Consider two Ordinary Differential Equations (ODEs):

P: $ \dfrac{dy}{dx} = \dfrac{x^4 + 3x^2 y^2 + 2y^4}{x^3 y} $

Q: $ \dfrac{dy}{dx} = -\dfrac{y^2}{x^2} $

Which one of the following options is CORRECT?

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having a heart attack by 50%. Medicine X is taken by 50 of these 100 patients. The probability that a randomly selected patient, out of the 100 patients, takes medicine X and has a heart attack is

Three vectors $\overrightarrow{p}$, $\overrightarrow{q}$, and $\overrightarrow{r}$ are given as

$ \overrightarrow{p} = \hat{i} + \hat{j} + \hat{k}$

$ \overrightarrow{q} = \hat{i} + 2\hat{j} + 3\hat{k}$

$ \overrightarrow{r} = 2\hat{i} + 3\hat{j} + 4\hat{k}$

Which of the following is/are CORRECT?

The expression for computing the effective interest rate $(i_{eff})$ using continuous compounding for a nominal interest rate of 5% is

$i_{eff} = \lim\limits_{m \to \infty} \left(1 + \frac{0.05}{m}\right)^m - 1$

The effective interest rate (in percentage) is ___________ (rounded off to 2 decimal places).