The equation is x^2 - (25/36)x + 4/36 = 0, which is 36x^2 - 25x + 4 = 0.