Victorian VCE · VCE Mathematical Methods

Functions, calculus and probability

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

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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?
Solve $lo g_{2} x + lo g_{2} (x - 2) = 3$ for $x$ .
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 $P (A \cap B) = 0.075$ . What is $P (A ∣ B)$ ?
If $f (x) = (5 x + 4)^{4}$ , find $f^{'} (2)$ .
Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 27 x$ .
What are the coordinates of the vertex of the parabola $y = - (x - 1)^{2} - 2$ ?
The function $f$ is defined by $f (x) = 2 x + 7$ . Find $f^{- 1} (- 1)$ .
A variable is normally distributed with mean 50 and standard deviation 15. Approximately what percentage of values lie between 20 and 80?
A set of scores has a mean of 50 and a standard deviation of 12. What is the $z$ -score of a score of 38?
A random variable $X$ follows a binomial distribution with $n = 30$ and $p = 0.4$ . What is $E (X)$ ?
How many real roots does the equation $1 x^{2} - 3 x + 3 = 0$ have?
Find the gradient of the tangent to $y = 4 x^{2} - 5 x$ at the point where $x = 3$ .
What is the area enclosed by $y = 2 x^{2}$ , the $x$ -axis and the line $x = 3$ (in square units)?
Differentiate $y = e^{3 x}$ with respect to $x$ .
Let $f (x) = x^{2} (3 x + 1)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 4 x^{3} + 5 x^{2} - 3$ , what is $f^{'} (3)$ ?
What is the value of $lo g_{3} (24 3^{2})$ ?
What is the sum to infinity of the geometric series with first term 8 and common ratio $\frac{2}{3}$ ?
If $α$ and $β$ are the roots of $x^{2} + 2 x - 24 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 21 0^{\circ}$ ?
What is the sum of the first 13 terms of an arithmetic progression with first term 9 and common difference 6?
A line is given by $y = 2 x + 6$ . What is the slope of a line perpendicular to it?
In how many ways can 4 books be chosen from 8 different books?
Find $lim_{x \to 5} \frac{x ^{2} - 25}{x - 5}$ .
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 160 g sample remains after 9 years?
For events $A$ and $B$ , $P (B) = 0.25$ and $P (A \cap B) = 0.1875$ . What is $P (A ∣ B)$ ?
If $f (x) = (4 x - 1)^{5}$ , find $f^{'} (2)$ .
Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 48 x$ .
What are the coordinates of the vertex of the parabola $y = - 2 (x + 2)^{2} + 6$ ?
The function $f$ is defined by $f (x) = 2 x - 1$ . Find $f^{- 1} (10)$ .
A variable is normally distributed with mean 100 and standard deviation 15. Approximately what percentage of values lie between 70 and 130?
A set of scores has a mean of 60 and a standard deviation of 10. What is the $z$ -score of a score of 70?
A random variable $X$ follows a binomial distribution with $n = 30$ and $p = 0.3$ . What is $Var (X)$ ?
An amount of 2000 dollars is invested at 10% per annum compounded annually. What is its value after 2 years, in dollars?
Solve $5^{x + 2} = 125$ for $x$ .
How many real roots does the equation $4 x^{2} - 6 x + 3 = 0$ have?
Find the gradient of the tangent to $y = 3 x^{2} + 4 x$ at the point where $x = 3$ .
What is the area enclosed by $y = 3 x^{2}$ , the $x$ -axis and the line $x = 4$ (in square units)?
Differentiate $y = e^{5 x}$ with respect to $x$ .
Let $f (x) = x^{2} (4 x + 4)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 4 x^{3} + x^{2} + 5$ , what is $f^{'} (3)$ ?
What is the value of $lo g_{2} (3 2^{3})$ ?
What is the sum to infinity of the geometric series with first term 12 and common ratio $\frac{2}{3}$ ?
What is the value of $cos 21 0^{\circ}$ ?
What is the sum of the first 20 terms of an arithmetic progression with first term 9 and common difference 5?
In how many ways can 2 books be chosen from 6 different books?
For events $A$ and $B$ , $P (B) = 0.3$ and $P (A \cap B) = 0.15$ . What is $P (A ∣ B)$ ?
If $f (x) = (4 x - 4)^{4}$ , find $f^{'} (0)$ .
What are the coordinates of the vertex of the parabola $y = - 2 (x - 2)^{2} - 5$ ?
The function $f$ is defined by $f (x) = 5 x + 5$ . Find $f^{- 1} (0)$ .
A variable is normally distributed with mean 100 and standard deviation 10. Approximately what percentage of values lie between 90 and 110?
A set of scores has a mean of 70 and a standard deviation of 5. What is the $z$ -score of a score of 65?
A random variable $X$ follows a binomial distribution with $n = 40$ and $p = 0.25$ . What is $Var (X)$ ?
What is the sum of the first 5 terms of the geometric sequence with first term 1 and common ratio 2?
An amount of 5000 dollars is invested at 10% per annum compounded annually. What is its value after 2 years, in dollars?
Solve $2^{x + 1} = 32$ for $x$ .
How many real roots does the equation $2 x^{2} + 7 x - 5 = 0$ have?
Find the gradient of the tangent to $y = 4 x^{2} + 2 x$ at the point where $x = 1$ .
What is the area enclosed by $y = x^{2}$ , the $x$ -axis and the line $x = 4$ (in square units)?
Differentiate $y = e^{2 x}$ with respect to $x$ .
Let $f (x) = x^{2} (1 x + 3)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = x^{3} + 5 x^{2} - 2$ , what is $f^{'} (2)$ ?
What is the value of $lo g_{5} (2 5^{4})$ ?
What is the sum to infinity of the geometric series with first term 8 and common ratio $\frac{1}{3}$ ?
If $α$ and $β$ are the roots of $x^{2} - 7 x + 6 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 13 5^{\circ}$ ?
What is the sum of the first 12 terms of an arithmetic progression with first term 9 and common difference 3?
A line is given by $y = 3 x - 5$ . What is the slope of a line perpendicular to it?
In how many ways can 2 books be chosen from 10 different books?
Find $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$ .

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