NSW HSC · Mathematics Advanced

Harder calculus, series and probability

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

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

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A particle moves along a line with displacement $x = t^{3} - 6 t^{2} + 9 t$ metres at time $t$ seconds. At what time $t > 0$ is the particle first at rest?
What is the area enclosed between the curve $y = x^{2}$ and the line $y = x$ ?
For events $A$ and $B$ , $P (B) = 0.3$ and $P (A \cap B) = 0.225$ . What is $P (A ∣ B)$ ?
A rectangle has a fixed perimeter of 28 units. What is the greatest possible area of the rectangle?
For events $A$ and $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.125$ . What is $P (A ∣ B)$ ?
A rectangle has a fixed perimeter of 24 units. What is the greatest possible area of the rectangle?
For events $A$ and $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.25$ . What is $P (A ∣ B)$ ?
A rectangle has a fixed perimeter of 48 units. What is the greatest possible area of the rectangle?
A radioactive substance has a half-life of 3 years. How much of a 320 g sample remains after 9 years?
Solve $lo g_{2} x + lo g_{2} (x - 2) = 3$ for $x$ .
A rectangle has a fixed perimeter of 32 units. What is the greatest possible area of the rectangle?
For events $A$ and $B$ , $P (B) = 0.4$ and $P (A \cap B) = 0.3$ . What is $P (A ∣ B)$ ?
For events $A$ and $B$ , $P (B) = 0.4$ and $P (A \cap B) = 0.1$ . What is $P (A ∣ B)$ ?
A rectangle has a fixed perimeter of 20 units. What is the greatest possible area of the rectangle?
A radioactive substance has a half-life of 3 years. How much of a 640 g sample remains after 9 years?
A rectangle has a fixed perimeter of 40 units. What is the greatest possible area of the rectangle?
For events $A$ and $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.375$ . What is $P (A ∣ B)$ ?
A radioactive substance has a half-life of 5 years. How much of a 320 g sample remains after 20 years?
A radioactive substance has a half-life of 2 years. How much of a 640 g sample remains after 6 years?
A radioactive substance has a half-life of 5 years. How much of a 800 g sample remains after 15 years?

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Harder calculus, series and probability

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