Quadratic equations practice · Medium

Given that $\tan A$ and $\tan B$ are the roots of ${x^2} - px + q = 0$, then the value of ${\sin ^2}\left( {A + B} \right)$ is

If $\alpha$ and $\beta$ are the roots of the equation $a{x^2} + bx + c = 0$ then, $\left( {1 + \alpha + {\alpha ^2}} \right)\;\left( {1 + \beta + {\beta ^2}} \right)$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$, then the equation whose roots are $\frac{1}{\alpha+\beta}, \frac{1}{\alpha}+\frac{1}{\beta}$ is

The value of $a$ for which the sum of the squares of the roots of the equation ${x^2} - \left( {a - 2} \right)x - a - 1 = 0$ assumes the least value, is

If $a_i > 0$ for $i=1,2, \ldots, n$ and $a_1 a_2 \ldots a_n=1$, then minimum value of $\left(1+a_1\right)\left(1+a_2\right) \ldots\left(1+a_n\right)$ is

If $a$ and $b$ are the non-zero distinct roots of ${x^2} + ax + b = 0,$ then the least value of ${x^2} + ax + b$ is

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

If $\mathrm{\;}\sec \,\,\mathrm{\alpha \;}\,\,\mathrm{and\;}\,\,\mathrm{cosec}\,\,\mathrm{\alpha \;}$ are the root of the equation ${x^2} - px + q = 0$,then

If $x_1, x_2, x_3, \ldots, x_n$ are the roots of $x^n+a x+b=0$, then the value of $\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_1-x_4\right) \ldots\left(x_1-x_n\right)$ is

If $\cos ^4 \theta+\alpha, \sin ^4 \theta+\alpha$ are the roots of the equation $x^2+2 b x+b=0$ and $\cos ^2 \theta+\beta, \sin ^2 \theta+\beta$ are the roots of $x^2+4 x+2=0$ the value of ${b}$ is

If $\alpha$ and $\beta$ are the roots of the equation $(5+\sqrt{2}) x^2-$ $(4+\sqrt{5}) x+8+2 \sqrt{5}=0$, then $\frac{2}{(1 / \alpha)+(1 / \beta)}$ is equal to

If the roots of $a_1 x^2+b_1 x+c_1=0$ are $\alpha_1, \beta_1$ and that of $a_2 x^2+b_2 x+c_2=0$ are $\alpha_2$, $\beta_2$ such that $\alpha_1 \alpha_2=\beta_1 \beta_2=1$, then

If $x^2+6 x-27 > 0$ and $x^2-3 x-4 < 0$, then

The number of solutions of the equation $\frac{{\left( {1 + {e^{{x^2}}}} \right)\sqrt {1 + {x^2}} }}{{\sqrt {1 + {x^4}} - {x^2}}} = 1 + \cos x,$ is

If the least value of the expression $y=x^2-a x+a$ is 1; then the value of $a$ is equal to

If $a, b$ and $c$ are the sides of a triangle, then $\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}$

The number of solutions of the equation $|x|^2-2|x|-8=0$ which belong to the domain of the function $f(x)=\sqrt{5-2 x}$ is

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

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

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

The value of '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

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

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

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

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 $m > 1$ and $n \in N$, then