Transformation of Roots: Linear Combination of Roots

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. $x^2-7x-12=0$

Q. The value of $k$ for which the equation
$3x^2+2x(k^2+1)+k^2-3k+2=0$
has roots of opposite signs, lies in the interval

1. $(1,2)$
2. $(\frac{3}{2},2)$
3. $(-\infty ,0)$
4. $(-\infty ,-1)$

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

Q. If one root of $4x^2-2x+(\lambda -4)=0$ is reciprocal of the other, then value of $\lambda$ is

1. 8

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3-3x+7=0$, then the equation whose roots are $\alpha +\beta -\gamma ,\beta +\gamma -\alpha ,\gamma +\alpha -\beta$ is

1. $x^3+12x+56=0$
2. $x^3-12x-56=0$
3. $x^3+12x-56=0$
4. $x^3-12x+56=0$

Q.

The number of quadratic equation which are unchanged by squaring their roots, is

1. $2$

2. $4$

3. $6$

4. None of these

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. $9x^2-42x+405=0$
3. $x^2-42x+405=0$

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

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

Q. Choose the correct pair of quadratic equations with roots with opposite sign.

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

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

4. $0$

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

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

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

Q.

If $\alpha ,\beta$ are the roots of the quadratic equation $x^2+3x+6=0$ , then find the equation whose roots are $\frac{1+\alpha }{1-\alpha }$, $\frac{1+\beta }{1-\beta }$

1. $5y^2-2y+5=0$

2. $5y^2+5y+2=0$

3. $5y^2+2y+5=0$

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

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

Q. If the graph of $y=x^2$ is given by


then graph of $y-2=3(x-5{)}^2$ will be given by:

1. 
2. 
3. 
4. 

Q. Let $\alpha ,\beta$ be the values of $m\in W$ for which the equation $(1-m)x^2-(2-3m)x-3m=0$ has distinct integer roots. Find the equation whose roots are $\alpha +5$ and $\beta +5$.

1. $x^2-11x+30=0$
2. $x^2-11x-30=0$
3. $x^2+11x+30=0$
4. $x^2+11x-30=0$

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

Q. If one root of $4x^2-2x+(\lambda -4)=0$ is reciprocal of the other, then value of $\lambda$ is

1. 8

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 $\alpha$ and $\beta$ are the roots of the equation $x^2+px+q=0$, then $-{\alpha }^{-1}$ and $-{\beta }^{-1}$ are the roots of which one of the following equations:

1. $q{x}^2-px+1=0$

2. $q^2+px+1=0$

3. $x^2+px-q=0$

4. $x^2-px+q=0$

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

Q. If one root of $4x^2-2x+(\lambda -4)=0$ is reciprocal of the other, then value of $\lambda$ is

1. 8

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

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

Q.

Find the equation whose roots are \( 2x _1 + 3\) and \(2x _2 + 3, \) if \( x_1\) and \(x _2\) are the roots of \(x^{2} + 6x + 7 = 0\).

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

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 $\alpha ,\beta$ 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. $x^2+35x-2=0$
2. $x^2+19x+2=0$
3. $x^2+35x+2=0$
4. $x^2+29x+2=0$

Q.

Find the equation whose roots are $2x_1+3$ and $2x_2+3,$ if $x_1$ and $x_2$ are the roots of $x^2+6x+7=0$.

1. ${(2x+3)}^2+6(2x+3)+7=0$

2. ${\left(\frac{x-2}{3}\right)}^2+6\left(\frac{x-2}{3}\right)+7=0$

3. ${\left(\frac{x-3}{2}\right)}^2+6\left(\frac{x-3}{2}\right)+7=0$

4. $(2(x-3){)}^2+6[2(x-3)]+7=0$

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Then the equation with roots $\alpha +k\&\beta +k\forall k\in R$ 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. $a(x-k{)}^2+b(x-k)+c=0$