Transformation of Roots: Algebraic Transformation

Q.

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

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

2. $\frac{9}{4}(9+q^2)$

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

4. $\frac{9}{4}(9-q^2)$

Q.

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

1. $\frac{27}{32}$

2. $\frac{1}{24}$

3. $\frac{27}{16}$

4. $\frac{3}{8}$

Q. Let $f\left(x\right)$ be a quadratic polynomial with leading coefficient $1$ such that $f\left(0\right)=p,p\ne 0$, and $f\left(1\right)=\frac{1}{3}$. If the equations $f\left(x\right)=0$ and $fofofof\left(x\right)=0$ have a common real root, then $f(-3)$ is equal to

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 $\alpha ,\beta$ 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 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. $Noneofthese$

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

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

4. $-\alpha ,-\beta$

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

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

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

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

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

Q. Which of the following can be the roots of the equation $x^2+px+45=0$ if the squared difference of its roots is equal to $144$ ?

1. $3,15$
2. $-3,-15$
3. $-3,15$
4. $3,-15$

Q. The sum of the squares of the roots of the equation $x^2-5x+p=0$ is $13$. Then what is the value of $p$?

1. $4$
2. $-4$
3. $6$
4. $-6$

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+3x+2=0$ , Find the equation whose roots are $(\alpha -\beta )(\alpha -\beta ),(\beta -\gamma )(\beta -\alpha ),(\gamma -\alpha )(\gamma -\beta )$.

1. $x^3-5x^2-36=0$

2. $x^3-9x^2-216=0$

3. $x^3-11x^2-100=0$

4. $x^3-9x^2-225=0$

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

Q.

If α and β are the roots of the equation \(a{x}^{2} + bx + c=0\). find the equation whose roots are - α and - β.

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

2. a$x^2$ + bx + c = 0

3. c$x^2$ + bx + a = 0

4. c$x^2$ - bx + a = 0

Q. Suppose $a^2=5a-8$ and $b^2=5b-8$. Then the equation whose roots are $\frac{a}{b}$ and $\frac{b}{a}$ is

1. $8x^2+9x+8=0$
2. $6x^2-5x+6=0$
3. $8x^2-9x+8=0$
4. $9x^2-8x+9=0$

Q.

If α and β are the roots of the equation \(a{x}^{2} + bx + c=0\). find the equation whose roots are - α and - β.

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

2. a$x^2$ - bx + c = 0

3. c$x^2$ + bx + a = 0

4. a$x^2$ + bx + c = 0

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

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

Q. A quadratic equation $a{x}^2+bx+c=0,a\ne 0$ have the roots $\alpha$ and $\beta$. Find the transformed equation with roots $-\alpha$ and $-\beta .$

1. $a{x}^2+bx+c=0$
2. $a{x}^2-bx+c=0$
3. None of the above
4. $b{x}^2-ax+c=0$

Q. If $m,n$ are the roots of the quadratic equation $x^2-3x+5=0,$ then the equation whose roots are $(m^2-3m+7)\&(n^2-3n+7)=$

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

Q. The sum of the squares of the roots of the equation $x^2-px+5=0$ is $6$. Then what is the value of $p$? $(p>0)$

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

Q. If $\alpha, \beta$ are the roots of the equation $3x^2 +5x -7 = 0$, then the equation whose roots are $\dfrac{1}{3\alpha+5}, \dfrac{1}{3\beta+5}$ 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

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

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

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+2x^2+3x+1=0$. Find the equation whose roots are $\frac{1}{{\alpha }^3},\frac{1}{{\beta }^3},\frac{1}{{\gamma }^3}$

1. $x^3+12x^2-7x+1=0$

2. $x^3-x^2+8x+2=0$

3. $x^3+15x^2-8x+1=0$

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

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

Q.

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

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

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

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

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

Q.

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

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

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

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

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

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