Transformation of Roots: Algebraic Transformation

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 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. Let $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0,(b\ne 0)$ and ${\alpha }^4,{\beta }^4$ are the roots of the equation $l{x}^2+mx+n=0$. If $\alpha ,\beta$ are real and distinct, then the roots of the equation $a^{2}l{x}^2-4aclx+2c^{2}l+a^{2}m=0$ are

1. always imaginary
2. opposite in sign
3. same in sign
4. always real

Q. If $\alpha ,\beta$ are the roots of the equation $m{x}^2+6x+(2m-1)=0\forall m\in R-\{0,\frac{1}{2}\},$ then the quadratic equation with roots as $\frac{1}{\alpha },\frac{1}{\beta }$ is:

1. $(2m-1)x^2-6x+m=0$
2. $6x^2-(2m-1)x-m=0$
3. $m{x}^2+(2m-1)x+6=0$
4. $(2m-1)x^2+6x+m=0$

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

1. $21x^2+5x-1=0$
2. $147x^2+35x-5=0$
3. $21x^2+5x-3=0$
4. $147x^2+35x-14=0$

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. $(p^3+q)x^2-(p^3+2q)x+(p^3+q)=0$
2. $(p^3-q)x^2-(5p^3+2q)x+(p^3-q)=0$
3. $(p^3+q)x^2-(5p^3-2q)x+(p^3-q)=0$
4. $(p^3+q)x^2-(p^3-2q)x+(p^3+q)=0$

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

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+3x+2=\mathrm{0...}\left(i\right)\text{and}(\alpha -\beta )(\alpha -\gamma ),(\beta -\gamma )(\beta -\alpha ),(\gamma -\alpha )(\gamma -\beta )$ are the roots of equation $y^3-9y^2-216=0...\left(ii\right)$.

Relation to transform the eq.(i) to eq.(ii) is given by .

1. $y={\alpha }^2-\frac{1}{\alpha }$

2. $y=2{\alpha }^2-\frac{1}{\alpha }$

3. $y=2{\alpha }^2-\frac{2}{\alpha }$

4. $y={\alpha }^2-\frac{2}{\alpha }$

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 $\alpha ,\beta$ 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. $x^2-5x+4=0$
2. $x^2-3x-2=0$
3. $x^2-13x+4=0$
4. $x^2+5x+4=0$

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

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

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

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

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

Q.

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

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

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

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

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

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

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Find the equation with roots ${\alpha }^2$ and ${\beta }^2.$

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

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

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

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

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 ${\alpha }^3$, ${\beta }^3$ and ${\gamma }^3$.

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

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

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

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

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

Q.

If α, β, γ and δ are the roots of $x^4+2x^3+3x^2+4x+5=0$.
Find the equation whose roots are $\frac{1}{\alpha },\frac{1}{\beta },\frac{1}{\gamma },\frac{1}{\delta }$.

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

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

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

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

Q. If, $\alpha$ and $\beta$ are the positive roots of the equation, $ax^2+bx+c=0$, find the equation whose roots are $\dfrac{1}{\sqrt\alpha}$ and $\dfrac{1}{\sqrt\beta}.$

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

2. $-\alpha ,-\beta$

3. $Noneofthese$

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

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

Q.

If $\alpha ,\beta$ are the roots of the equation $2x^2+5x+6=0$, then find the equation whose roots are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$.

1. $5x^2-6x+2=0$

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

3. $6x^2-5x+2=0$

4. $5x^2+6x+2=0$

Q. If the roots of $x^2-17x+16=0$ are $p,q$, then which of the following can be the equation with the roots $\frac{1}{\sqrt{p}},\frac{1}{\sqrt{q}}?$

1. $x^4+17x^2+16=0$
2. $x^4-17x^2-16=0$
3. $16x^4-17x^2+1=0$
4. $16x^4+17x^2+16=0$

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

Q. The equation formed by decreasing the roots of the quadractic equation $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0$, then which of the following statement(s) is/are true?

1. $a:b:c=1:2:-2$
2. $a=-c$
3. $b=-c$
4. $a:b:c=2:1:-2$

Q. The sum of the squares of the roots of the equation $x^2-9x+q=0$ is $53.$ Then the value of $q$ is