Descartes' Rule

Q. The number of real roots of the cubic equation $x^3-3x+1=0$ is

Q. The sum of the coefficients of all the even powers of $x$ in the expansion of $(2x^2–3x+1{)}^{11}$ is $P_1^m{P}_2^n,$ where $P_1$ and $P_2$ are prime numbers such that $P_1<P_2$ and $m$ and $n$ are positive integers. Then the value of $\frac{P_1+P_2}{n-m}$ is

1. $2$
2. $5$
3. $-2$
4. $7$

Q. If the roots of the equation $z^2+(a+ib)z+c+id=0$ are real, then

1. d² - b²c = abd
2. d² + b²c = acd
3. d² + b²c = abd
4. d² + b²a = abd

Q.

If the sum of the two roots of the equation $4x^3+16x^2-9x-36=0$ is zero, then the roots are

1. 1, 2 -2

2. 

3. 

4. 

Q. The roots of equation $a{x}^2+bx+c=0$ are $\alpha =z^2+z^4+.....+z^{2n}$ and $\beta =z+z^3+.....+z^{2n-1}$, where $z=e^{i\left(\frac{2\pi }{2n+1}\right)},n\in N,$ then

1. $\alpha +\beta =-1$
2. $\alpha +\beta =1$
3. Given equation can be written as $x^2+x+\frac{1}{4}{\sec }^2\left(\frac{\pi }{2n+1}\right)=0$
4. Given equation can be written as $x^2+x+\frac{1}{4}{\cos }^2\left(\frac{\pi }{2n+1}\right)=0$

Q. The nature of the roots of the equation $8x^3+36x^2+54x+27=0$ is:

1. One real and two imaginary
2. All real and distinct
3. Real and equal
4. Real and positive

Q.

The number of real roots of the equation $(x^2+2x{)}^2-(x+1{)}^2-55=0$

1. none of these

2. 1

3. 4

4. 2

Q. Two sets A and B are given as
$A=\{x|x\text{is an integer root of the equation}{x}^5-6x^4+11x^3-6x^2=0\}$
$B=\{x|x\text{is a real root of the equation}a{x}^5+2a{x}^3+2b{x}^2+b=0,a,b\in R\text{such that the given equation have}\text{maximum number of real roots.}\}$
Then, which of the following is correct?

1. $n\left(A\right)>n\left(B\right)$
2. $n\left(A\right)<n\left(B\right)$
3. $n\left(A\right)=n\left(B\right)$
4. cannot comment because $n\left(A\right)$ varries as the values of $a$ and $b$ vary

Q.

Find the equation whose roots are $2x_1+3and2x_2+3,if{x}_{1}and{x}_2$ are the roots of $x^2+6x+7=0$.

1. 

2. 

3. 

4. 

Q. Given, $x=20$ is a root of the equation $\ln (x^2+20a^2)-\ln (a-4)=\ln \left(\frac{9ax}{a-4}\right).$ Then the sum of all distinct roots of the equation is

1. $36$
2. $41$
3. $45$
4. $61$

Q. If $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$, then show that the matrix A is a root of the polynomial $f\left(x\right)=x^3-6x^2+7x+2$.

Q. The quadratic equation $3a{x}^2+2bx+c=0$ has at least one root between $0$ and $1$ if

1. $a+b+c=0$
2. $c=0$
3. $3a+2b+c=0$
4. $a+b=c$

Q. Find the value of p if the equation $3x^2-2x+p=0$ and $6x^2-17x+12=0$ have a common root.

1. $-8/3,-15/4$
2. $-7/3,-14/3$
3. $2,4$
4. none of these

Q. If $g\left(x\right)=\frac{f\left(x\right)}{(x-a)(x-b)(x-c)}$, where $f\left(x\right)$ is a polynomial of degree $<3$, then

1. $\int g\left(x\right)dx=\left|\begin{array}{ccc}1& a& f\left(a\right)\log |x-a|\\ 1& b& f\left(b\right)\log |x-b|\\ 1& c& f\left(c\right)\log |x-c|\end{array}\right|÷\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|+k$
2. $\frac{dg\left(x\right)}{dx}=\left|\begin{array}{ccc}1& a& f\left(a\right){(x-a)}^{-2}\\ 1& b& f\left(b\right){(x-b)}^{-2}\\ 1& c& f\left(c\right){(x-c)}^{-2}\end{array}\right|÷\left|\begin{array}{ccc}{a}^2& a& 1\\ b^2& b& 1\\ c^2& c& 1\end{array}\right|$
3. $\frac{dg\left(x\right)}{dx}=\left|\begin{array}{ccc}1& a& f\left(a\right){(x-a)}^{-2}\\ 1& b& f\left(b\right){(x-b)}^{-2}\\ 1& c& f\left(c\right){(x-c)}^{-2}\end{array}\right|÷\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|$
4. $\int g\left(x\right)dx=\left|\begin{array}{ccc}1& a& f\left(a\right)|x-a|\\ 1& b& f\left(b\right)|x-b|\\ 1& c& f\left(c\right)|x-c|\end{array}\right|÷\left|\begin{array}{ccc}{a}^2& a& 1\\ b^2& b& 1\\ c^2& c& 1\end{array}\right|+k$

Q. Find the roots of equation :
$81y^2-1=0$

1. $\frac{1}{9},\frac{1}{9}$
2. $\frac{-1}{9},\frac{-1}{9}$
3. $\frac{-1}{9},\frac{1}{9}$
4. None

Q. Two sets A and B are given as
$A=\{x|x\text{is an integer root of the equation}{x}^5-6x^4+11x^3-6x^2=0\}$
$B=\{x|x\text{is a real root of the equation}a{x}^5+2a{x}^3+2b{x}^2+b=0,a,b\in R\text{such that the given equation have}\text{maximum number of real roots.}\}$
Then, which of the following is correct?

1. $n\left(A\right)>n\left(B\right)$
2. $n\left(A\right)<n\left(B\right)$
3. $n\left(A\right)=n\left(B\right)$
4. cannot comment because $n\left(A\right)$ varries as the values of $a$ and $b$ vary

Q. From the roots of $f\left(x\right)\cdot g\left(x\right)\cdot h\left(x\right)=0,$ the maximum number of different matrices is formed (where $f\left(x\right),g\left(x\right),h\left(x\right)$ are quadratic polynomials). If no roots in common for these quadratic equations, then which of the following statement(s) is(are) always correct

1. All quadratic equation have two different real roots.
2. Atleast one quadratic equations have equal roots.
3. Atleast one quadratic equation have no real roots.
4. All quadratic equations have non-real roots.

Q. For the equation
$2x^4-3x^3-x^2-3x+2=0$,
Which among the following statement(s) is/are correct?

1. All roots of the equation are real
2. Exactly two roots of the equation are real
3. Product of real roots = $1$.
4. Sum of the real roots = $\frac{5}{2}$

Q. Discriminant of the equation $-3x^2+2x-8=0$ is

1. $-92$
2. $-29$
3. $39$
4. $49$

Q. Two sets A and B are given as
$A=\{x|x\text{is an integer root of the equation}{x}^5-6x^4+11x^3-6x^2=0\}$
$B=\{x|x\text{is a real root of the equation}a{x}^5+2a{x}^3+2b{x}^2+b=0,a,b\in R\text{such that the given equation have}\text{maximum number of real roots.}\}$
Then, which of the following is correct?

1. $n\left(A\right)>n\left(B\right)$
2. $n\left(A\right)<n\left(B\right)$
3. $n\left(A\right)=n\left(B\right)$
4. cannot comment because $n\left(A\right)$ varries as the values of $a$ and $b$ vary

Q. For what values of $k$ will the quadratic equation : ${2x}^2-kx+1=0$ have real and equal roots?

1. $\pm 2\sqrt{2}$
2. $\pm \sqrt{2}$
3. $\pm 3\sqrt{2}$
4. $\pm \sqrt{3}$

Q. What number should be added to $\text{f(x)}=12{\text{x}}^3+16\text{x}2-5\text{x}-8$ so that the resulting polynomial is exactly divisible by $(2\text{x}-1)$.​

Q. For what values of k, will quadratic equation ${9x}^2+3kx+4=0$ have real and equal roots?

1. $\pm 4$
2. $\pm 2$
3. $\pm 1$
4. $\pm 3$

Q. For the equation
$2x^4-3x^3-x^2-3x+2=0$,
Which among the following statement(s) is/are correct?

1. All roots of the equation are real
2. Exactly two roots of the equation are real
3. Product of real roots = $1$.
4. Sum of the real roots = $\frac{5}{2}$

Q. Find the real roots of the equation $7^{{\log }_7(x^2-4x+5)}=(x-1)$.

Q. The product of non-real roots of the equation $(x^2+x-2)(x^2+x-3)=12$ is

1. $1$
2. $2$
3. $3$
4. $6$