(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $$\frac{3}{5}$$ be the probability that he knows the answer and $$\frac{2}{5}$$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $$\frac{1}{3}$$. What is the probability that the student knows the answer, given that he answered it correctly?

OR

(b) A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.

Case Study I

An organization conducted bike race under two different categories-Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets $$B$$ and $$G$$ with these participants for his college project.

let $$B=\left\{b_1, b_2, b_3\right\}$$ and $$G=\left\{g_1, g_2\right\}$$, where B represents the set of Boys selected and G the set of Girls selected for the final race.

Based on the above information, answer the following questions:

(i) How many relations are possible from $$B$$ to $$G$$ ?

(ii) Among all the possible relations from $$B$$ to $$G$$, how many functions can be formed from $$B$$ to $$\mathrm{G}$$ ?

(iii) Let $$\mathrm{R}: \mathrm{B} \rightarrow \mathrm{B}$$ be defined by $$R=\{(x, y): x$$ and $$y$$ are students of the same sex $$\}$$. Check if $$R$$ is an equivalence relation.

OR

(iii) A function $$f: \mathrm{B} \rightarrow \mathrm{G}$$ be defined by $$f=\left\{\left(b_1, g_1\right)\right.$$, $$\left.\left(b_2, g_2\right),\left(b_3, g_1\right)\right\}$$.

Check if $$f$$ is bijective. Justify your answer.

Case Study II

Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop. Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250 .

Based on the above information, answer the following questions:

(i) Convert the given above situation into a matrix equation of the form $$A X=B$$.

(ii) Find $$|\mathrm{A}|$$.

(iii) Find $$\mathrm{A}^{-1}$$.

OR

(iii) Determine $$P=A^2-5 A$$.

Case Study III

An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $$\frac{d y}{d x}=F(x, y)$$ is said to be homogeneous if $$\mathrm{F}(x, y)$$ is a homogeneous function of degree zero, whereas a function $$\mathrm{F}(x, y)$$ is a homogenous function of degree $$n$$ if $$\mathrm{F}(\lambda x, \lambda y)=\lambda^n F(x, y)$$. To solve a homogeneous differential equation of the type $$\frac{d y}{d x}=\mathrm{F}(x, y)= g\left(\frac{y}{x}\right)$$, we make the substitution $$y=v x$$ and then separate the variables.

Based on the above, answer the following questions:

(i) Show that $$\left(x^2-y^2\right) d x+2 x y d x=0$$ is a differential equation of the type $$\frac{d y}{d x}=g\left(\frac{y}{x}\right)$$.

(ii) Solve the above equation to find its general solution.