Higher Order Equations

Q.

Explain real root with examples.

Q.

Factorise: $x^3-3x^2+3x+7$

Q. The value of $\sqrt{2+\sqrt{2+\sqrt{2+\dots \dots \infty }}}$ is

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

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

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

Q. Find the equation whose roots are the cubes of the roots of $x^3+3x^2+2=0$

1. $x^3+33x^2+12x-8=0$
2. $x^3+33x^2+12x+8=0$
3. $x^3+33x^2-12x+8=0$
4. $x^3-33x^2+12x-8=0$

Q. If $p\left(x\right)$ is a polynomial of degree greater than $2$ such that $p\left(x\right)$ leaves remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively. If $p\left(x\right)$ is divided by $x^2-a^2$ then remainder is

1. $2x$
2. $-2x$
3. $x$
4. $-x$

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 difference of roots of the equation $x^2-px+q=0$ is $1$ then :

1. $p^2+4q^2={\left(1+2q\right)}^2$

2. $4p^2+q^2={\left(1+2q\right)}^2$

3. $p^2-4q^2={\left(1+2q\right)}^2$

4. $4p^2+q^2={\left(1-2q\right)}^2$

Q.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3-2x+1=0$ then the value of $\sum \frac{1}{\alpha +\beta -\gamma }$ is

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

2. $-1$

3. $0$

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

Q. The least positive value of $a$ for which $4^x-a\cdot 2^x-a+3\le 0$ is satisfied by atleast one real value of $x$ is

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. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=38$

Q.

If one root of the quadratic equation $2x^2+kx-6=0$ is $2$, then the value of $k$ is

1. 1

2. -1

3. 0

4. -2

Q. The number of integral roots of the equation $x^4+\sqrt{x^4+20}=22$ is

1. $0$
2. $8$
3. $2$
4. $4$

Q. If the two equations $x^3+3p{x}^2+3qx+r=0$ and $x^2+2px+q=0$ have a common root, then the value of $4(p^2-q)(q^2-pr)$ is

1. $(pq+r{)}^2$
2. $(pq-r{)}^2$
3. $(p-qr{)}^2$
4. $(p+qr{)}^2$

Q. If one root of the equation $x^4-9x^3+27x^2-29x+6=0$ is $2-\sqrt{3}$, then the number of rational root(s) of the equation are

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

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

Q. If the product of two roots of the equation $x^4-5x^3+5x^2+5x-6=0$ is $3$, then which of the following is/are correct?

1. The equation has only one negative root.
2. The equation has three negative roots.
3. The product of all positive roots will be $6.$
4. The product of all negative roots will be $6.$

Q. The number of integral values of $a$ for which $y=\frac{a{x}^2-7x+5}{5x^2-7x+a}$ can take all real values, where $x\in R$ (wherever the function is defined), is

Q. If all the roots of the equation $x^3+px+q=0,p,q\in R,q\ne 0$ are real, then which the following is correct?

1. $p=0$
2. $p>0$
3. $p\le 0$
4. $p<0$

Q. The sum of values of $x$ satisfying the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$ is

1. $1$
2. $\frac{5}{13}$
3. $-1$
4. $-\frac{5}{13}$

Q. If $\alpha ,\beta ,\gamma$ are non zero roots of $x^3+p{x}^2+qx+r=0$, then the equation whose roots are $\alpha (\beta +\gamma ),\beta (\gamma +\alpha ),\gamma (\alpha +\beta )$

1. $x^3-2q{x}^2+(pr+q^2)x+(r^2-pqr)=0$
2. $x^3-2q{x}^2+(pr+q^2)x+(r^2+pqr)=0$
3. $x^3+2q{x}^2+(pr+q^2)x+(r^2+pqr)=0$
4. $x^3+2q{x}^2+(pr+q^2)x+(r^2-pqr)=0$

Q. Let $m_1,m_2,m_3$ be the slopes of all the three straight lines represented by an equation $y^3+(2a+5)x{y}^2-6x^{2}y-2a{x}^3=0.$ If $a,m_1,m_2,m_3$ are all integers, then which of the following holds good?

1. $a+\sum _{i=1}^3{m}_i=-1$
2. $a+\sum _{i=1}^3{m}_i=-5$
3. $a+\prod _{i=1}^3{m}_i=0$
4. $a+\prod _{i=1}^3{m}_i=4$

Q. If the polynomial equation $(x^2+x+1{)}^2-(m-3\left)\right(x^2+x+1)+m=0,m\in R$ has two distinct real roots, then $m$ lies in the interval

1. $(\frac{45}{4},\infty )$
2. $(\frac{9}{2},\infty )$
3. $(-\infty ,\frac{-45}{4})$
4. $(9,\infty )$

Q. The linear factor(s) of the equation $9x^2-24xy+16y^2-12x+16y-12=0$ is/are

1. $3x-2y+6=0$
2. $3x-2y+2=0$
3. $3x-4y+2=0$
4. $3x-4y-6=0$

Q. The absolute difference between the roots of the equation $4^x-3\left(2^{x+3}\right)+108=0$ is

1. $12$
2. $1$
3. ${\log }_{3}2$
4. ${\log }_{2}3$

Q. For the equation $\frac{4x^2+x+4}{x^2+1}+\frac{x^2+1}{x^2+x+1}=\frac{31}{6}$
Which of the following statement(s) is/are correct?

1. The equation has $4$ real and distinct roots.
2. The equation has $3$ real and distinct roots.
3. The sum of all real and distinct roots is $-2.$
4. The sum of all real and distinct roots is $-1.$

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. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=38$

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2+3x+1=0$, then the value of ${\left(\frac{\alpha }{1+\beta }\right)}^2+{\left(\frac{\beta }{\alpha +1}\right)}^2$ is equal to

1. $18$
2. $19$
3. $20$
4. $21$

Q. Find the equation whose roots are the cubes of the roots of $x^3+3x^2+2=0$

1. $x^3-33x^2+12x-8=0$
2. $x^3+33x^2+12x-8=0$
3. $x^3+33x^2+12x+8=0$
4. $x^3+33x^2-12x+8=0$