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

Solve the following equation: $x^2-7x+12=0$
  1. $x_1=1,x_2=2$
  2. $x+1=2,x+2=3$
  3. $x_1=3,x_2=4$
  4. $x_1=5,x_2=6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The quadratic equation x^2 - 7x + 12 = 0 can be factored as (x-3)(x-4) = 0. Thus, the roots are x = 3 and x = 4.

Multiple choice

The correct statement is / are-
  1. If $x _ { 1 }$ $\&$ $x _ { 2 }$ are roots of the equation $2 x ^ { 2 } - 6 x - b = 0 ( b > 0 ) ,$ then
    $\dfrac { x _ { 1 } } { x _ { 2 } } + \dfrac { x _ { 2 } } { x _ { 1 } } < - 2$
  2. Equation $a x ^ { 2 } + b x + c = 0$ has real roots if $a < 0 , c > 0$ and $b \in R$
  3. If $P ( x ) = a x ^ { 2 } + b x + c$ and $Q ( x ) = - a x ^ { 2 } + b x + c ,$ where $a c \neq 0$ and ${ a },{ b },{ c }\in { R },$ then ${ P }({ x })\cdot { Q }({ x })$ has at least two real roots.

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

If $\alpha $ and $\beta $ are the roots of the equation ${ x }^{ 2 }+px+q=0$ and if the sum $\left( \alpha +\beta \right) x-\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } .{ x }^{ 2 }+\frac { { \alpha }^{ 3 }+{ \beta }^{ 3 } }{ 3 } .{ x }^{ 3 }-...$ exists then it is equal to
  1. $log\left( { x }^{ 2 }+px+q \right) $
  2. $log\left( { x }^{ 2 }-px+q \right) $
  3. $log\left( 1+px+{ qx }^{ 2 } \right) $
  4. $log\left( 1-px+{ qx }^{ 2 } \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

Equations ${ ax }^{ 2 }+bx+c=0$ and ${ cx }^{ 2 }+bx+a=0$have a common root and the difference of the other roots is 1. Maximum value of $\left| \frac { a }{ c } \right| $ is _______________.
  1. $\frac { \sqrt { 5 } +1 }{ 2 } $
  2. $\frac { \sqrt { 5 } -1 }{ 2 } $

  3. 1

  4. 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let alpha be the common root. Then a*alpha^2 + b*alpha + c = 0 and c*alpha^2 + b*alpha + a = 0. Subtracting gives (a-c)(alpha^2 - 1) = 0. If a != c, alpha = 1 or -1. If alpha = 1, a+b+c=0. If alpha = -1, a-b+c=0. Using the difference of other roots = 1, the maximum value of |a/c| is (sqrt(5)+1)/2.