Quadratic equations practice · Mixed difficulty

The number of integral values of $m$ for which the equation $(1 + {m^2}){x^2} - 2(1 + 3m)x + (1 + 8m) = 0,$ has no real root is

The number of integral solutions of $\frac{{x + 2}}{{{x^2} + 1}} > \frac{1}{2}$ is

If the roots of the equation ${x^2} + px + q = 0$ are $\tan {30^0}$ and $\tan {15^0}$ respectively, then the value of $2 + q - p$ is

The roots of the equation $\left( {q - r} \right){x^2} + \left( {r - p} \right)x + \left( {p - q} \right) = 0$ are

If $0< c < a < b < d$, then roots of the equation $b x^2+\{1-b(c+d)\} x+b c d-a=0$ are

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - ax + b = 0$ and ${A_n} = {\alpha ^n} + {\beta ^n},$ then which one of the following is true?

Let $\alpha$ and $\beta$ be the roots of ${x^2} - 6x - 2 = 0$. If ${a_n} = {\alpha ^n} - {\beta ^n}$ for $n \ge 1$, then the value of $\frac{{{a_{10}} - 2{a_8}}}{{3{a_9}}}$ is

Let $\left( {\sin a} \right){x^2} + \left( {\sin a} \right)x + \left( {1 - \cos a = 0} \right)$. The value of $a$ for which roots of this equation are real and distinct, is

$\sin A,\sin B,\cos A$ are in GP. Roots of ${x^2} + 2x\cot B + 1 = 0\;$ are always

If $a{\left( {p + q} \right)^2} + 2b\;pq + c = 0$ and $a{\left( {p + r} \right)^2} + 2b\;pr + c = 0,$ then $qr =$

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

The value of 'a' for which one root of quadratic equation $\left(a^2-5 a+3\right) x^2+(3 a-1) x+2=0$ is twice as large as other is

The solution set of inequation ${\log _{1/3}}\left( {{2^{x + 2}} - {4^x}} \right) \ge - 2,$ is

The roots of $a{x^2} + 2\;bx + c = 0$ and $b{x^2} - 2\sqrt {ac} \;x + b = 0$ are simultaneously real, then

If $0 < a < b < c$ and the roots $\alpha$ and $\beta$ of the equation $a x^2+b x+c=0$ are imaginary, then

The value of $p$ for which the quadratic equation $x^2-p x+p+3=0$ has reciprocal roots is

In writing an equation of the form $a{x^2} + bx + c = 0;$ the coefficient of $x$ is written incorrectly and roots are found to be equal. Again in writing the same equation the constant term is written incorrectly and it is found that one root is equal to those of the previous wrong equation while the other is double of it. If $\alpha$ and $\beta$ be the roots of correct equation, then ${\left( {\alpha - \beta } \right)^2}$ is equal to

If $\alpha, \beta$ are roots of $x^2-2 x-1=0$, then value of $5 \alpha^4+12 \beta^3$ is

If ${x_1},{x_2},{x_3}$ are distinct roots of the equation $a{x^2} + bx + c = 0$, then

The numbers of pairs $(a,\,\,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation ${x^2} + ax + b = 0,\,\,\,{\alpha ^2} - 2$ is also a root of this equation, is

The real values of $a$ for which the quadratic equation $3 x^2+2\left(a^2+1\right) x+\left(a^2-3 a+2\right)=0$ possesses roots of opposite signs lie in

If one root is square of the other root of the equation $x^2+p x+q=0$, then the relation between $p$ and $q$ is

If $\alpha ,\beta$ are the roots of the equation $a{x^2} + bx + c = 0$, then the equation whose roots are $2\alpha + 3\beta$ and $3\alpha + 2\beta$, is

The coefficients of $x$ in the quadratic equation ${x^2} + bx + c = 0$ was taken as 17 in place of 13, its roots were found to be $- 2$ and $- 15$. The correct root of the original equation are

If both the roots of the equation $x^2+a x+b+1=0$ are positive integers, then a possible value of $a^2+b^2$ is

Let $\alpha$ and $\beta$ be the roots of equation $p{x^2} + qx + r = 0,\,\,\,p \ne 0.$ If $p,\,\,q$ and $r$ are in AP and $\frac{1}{\alpha } + \frac{1}{\beta } = 4,$ then the value of $\left| {\alpha - \beta } \right|$ is

A solution of the set of equations $x+y+z=x y+y z+z x=0$, $x^3+y^3+z^3=-3$ is-

Let $f\left( x \right)$ be a quadratic polynomial such that $f\left( { - 1} \right) + f\left( 2 \right) = 0.$ If one of the roots of $f\left( x \right) = 0$ is 3, then its other root lies in

If the sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is 5, then the solution set for $x$ is

If the roots of the equation $a{x^2} + bx + c = 0\;\,\mathrm{be}\,\;\alpha \;\mathrm{and}\;\beta \;,$ then the roots of the equation $c{x^2} + bx + a = 0$ are