Board Paper of Class 12-Commerce Term-II 2022 Math Delhi(Set 3) - Solutions

The Cartesian equation of a line AB is:

$\frac{2x-1}{12}=\frac{y+2}{2}=\frac{z-3}{3}$

Find the direction cosines of a line parallel to line AB. VIEW SOLUTION

Find: $\int \frac{dx}{\sqrt{4x-x^2}}$ VIEW SOLUTION

Events A and B are such that 
$\mathrm{P}\left(\mathrm{A}\right)=\frac{1}{2}, \mathrm{P}\left(\mathrm{B}\right)=\frac{7}{12} \mathrm{and} \mathrm{P}\left(\overline{\mathrm{A}}\cup \overline{\mathrm{B}}\right)=\frac{1}{4}$
Find whether the events A and B are independent or not.

OR

A box ${\mathrm{B}}_1$ contains 1 white ball and 3 red balls. Another box ${\mathrm{B}}_2$ contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes ${\mathrm{B}}_1 \mathrm{and} {\mathrm{B}}_2$, then find the probability that the two balls drawn are of the same colour. VIEW SOLUTION

Find the general solution of the following differential equation:
$\frac{dy}{dx}=e^{x-y}+x^2{e}^{-y}$ VIEW SOLUTION

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{a}.\overrightarrow{b}=1$ and $\overrightarrow{a}\times \overrightarrow{b}=\hat{j}–\hat{k}, \mathrm{then} \mathrm{find} \left|\overrightarrow{\mathit{b}}\right|$ VIEW SOLUTION

Let X be a random variable which assumes values x₁, x₂, x₃, x₄ such that 2P(X = x₁) = 3P(X = x₂) = P(X = x₃) = 5P(X = x₄).
Find the probability distribution of X. VIEW SOLUTION