Relations between Roots and Coefficients : Higher Order Equations

Q. Let $\alpha ,\beta$ be two roots of the equation $x^2+(20{)}^{\frac{1}{4}}x+(5{)}^{\frac{1}{2}}=0$. Then ${\alpha }^8+{\beta }^8$ is equal to

1. 160
2. 10
3. 50
4. 100

Q. If alpha and beta are the zeroes of the polynomial x2+7x+3, then the value of (alpha-beta)2

Q.

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

1. $12$

2. $13$

3. $14$

4. $15$

Q. If the equation $a{x}^2+2bx-3c=0$ has non real roots and $\left(3c/4\right)<(a+b);$ then c is always

1. < 0
2. > 0
3. zero
4. $\ge 0$

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

1. 0
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, b, c are positive numbers such that a>b>c and the equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root in the interval (-1, 0), then

1. b cannot be the G.M. of a, c

2. b may be the G.M. of a, c

3. b is the G.M. of a, c

4. none of these

Q. If the roots of the equation $x^4+a{x}^3+b{x}^2+cx+d=0$ are in geometric progression, then

1. $b^2=ac$
2. $a^2=b$
3. $a^2{b}^2=c^2$
4. $c^2=a^{2}d$

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2+5x-7=0$. Then a equation with roots
$\frac{1}{\alpha }and\frac{1}{\beta }$
is .

1. $7x^2-5x-1=0$
2. $7x^2+5x+1=0$
3. $7x^2-5x+1=0$
4. $7x^2+5x-1=0$

Q. Consider the polynomial $f\left(x\right)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of \f(x)=0\) and let $t=|s|$.

1. B
2. A
3. D
4. C

Q. If sum of the squares of zeros of the quadratic polynomial f(x)=xsquare -8x+k is 40 find the value of K

Q. Find the value of $m$ such that one root is greater than $2$ and the other root is smaller than $1$ of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$.

1. $(-\infty ,1)$
2. $(10,\infty )$
3. No solution
4. $(9,\infty )$

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. If a and b are the zeroes of the polynomial f(x)=x*2 - 5x+k such that a-b=1 then find the value of k

Q. For the cubic equation $x^3-3x^2+3x-1=0$, which of the following is/are true

1. All $3$ roots are equal
2. Only one root is repeated
3. It is increasing in $x\in R$
4. It is decreasing in $x\in R$

Q. Let $D_1=\left|\begin{array}{ccc}x& a& b\\ -1& 0& x\\ x& 2& 1\end{array}\right|$ and $D_2=\left|\begin{array}{ccc}c{x}^2& 2a& -b\\ x& 2& 1\\ -1& 0& x\end{array}\right|.$ If all the roots of $(x^2-4x-7)(x^2-2x-3)=0$ satisfies the equation $D_1+D_2=0,$ then the value of $a+4b+c$ is

1. $-1$
2. $4$
3. $0$
4. $42$

Q. If alpha and beta are zeroes of polynomial x-ax+b then value of alpha ( alpha/beta -beta) + beta (beta/alpha-alpha) is?

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.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+8=0$, then the equation whose roots are ${\alpha }^2,{\beta }^{2}and{\gamma }^2$ is

1. 

2. 

3. 

4. 

Q.

If $\alpha ,\beta$ are the roots of the equation $x^2+px+1=0;\gamma ,\delta$ the roots of the equation $x^2+px+1=0;$ then $(\alpha -\gamma )(\alpha +\delta )(\beta -\gamma )(\beta +\delta )$

1. none of these

2. $p^2-q^2$

3. $p^2+q^2$

4. $q^2-p^2$

Q. If $\alpha ,\beta$ are zeroes of a polynomial $6x^2+x–2$, then the polynomial whose zeroes are $2\alpha +3\beta$ and $3\alpha +2\beta$ , is .

1. $6x^2+5x-1$
2. $6x^2-5x+1$
3. $6x^2-5x-1$
4. $6x^2+5x+1$

Q. The sum of all non-integer roots of the equation $x^5–6x^4+11x^3–5x^2–3x+2=0$ is

1. 6
2. -11
3. -5
4. 3

Q. If $\alpha ,\beta ,\gamma$ are the roots of $a{x}^3+b{x}^2+cx+d=0$ and $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|=0$, $\alpha \ne \beta \ne \gamma$, then find the equation whose roots are $\alpha +\beta -\gamma ,\gamma +\alpha -\beta ,\beta +\gamma -\alpha$

1. $a{x}^3-2b{x}^2+4cx+8d=0$
2. $a{x}^3-2b{x}^2+4cx-8d=0$
3. $a{y}^3-2b{y}^2-4cy-8d=0$
4. $a{y}^3-2b{y}^2+4cy-8d=0$

Q.

If $\alpha ,\beta$ are roots of the equation $4x^2+3x+7=0,$ then~,

$\frac{1}{\alpha }+\frac{1}{\beta }$ 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. If $\alpha ,\beta$ are zeroes of a polynomial $6x^2+x–2$, then the polynomial whose zeroes are $2\alpha +3\beta$ and $3\alpha +2\beta$ , is .

1. $6x^2+5x-1$
2. $6x^2-5x+1$
3. $6x^2-5x-1$
4. $6x^2+5x+1$

Q. Let $\alpha$ and $\beta$ be the roots of equation $p{x}^2+qx+r=0,p\ne 0.$ If $p,q,r$ are in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4,$ then the value of $|\alpha -\beta |$ is

1. $\frac{\sqrt{61}}{9}$
2. $\frac{\sqrt{34}}{9}$
3. $\frac{2\sqrt{13}}{9}$
4. $\frac{2\sqrt{17}}{9}$

Q. Let $A\left(z_1\right)$ and $B\left(z_2\right)$ are points on Argand plane. If $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$, then

1. origin, $A$ and $B$ are collinear.
2. origin, $A$ and $B$ are vertices of an equilateral triangle.
3. origin, $A$ and $B$ are vertices of a right angled triangle.
4. origin, $A$ and $B$ are vertices of an isosceles triangle.

Q. Find the values of k for which the zeroes of the polynomial f(n)=n^3 +12n^2+39n+k are in A.P

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