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

$x^{2}+x+1$ is a factor of $ax^{3}+bx^{2}+cx+d=0$ where $(a,b,c,d, \in \ R)$, then the real root of above equation is
  1. $-d/a$
  2. $d/a$
  3. $(b-a)/a$
  4. $(a-b)/a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since x^2 + x + 1 is a factor of ax^3 + bx^2 + cx + d = 0, we can write the cubic as (x^2 + x + 1)(ax + k) = 0. Expanding this gives ax^3 + (a+k)x^2 + (a+k)x + k = 0. Comparing the constant term, k = d. The real root comes from the linear factor ax + d = 0, which gives x = -d/a.

Multiple choice

If m and n are the roots of the equation $x^2+ax+b=0$ and $m^2$, $n^2$ are the roots of the equation $x^2-cx+d=0$, then which of the following is/are correct? I. $2b-a^2=-c$ II. $b^2=d$ Select the correct answer using the codes given.

  1. Only I

  2. Only II

  3. Both I and II

  4. Neither I nor II
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given roots m, n for x^2+ax+b=0, we have m+n=-a and mn=b. For x^2-cx+d=0 with roots m^2, n^2, we have m^2+n^2=c and m^2n^2=d. Using (m+n)^2 = m^2+n^2+2mn, we get (-a)^2 = c + 2b, so a^2 = c + 2b or 2b-a^2 = -c (I is correct). Also, (mn)^2 = d, so b^2 = d (II is correct).

Multiple choice

If one root of quadratic equation ${ x }^{ 2 }-x+3m=0$ is 4 times the root of the equation ${ x }^{ 2 }-x+m=,0$ where m$\neq 0,$ then m equals
  1. $\dfrac { 12 }{ 196 } $
  2. $\dfrac { 12 }{ 169 } $
  3. $\dfrac { 12 }{ 256 } $
  4. $\dfrac { 12 }{ 189 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let r be the relevant root of x^2 - x + m = 0. The corresponding root of x^2 - x + 3m = 0 is 4r. Using r^2 - r + m = 0 and 16r^2 - 4r + 3m = 0 gives r = 1/13, so m = r - r^2 = 12/169.

Multiple choice

If $\alpha$ and $\beta$ are the roots of quadratic equation, then the value of ${\left(\dfrac{\alpha}{\beta}-\dfrac{\beta}{\alpha}\right)}^{2}$ is
  1. $\dfrac{b^{2}(b^{2}-4ac)}{c^{2}a^{2}}$
  2. $\dfrac{b^{2}(b^{2}+4ac)}{c^{2}a^{2}}$
  3. $\dfrac{b^{2}(b^{2}-2ac)}{c^{2}a^{2}}$
  4. $\dfrac{b^{2}(b^{2}+2ac)}{c^{2}a^{2}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(alpha/beta - beta/alpha)^2 = ((alpha^2 - beta^2) / (alpha*beta))^2 = ((alpha+beta)^2 * (alpha-beta)^2) / (alpha*beta)^2. Since (alpha-beta)^2 = (alpha+beta)^2 - 4*alpha*beta, we have ((-b/a)^2 * (b^2/a^2 - 4c/a)) / (c^2/a^2) = (b^2/a^2 * (b^2 - 4ac)/a^2) / (c^2/a^2) = b^2(b^2 - 4ac) / (a^2*c^2).

Multiple choice

Use the quadratic formula to solve $x^2-4x-8=0$.
  1. $2 \pm 2\sqrt 3$
  2. $1 \pm \sqrt 3$
  3. $3-3\sqrt 3$

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

Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a for x^2-4x-8=0, we get x = (4 +/- sqrt(16 - 4(1)(-8))) / 2 = (4 +/- sqrt(48)) / 2 = (4 +/- 4sqrt(3)) / 2 = 2 +/- 2sqrt(3).