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

Let one root of the equation $x^2 + \ell x + m = 0$ is square of other root. If $m \in R$ then
  1. $\ell \in \left(- \infty, \dfrac{1}{4} \right] \cup \{1\}$
  2. $\ell \in (-\infty , o]$
  3. $\ell \in \left(- \infty, \dfrac{1}{9} \right]$
  4. $\ell \in \left(\dfrac{1}{4} , 1 \right]$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let roots be a and a^2. Then a + a^2 = -l and a^3 = m. For real roots, the discriminant D = l^2 - 4m >= 0. Substituting m = a^3 and l = -(a + a^2), we get (a + a^2)^2 - 4a^3 >= 0, which simplifies to a^2(a-1)^2 >= 0. This is always true for real a. The condition for the roots to be real is l^2 >= 4m. Analysis of the range of l leads to the given set.

Multiple choice

In solving a problem, one student makes a mistake in the coefficient of the first degree term and obtains $-9$ and $-1$ for the roots. Another student makes a mistake in the constant term of the equation and obtains $8$ and $2$ for the roots. The correct equation was?
  1. $x^2+10x+9=0$
  2. $x^2-10x+16=0$
  3. $x^2-10x+9=0$

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

First student (mistake in linear coefficient): Roots -9, -1. Product = 9. Correct constant term is 9. Second student (mistake in constant term): Roots 8, 2. Sum = 10. Correct linear coefficient is -10. Equation: x^2 - 10x + 9 = 0.

Multiple choice

If $- 4$ is a root of the equation $x ^ { 2 } + p x - 4 = 0$ and the equation $x ^ { 2 } + p x + q = 0$ has equal roots, then the values of $p$ and $q$ are, respectively
  1. $- 3$ and $\dfrac { 9 } { 4 }$
  2. $3$ and $\dfrac { 9 } { 4 }$
  3. $\dfrac { 9 } { 4 }$ and $3$
  4. $4$ and $3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since -4 is a root of x^2 + px - 4 = 0, (-4)^2 + p(-4) - 4 = 0, so 16 - 4p - 4 = 0, 4p = 12, p = 3. For x^2 + px + q = 0 to have equal roots, the discriminant D = p^2 - 4q = 0. With p = 3, 9 - 4q = 0, so q = 9/4.

Multiple choice

The imaginary roots of the equation $\left( x ^ { 2 } + 2 \right) ^ { 2 } + 8 x ^ { 2 } = 6 x \left( x ^ { 2 } + 2 \right)$ are
  1. $1 \pm i$
  2. $2 \pm i$
  3. $-1 \pm i$
  4. $-2 \pm i$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let u = x^2+2. The equation is u^2 + 8x^2 = 6xu. u^2 - 6xu + 8x^2 = 0. (u-4x)(u-2x) = 0. Case 1: x^2+2 = 4x => x^2-4x+2=0 => x = (4 +/- sqrt(16-8))/2 = 2 +/- sqrt(2). Case 2: x^2+2 = 2x => x^2-2x+2=0 => x = (2 +/- sqrt(4-8))/2 = 1 +/- i. The imaginary roots are 1 +/- i.

Multiple choice

If r,s,t are the roots of the equation ${ 8x }^{ 3 }+1001x+2008=0$ . the value of ${ \left( r+s \right) }^{ 3 }+{ \left( s+t \right) }^{ 3 }+{ \left( t+r \right) }^{ 3 }$ is
  1. $751$
  2. $752$
  3. $753$
  4. $754$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given r, s, t are roots of 8x^3 + 1001x + 2008 = 0. By Vieta's formulas, r+s+t = 0. Thus r+s = -t, s+t = -r, t+r = -s. The expression becomes (-t)^3 + (-r)^3 + (-s)^3 = -(r^3 + s^3 + t^3). Using the identity for roots of cubic equations, the result is 753.

Multiple choice

If ${ \alpha },\beta $ are roots of the equation ${ x }^{ 2 }+px-q=0$ and $\gamma ,\delta $ are roots of ${ x }^{ 2 }+px+r=0,$ then the value of $(\alpha -\gamma )(\alpha -\delta )$ is-

  1. p+r

  2. p-r

  3. q-r

  4. q+r
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

x^2 + px - q = 0 roots alpha, beta. alpha + beta = -p, alpha*beta = -q. x^2 + px + r = 0 roots gamma, delta. gamma + delta = -p, gamma*delta = r. (alpha - gamma)(alpha - delta) = alpha^2 - alpha(gamma + delta) + gamma*delta = alpha^2 - alpha(-p) + r = alpha^2 + p*alpha + r. Since alpha is a root of x^2 + px - q = 0, alpha^2 + p*alpha = q. Substituting, q + r.