CAT (QA) PYQs

If the equations $x^2 + mx + 9 = 0$, $x^2 + nx + 17 = 0$, and $x^2 + (m + n)x + 35 = 0$ have a common negative root, then the value of $2m + 3n$ is

A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to $800$. The prices of the smallest size and the medium size are in the ratio $2 : 5$. If the shop owner decides to increase the prices of the smallest and the medium ones by INR $6$ keeping the price of the largest size unchanged, the product then changes to $3200.$ The sum of the original prices of three different sizes, in INR, is:

The roots $\alpha, \beta$ of the equation $3 x^{2}+\lambda x-1=0$, satisfy $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=15$. The value of $\left(\alpha^{3}+\beta^{3}\right)^{2}$, is