Higher Order Equations

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a facter of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then value of $P\left(1\right)$ is -

1. $4$
2. $8$
3. $10$
4. $12$

Q.

Given that - 4 is a root of the equation $2x^3+6x^2+7x+60=0$. Find the other roots.

1. 

2. 

3. 

4. 

Q.

If the equation $x^4-3x^3+m{x}^2-7x+3=0$ has four positive real roots. Then the value of m should always be greater than zero.

1. True

2. False

Q.

If the equation $x^4-4x^3+a{x}^2+bx+1=0$ has four roots and all of them are positive real roots then the value of a and b are

1. a = - 4, b = 8

2. a = 6, b = - 4

3. a = - 8, b = - 6

4. a = 8, b = 6

Q.

Divide $4x^3+12x^2+11x+3byx+1$ and then find the quotient.

1. $2x^2+3x+3$

2. 

3. $8x^2+11x+3$

4. $4x^2+6x+1$

Q. The equation $x^{\frac{3}{4}(lo{g}_{2}x{)}^2+lo{g}_{2}x-\frac{5}{4}}=\sqrt{2}$ has

1. atleast one real solution
2. exactly three real solutions
3. exactly one irrational solution
4. complex roots

Q. If $x^2+px+1$ is a factor of 2 $co{s}^2\theta x^3+2x+sin2\theta$, then

1. $\theta =n\pi ,n\epsilon I$
2. $\theta =n\pi +\frac{\pi }{2},n\epsilon I$
3. $\theta =\frac{n\pi }{2},n\epsilon I$
4. $\theta =2n\pi ,n\epsilon I$

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a facter of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then value of $P\left(1\right)$ is -

1. $4$
2. $8$
3. $10$
4. $12$

Q. Exhaustive values of $x$ satisfying the equation $|x^4-x^2-12|=|x^4-9|-|x^2+3|$ is -

1. $x\in [1,\infty )$
2. $x\in (-\infty ,-2]\cup [2,\infty )$
3. $x\in (-\infty ,-1]\cup [1,\infty )$
4. $x\in [-2,2]$

Q. Let $p,q,r$ be roots of cubic equation $x^3+2x^2+3x+3=0$, then

1. $\frac{p}{p+1}+\frac{q}{q+1}+\frac{r}{r+1}=5$
2. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=44$
3. $\frac{p}{p+1}+\frac{q}{q+1}+\frac{r}{r+1}=6$
4. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=38$