Transformation of Roots: Algebraic Transformation

Q.

If $\alpha ,\beta$ are the roots of the equation $2x^2-3x-5=0$, then the equation whose roots are $\frac{5}{\alpha }$ , $\frac{5}{\beta }$ is :

1. $x^2-3x-10=0$

2. $x^2+3x+10=0$

3. $x^2-3x+10=0$

4. $x^2+3x-10=0$

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

1. $ab{x}^2+(a+c)bx+(a+c{)}^2=0$
2. $ac{x}^2+(a+c)bx+(a+c{)}^2=0$
3. $ac{x}^2+(a+c)cx+(a+c{)}^2=0$
4. None of these

Q. If $\alpha ,\beta$ are the root of a quadratic equation $x^2-3x+5=0$, then the equation whose roots are $({\alpha }^2-3\alpha +7)\text{and}({\beta }^2-3\beta +7)$ is

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

Q. The equation whose roots are the square of the roots of $x^2+4x+5=0$ is

1. $x^2-6x+25=0$
2. $x^2+10x+25=0$
3. $x^2+6x+25=0$
4. $x^2-26x+25=0$

Q.

Suppose $a,b\in R$ and $a\ne 0,b\ne 0$. Let $\alpha ,\beta$ be the roots of $x^2+ax+b=0$. Find the equation whose roots are ${\alpha }^2,{\beta }^2$.

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

2. $x^2+(2b-a^2)x+a^2=0$

3. $x^2+(a^2-2b)x+b^2=0$

4. $x^2+(2b-a^2)x+b^2=0$

Q. If $\alpha ,\beta$ are the roots of the equation $2x^2-3x-6=0$, then the equation whose roots are ${\alpha }^2-1$ and ${\beta }^2-1$ is

1. $2x^2-25x-7=0$
2. $2x^2-17x+7=0$
3. $4x^2-17x+7=0$
4. $4x^2-25x+7=0$

Q.

Find the equation whose roots are reciprocals of the roots of $2x^2+5x+4=0$ .

1. $2x^2+4x+5=0$

2. $5x^2+4x+2=0$

3. $4x^2+5x+2=0$

4. $4x^2+2x+5=0$

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. The number of quadratic equation(s), with real roots which remain unchanged even after squaring its roots, is

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

Q. If $\alpha ,\beta$ are the root of a quadratic equation $x^2-3x+5=0$, then the equation whose roots are $({\alpha }^2-3\alpha +7)\text{and}({\beta }^2-3\beta +7)$ is

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

Q. If $\alpha ,\beta$ are the roots of $a{x}^2+c=bx$, then the equation $(a+cy{)}^2=b^{2}y$ in $y$ has the roots

1. ${\alpha }^{-1},{\beta }^{-1}$
2. ${\alpha }^{-2},{\beta }^{-2}$
3. ${\alpha }^2,{\beta }^2$
   
4. $\alpha {\beta }^{-1},{\alpha }^{-1}\beta$

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-2x+4=0$, then the value of ${\alpha }^6+{\beta }^6$ is

Q. If \(\alpha, \beta\) are the roots of \(2x^2-3x+4=0\), then the equation whose roots are \(\alpha^2\) and \(\beta^2\) is

Q.

If $a,b,c$ are the roots of the equation $x^3+2x^2+1=0$ . Find the equation whose roots are $b+c-a,c+a-b,a+b-c.$

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

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

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

4. $x^3$ + $x^2+4x+9=0$

Q. If $\alpha ,\beta$ are the roots of $2x^2-3x+4=0$, then the equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is

1. $4x^2-7x+16=0$
2. $4x^2-7x-16=0$
3. $4x^2+7x+16=0$
4. $4x^2+7x-16=0$

Q.

If $\alpha ,\beta$ are the roots of equation $a(x^2-1)+2bx=0$, then the equation whose roots are $2\alpha -\frac{1}{\beta }\&2\beta -\frac{1}{\alpha }$ is

1. $a{x}^2+2bx-a=0$

2. $a{x}^2+6bx+9a=0$

3. $b{x}^2+6ax-9b=0$

4. $a{x}^2+6bx-9a=0$

Q.

Suppose $a,b\in R$ and $a\ne 0,b\ne 0$. Let $\alpha ,\beta$ be the roots of $x^2+ax+b=0$. Find the equation whose roots are ${\alpha }^2,{\beta }^2$.

1. $x^2+(2b-a^2)x+a^2=0$

2. $x^2+(2b-a^2)x+b^2=0$

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

4. $x^2+(a^2-2b)x+b^2=0$

Q. Let $p$ and $q$ be real numbers such that $p\ne 0,p^3\ne q$ and $p^3\ne -q$ . If $\alpha$ and $\beta$ are nonzero 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. $(p^3+q)x^2-(5p^3-2q)x+(p^3-q)=0$
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. $(p^3-q)x^2-(5p^3+2q)x+(p^3-q)=0$

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

1. $x^2-13x+4=0$
2. $x^2-5x+4=0$
3. $x^2+5x+4=0$
4. $x^2-3x-2=0$

Q. If the roots of $x^2-5x+4=0$ are $p,q$, then which of the following CANNOT be the equation with the roots $1/\sqrt{p},1/\sqrt{q}?$

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

Q. If the roots of $x^2-5x+4=0$ are $p,q$, then which of the following CANNOT be the equation with the roots $1/\sqrt{p},1/\sqrt{q}?$

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

Q.

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

1. $4x^2+7x+1=0$

2. $4x^2+7x+16=0$

3. $4x^2+7x+6=0$

4. $4x^2-7x+16=0$

Q. If the roots of the equation $a{x}^2+bx+c=0$ are reciprocal of the roots of the equation $p{x}^2+qx+r=0$, then which of the following options is always correct?

1. $ap=rc$
2. $aq=cr$
3. $aq=bp$
4. $bp=aq$

Q. Let $\alpha ,\beta$ be the roots of the quadratic equation $3x^2+10x+2=0$, then the quadratic equation whose roots are $\frac{\alpha }{\alpha +5},\frac{\beta }{\beta +5}$, is

1. $27x^2+48x+2=0$
2. $27x^2+46x+2=0$
3. $27x^2+2x+48=0$
4. $46x^2+27x+2=0$

Q. If the equation $x^2+x-12=0$ has roots as $\alpha ,\beta$, then the equation with the roots ${\alpha }^2,{\beta }^2$ is:

1. $x^2-25x+144=0$
   
2. $x^2+25x+144=0$
   
3. $x^2+25x-144=0$
   
4. $x^2-25x-144=0$
   

Q. If $\alpha ,\beta$ are roots of the equation $a{x}^2+bx+c=0,$ then the quadratic equation whose roots are $\frac{1}{(a\alpha +b{)}^2},\frac{1}{(a\beta +b{)}^2}$, is

1. $\left(a^2{c}^2\right)x^2+(2ac-b^2)x+1=0$
2. $\left(c^2\right)x^2+(2ac-b^2)x+1=0$
3. $\left(a^2\right)x^2+(2ac-b^2)x+1=0$
4. $\left(a^2{c}^2\right)x^2+(2ac+b^2)x+1=0$

Q.

If $a,b,c$ are the roots of the equation $x^3+2x^2+1=0$ . Find the equation whose roots are $b+c-a,c+a-b,a+b-c.$

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

2. $x^3+2x^2+4x+9=0$

3. $x^3$ + $x^2+4x+9=0$

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

Q. Let $\alpha ,\beta$ be the roots of the quadratic equation $3x^2+10x+2=0$, then the quadratic equation whose roots are $\frac{\alpha }{\alpha +5},\frac{\beta }{\beta +5}$, is

1. $46x^2+27x+2=0$
2. $27x^2+2x+48=0$
3. $27x^2+46x+2=0$
4. $27x^2+48x+2=0$

Q.

Let $\alpha ,\beta$ be the values of m for which the equation $(1+m)x^2-2(1+3m)x+(1+8m)$ has equal roots. Find the equation whose roots are $\alpha +2$ and $\beta +2$.

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

2. $x^2-5x+6=0$

3. $x^2-3x=0$

4. $x^2-7x+10=0$

Q. Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$. The roots of $a(x-2{)}^2-b(x-2\left)\right(x-3)+c(x-3{)}^2=0$, where $a\ne 0$ are

1. $\frac{2+3\alpha }{2+\alpha },\frac{2+3\beta }{2+\beta }$
2. $\frac{2+\alpha }{1+\beta },\frac{2+\beta }{1+\alpha }$
3. $\frac{2+\alpha }{3+\beta },\frac{3+\alpha }{2+\beta }$
4. $\frac{2+3\alpha }{1+\alpha },\frac{2+3\beta }{1+\beta }$