Relation between Roots and Coefficients for Quadratic

Q. If $\alpha ,\beta$ are the roots of $2x^2+6x+b=0$ and $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }<2$ then b lies in the interval

1. ( 0, ∞)
2. (- ∞, 0)
3. ( -∞ , 0]
4. ( -1, ∞)

Q. If $\alpha ,and\beta$ are the roots of $x^2+px+q=0$ , the quadratic equation having roots ${\alpha }^3,{\beta }^3$ is given by

1. 
2. 
3. 
4. 

Q. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having $\sqrt[3]{37}+\sqrt[3]{349}$ as one of its roots. Then the product of all the roots of p(x) = 0 is

1. 7
2. 49
3. 56
4. 63

Q. If the roots of the quadratic equation $x^2+px+q=0$ are $tan{30}^{\circ }$ and $tan{15}^{\circ }$, respectively, then the value of $2+q-p$ is :

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

Q.

If α and β are the roots of the equation $x^2+6x+\lambda =0$ and $3\alpha +2\beta =-20$, then λ =

1. -16

2. -8

3. 16

4. 8

Q.

If the sum of the roots of the equation $\lambda x^2+2x+3\lambda =0$ be equal to their product, then λ =

1. -4

2. 6

3. 

4. 4

Q.

α, β are the roots of the equation $x^2$ + 4x + 5. Find ${\alpha }^3$ + ${\beta }^3$

1. 4

2. -6

3. -4

4. 6

Q.

If α and β are the roots of the equation $x^2-a(x+1)-b=0$, then $(\alpha +1)(\beta +1)=$

1. 1-b

2. -b

3. b

4. b-1

Q.

If a root of the equation $a{x}^2+bx+c=0$ be reciprocal of the equation then $a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$, then

1. $(c{c}^{\prime }-a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

2. $(b{b}^{\prime }-a{a}^{\prime }{)}^2=(c{a}^{\prime }-b{c}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

3. $(c{c}^{\prime }+a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

4. $(c{c}^{\prime }-a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

Q.

If the roots of the equation $a{x}^2+bx+c=0$ be$\alpha and\beta$, , then the roots of the equation $c{x}^2+bx+a=0$ are

1. 

2. 

3. 

4.