Biquadratic Equations of the form: ax^4+bx^3+cx^2+bx+a=0

Q.

The roots of the equation $x^4-4x^3+6x^2-4x+1=0$ are

1. 2, 2, 2, 2

2. 3, 1, 3, 1

3. 1, 2, 1, 2

4. 1, 1, 1, 1

Q.

The equation $12x^4-56x^3+89x^2-56x+12=0$

1. has all roots integral

2. has all roots rational

3. two roots real, two roots imaginary

4. has all roots irrational

Q.

Solve the equation $x^4-16x^3+86x^2-176x+105=0$. If two roots being $1$ and $7$, Find the sum of the square of other two roots.

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Q. If $x=\sqrt{2}+\sqrt{3}+\sqrt{6}$ is a root of $x^4+a{x}^3+b{x}^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$ ?

Q. The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

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

Q. If the equation $x^4+k{x}^2+k=0$ has exactly two distinct real roots, then the smallest integral value of $|k|$ is

Q. If the equation ${\cot }^{4}x-2{\text{cosec}}^{2}x+a^2=0$ has at least one real solution in $x$, then the number of possible integral values of $a$ is

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

Q. Assume the biquadratic $x^4-a{x}^3+b{x}^2-ax+d=0$ has four real roots with $\frac{1}{2}<\alpha ,\beta ,\gamma ,\delta \le 2.$ Maximum possible value of $\frac{(\alpha +\beta )(\alpha +\gamma )\delta }{(\delta +\beta )(\delta +\gamma )\alpha }$ is

1. $\frac{4}{5}$
2. $1$
3. $\frac{5}{4}$
4. $2$

Q. The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has only $2$ real roots which are negative is

1. $(-\frac{1}{2},\frac{5}{2}]$
2. $\varphi$
3. $(-\infty ,-\frac{1}{2}]$
4. $[\frac{5}{2},\infty )$

Q.

If the equation $x^4-p{x}^2+3x+5=0$ has $2$ as a one root. Find the value of $p$.

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Q. The number of real roots of the equation $2x^4-4x^3-4x+2=0$ is

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

Q.

If bi-quadratic equation $x^4-2x^3+4x^2+6x-21=0$ can also be written as $(x^2+p)(x^2-2x+q)=0$. Find the sum of $p+q$.

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Q. The number of distinct positive real roots of the equation $(x^2+6{)}^2-35x^2=2x(x^2+6)$ is

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

Q. If all the roots of the equation $p{x}^4+q{x}^2+r=0,p\ne 0,q^2\ge 9pr$ are real, then which of the following option(s) is/are correct?

1. $q>0,p>0,r>0$
2. $q>0,p<0,r>0$
3. $q<0,p>0,r>0$
4. $q>0,p<0,r<0$

Q. The number of real roots of the polynomial equation $x^4-x^2+2x-1=0$ is

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

Q. The product of all roots of the equation $(x^2+7x+3{)}^2-(x-6\left)\right(x-1\left)\right(x-9)=5$ is

Q. Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+ax+1$ be a polynomial, where $a,b\in R$. If $b=-1$, then the range of $a$ for which $f\left(x\right)=0$ does not have real roots is

1. $(-\frac{1}{2},\frac{1}{2})$
2. $(-3,3)$
3. $(-\frac{5}{2},\frac{5}{2})$
4. $(-1,1)$

Q. The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

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

Q. The roots of the equation, \(6x^4-25x^3+12x^2+25x+6=0\) are

Q. Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+ax+1$ be a polynomial, where $a,b\in R$. If $b=-1$, then the range of $a$ for which $f\left(x\right)=0$ does not have real roots is

1. $(-\frac{5}{2},\frac{5}{2})$
2. $(-\frac{1}{2},\frac{1}{2})$
3. $(-3,3)$
4. $(-1,1)$

Q. Among the given polynomials equations, select the biquadratic polynomial equation(s).

1. $-4x^4-x^3+2x-9=0$
2. $x^2-x=0$
3. $(x^2-4)(2x-x^2)=0$
4. $(x-2)(2x-4)(x-4)(x+1)=0$

Q. The number of distinct positive real roots of the equation $(x^2+6{)}^2-35x^2=2x(x^2+6)$ is

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

Q. If the equation $x^4-(k-1)x^2+(2-k)=0$ has three distinct real roots, then the possible value(s) of $k$ is/are

1. $\{2\sqrt{2},\sqrt{3}-\sqrt{2}\}$
2. $\{\sqrt{2}-1,2\}$
3. $\left\{2\right\}$
4. $\{\sqrt{5}-1\}$

Q. The roots of the equation $2x^4+x^3-11x^2+x+2=0$ is/are

1. $\frac{-3-\sqrt{5}}{2}$
2. $\frac{-5+\sqrt{3}}{2}$
3. $\frac{1}{2}$
4. $\frac{1}{4}$

Q. The product of all roots of the equation \((x^2-5x+7)^2 - (x-2)(x-3)=1\) is

Q. The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has $2$ positive and $2$ negative roots is

1. $(\frac{5}{2},\infty )$
2. $(-\infty ,-\frac{1}{2})$
3. $\varphi$
4. $(-\frac{1}{2},\frac{5}{2})$

Q. If the equation $x^4-(k-1)x^2+(2-k)=0$ has three distinct real roots, then the possible value(s) of $k$ is/are

1. $\{\sqrt{5}-1\}$
2. $\left\{2\right\}$
3. $\{\sqrt{2}-1,2\}$
4. $\{2\sqrt{2},\sqrt{3}-\sqrt{2}\}$

Q. A polynomial equation with a degree $4$ will have roots.

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

Q. Among the given polynomials equations, select the biquadratic polynomial equation(s).

Q. If all the roots of the equation $p{x}^4+q{x}^2+r=0,p\ne 0,q^2\ge 9pr$ are real, then which of the following option(s) is/are correct?

1. $q>0,p>0,r>0$
2. $q>0,p<0,r<0$
3. $q<0,p>0,r>0$
4. $q>0,p<0,r>0$