Quadratic Equations - Algebra PYQ | Mathematics

If \(\alpha\) and \(\beta\) are the roots of \(11 x^2+12 x-13=0\), then \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}\) is equal to (approximately close to)

The value of \(a\) for which the equations \(x^3+a x+1=0\) and \(x^4+a x^2+1=0\) have a common root is

If \(a\) is a positive integer such that roots of the equation \(7 x^2-13 x+a=0\) are rational numbers, then the smallest possible value of \(a\) is

The sum of the roots of the equation \(e^{4 t}-10 e^{3 t}+29 e^{2 t}-22 e^t+4=0\) is

If the product of the roots of \(9x^3+112x^2-120x+a=0\) is 12, then the value of \(a\) is

\(2+\sqrt{5}, 1\) are roots of the cubic equation given by

For \(a\ne b\), if the equation \(x^2+ax+b=0\) and \(x^2+bx+a=0\) have a common root, then the value of \(a+b\) is equal to

If \(f(x)=2x^3+mx^2-13x+n\) and 2, 3 are the roots of the equation \(f(x)=0\), then the values of m and n are

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2+x+1=0\), then the equation whose roots are \(\alpha^{2021}, \beta^{2021}\) is given by

If \(2, 1\) and \(1\) are roots of the equation \(x^3-4 x^2+5 x-2=0\), then the roots of \(\left(x+\frac{1}{3}\right)^3-4\left(x+\frac{1}{3}\right)^2+5\left(x+\frac{1}{3}\right)-2=0\)

The sum of the real roots of the equation \(|x-2|^2+|x-2|-2=0\) is

If \(\frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6}\), then \(A^2+B^2=\)

If \(f(f(0))=0\), where \(f(x)=x^2+a x+b, b \neq 0\), then \(a+b=\)

If the difference between the roots of \(x^2+a x+b=0\) and that of the roots of \(x^2+b x+a=0\) is same and \(a \neq b\), then

For what values of \(a \in Z\), the quadratic expression \((x+a)(x+1991)+1\) can be factorised as \((x+b)(x+c)\), where \(b, c \in Z\) ?

If \(f(x)=a x^2+b x+c\) for some \(a, b, c \in R\) with \(a+b+c=3\) and \(f(x+y)=f(x)+f(y)+x y, \forall x, y \in R\). Then, \(\sum_\limits{n=1}^{10} f(n)=\)

The number of positive real roots of the equation \(3^{x+1}+3^{-x+1}=10\) is

The number of real roots of the equation \(\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}\) is

The sum of the real roots of the equation \(x^4-2 x^3+x-380=0\) is

If one root of the cubic equation \(x^3+36=7 x^2\) is double of another, then the number of negative roots are