Relations between Roots and Coefficients : Higher Order Equations

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

Q.

Let $\alpha$ and $\beta$ be the roots of $x^2-3x+p=0$ and $\frac{1}{\gamma }$ and $\frac{1}{\delta }$ be the roots of $x^2-6x+q=0$. If $\alpha ,\beta ,\gamma ,\delta$ form a geometric progression. Then ratio $(2q+p):(2q-p)$ is :

1. $33:31$

2. $9:7$

3. $3:1$

4. $5:3$

Q.

Which option is equal to the square of $3x-4y$.

1. $9x^2–16y^2$

2. $6x^2–8y^2$

3. $9x^2+16y^2+24xy$

4. $9x^2+16y^2–24xy$

Q. If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $2^{33x-2}+2^{11x+2}=2^{22x+1}+1,$ then $11(\alpha +\beta +\gamma )$ is equal to

Q. All the roots of the equation $x^5-40x^4+A{x}^3+B{x}^2+Cx+D=0$ are positive and are in geometric progression with common ratio $r$. If sum of the reciprocal of roots is $10,$ then

1. mid term of geometric progression is $2$
2. $D=32$
3. $\frac{1}{r}+r=\frac{-1\pm \sqrt{85}}{2}$
4. $\frac{1}{r}+r=\frac{\sqrt{85}-1}{2}$

Q. Let $f\left(x\right)=Ax+B,A,B\in R$ and $y=f\left(x\right)$ passes through the points $(A,2A-B^2)$ and $(2B+3,(A+B{)}^2-1)$. If $B_1,B_2\cdots B_n,n\in N,$ are different possible value(s) of $B$, then the value of $\sum _{r=1}^n{B}_r$ is

1. $-\frac{9}{5}$
2. $-\frac{9}{10}$
3. $-\frac{18}{5}$
4. $-\frac{9}{20}$

Q. If 1 and –2 are the roots of the equation $x^3–4x^2–px+r=0$, then its third root will be .

1. -1
2. 2
3. -5
4. 5

Q.

If the roots of the equation $8x^3-14x^2+7x-1=0$ are in G.P., then the roots are

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

2. 2, 4, 8

3. 3, 6, 12

4. 1, 2, -4

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

Q. The number of distinct real roots of
$x^4-4x^3+12x^2+x-1=0$ is

1. \N
2. 1
3. 2
4. 3

Q. If $a^2(a+p)=b^2(b+p)=c^2(c+p),$ where $a,b,c,p\in R,$ then value of $ab+bc+ca$ is

1. $-p$
2. $k$
3. $0$
4. $p$

Q.

If the sum of two of the roots of $x^3+p{x}^2+qx+r=0$ is zero, then pq =

1. r

2. 2 r

3. - 2 r

4. - r

Q.

The sum of roots of the polynomial equation $(x-1)(x-2)(x-3)=2(x-2)(x-3)$ is

1. 3

2. 5

3. 1

4. 8

Q.

If α, β, γ are the roots of the equation $x^3+4x+1=0$, then $(\alpha +\beta {)}^{-1}+(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}=$

1. 2

2. 3

3. 4

4. 5

Q. The equation $e^x-x-1=0$ has

1. Only one real root x=0
2. At least two real roots
3. Exactly two real roots
4. Infinitely many real roots

Q. If $a^2(a+p)=b^2(b+p)=c^2(c+p),$ where $a,b,c,p\in R,$ then value of $ab+bc+ca$ is

1. $-p$
2. $k$
3. $0$
4. $p$

Q. The maximum possible number of real roots of equation $x^5-6x^2-4x+5=0$ is

1. \N
2. 3
3. 4
4. 5

Q. The number of distinct real roots of
$x^4-4x^3+12x^2+x-1=0$ is

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

Q.

If $x^2-3x+2$ be a factor of $x^4-p{x}^2+q$, then (p, q) =

1. (3, 4)

2. (4, 5)

3. (4, 3)

4. (5, 4)

Q. Let $f\left(x\right)=Ax+B,A,B\in R$ and $y=f\left(x\right)$ passes through the points $(A,2A-B^2)$ and $(2B+3,(A+B{)}^2-1)$. If $B_1,B_2\cdots B_n,n\in N,$ are different possible value(s) of $B$, then the value of $\sum _{r=1}^n{B}_r$ is

1. $-\frac{9}{10}$
2. $-\frac{18}{5}$
3. $-\frac{9}{20}$
4. $-\frac{9}{5}$