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Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

Suppose $a$ and $b$ are real no. such that the roots of the cubic equation $ax^{3}-x^{2}+bx+1=0$ are all positive real no. then
$0 < 3ab \le 1$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The given equation is $ax^3-x^2+bx+1=0$
Let $\alpha,\,\beta,\,\gamma$ be the roots of the given equation.
We have
$\alpha+\beta+\gamma=\dfrac{1}{a}$
$\alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{b}{a}$
$\alpha\beta\gamma=\dfrac{1}{a}$
It follows that $a,b$ are positive. we obtain
$\dfrac{3b}{a}=3(\alpha\beta+\beta\gamma+\gamma\alpha)\le(\alpha+\beta+\gamma)^2=\dfrac{1}{a^2}$
Which gives, $0<3ab\le1.$
Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

If the sum of two roots of the equation $\displaystyle x^{3}+ax^{2}+bx+c= 0 $ is zero, then value of $ab$ equals

  1. $c$
  2. $2c$
  3. $-2c$
  4. $-c$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given equation is $x^{3}+ax^{2}+bx+c= 0$

Let the roots be $\alpha, -\alpha, \beta$

Then $\alpha-\alpha+\beta=-a$

$\Rightarrow \beta=-a$       ....(1)

Also, $-{\alpha}^{2}+{\alpha}\beta-\alpha\beta=b$

$\Rightarrow -{\alpha}^{2}=b$       .....(2)

Also, $-{\alpha}^{2} \beta=-c$  ....(by (1)and (2))

$\Rightarrow -ab=-c$

$\displaystyle \Rightarrow ab= c $ 

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

If $\displaystyle x^{3}-mx^{2}-3x+2=0$ has two roots equal in magnitude but opposite in sign, then $m$ is:

  1. $\displaystyle \frac{3}{2}$
  2. $\displaystyle \frac{2}{3}$
  3. $\displaystyle -\frac{2}{3}$

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

Let $\alpha ,-\alpha ,\beta $ be the roots of $x^{ 3 }-mx^{ 2 }-3x+2=0$
Then
${ s } _{ 1 }=\alpha -\alpha +\beta =m\ \Rightarrow \beta =m$
Substituting $x=m$ in equation, we get
$m^{ 3 }-m.m^{ 2 }-3.m+2=0\ \Rightarrow m=\cfrac { 2 }{ 3 } $
Hence, option 'B' is correct.

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

The equation $\displaystyle x^{4} - x^{3} + 1 = 0$, has

  1. all imaginary roots

  2. all four real roots

  3. two real and two imaginary roots

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

Let $f(x)=x^4-x^3+1$

our first case is the positive-root case: 
In $f(x),$ there are two sign changes in the positive-root case. 
This number "two" is the maximum possible number of positive zeroes (that is, all the positive x-intercepts) for the given polynomial.
I've finished the positive-root case, so now I look at $f(-x)$. That is, having changed the sign on $x$, I'm now doing the negative-root case:
$f(-x)=x^4+x^3+1$
There is zero sign change in this negative-root case, so there is no negative root.
Therefore, there are max two positive roots and no negative roots, therefore remaining two roots are imaginary.
Hence, option C is correct. 

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

The equation $x-\dfrac{2}{x-1}=1-\dfrac{2}{x-1}$ has

  1. no root

  2. one root

  3. two equal roots

  4. infinitely many roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For equation $x-\dfrac { 2 }{ x-1 } =1-\dfrac { 2 }{ x-1 } $, the term $'x-1'$ is in the denominator. Hence the solution isn't defined. For $x=1$ $\Rightarrow $ $x\neq 1$

We have our equation as $x-\dfrac { 2 }{ x-1 } =1-\dfrac { 2 }{ x-1 } $
cancelling the common term on both sides,we get $x=1$. 
But for well defined solution $x\neq 1$. Hence,this equation has no solution.

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

The equation $\displaystyle x - \frac{5}{x - 2} = 2 - \frac{5}{x - 2}$ has

  1. No real roots

  2. Only one real root

  3. Two real roots

  4. Infinitely many roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x-\dfrac { 5 }{ x-2 } =2-\dfrac { 5 }{ x-2 } $       ...(1)
Equation (1) is valid when $x\neq 2$
Rewriting eq. (1), we get $x=2$
But $x\neq 2$
Therefore, number of roots satisfying eq. (1) are zero.

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

Number of real roots of equation $\displaystyle 2x^{99}+3x^{98}+2x^{97}+3x^{96}+........+2x+3=0$ are

  1. $99$
  2. $49$
  3. $1$
  4. $3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

 $2x^{99}+3x^{98}+2x^{97}+3x^{96}+........+2x+3=0$ .... $(i)$

Taking $2x+3$ common, we get
$(2x+3)(x^{98}+x^{97}+...1)=0$
Since $x^{98}+x^{97}+...1$ can not be equal to zero
Therefore only $2x+3 =0$ or $x=-\dfrac{3}{2}$ is real root
Hence, number of real roots of equation $(i)$ is $1$.

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation
If the equation $\displaystyle 5x^{5}-25x^{4}+ax^{3}+bx^{2}+cx-5=0$ has five positive roots, then the value of $2a + 3b + 2c$ is 

  1. 60

  2. 300

  3. 0

  4. cannot be determine
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If a polynomial has five positive roots, by Vieta's formulas, the coefficients must satisfy specific relations. For 5x^5 - 25x^4 + ax^3 + bx^2 + cx - 5 = 0, the product of roots is 5/5 = 1. If all roots are positive, the sum of roots is 25/5 = 5. Using these, the coefficients a, b, c are determined, and 2a + 3b + 2c evaluates to 0.

Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

Find the number of rational roots of 
$\displaystyle P(x)=2x^{98}+3x^{97}+2x^{96}+.....+2x+3=0$

  1. $2$
  2. $3$
  3. $4$
  4. $5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$P(x)=2x^{98}+3x^{97}+2x^{96}+.....+2x+3=0$

Carrying out the $2x+3$ and $x+1$ as common term
$(2x+3)(x^{97}+x^{96}+....1)=(2x+3)(x+1)(x^{96}+x^{94}+....1)$  
Hecne two rational roots are $x=\dfrac{-3}{2},-1$