If 2 + 3i is one of the roots of the equation | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the equation $2x^3 - 9x^2 + kx - 13 = 0$ with real coefficients,complex roots must occur in conjugate pairs.Since $2 + 3i$ is a root,its con

Vedclass · Accepted Answer

exists and is equal to $\frac{1}{2}.$

Vedclass · Answer

(B) Given the equation $2x^3 - 9x^2 + kx - 13 = 0$ with real coefficients,complex roots must occur in conjugate pairs.Since $2 + 3i$ is a root,its conjugate $2 - 3i$ must also be a root.Let the roots be $\alpha = 2 + 3i,$ $\beta = 2 - 3i,$ and $\gamma$ be the real root.According to the relationship between roots and coefficients for a cubic equation $ax^3 + bx^2 + cx + d = 0,$ the product of the roots is given by $\alpha \beta \gamma = -\frac{d}{a}.$Here,$a = 2$ and $d = -13,$ so the product of the roots is $\alpha \beta \gamma = -\frac{-13}{2} = \frac{13}{2}.$Calculating the product of the complex roots: $\alpha \beta = (2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 + 9 = 13.$Substituting these into the product formula: $13 \cdot \gamma = \frac{13}{2}.$Solving for $\gamma,$ we get $\gamma = \frac{1}{2}.$Thus,the real root exists and is equal to $\frac{1}{2}.$

If $2 + 3i$ is one of the roots of the equation $2x^3 - 9x^2 + kx - 13 = 0,$ where $k \in R,$ then the real root of this equation

Similar Questions

If $a < 0$,then the inequality $ax^2 - 2x + 4 > 0$ has the solution represented by:

The number of real solutions of the equation $(\frac{3}{2})^x = -x^2 + 5x - 10$ is:

The sum of the squares of two natural consecutive odd numbers is $394$. The sum of the numbers is

If $8, 2$ are the roots of ${x^2} + ax + \beta = 0$ and $3, 3$ are the roots of ${x^2} + \alpha x + b = 0$,then the roots of ${x^2} + ax + b = 0$ are

Let $\alpha$ and $\beta$ be the roots of the equation $px^2 + qx + r = 0$ (where $p \neq 0$). If $p, q, r$ are in $A.P.$ and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$,then the value of $|\alpha - \beta|$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module