Quadratic equations practice · Mixed difficulty

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

Let $\alpha ,\beta$ be the roots of ${x^2} + bx + 1 = 0.$ Then the equation whose roots are $- \left( {\alpha + \frac{1}{\beta }} \right)$ and $- \left( {\beta + \frac{1}{\alpha }} \right),$ is

Suppose that two persons $A$ and $B$ solve the equation ${x^2} + ax + b = 0$. While solving $A$ commits a mistake in the coefficient of $x$ was taken as 15 in place of $- 9$ and finds the roots as $- 7$ and $- 2$. Then the equation is

If the equations $x^{2}+a x+b=0$ and $x^{2}+a^{\prime} x+b^{\prime}=0$ have a common root, then this common root is equal to

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

The value of '$k$' for which one of the roots of ${x^2} - x + 3\;k = 0,$ is double of one of the roots of ${x^2} - x + k = 0$ is

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

The value of
$x = 4 + \frac{1}{{5 + \frac{1}{{4 + \frac{1}{{5 + \frac{1}{{4 + ...\infty }}}}}}}}$

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

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

Two students while solving a quadratic equation in $x$, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term coefficient of ${x^2}$ correctly as $- 6$ and 1 respectively the correct roots are

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

The sum of all values of $x$, so that $16^{\left(x^2+3 x-1\right)}=8^{\left(x^2+3 x+2\right)}$, is

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2+p x+q=0$, then the value of $\left(\omega \alpha+\omega^2 \beta\right)\left(\omega^2 \alpha+\omega \beta\right)$, where $\omega$ is an imaginary cube root of unity, is

The value of $x=\sqrt{2+\sqrt{2+\sqrt{2+\ldots \ldots}}}$ is;

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 $\alpha$ and $\beta$ are the distinct roots of the equation ${x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0$ then the value of ${\alpha ^{96}}\left( {{\alpha ^{12}} - 1} \right) + {\beta ^{96}}\left( {{\beta ^{12}} - 1} \right)$ is equal to

Let $\left[ x \right]$ denote the greatest integer less than or equal to $x$. Then, in $\left[ {0,3} \right]$ the number of solutions of the equation ${x^2} - 3x + \left[ x \right] = 0,$ is

If the equation $5 x^{2}-10 x+\log _{1 / 5} a=0$ has real roots, then the minimum value of $a$ is

Let $f(x)=\int_1^x \sqrt{\left(2-t^2\right)} d t$, the real roots of the equation $x^2-f^{\prime}(x)=0$ are

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

If $a, b \in R, a \neq 0$ and the quadratic equation $a x^{2}-b x+1=0$ has imaginary roots, then $a+b+1$ is

The product of the roots of the equation $x^{\log _{10} x}=\frac{10^4}{x^3}$, is

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

$a x^2-b x(x-1)+c(x-1)^2=0$ (1)

are

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

A real root of the equation $\log _4\left\{\log _2(\sqrt{x+8}-\sqrt{x})\right\}=0$ is:

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