Relations Between Roots and Coefficients

Q.

${\text{a}}^2{\text{+b}}^2\text{is equal to ?}$

Q.

If $\alpha$ is a root of $4x^2+2x-1=0$ then root is:

1. $4{\alpha }^3+3\alpha$

2. $4{\alpha }^3-3\alpha$

3. None of these

4. $3{\alpha }^3-4\alpha$

Q. If $x_1,x_2$ are the roots of the equation $5x^2+bx-28=0\&5x_1+2x_2=1\forall b\in Z.$ Then find the integral values of $b.$

1. $-13$
2. $13$
3. None of the above.
4. $-31$

Q. Given that $2$ is a root of the equation $3x^2-p(x+1)=0$ and that the equation
$p{x}^2-qx+9=0$ has equal roots, find the values of $p$ and $q.$

Q.

If the roots of equation $a{x}^2+bx+c=0$ are equal in magnitude but opposite in sign then the value of $b$ will be.

Q.

If $\alpha$ and $\beta$are the roots of the equation $x^2+ax+b=0,$ then $(1/{\alpha }^2)+(1/{\beta }^2)=$

1. $\frac{(a^2-2b)}{b^2}$

2. $\frac{(b^2-2a)}{b^2}$

3. $\frac{(a^2+2b)}{b^2}$

4. $\frac{(b^2+2a)}{b^2}$

Q.

If$\alpha$ and$\beta$ are the roots of the equation$a{x}^2+bx+c=0$ and$S_n={\alpha }^n+{\beta }^n$, $a{S}_{n+1}+b{S}_n+c{S}_{n-1}$ is equal to

1. $0$

2. $abc$

3. $a+b+c$

4. $Noneofthese$

Q. $x^2-6x-2=0$ is a quadratic equation with the roots $\alpha$ and $\beta$.
Also $a_n={\alpha }^n-{\beta }^n,n\in Z^{+}$.
Then the value of $\frac{a_{10}-2a_8}{2a^9}$ is :

1. $6$
2. $9$
3. $2$
4. $3$

Q.

If one root of the equation $a{x}^2+bx+c=0$ the square of the other, then $a(c-b{)}^3=cX,$ where $X$ is

1. $(a-b{)}^3$

2. $a^3+b^3$

3. None of these

4. $a^3-b^3$

Q.

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

1. $x^3–4x–1=0$

2. $x^3–4x+1=0$

3. $x^3+4x–1=0$

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

Q.

If $\alpha ,\beta$ be the roots of the equation
$2x^2-2(m^2+1)x+(m^4+m^2+1)=0$
then $({\alpha }^2+{\beta }^2)=$

1. $1$

2. $m^2$

3. $0$

4. $m$

Q. For the equation $2x^2+(a+1)x+(a-1)=0$, the difference between the roots is equal to their product. Then the value of $a$ is

1. $1$
2. $2$
3. $4$
4. $3$

Q. If the ratio of the roots of $l{x}^2+nx+n=0$ is $p:q$, then

1. $\sqrt{\frac{q}{p}}+\sqrt{\frac{p}{q}}+\sqrt{\frac{l}{n}}=1$
2. $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=1$
3. $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$
4. $\sqrt{\frac{q}{p}}+\sqrt{\frac{p}{q}}+\sqrt{\frac{l}{n}}=0$

Q. The sum of the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$ is , while the product of the roots of the quadratic equation is .

1. $\frac{b}{a}$
2. $\frac{-c}{a}$
3. $\frac{c}{a}$
4. $\frac{-b}{a}$

Q. Find the value of k such that the sum of the squares of the roots of the quadratic equation $x^2-8x+k=0$ is 40

Q.

If the roots of the equations $x^2-bx+c=0$ and $x^2-cx+b=0$ differ by the same quantity, then $b+c$ is equal to

1. 4

2. -4

3. 0

4. 1

Q. If the roots of the quadratic equation $k^2{x}^2+(kx+1)(x+k)+1=0\forall k\ne 0,-1$ are $\alpha ,\beta$, then the value of ${\alpha }^2{\beta }^2+(\alpha \beta +1)(\alpha +\beta )+1$ will be

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

Q. The quadratic equation whose roots are $-4$ and $6$ is given by

1. $x^2-2x-24=0$
2. $x^2-2x+24=0$
3. $x^2+2x-24=0$
4. $x^2-4x-24=0$

Q. A quadratic polynomial $p\left(x\right)$ has $1+\sqrt{5}$ and $1-\sqrt{5}$ as its zeros and $p\left(1\right)=2$. Then the value of $p\left(0\right)$ is

1. $-\frac{4}{5}$
2. $\frac{8}{5}$
3. $-\frac{8}{5}$
4. $\frac{4}{5}$

Q. If $\alpha ,\beta$ are the roots of the equation $5x^2+3x-2=0,$ then the equation whose roots are $7\alpha ,7\beta$ is $a{x}^2+bx+c=0.$ Then the value of $\frac{-c-3b}{a}$ is

1. 7

Q. Suppose $a,c$ are the roots of the equation $p{x}^2-3x+2=0$ and $b,d$ are the roots of the equation $q{x}^2-4x+2=0$. Find the value of $p$ and $q$ such that $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ and $\frac{1}{d}$ are in A.P.

1. $p=0,q=\frac{3}{4}$
2. $p=\frac{15}{8},q=\frac{5}{4}$
3. $p=3,q=-1$
4. $p=1,q=\frac{15}{8}$

Q.

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

Q. Choose the quadratic equations whose sum and product of roots are given as $-5$ and $10,$ respectively.

1. $x^2+5x-10=0$
2. $x^2+10x+5=0$
3. $3x^2+15x+30=0$
4. $x^2+5x+10=0$

Q. If $p,q$ are the roots of the equation $3x^2+7x-2=0,$ then the equation whose roots are $\frac{p}{5}+3,\frac{q}{5}+3$ is $a{x}^2+bx+c=0.$ Now the value of $\frac{c-b}{a}=\frac{k}{75},$ then $k$ is

Q.

Construct the quadratic equation whose roots are $3$ times the roots $5x^2-3x+3=0$

1. $5x^2-9x+27=0$

2. $5x^2+9x+27=0$

3. $5x^2-9x-27=0$

4. $5x^2+9x-27=0$

Q. If $\alpha ,\beta$ be the roots of the equation $x^2+px+q=0,$ then the equation whose roots are $\frac{1}{\alpha +\beta }$ and $\frac{1}{\alpha }+\frac{1}{\beta }$ will be:

1. None of the above
2. $pq{x}^2+(p^2+q)x+p=0$
3. $pq{x}^2+(p^2+q^2)x+p=0$
4. $pq{x}^2+(q^2+p)x+p=0$

Q.

Find the value of 'a' if one root of the quadratic equation ($a^2$ - 5a + 3)$x^2$ + (3a - 1) x + 2 = 0 is twice as large as the other.

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

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

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

4. $\frac{-2}{3}$

Q. If $p,q$ are the roots of the equation $a{x}^2+bx+c=0,$ then the value of $(ap+b{)}^{-2}+(aq+b{)}^{-2}$ is

1. $\frac{b^2+2ac}{a^2{c}^2}$
2. $\frac{b^2-2ac}{a^2{c}^2}$
3. $\frac{b^2-2ac}{ac}$
4. $\frac{b^2-4ac}{a^2{c}^2}$

Q. Let $\alpha$ and $\beta$ are two real roots of the equation $(k+1){\tan }^{2}x-\sqrt{2}\lambda \tan x=1-k,$ where $k\ne -1$ and $\lambda$ are real numbers. If ${\tan }^2(\alpha +\beta )=50,$ then the value of $\lambda$ is

1. $10$
2. $5$
3. $10\sqrt{2}$
4. $5\sqrt{2}$

Q. Which of the following is the quadratic equation with roots $7,-12$ ?

1. $x^2+5x-84=0$
2. $x^2+5x+84=0$
3. $x^2-5x-84=0$
4. $x^2-5x+84=0$