Relations between Roots and Coefficients : Higher Order Equations

Q.

One root of the following given equation $2x^5-14x^4+31x^3-64x^2+19x+130=0$ is

1. 1

2. 5

3. 7

4. 3

Q.

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

1. 

2. 2, 4, 8

3. 3, 6, 12

4. 1, 2, -4

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.

If x, y, z are real and distinct, then $x^2+4y^2+9z^2-6yz-3zx-2xy$

1. Non-negative

2. Non-positive

3. positive only

4. Zero

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. 5

3. 3

4. 4

Q.

One root of the following given equation $2x^5-14x^4+31x^3-64x^2+19x+130=0$ is

1. 1

2. 3

3. 5

4. 7

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 roots of equation $x^3-x-1=0$, then the equation whose roots are $\frac{1}{\beta +\gamma },\frac{1}{\gamma +\alpha },\frac{1}{\alpha +\beta }$ is -

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

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. 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.

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. (5, 4)

4. (4, 3)

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. r

3. 2 r

4. - 2 r

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. 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 $\alpha ,\beta$ 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.

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)