Transformation of Roots: Linear Combination of Roots

Q. If the roots of $x^2+mx+n=0$ are twice the roots of $x^2+px+m=0,p\ne 0$, then the value of $\frac{n}{p}$ is

Q. If the roots of $a{x}^2-bx-c=0$ are increased by same quantity, then which of the following expressions does not change?

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

Q. If $\alpha ,\beta$ are zeroes of a polynomial $6x^2+x–2$, then the polynomial whose zeroes are $2\alpha +3\beta$ and $3\alpha +2\beta$ , is .

1. $6x^2-5x+1$
2. $6x^2+5x-1$
3. $6x^2-5x-1$
4. $6x^2+5x+1$

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 $k\alpha +s\&k\beta +s,$ where $k\&s$ are constants and $k\ne 0.$

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

Q. Let $\alpha ,\beta$ be the roots of the quadratic equation $(1-m)x^2-(2-3m)x-3m=0,m\in W$ has distinct integer roots. Then find the equation whose roots are $\alpha +5\&\beta +5$.

1. $x^2+12x-35=0$
2. $x^2-12x-35=0$
3. $x^2+12x+35=0$
4. $x^2-12x+35=0$

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

1. $x^2+3x+4=0$
2. $4x^2+3x+2=0$
3. $x^2+3x+8=0$
4. $4x^2+3x+4=0$

Q. The value of $a$ for which one root of the quadratic equation $(a^2-5a+3)x^2+(3a-1)x+2=0$ is twice as large as the other is:

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

Q.

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

1. $\frac{7}{4}$

2. $\frac{-7}{8}$

3. $\frac{7}{8}$

4. $\frac{-7}{4}$

Q.

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ and $\alpha +k,\beta +k$ are the roots of $p{x}^2+qx+r=0$.
Then $\frac{b^2-4ac}{q^2-4pr}$ is equal to

1. $1$

2. ${\left(\frac{a}{p}\right)}^2$

3. $0$

4. ${\left(\frac{p}{a}\right)}^2$

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

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

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. The equation with roots as $k\alpha \&k\beta \forall k\in R$ is:

1. $a{\left(\frac{x}{k}\right)}^2+b\left(\frac{x}{k}\right)+c=0$
2. $a(x-k{)}^2+b(x-k)+c=0$
3. $ka{x}^2+bkx+c=0$
4. $a(kx{)}^2+b(kx)+c=0$

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

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

Q.

If $\alpha ,\beta$ are the roots of the equation $3x^2+5x+4=0$, then the quadratic equation whose roots are ${\alpha }^2,{\beta }^2$.

1. $9x^2+x+24=0$

2. $9x^2-x+16=0$

3. $9x^2-x+24=0$

4. $9x^2+x+16=0$

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-(3+\sqrt{2^{5{\log }_{2}4}-{\log }_2({\log }_3({\log }_{6}216)})x-2(3^{{\log }_{3}2}-2^{{\log }_{2}3})=0$, then the equation whose roots are $-\alpha ,-\beta$ is

1. $x^2+19x+2=0$
2. $x^2+35x-2=0$
3. $x^2+29x+2=0$
4. $x^2+35x+2=0$

Q. What is the equation of new parabola formed if parabola $y=4x^2$ is shifted by 3 units horizontally?

1. $y=4(x-3{)}^2$
2. Neither (a) nor (b)
3. (a) and (b) above.
4. $y=4(x+3{)}^2$

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

1. $4x^2+3x+4=0$
2. $x^2+3x+8=0$
3. $x^2+3x+4=0$
4. $4x^2+3x+2=0$

Q.

Find the equation whose roots are equal in magnitude but opposite in sign to the roots of the equation $x^5$- 3 $x^3$ + 2 $x^2$ + 4x + 1 = 0.

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

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

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

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

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $x^2-14x+45=0,$ then the quadratic equation with roots as $3\alpha ,3\beta$ is

1. $\frac{x^2}{9}-\frac{14x}{3}+5=0$
2. $x^2-42x+405=0$
3. $9x^2-42x+405=0$

Q. If $\alpha ,\beta$ are the roots of quadratic equation $x^2+7x+10=0$. Then the quadratic equation with roots as $\alpha +2$ and $\beta +2$ is

1. $x^2+7x+18=0$
2. $x^2+3x=0$
3. $x^2+3x+10=0$
4. $x^2+11x+18=0$

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

1. $x^2+7x+13=0$
2. $x^2+5x+1=0$
3. $x^2+9x+13=0$
4. $x^2+5x-1=0$

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 $\alpha -k$ and $\beta -k\forall k\in R$ is

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

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

1. $27x^2+6x-1=0$
2. $27x^2-6x+1=0$
3. $27x^2+6x+1=0$
4. $9x^2+6x+1=0$

Q. If roots of the equation $(x-12{)}^2+(2x-24\left)\right(x+12)+(x+12{)}^2=0$ are decreased by $12$, then the roots of the transformed equation are

1. irrational roots
2. rational and distinct roots
3. rational and equal roots
4. non real roots

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

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

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $x^2-7x+12=0$. Then the equation with roots $-\alpha ,-\beta$ is

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