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

1 Limits / 4 pts

1. \( \dfrac{(x-5)(x+5)}{x-5}=x+5\to10 \). (1 pt)
2. Equal degrees: \( \dfrac63=2 \). (1 pt)
3. Left: \( 2+3=5 \) (0.5 pt). Right: \( 2^2-1=3 \) (0.5 pt). They differ, so the limit does not exist (1 pt).

2 Asymptotes / 3 pts

Factor the denominator: \( (x-3)(x+3) \). The numerator \( 4(x-2) \) shares no factor. (1 pt)

Vertical asymptotes: \( x=3 \) and \( x=-3 \). (1 pt)

Horizontal asymptote: degree 1 below degree 2, so \( y=0 \). (1 pt)

3 Counting / 3 pts

1. Order matters: \( P(9,3)=9\cdot8\cdot7=504 \). (1.5 pts)
2. Order does not matter: \( C(9,3)=\dfrac{504}{6}=84 \). (1.5 pts)

4 A café survey / 3 pts

\( P(P)=0.30+0.20=0.50 \). (0.5 pt)

1. \( P(P\mid C)=\dfrac{0.30}{0.55}=\dfrac6{11}\approx0.545 \). (0.75 pt)
2. \( P(C\mid P)=\dfrac{0.30}{0.50}=0.6 \). (0.75 pt)
3. \( P(C)P(P)=0.55\cdot0.50=0.275\neq0.30 \), so they are not independent. (1 pt)

5 Seeds and scores / 4 pts

1. \( P(X=5)=C(6,5)(0.8)^5(0.2)=6\cdot0.32768\cdot0.2=0.393216\approx0.393 \) (1.5 pts). Mean \( =6\cdot0.8=4.8 \) seeds (0.5 pt).
2. \( z=\dfrac{150-120}{15}=2 \) (1 pt). About 95% lie within \( 2\sigma \), so about \( 2.5\% \) are faster (1 pt).

6 Regression / 3 pts

1. \( m=-1.7 \), \( b=5-(-1.7)(3)=10.1 \), so \( \hat y=-1.7x+10.1 \). (1.5 pts)
2. \( r=\dfrac{-17}{\sqrt{300}}\approx-0.98 \): a very strong negative linear relationship. (1 pt)
3. \( \hat y=-1.7(2.5)+10.1=5.85 \). (0.5 pt)