Relation between Roots and Coefficients for Quadratic

Q. Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation, $x^2-x-1=0,$ where $\alpha \ne \beta .$ For $n=0,1,2,....,$ let $a_n=p{\alpha }^n+q{\beta }^n.$

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0.$ then $a=0=b.$

If $a_4=28,$ then $p+2q=$

1. $21$
2. $14$
3. $7$
4. $12$

Q.

If $\alpha$ and $\beta$ are the roots of$x^2-ax+b^2=0$, then ${\alpha }^2+{\beta }^2$is equal to

1. $a^2+2b^2$

2. $a^2-2b^2$

3. $a^2-2b$

4. $a^2+2b$

Q.

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to the $n\text{th}$ power of the other root, then the value of ${\left(a^{n}c\right)}^{\frac{1}{n+1}}+{\left(a{c}^n\right)}^{\frac{1}{n+1}}$ is equal to

1. $b$

2. $-b$

3. $\frac{1}{\left(b{x}^{n+1}\right)}$

4. $-\frac{1}{\left(b{x}^{n+1}\right)}$

Q.

If $\alpha and\beta$ are the roots of the equation $4x^2+3x+7=0$, then $\frac{1}{\alpha }+\frac{1}{\beta }=$

1. $-\frac{3}{7}$

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

3. $-\frac{3}{5}$

4. $\frac{3}{5}$

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 integral values of $a$ such that $x^2+ax+a+1=0$ has integral roots is

Q. If one root is $n$ times the other for the equation $a{x}^2+bx+c=0,$ then

1. $n{b}^2=ac(n+1{)}^2$
2. $n^{2}b=ac(n+1{)}^2$
3. $(nb{)}^2=a^{2}c(n+1)$
4. $nb=ac(n+1)$

Q. Difference between the corresponding roots of $x^2+ax+b=0$ and $x^2+bx+a=0$ is same and a $\ne$ b, then

1. a + b - 4 = 0
2. a + b + 4 = 0
3. a - b + 4 = 0
4. a - b - 4 = 0

Q. If the roots of the equation $12x^2-mx+5=0$ are in the ratio 2 : 3, then m=

1. $5\sqrt{10}$
2. $2\sqrt{10}$
3. $3\sqrt{10}$
4. None of these

Q. The sum of real roots of the equation $\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0,$ is equal to :

1. $-4$
2. $0$
3. $1$
4. $6$

Q. The graph of quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is shown below


Which of the following statement(s) is/are correct?

1. $\frac{c}{a}(\alpha -\beta )>2$
2. $f\left(x\right)>0;\forall x>\beta$
3. $ac>0$
4. $\frac{c}{a}<-1$

Q.

Let $\alpha ,\beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha }{2},2\beta$ be the roots of the equation $x^2-qx+r=0$. Then, the value of $r$ is:

1. $\frac{2}{9}(2p-q)(2q-p)$

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

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

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

Q. Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots (4, 3). Rahul made a mistake in writing down coefficient of x to get roots (3, 2). The correct roots of equation are:

1. 4, 3
2. -6, -1
3. -4, -3
4. 6, 1

Q. Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0,$with $\alpha$ > $\beta$, For all positive integers n, define,
$a_n=\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta },n\ge 1$
$b_1=1$ and $b_n=a_{n-1}+a_{n+1},n\ge 2$.
Then which of the following option is/are correct?

1. $a_1+a_2+a_3+......+a_n=a_{n+2}-1\text{such that}n\ge 1$
2. $\sum _{n=1}^{\infty }\frac{a_n}{{10}^n}=\frac{10}{89}$
3. $\sum _{n=1}^{\infty }\frac{b_n}{{10}^n}=\frac{8}{89}$
4. $b_n={\alpha }^n+{\beta }^n$ such that $n\ge 1$

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

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

Q.

The quadratic equations having the roots $\frac{1}{10-\sqrt{72}}$ and $\frac{1}{10+6\sqrt{2}}$ is

1. $28x^2-20x+1=0$

2. $20x^2-28x+1=0$

3. $x^2-20x+28=0$

4. $x^2-28x+20=0$

Q. Let $C:x^2+y^2-3x-4y-4=0.$ A chord of $C$ passes through the origin such that the origin divides it in the ratio $4:1$. Then the equation(s) of chord is/are

1. $x=0$
2. $y=0$
3. $7x+24y=0$
4. $24x+7y=0$

Q. If polynomial $P\left(x\right)=x^2+ax+b$ has factors $(x-a)\text{and}(x-b),\text{where}a,b\in R,$ then the value of $P\left(2\right)$ is

1. $8$
2. $7$
3. $6$
4. $4$

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

Q. A quadratic equation whose difference of roots is $3$ and the sum of the squares of the roots is $29,$ is given by

1. $x^2+9x+14=0$
2. $x^2+7x+10=0$
3. $x^2-7x-10=0$
4. $x^2-7x+10=0$

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

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

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

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

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

1. $x^2-8x+64=0$
2. $x^2+8x+64=0$
3. $x^2+16x+64=0$
4. $x^2-16x+64=0$

Q. Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0,a\ne 0$ and ${\alpha }_1,-\beta$ be the roots of $a_1{x}^2+b_{1}x+c_1=0,a_1\ne 0$. Then the quadratic equation whose roots are $\alpha ,{\alpha }_1$ is

1. $\frac{x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}-x+\frac{1}{(\frac{b}{c}+\frac{b_1}{c_1})}=0$
2. $\frac{x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}+2x+\frac{1}{(\frac{b}{c}-\frac{b_1}{c_1})}=0$
3. $\frac{2x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}+x+\frac{1}{(\frac{b}{c}+\frac{b_1}{c_1})}=0$
4. $\frac{x^2}{(\frac{b}{a}+\frac{b_1}{a_1})}+x+\frac{1}{(\frac{b}{c}+\frac{b_1}{c_1})}=0$

Q. If $m$ is choosen in the quadratic equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :

1. $8\sqrt{5}$
2. $4\sqrt{3}$
3. $10\sqrt{5}$
4. $8\sqrt{3}$

Q. For $x^2-(a+3)|x|+4=0$ to have real solutions, the range of $a$ is

1. $(-3,\infty )$
2. $(-\infty ,-7]$
3. $[1,\infty )$
4. $(-\infty ,-7]\cup [1,\infty )$

Q. If $\alpha$ and $\beta$ are the roots of the quadratic equation $(x-2)(x-3)+(x-3)(x+1)+(x+1)(x-2)=0$, then the value of $\frac{1}{(\alpha +1)(\beta +1)}+\frac{1}{(\alpha -2)(\beta -2)}+\frac{1}{(\alpha -3)(\beta -3)}$ is

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

Q. The number of quadratic equations (consider leading coefficient as $1$) with real roots which remain unchanged when their roots are squared, is

Q. If $\alpha$ and $\beta$ are the roots of a quadratic equation satisfying the conditons $\alpha \beta =4$ and $\frac{\alpha }{\alpha -1}+\frac{\beta }{\beta -1}=\frac{a^2-7}{a^2-4},\alpha ,\beta ,a\in R$. For what values of ${}^{\prime }{a}^{\prime }$ will the quadratic equation have equal roots?

1. $\sqrt{3}$
2. $\sqrt{5}$
3. $-\sqrt{3}$
4. $-\sqrt{5}$

Q. If the roots of a quadratic equation are $-6$ and $7$, then the quadratic equation is

1. $x^2-x+42=0$
2. $x^2+x+42=0$
3. $x^2-x-42=0$
4. $x^2-x+21=0$

Q. If the vertex of the curve $y=-2x^2-4ax-k$ is $(-2,7),$ then the value of $k$ is