Relations Between Roots and Coefficients

Q.

Let $f\left(x\right)$ be a quadratic polynomial such that $f(-1)+f\left(2\right)=0$. If one of the roots of $f\left(x\right)=0$ is $3$, then its other roots lies in

1. $\left(0,1\right)$

2. $\left(1,3\right)$

3. $\left(-1,0\right)$

4. $\left(-3,-1\right)$

Q.

Factorize the following: $64a^3-27b^3-144a^{2}b+108a{b}^2$

Q. If the roots of the equation $a{x}^2+bx+c=0$ are in the ratio $m:n,$ then

1. $c^2\left(mn\right)=ab(m+n{)}^2$
2. $b^2(m+n)=mn$
3. $mn{b}^2=ac(m+n{)}^2$
4. $m+n=b^{2}mn$

Q.

If $a+b=5,ab=2,$ then \(a^{3} +b^{3}=\)

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. 0

3. 1

4. -4

Q.

If the roots of the equation 6 $x^2$ -5x + k = 0 are in the ratio 2:3, then k

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Q. If one root of $6x^2+13x+b+1=0$ is the reciprocal of the other root, then the value of $b$ is

Q. Which of the following is the quadratic equation with roots $-4,6$ ?

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

Q.

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

1. $\frac{(-16)}{21}$

2. $\frac{16}{21}$

3. $\frac{21}{16}$

4. $\frac{(-21)}{16}$

Q. If the product of the roots of the quadratic equation $m{x}^2-2x+(2m-1)=0$ is $3$, then the value of $m$ is

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

Q.

If the ratio of the roots of the equation $x^2$+px+q=0 be equal to the ratio of the roots of $x^2$+lx+m=0. Then

1. p$m^2$ = $q^2$ l

2. $p^2$ l = $q^2$ m

3. $p^2$m = $l^2$ q

4. $p^2$ m = $q^2$ l

Q.

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

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

2. $5$

3. $0$

4. $6$

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+2=0$ and $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ are the roots of the equation $2x^2+2qx+1=0$, then $(\alpha -\frac{1}{\alpha })(\beta -\frac{1}{\beta })(\alpha +\frac{1}{\beta })(\beta +\frac{1}{\alpha })$ is equal to

1. $\frac{9}{4}(9-q^2)$
2. $\frac{9}{4}(9+p^2)$
3. $\frac{9}{4}(9+q^2)$
4. $\frac{9}{4}(9-p^2)$

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. $x^2+5x+84=0$

Q. If $1-p$ is a root of the equation $x^2+px+1-p=0$, then the roots of the equation are

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

Q.

If $tan\alpha ,tan\beta$ are the roots of the equation $x^2$ + ax + b = 0 then the value of $si{n}^2(\alpha +\beta )+asin(\alpha +\beta )cos(\alpha +\beta )+bco{s}^2(\alpha +\beta )=$

1. a

2. $\frac{1}{a}+b$

3. ab

4. b

Q. If the ratio of the roots of $x^2+ax+b=0$ and that of $x^2+cx+d=0$ are equal, then

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

Q. If the roots of a quadratic equation are $-6$ and $7$, then the quadratic equation is

1. $x^2-x-42=0$
2. $x^2+x+42=0$
3. $x^2-x+42=0$
4. $x^2-x+21=0$

Q. If the equation ${(3-{\log }_{\sqrt[12]{124}}(x-2))}^2-4|3-{\log }_{\sqrt[12]{124}}(x-2)|+3=0$ has integral roots $\alpha$ and $\beta$, then $|\alpha +\beta |$ is equal to

1. $5$
2. $7$
3. $3$
4. $6$

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.

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.

The coefficient of $x$ in the quadratic equation $x^2+px+q=0$ was taken as $-14$ in place of $-10$, its roots were found to be $6,4$. If $\alpha ,\beta$ are the roots of correct equation, then the value of ${\alpha }^2+{\beta }^2$ must be equal to

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Q. If $\alpha$, $\beta$ are the roots of $3x^2+7x+5=0$, then the value of ${\left(\frac{1}{3\alpha +7}\right)}^3+{\left(\frac{1}{3\beta +7}\right)}^3$ is

1. $\frac{-28}{3375}$
2. $\frac{-672}{3375}$
3. $\frac{672}{3375}$
4. $\frac{28}{3375}$
   

Q.

If tan A, tan B are the roots of $x^2-Px+Q=0$ the value of $si{n}^2$ (A+B)=(where P, Q $ϵ$ R)

1. $\frac{P^2}{P^2+Q^2}$

2. $\frac{P^2}{(P+Q{)}^2}$

3. $\frac{P^2}{P^2+(1-Q{)}^2}$

4. $\frac{Q^2}{P^2+(1-Q{)}^2}$

Q. If one root of the quadratic equation $a{x}^2-bx-c=0;a,b,c\in R$ is reciprocal of the other, then which one of the following is correct?

1. $a=c$
2. $b=c$
3. $a+c=0$
4. $a+b=0$

Q.

If ∝, β are the roots of the equation $x^2-ax+b=0$, then ${\alpha }^4$+${\alpha }^3$β+${\alpha }^2$${\beta }^2$+α${\beta }^3$+${\beta }^4$

1. None of these

2. $a^3+3a^{2}b$ + $b^2$

3. $a^4+3a^{2}b$ + $b^2$

4. $a^4-3a^{2}b$ + $b^2$

Q.

If $k_1,k_2,k_3$ are real numbers and the equation $k_1{x}^2+k_{2}x+k_3=0$ have three roots, then value of $k_1+k_2+k_3$ is

1. $1$

2. $3$

3. $0$

4. $2$

Q. If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+p{x}^2+qx+r=0,$ $r\ne 0$ and $\beta \gamma +\frac{1}{\alpha },$ $\alpha \gamma +\frac{1}{\beta },$ $\alpha \beta +\frac{1}{\gamma }$ are the roots of the equation $x^3+a{x}^2+bx+c=0,$ $c\ne 0,$ then $\frac{ab}{c}$ is equal to

1. $\frac{pq}{r}$
2. $pq$
3. $\frac{q}{r}$
4. $\frac{p}{r}$

Q.

If the ratio of the roots of the equation $a{x}^2+bx+c=0$ be $p:q$, then

1. $pq{b}^2-(p+q{)}^{2}ac=0$

2. $pq{a}^2-(p+q{)}^{2}bc=0$

3. $pq{b}^2+(p+q{)}^{2}ac=0$

4. $pq{b}^2-(p-q{)}^{2}ac=0$

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0.$ Then find the equation with roots as $\frac{1}{\alpha }\&\frac{1}{\beta }.$

1. $a{\left(\frac{1}{x}\right)}^2+b\left(\frac{1}{x}\right)+cx=0$
2. ${\left(\frac{a}{x}\right)}^2+\left(\frac{b}{x}\right)+c=0$
3. None of the above
4. $a{\left(\frac{1}{x}\right)}^2+b\left(\frac{1}{x}\right)+c=0$