Relation between Roots and Coefficients for Quadratic

Q.

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

1. $x^2-2abx-(a^2-b^2{)}^2=0$

2. $x^2-4abx-(a^2-b^2{)}^2=0$

3. $x^2-4abx+(a^2-b^2{)}^2=0$

4. None of these

Q. let α, β are roots of quadratic equation ax^2 + bx + c = 0, if a, b, c are in GP and αβ = 9, the possible value of |α+ β| is

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. -4

3. 6

4. 

Q. Let $\alpha ,\beta$ are the roots of the equation $2x^2-3x-7=0$, then the quadratic equation whose roots are $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ is

1. $14x^2+37x-14=0$
2. $14x^2-37x-14=0$
3. $14x^2-37x+14=0$
4. $14x^2+37x+14=0$

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. 3
2. 0
3. 1
4. 2

Q. If $\alpha$ and $\beta$ are the roots of the equation, $x^2+x\sin \theta -2\sin \theta =0,\theta \in (0,\frac{\pi }{2})$ , then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12})(\alpha -\beta {)}^{24}}$ is equal to

1. $\frac{2^6}{(\sin \theta +8{)}^{12}}$
2. $\frac{2^{12}}{(\sin \theta -8{)}^6}$
3. $\frac{2^{12}}{(\sin \theta +8{)}^{12}}$
4. $\frac{2^{12}}{(\sin \theta -4{)}^{12}}$

Q. The sum and the product of roots of the quadratic equation $2x^2-17x+17=0$ is equal.

1. True
2. False

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. -8

2. -16

3. 16

4. 8

Q. If alpha, beta and gamma are the roots of the equation x^3+4x+1=0, then (alpha+beta)^-1 + (beta+gamma)^-1 + (gamma+alpha)^-1 is

Q. If α, β are roots of the equation 2x2 – 35x + 2 = 0, then the value of (2α– 35)3 × (2β– 35)3 is equal to

Q. Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0,a\ne 0$ and ${\alpha }_1,-\beta$ be the roots of $a_1{x}^2+b_{1}x+c_1=0,a_1\ne 0$. Then the quadratic equation whose roots are $\alpha ,{\alpha }_1$ is

1. $\frac{x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}-x+\frac{1}{(\frac{b}{c}+\frac{b_1}{c_1})}=0$
2. $\frac{x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}+2x+\frac{1}{(\frac{b}{c}-\frac{b_1}{c_1})}=0$
3. $\frac{2x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}+x+\frac{1}{(\frac{b}{c}+\frac{b_1}{c_1})}=0$
4. $\frac{x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}+x+\frac{1}{(\frac{b}{c}+\frac{b_1}{c_1})}=0$

Q. If a root of the equation $x^2+px+12=0$ is 4, while the roots of the equation $x^2+px+q=0$ are same, then the value of q will be

1. $4$
2. $\frac{4}{49}$
3. $\frac{49}{4}$
4. None of these

Q. If p and q are the roots of the quadratic equation
ax2+bx+c=0, $\mathrm{(c}\ne 0)$
then $\frac{1}{ap+b}+\frac{1}{aq+b}$=
.

1. $\frac{c}{ab}$
2. $\frac{b}{ac}$
3. $\frac{a}{bc}$
4. 1

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3-3x+7=0$, then the equation whose roots are $\alpha +\beta -\gamma ,\beta +\gamma -\alpha ,\gamma +\alpha -\beta$ is

1. $x^3-12x-56=0$
2. $x^3+12x-56=0$
3. $x^3+12x+56=0$
4. $x^3-12x+56=0$

Q.

If one root of $5x^2+13x+k=0$ is reciprocal of the other, then k=

1. 

2. 

3. 

4. 

Q. If $\lambda$ be the ratio of the roots of the quadratic equation in $x,3m^2{x}^2+m(m-4)x+2=0,$ then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :

1. $4-3\sqrt{2}$
2. $4-2\sqrt{3}$
3. $2-\sqrt{3}$
4. $-2+\sqrt{2}$

Q. If $\alpha$ and $\beta$ are the roots of the quadratic equation $(x-2)(x-3)+(x-3)(x+1)+(x+1)(x-2)=0$, then the value of $\frac{1}{(\alpha +1)(\beta +1)}+\frac{1}{(\alpha -2)(\beta -2)}+\frac{1}{(\alpha -3)(\beta -3)}$ is

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

Q. If r be the root of x ^ 2 + 2x + 6 = 0 then find the value of (r + 2)(r + 3)(r + 4)(r + 5)

Q.

If one root of $5x^2+13x+k=0$ is reciprocal of the other, then k=

1. $0$

2. $5$

3. $\frac{1}{6}$

4. $6$

Q.

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

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

Q. If one root is $n$ times the other for the equation $a{x}^2+bx+c=0,$ then

1. $n{b}^2=ac(n+1{)}^2$
2. $n^{2}b=ac(n+1{)}^2$
3. $(nb{)}^2=a^{2}c(n+1)$
4. $nb=ac(n+1)$

Q.

If $\alpha and\beta$ are the roots of the equation $4x^2+3x+7=0$, then $\frac{1}{\alpha }+\frac{1}{\beta }=$

1. $-\frac{3}{7}$

2. $\frac{3}{7}$

3. $-\frac{3}{5}$

4. $\frac{3}{5}$

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. $-\alpha ,-\beta$

2. $\alpha ,\frac{1}{\beta }$

3. $\frac{1}{\alpha },\frac{1}{\beta }$

4. $Noneofthese$

Q. If $\alpha$ and $\beta$ are the equation, $7x^2-3x-2=0$, then the value of $\frac{\alpha }{1-{\alpha }^2}+\frac{\beta }{1-{\beta }^2}$ is equal to :

1. $\frac{27}{32}$
2. $\frac{1}{24}$
3. $\frac{27}{16}$
4. $\frac{3}{8}$

Q.

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to the n^th power of the other root, then the value of $(a{c}^n{)}^{\frac{1}{n+1}}$ + $(a^{n}c{)}^{\frac{1}{n+1}}$

1. b

2. -b

3. $b^{\frac{1}{n+1}}$

4. -$b^{\frac{1}{n+1}}$

Q. If α, β are roots of the equation $x^2+lx+m=0$, write an equation whose roots are $-\frac{1}{\alpha }\mathrm{and}-\frac{1}{\beta }$.

Q. If $\alpha ,\beta$ are the roots of $3x^2-2x+1=0$, then the equation whose roots are $-\frac{\alpha }{3},-\frac{\beta }{3}$ is

1. $27x^2+6x-1=0$
2. $9x^2+6x+1=0$
3. $27x^2+6x+1=0$
4. $27x^2-6x+1=0$

Q.

Given f(x) = $a{x}^2+bx+c$

If f(m) = a - b + c

f(n) = 4a + 2b + c

f(p) = a + b + c

Then, m + n + p = __

Q. Let p and q be real numbers such that $p\ne 0$, $p^3\ne 2q$ and $p^3\ne -q$. If $\alpha$ and $\beta$ are non zero complex numbers satisfying $\alpha +\beta =-p$ and ${\alpha }^3+{\beta }^3=q,$ then a quadratic equation having $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ as its roots is

1. $(p^3+q)x^2-(p^3+2q)x+(p^3+q)=0$
2. $(p^3+q)x^2-(p^3-2q)x+(p^3+q)=0$
3. $(p^3-q)x^2-(p^3-2q)x+(p^3+q)=0$
4. $(p^3-q)x^2-(p^3+2q)x+(p^3-q)=0$

Q.

If $p{x}^2$ + qx + r = 0 has no real roots and p, q, r are real such that p + r > 0 then

1. All of these

2. p - q + r > 0

3. p + r = q

4. p - q + r < 0