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

Find the value of p if the equation $3x^2-2x+p=0$ and $6x^2-17x+12=0$ have a common root.
  1. $-8/3,-15/4$
  2. $-7/3,-14/3$
  3. $2,4$

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

Solve 6x^2 - 17x + 12 = 0 using the quadratic formula or factoring: (2x-3)(3x-4)=0, so roots are 3/2 and 4/3. Substitute these into 3x^2 - 2x + p = 0. For x=3/2: 3(9/4) - 2(3/2) + p = 0 => 27/4 - 3 + p = 0 => p = -15/4. For x=4/3: 3(16/9) - 2(4/3) + p = 0 => 16/3 - 8/3 + p = 0 => p = -8/3.

Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$ $(a\neq 0,$ a, b, c being different), then the sum of the roots of the equation $a^{2}x^{2}+(b^{2}-2ac)x+b^{2}-4ac=0$ is

  1. Zero

  2. Positive

  3. Negative

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

For ax^2 + bx + c = 0, alpha + beta = -b/a. For the second equation, the sum of roots is -(b^2 - 2ac) / a^2. Since a, b, c are different and the roots of the first are real (implied by context), the expression -(b^2 - 2ac) / a^2 is negative.

Multiple choice

If $\alpha ,\beta $ are roots of the equation $2x^{2}+6x+b=0$ where $b<0$, then find least integral value of $\displaystyle \left ( \dfrac{\alpha ^{2}}{\beta }+\dfrac{\beta ^{2}}{\alpha } \right )$.
  1. $10$
  2. $11$
  3. $12$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

alpha + beta = -3, alpha*beta = b/2. Expression = (alpha^3 + beta^3) / (alpha*beta) = ((alpha+beta)^3 - 3*alpha*beta*(alpha+beta)) / (alpha*beta) = (-27 - 3*(b/2)*(-3)) / (b/2) = (-27 + 4.5b) / (b/2) = -54/b + 9. Since b < 0, let b = -k (k>0). Expression = 54/k + 9. To minimize this, k should be large, but b is a coefficient. With integer roots or constraints, 10 is the least value.

Multiple choice

If $\displaystyle \alpha $ is a root repeated twice of the quadratic equation $\displaystyle \left ( a-d \right )x^{2}+ax+\left ( a+d \right )=0 $ then $\displaystyle \frac{d^{2}}{a^{2}}$ has the value equal to
  1. $\displaystyle \sin ^{2}90^{\circ}$
  2. $\displaystyle \cos ^{2}60^{\circ}$
  3. $\displaystyle \sin ^{2}45^{\circ}$
  4. $\displaystyle \cos ^{2}30^{\circ}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If alpha is a repeated root, the discriminant must be zero. For (a-d)x^2 + ax + (a+d) = 0, D = a^2 - 4(a-d)(a+d) = a^2 - 4(a^2 - d^2) = 0. So a^2 - 4a^2 + 4d^2 = 0, which means 4d^2 = 3a^2, or d^2/a^2 = 3/4. This equals cos^2(30 degrees) = (sqrt(3)/2)^2 = 3/4.

Multiple choice

If $\displaystyle \alpha : and: \beta $ are the roots of $\displaystyle x^{2}+p=0 $ where p is a prime then which equation has the roots $\displaystyle \frac{1}{\alpha }: &amp;: \frac{1}{\beta } $?
  1. $\displaystyle \frac{1}{x^{2}}+\frac{1}{p}=0 $
  2. $\displaystyle px^{2}+1=0$
  3. $\displaystyle px^{2}-1=0$
  4. $\displaystyle \frac{1}{x}+\frac{1}{p^2}=0 $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots of x^2 + p = 0 are alpha and beta. alpha + beta = 0, alpha * beta = p. New roots are 1/alpha and 1/beta. New sum = (alpha + beta) / (alpha * beta) = 0/p = 0. New product = 1 / (alpha * beta) = 1/p. Equation: x^2 - (sum)x + product = 0 => x^2 + 1/p = 0 => px^2 + 1 = 0.

Multiple choice

If $\alpha ,\beta $ are the roots of the equation ${ x }^{ 2 }-3x+a=0,a\in R$ and $\alpha <1<\beta $, then $a$ belong to
  1. $\displaystyle \left( -\infty ,\frac { 9 }{ 4 } \right) $
  2. $\left( -\infty ,2 \right) $
  3. $\left( 2,\infty \right) $
  4. $\displaystyle \left( \frac { 9 }{ 4 } ,\infty \right) $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the quadratic f(x) = x^2 - 3x + a, if the roots alpha and beta satisfy alpha < 1 < beta, then f(1) must be less than 0. f(1) = 1 - 3 + a = a - 2. So, a - 2 < 0, which means a < 2.