If the sum of the roots of the equation $\lambda x^2 + 2x + 3\lambda = 0$ is equal to their product,then $\lambda = $

If $\alpha, \beta$ are the roots of the equation $x^2+bx+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$,then $3c+2=$ (in $b$)

Let $\tan 30^{\circ}$ and $\tan 15^{\circ}$ be the roots of the quadratic equation $x^2+ax+b=0$,then $1+a-b=$

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1$. Let $p_{n}=\alpha^{n}+\beta^{n}$,$p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n \geq 1$. Then,the value of $p_{n}^{2}$ is .... .

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+ax^2+bx+c=0$,then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}$ is equal to

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 2x - 5 = 0$ and the equation $x^3 + bx^2 + cx + d = 0$ has roots $2\alpha + 1, 2\beta + 1, 2\gamma + 1$,then the value of $|b + c + d|$ is (where $b, c, d$ are constants):