Which of the following six statements are tru | vedclass

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

Question

(C) Given the polynomial $P(x) = 2x^3 + x^2 + 3x - 2$.First,we check the derivative $P'(x) = 6x^2 + 2x + 3$.The discriminant of $P'(x)$ is $D = (2)^2

Vedclass · Accepted Answer

only $(i)$ and $(iii)$

Vedclass · Answer

(C) Given the polynomial $P(x) = 2x^3 + x^2 + 3x - 2$.First,we check the derivative $P'(x) = 6x^2 + 2x + 3$.The discriminant of $P'(x)$ is $D = (2)^2 - 4(6)(3) = 4 - 72 = -68 Since the leading coefficient is positive and the discriminant is negative,$P'(x) > 0$ for all $x \in \mathbb{R}$.This implies that $P(x)$ is a strictly increasing function.As a strictly increasing cubic polynomial,it must have exactly one real root and two complex conjugate roots.Evaluating at endpoints: $P(0) = -2$ and $P(1) = 2(1)^3 + (1)^2 + 3(1) - 2 = 4$.Since $P(0)  0$,by the Intermediate Value Theorem,there exists a root in the interval $(0, 1)$.Since there is only one real root and it is positive,statements $(i)$ and $(iii)$ are true.Statement $(ii)$ is false because there are no negative roots.Statement $(iv)$ is false because a cubic polynomial with real coefficients must have either one or three real roots.Statement $(v)$ is false because the only real root is positive.Statement $(vi)$ is false because a cubic polynomial with one real root must have two complex roots.Therefore,only $(i)$ and $(iii)$ are true.

Similar Questions

If $S = \{a \in R : |2a - 1| = 3[a] + 2\{a\}\}$,where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$,then $72 \sum_{a \in S} a$ is equal to:

If both roots of the equation $x^2 - (p - 4)x + 2e^{2 \ln p} - 4 = 0$ are negative,then in which interval does $p$ lie?

What is the number of real solutions to the equation $(5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10$?

Let $\alpha \neq 1$ be a real root of the equation $x^3-a x^2+a x-1=0$,where $a \neq -1$ is a real number. Then,a root of this equation,among the following,is

For which value of $a$ is the expression $x^2 - ax + 1 - 2a^2$ always positive for all real values of $x$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module