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 $\alpha, \beta, \gamma,\delta$ are the roots of the equation $x^4 - Kx^3 + Kx^2 + Lx + M = 0$, where $K, L$ & $M$ are real numbers, then the minimum value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is
  1. $0$
  2. $-1$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sum of the squares of the roots is given by (sum of roots)^2 - 2(sum of roots taken two at a time). For x^4 - Kx^3 + Kx^2 + Lx + M = 0, the sum of roots is K and the sum of roots taken two at a time is K. Thus, the sum of squares is K^2 - 2K, which is (K-1)^2 - 1. The minimum value is -1 when K = 1.

Multiple choice

The equation having roots $tan(A+B)$ and $tan(A+B)$ is given by:
  1. ${x^2}\cos 2A - x\left( {2\sin 2A} \right)\cos 2B = 0$
  2. ${x^2}\left( {\cos 2A + \cos 2B} \right) - x\left( {2\sin 2A} \right) + \cos 2B - \cos 2A = 0$
  3. ${x^2}\left( {\cos 2A + \cos 2B} \right) - x\sin 2A + \cos 2A - \cos 2B = 0$

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

The question asks for the quadratic equation with roots tan(A+B) and tan(A-B) (the duplicate in the prompt is a typo in the source). Using sum and product of roots where sum = 2*sin(2A)/(cos(2A)+cos(2B)) and product format leads to option B.

Multiple choice

If $\alpha , \, \beta$ are roots of the equation $ax^2 + bx + c = 0$, where $a, b, c$ are distinct real values, then $(1 + \alpha + \alpha^2) ( 1 + \beta + \beta^2)$ is
  1. $1-\left [ \left ( \dfrac{b+c-2c}{a} \right )-\left ( \dfrac{b^2+c^2+bc}{a^2} \right ) \right ]$
  2. $1-\left [ \left ( \dfrac{b+c-2c}{a} \right )-\left ( \dfrac{b^2+c^2+bc}{a^3} \right ) \right ]$
  3. $1-\left [ \left ( \dfrac{a+c-2c}{a} \right )-\left ( \dfrac{a^2+c^2+bc}{a^2} \right ) \right ]$
  4. $1-\left [ \left ( \dfrac{a+b-2c}{a} \right )-\left ( \dfrac{c^2+b^2+ac}{a^2} \right ) \right ]$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

If $\alpha $ and $\beta $ are the roots of equation ${x^2} - 3x + 1 = 0$ and ${a_n} = {\alpha ^n} + {\beta ^n},n \in N$ then the value of $\dfrac{{{a_7} + {a_5}}}{{{a_6}}}$
  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots of x^2 - 3x + 1 = 0 satisfy alpha^2 - 3alpha + 1 = 0, so alpha^2 + 1 = 3alpha. Multiply by alpha^(n-1): alpha^(n+1) + alpha^(n-1) = 3 * alpha^n. Similarly for beta. Adding them: a_(n+1) + a_(n-1) = 3 * a_n. For n=6, a_7 + a_5 = 3 * a_6. Thus (a_7 + a_5) / a_6 = 3.

Multiple choice

If $4-i\sqrt{3}$ is a root of quadratic equation with real coefficients then the equation is?
  1. $x^2-8x+13=0$
  2. $x^2-8x+19=0$
  3. $x^2-8x-13=0$
  4. $x^2-8x-19=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If a quadratic equation has real coefficients and a complex root 4-i*sqrt(3), the other root must be the conjugate 4+i*sqrt(3). The equation is (x - (4-i*sqrt(3))) * (x - (4+i*sqrt(3))) = 0. This is (x-4)^2 - (i*sqrt(3))^2 = 0 -> x^2 - 8x + 16 + 3 = 0 -> x^2 - 8x + 19 = 0.

Multiple choice

If the roots of the equation $\cfrac { { x }^{ 2 }-bx }{ ax-c } =\cfrac { \lambda -1 }{ \lambda +1 } $ are equal and opposite then the value of $\lambda$ is
  1. $\dfrac{a-b}{a+b}$
  2. $c$
  3. $\cfrac{1}{c}$
  4. $\cfrac{a+b}{a-b}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If roots are equal and opposite, their sum is 0. For the equation (x^2 - bx)/(ax - c) = (lambda - 1)/(lambda + 1), cross-multiply to get a quadratic. The sum of roots -B/A = 0 implies the coefficient of x is 0. Solving this leads to lambda = (a - b)/(a + b).

Multiple choice

If $p, q, r$ are the roots of the cubic equation $x^3-3x+4=0$, then value of $\dfrac{1}{p^3+q^3+8}+\dfrac{1}{q^3+r^3+8}+\dfrac{1}{r^3+p^3+8}$ is equal to
  1. maximum value of $f(x)=-x^2+2x-\frac{1}{4}\forall x\in \begin{bmatrix}
    -2,2
    \end{bmatrix}$
  2. minimum value of $g(x)=2x^2-3x+\frac{7}{8}\forall x\in \begin{bmatrix}
    -1,1
    \end{bmatrix}$
  3. $-\frac{1}{\alpha }$ where $\alpha $ is characteristic of $\log_9(6569)$
  4. value of $\log_{(antilog_{128}\left(\frac{4}{7}\right)}\left(\dfrac {1}{2}\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

Product of real roots of the equation $t^{2}x^{2}+|x|+9=0$

  1. is always positve

  2. is always negative

  3. does not exists

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

For real x, t^2x^2 is nonnegative and |x| is nonnegative, while 9 is positive. Therefore the expression cannot equal zero, so there are no real roots and their product does not exist.

Multiple choice

Let $a,b,c$ be positive real numbers. The number of real roots the equation $(a{x}^{2}+bx-c)(b{x}^{2}+cx-a)(c{x}^{2}+ax-b)=0$ may have, is

  1. 2

  2. 0

  3. 4

  4. 6
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Each quadratic factor (ax^2 + bx - c), (bx^2 + cx - a), and (cx^2 + ax - b) has a discriminant D = b^2 - 4ac. For the first: D1 = b^2 + 4ac. Since a, b, c > 0, D1 > 0, so it has 2 real roots. Similarly, D2 = c^2 + 4ab > 0 and D3 = a^2 + 4bc > 0. Each factor provides 2 distinct real roots, totaling 6.