Descartes' Rule

Q. The maximum number of non-negative real roots of $2^x-x-1=0$ equals (CAT 2003)

1. 0
2. 1
3. 2
4. 3

Q. For the equation$x^8+6x^7-5x^4-3x^2+2x-5=0$, determine the maximum number of real roots possible.

1. 4
2. 3
3. 8
4. 7
5. can’t be determined

Q. For the equation $x^8+6x^7-5x^4-3x^2+2x-5=0$, determine the maximum number of real roots possible.

1. 4
2. 3
3. 8
4. 7

Q. How many real roots will be there for the equation $x^{10}+x^8+x^6+x^4+x^2+8=0?$ ___

Q. How many real roots are possible for the below equation?
$(x+1)(x+2)(x+3)(x+4)=24$

1. 0, 1, 2, 3
2. 0, -5
3. 0, 5
4. 0, 3, 2, -1

Q. For the given equation $x^{12}+x^{10}-x^9+x^7-x^6+x^5+x^4-x^2+x-2=0$, how many maximum imaginary roots are possible?___

Q. $x^2+9x-36=0,findthepositiverootoftheequation.Ans.$_____________.
___

Q. If $\alpha$ and $\beta$ are the distinct roots of $a{x}^2+bx+c=0,$ where $a,b$ and $c$ are non-zero real numbers then
$\frac{a{\alpha }^2+b\alpha +6c}{a{\beta }^2+b{\beta }^2+9c}+\frac{a{\beta }^2+b{\beta }^2+19c}{a{\alpha }^2+b\alpha +13c}$ is equal to :

1. $18c$
2. $27c$
3. $\frac{19}{13}$
4. $\frac{17}{8}$
5. $\frac{36}{27}$

Q. What is the maximum number of negative real roots which can be formed for the following equation
$x^8+9x^6+18x^5-9x^3+4x^2+2x+1=0$?

1. 1
2. 2
3. 3
4. 4