Q$1)$ In operations management, the Critical Path Method $($CPM$)$ is used to:

A$)$ Analyze the financial viability of a new project

B$)$ Estimate the shortest possible time to complete a project

C$)$ Determine the maximum storage space required for inventory

D$)$ Calculate the most efficient layout of a manufacturing plant

Q$2)$ Calculus Challenge: Find the derivative of the function f$($x$)=3$x$^4-2$x$^3+5$x$^2-7$x $+9.$

Number Theory Puzzle: Determine the number of positive integer solutions to the equation $2$x $+3$y $=100.$

Geometry Stumper: In triangle ABC, angle A measures $50$ degrees, and angle B measures $70$ degrees. Find the measure of angle C$.$

Probability Challenge: You have a deck of $52$ cards. What is the probability of drawing two aces $($without replacement$)$ in two consecutive draws?

Algebraic Riddle: Solve for x and y in the system of equations:

$2$x $+3$y $=11$

$5$x $-2$y $=7$

Q$3)$ Calculus: Find the derivative of the function f$($x$)=$ ln$($x$^2+3$x $+1).$

Linear Algebra: Given the matrices A $=[123$; $456$; $789]$ and B $=[987$; $654$; $321],$ calculate the product AB$.$

Number Theory: Find the largest prime factor of $1234567890.$

Geometry: In triangle ABC, angle A measures $50$ degrees, and side AC is $12$ cm long. If side BC is $8$ cm long, find the length of side AB using the Law of Sines.

Probability: A standard deck of playing cards is shuffled thoroughly. What is the probability of drawing two consecutive cards of the same suit from the deck $($e$.$g$.,$ two consecutive hearts$)?$