Limits, Probability, and Statistics: math test, Grade 12 – download the PDF

1 Limits / 4 pts

1. Find \( \lim_{x\to5}\dfrac{x^2-25}{x-5} \).
2. Find \( \lim_{x\to\infty}\dfrac{6x^2-x}{3x^2+2} \).
3. Let \( f(x)=\begin{cases}x+3 & x\lt2\\ x^2-1 & x\ge2\end{cases} \). Find both one-sided limits at 2 and say whether \( \lim_{x\to2}f(x) \) exists.

2 Asymptotes / 3 pts

Find every vertical and horizontal asymptote of \( g(x)=\dfrac{4x-8}{x^2-9} \).

3 Counting / 3 pts

A coach has 9 players. (a) In how many ways can a captain, a co-captain, and a treasurer be named? (b) In how many ways can a group of 3 players be chosen to run an errand?

4 A café survey / 3 pts

At a café, 55% of customers order coffee (C), 30% order coffee and a pastry, and 20% order a pastry without coffee. Let P be "orders a pastry". (a) Find \( P(P\mid C) \). (b) Find \( P(C\mid P) \). (c) Are C and P independent?

5 Seeds and scores / 4 pts

(a) A seed sprouts with probability 0.8. Six seeds are planted independently. Find the probability that exactly 5 sprout and the expected number that sprout. (b) A typing speed is approximately normal with mean 120 and standard deviation 15 (characters per minute). Find the z-score of 150 and the percent of typists faster than 150.

6 Regression / 3 pts

The data \( (1,8),(2,7),(3,5),(4,4),(5,1) \) give \( \bar x=3 \), \( \bar y=5 \), \( \sum(x-\bar x)(y-\bar y)=-17 \), \( \sum(x-\bar x)^2=10 \), \( \sum(y-\bar y)^2=30 \). Calculator allowed. (a) Find the least-squares line. (b) Find \( r \) and interpret it. (c) Predict \( y \) when \( x=2.5 \).