NSW HSC · Mathematics Advanced

Integral Calculus, Probability and Statistics

21 practice questions, each with the answer and an explanation.

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Evaluate the indefinite integral $\int \frac{2 x}{x ^{2} + 5} d x$ .
Calculate the area enclosed by the curve $y = x^{2}$ and the line $y = 4$ .
Evaluate $\int_{0}^{π /4} sec^{2} (x) d x$ .
A continuous random variable $X$ has probability density function $f (x) = k x$ for $0 \leq x \leq 4$ , and $f (x) = 0$ elsewhere. What is the value of the constant $k$ ?
For the random variable $X$ with PDF $f (x) = \frac{1}{8} x$ on $[0, 4]$ , calculate the expected value $E (X)$ .
Scores on a statewide exam are normally distributed with mean $μ = 70$ and standard deviation $σ = 8$ . What is the z-score of a student who scored 86?
According to the empirical 68-95-99.7 rule for normal distributions, approximately what percentage of scores lie within two standard deviations of the mean?
Using the trapezoidal rule with 2 subintervals ( $n = 2$ ), approximate $\int_{1}^{3} x^{2} d x$ .
Evaluate $\int e^{- 3 x} d x$ .
What is the volume of the solid generated when the region under $y = x$ from $x = 0$ to $x = 4$ is rotated about the x-axis?
If events $A$ and $B$ are mutually exclusive, and $P (A) = 0.35, P (B) = 0.40$ , find $P (A \cup B)$ .
Find $\int_{1}^{4} \frac{1}{x} d x$ .
Find $\int cos (2 x) d x$ .
Evaluate $\int_{0}^{1} e^{2 x} d x$ .
Find the area enclosed between the curves $y = x$ and $y = x^{2}$ .
A fair coin is tossed 5 times. What is the probability of exactly 3 heads?
Scores are normally distributed with mean 60 and standard deviation 10. Using $Pr (Z > 1.5) = 0.0668$ , find the probability that a score exceeds 75.
The region between $y = x$ and $y = x^{2}$ is rotated about the $x$ -axis. What is the volume of the solid formed?
For events $A$ and $B$ : $Pr (A) = 0.6$ , $Pr (B) = 0.5$ and $Pr (A \cap B) = 0.3$ . Which statement is correct?
A continuous random variable has pdf $f (x) = \frac{x ^{2}}{9}$ for $0 \leq x \leq 3$ and $0$ elsewhere. The median $m$ satisfies:
Test marks are normally distributed with mean 50. Given that $Pr (X > 65) = 0.0228$ and $Pr (Z > 2) = 0.0228$ , the standard deviation is:

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Integral Calculus, Probability and Statistics

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