Relation Between Roots and Coefficients for Higher Order Equations

Q.

The two roots of an equation $x^3-9x^2+14x+24=0$ are in the ration 3 : 2. The roots will be

1. -6, 4, 1

2. 6, 4, 1

3. 6, 4, -1

4. -6, -4, 1

Q. If \(b^{2}<2ac\) and \(a, b, c, d \in \mathbb{R}, \) then the number of real roots of the equation \(ax^{3}+bx^{2}+cx+d=0\) are

Q. If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+p{x}^2+qx+r=0,r\ne 0$ and $\beta \gamma +\frac{1}{\alpha },\alpha \gamma +\frac{1}{\beta },\alpha \beta +\frac{1}{\gamma }$ are the roots of the equation $x^3+a{x}^2+bx+c=0,c\ne 0,$ then

1. $\frac{a}{b}=\frac{p}{q(1-r)}$
2. $c=\frac{(1-r{)}^3}{r}$
3. $b=\frac{p(1-r{)}^2}{r}$
4. $a=\frac{q(1-r)}{r}$

Q.

For equation $x^4-2x^3-2x^2+18x-63=0$ if two of its roots are equal in magnitude but opposite in sign. Then other two roots are

1. None of these

2. Imaginary

3. Real and negative

4. Real and positive

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3+3x^2+4x+5=0$ , then which of the following is/ are true.

1. $\sum \alpha =3$
2. $\sum {\beta }^2{\gamma }^2=-14$
3. $\sum \frac{1}{\alpha }=-\frac{4}{5}$
4. $\sum {\alpha }^2=1$

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, $\frac{3}{2},-\frac{3}{2}$

3. -4, $\frac{3}{2},-\frac{3}{2}$

4. -2, $\frac{2}{3},-\frac{2}{3}$

Q. For the equation $p{x}^4+q{x}^3+r{x}^2+sx+t=0;p>0$, all the roots are positive real numbers, then which of the following is/are true?

1. $s<0$
2. $q<0$
3. $r>0$
4. $t>0$

Q.

If roots of the equation $a{x}^3+b{x}^2+cx+d=0$ remain unchanged by increasing each coefficient by one unit, then

1. a = b $\ne$ c = d $\ne$ 0

2. $a\ne 0,b+c=0,d\ne 0$

3. $a\ne b\ne c\ne d\ne 0$

4. a = b = c = d $\ne$ 0

Q. Which of the following is/are true for the equation, $a_0{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+...+a_n=0$, if $S_1,S_2,S_3\cdots S_n$ are the sum of the roots taken $1,2,3,\cdots ,n$ at a time respectively?

1. $S_2=-\frac{a_2}{a_0}$
2. $S_3=\frac{a_3}{a_0}$
3. $S_n=(-1{)}^n\frac{a_n}{a_0}$
4. $S_1=-\frac{a_1}{a_0}$

Q. If $a_1,a_2,a_3,a_4,a_5$ are the roots of the equation $6x^5-41x^4+97x^3-97x^2+41x-6=0,$ such that $|a_1|\le |a_2|\le |a_3|\le |a_4|\le |a_5|,$ then which of the following is/are correct?

1. $a_1,a_2,a_3$ are in G.P.
2. $a_1,a_2,a_3$ are in H.P.
3. $a_3,a_4,a_5$ are in A.P.
4. the equation has three real roots and two imaginary roots.

Q. The number of real roots of the equation $3x^4+25x^3+6x^2+25x+3=0$ is

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

Q. If the sum of two roots of $x^4-2x^3+4x^2+6x-21=0$ is zero, then which of the following is/are true?

1. One of the roots of the equation is $1+i\sqrt{6}$
2. The equation has only two real roots
3. Sum of all the real roots of the equation is $0$
4. All roots of the equation are real

Q. For the equation, $x^7+3x^6-2x^5+4x^4-4x^3+3x^2-2x+1=0,$
the value of the sum of the roots taken $3$ at a time is:

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

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

Q. Which of the following is/are the roots of the equation, $3x^4+25x^3+6x^2+25x+3=0$ ?

1. $0$
2. $\frac{-25-\sqrt{589}}{6}$
3. All of the above
4. $\frac{-25+\sqrt{589}}{6}$

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3$ + 3$x^2$ + 5x - 6 = 0, find the value of $(\alpha -\frac{1}{\beta \gamma })(\beta -\frac{1}{\gamma \alpha })(\gamma -\frac{1}{\alpha \beta }){(\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma })}^{-1}$

1. $\frac{36}{25}$

2. $\frac{25}{6}$

3. $\frac{36}{125}$

4. $\frac{125}{36}$

Q. The product of two of the four roots of the quadratic equation $x^4-{18}^3+k{x}^2+200x-1984=0$ is $-32.$ Determine the value of $k.$
(correct answer + 3, wrong answer 0)

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

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

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3+2x^2-3x+1=0$, then

1. $\frac{\alpha \beta }{\alpha +\beta }+\frac{\alpha \gamma }{\alpha +\gamma }+\frac{\beta \gamma }{\beta +\gamma }=-\frac{11}{7}$
2. $\frac{\alpha \beta }{\alpha +\beta }+\frac{\alpha \gamma }{\alpha +\gamma }+\frac{\beta \gamma }{\beta +\gamma }=\frac{11}{7}$
3. $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=-3$
4. $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=3$

Q. For an equation $x^3-x^2-6x+6=0$, what is the value of $\sum a^2$, if $a,b,c$ are its roots ?

1. $13$
2. $14$
3. $15$
4. $16$

Q. If the equation $x^4-3x^3+m{x}^2-7x+3=0$ has four positive real roots. Then,

1. the value of $m$ should always be greater than zero.
2. None of these
3. the value of $m$ should be zero
4. the value of $m$ should always be less than zero.

Q. The number of real roots of the equation $(x+1)(x+2)(x+3)(x+4)=120$ is

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