Quadratic Equations

If $f(x)$ is a second degree polynomial such that $f(x) \geq 0 \forall x \in R, f(-3)=0$ and $f(0)=18$, then $f(3)=$

If $f(x)=x^2+b x+c$ and $f(1+k)=f(1-k) \forall k \in R$, for two real numbers $b$ and $c$ then

If $\alpha, \beta$ are the roots of the equation $x^2+3 x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4 x^2+p x+18=0$, then $k$ satisfies the equation

If one of the roots of the equation $6 x^3-25 x^2+2 x+8=0$ is an integer and $\alpha>0, \beta<0$ are the other two roots, then $\frac{4}{\alpha}+\frac{1}{\beta}=$

If $\alpha, \beta, \gamma, \delta$ and $\varepsilon$ are the roots of the equation $x^5+x^4-13 x^3-13 x^2+36 x+36=0$ and $\alpha<\beta<\gamma<\delta<\varepsilon$ then $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+\frac{a}{2} x+b=0$ and $(\alpha-\beta)(\alpha-\gamma),(\beta-\alpha)(\beta-\gamma),(\gamma-\alpha),(\gamma-\beta)$ are the roots of the equation
$(y+a)^3+K(y+a)^2+L=0$, then $\fr...

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4-4 x^3+3 x^2+2 x-2=0$ such that $\alpha$ and $\beta$ are integers and $\gamma, \delta$ are irrational numbers, then $\alpha+2 \beta+\gamma^2+\delta^2=$

The equation having the multiple root of the equation $x^4+4 x^3-16 x-16=0$ as its roots is

If both roots of the equation $x^2-5 a x+6 a=0$ exceed 1 , then the range of ' $a$ ' is

If the equations $x^2+p x+2=0$ and $x^2+x+2 p=0$ have a common root, then the sum of the roots of the equation $x^2+2 p x+8=0$ is

If $\tan \theta$ and $\cot \theta$ are two distinct roots of the equation $a x^2+b x+c=0, a \neq 0, b \neq 0$, then

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $12 x^4-56 x^3+89 x^2-56 x+12=0$ such that $\alpha \beta=\gamma \delta=1$ and $\frac{\alpha+\beta}{\gamma+\delta}>1$, then $\frac{\alpha+\beta}{\gamma+\delta}=$

Sum of all the roots of the equation $||2 x-3|-4|=2$ is

If the quotient and remainder obtained when the expression $3 x^5-6 x^4+2 x^3+4 x^2-5 x+8$ is divided by the expression $x^2-2 x+3$ are $a x^3+b x^2+c x+d$ and $p x+q$ respectively, then $a b+c d=$

If the sum of two roots of the equation $x^4-2 x^3+x^2+4 x-6=0$ is zero, then the sum of the squares of the other two roots is

If the roots of the equation $32 x^3-48 x^2+22 x-3=0$ are in arithmetic progression, then the square of the common difference of the roots is

The number of integral values of ' $a$ ' for which the quadratic equation $a x^2+a x+5=0$ cannot have real roots is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-P x^2+Q x-R=0$ and $(\alpha-2)^2,(\beta-2)^2,(\gamma-2)^2$ are the roots of the equation $x^3-5 x^2+4 x=0$, then the possible least value of $P+Q+R$ is

The number of all common roots of the equation $x^4-10 x^3+37 x^2-60 x+36=0$ and the transformed equation of it obtained by increasing any two distinct roots of it by 1 , keeping the other two roots fixed, is

If the equation $x^2-3 a x+a^2-2 a-k=0$ has different real roots for every rational number $a$, then $k$ lies in the interval

The sum of two roots of the equation $x^4-x^3-16 x^2+4 x+48=0$ is zero. If $\alpha, \beta, \gamma$ and $\delta$ are the roots of this equation, then $\alpha^4+\beta^4+\gamma^4+\delta^4=$

The equations $2 x^2+a x-2=0$ and $x^2+x+2 a=0$ have exactly one common root. If $a \neq 0$, then one of the roots of the equation $a x^2-4 x-2 a=0$ is

If the quadratic equation $3 x^2+(2 k+1) x-5 k=0$ has real and equal roots, then the value of $k$ such that
$\frac{1}{2}$ < $k$ < 0 is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2 x^3-3 x^2+5 x-7=0$, then $\sum \alpha^2 \beta^2=$

$\alpha, \beta, \gamma, 2$ and $\varepsilon$ are the roots of the equation
$$ \begin{aligned} & \alpha, \beta, \gamma+4 x^4-13 x^3-52 x^2+36 x+144=0 . \text { If } \\ & \alpha<\beta<\gamma<2<\varepsilon \text {, then } \alpha+2 \beta+3 \gam...

$\alpha$ is a root of the equation $\frac{x-1}{\sqrt{2 x^2-5 x+2}}=\frac{41}{60}$. If $-\frac{1}{2}<\alpha<0$, then $\alpha$ is equal to

$\alpha, \beta$ are the real roots of the equation $x^{2}+a x+b=0$. If $\alpha+\beta=\frac{1}{2}$ and $\alpha^{3}+\beta^{3}=\frac{37}{8}$, then $a-\frac{1}{b}=$

If $f(x)$ is a quadratic function such that $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$, then $\sqrt{f\left(\frac{2}{3}\right)+f\left(\frac{3}{2}\right)}=$

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

The equation $16 x^{4}+16 x^{3}-4 x-1=0$ has a multiple root. If $\alpha, \beta, \gamma, \delta$ are the roots of this equation, then $\frac{1}{\alpha^{4}}+\frac{1}{\beta^{4}}+\frac{1}{\gamma^{4}}+\frac{1}{\delta^{4}}=$

If $\alpha$ is a root of the equation $x^{2}-x+1=0$, then $\left(\alpha+\frac{1}{\alpha}\right)^{3}+\left(\alpha^{2}+\frac{1}{\alpha^{2}}\right)^{3}+\left(\alpha^{3}+\frac{1}{\alpha^{3}}\right)^{3}+\left(\alpha^{4}+\frac{1}{\alpha^{4}}\righ...

$\alpha, \beta$ and $\gamma$ are the roots of the equation $8 x^3-42 x^2+63 x-27=0$. If $\beta<\gamma<\alpha$ and $\beta, \gamma$ and $\alpha$ are in geometric progression, then the extreme value of the expression $\gamma x^2+4 \beta x+\alp...

$\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3 x^2-10 x-24=0$. If $\alpha>\beta>\gamma$ and $\alpha^3+3 \beta^2-10 \gamma-24=11 k$, then $k=$

If $\alpha, \beta$ are the roots of the equation $x+\frac{4}{x}=2 \sqrt{3}$, then $\frac{2}{\sqrt{3}}\left|\alpha^{2024}-\beta^{2024}\right|=$

If $\frac{2 x^3+1}{2 x^2-x-6}=a x+b+\frac{A}{P x-2}+\frac{B}{2 x+q}$, then 51 apB $=$

$\alpha, \beta$ are the real roots of the equation $12 x^{\frac{1}{3}}-25 x^{\frac{1}{6}}+12=0$. If $\alpha>\beta$, then $6 \sqrt{\frac{\alpha}{\beta}}=$

The common solution set of the inequations $x^{2}-4 x \leq 12$ and $x^{2}-2 x \geq 15$ taken together is

With respect to the roots of the equation $3 x^{3}+b x^{2}+b x+3=0$, match the items of List I with those fo List II


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The solution set of the equation $3^{x}+3^{1-x}-4 < 0$ contained in $R$ is

If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^2+a x-c=0$, then

If $R-(\alpha, \beta)$ is the range of $\frac{x+3}{(x-1)(x+2)}$, then the sum of the intercepts of the line $\alpha x+\beta y+1=0$ on the coordinate axes is

If $x^2+2 p x-2 p+8>0$ for all real values of $x$, then the set of all possible values of $p$ is

The roots of the equation $x^4+x^3-4 x^2+x+1=0$ are diminished by $h$ so that, the transformed equation does not contain $x^2$ term. If the values of such $h$ are $\alpha$ and $\beta$, then $12(\alpha-\beta)^2=$

The sum of all the real values of $x$ satisfying the equation $\left(x^2-7 x+11\right)^{x^2-6 x-7}=1$ is

If $\alpha, \beta$ are the roots of $x^2+a x+2=0$ and $1 / \alpha, 1 / \beta$ are the roots of $x^2-b x+c=0$, then
$$ \left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\fra...

If $x^2+3 x-2 k=0$ and $x^2-2 x-7 k=0$ have a non-zero common root, then the positive root of the equation $k x^2+(k+2) x-(k+1)=0$ is

If the sum of two particular roots of the equation $x^4-4 x^3-7 x^2+22 x+24=0$ is equal to the sum of the remaining two roots, then the sum of the cubes of all the roots of this equation is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$

The values of $\frac{x^2-2 x+1}{x^2+x-1}$ do not lie in the interval

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$