NSW HSC · Mathematics Advanced

Calculus, functions and statistics

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

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Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 3 x$ .
What are the coordinates of the vertex of the parabola $y = 3 (x + 5)^{2} + 5$ ?
The function $f$ is defined by $f (x) = 3 x - 3$ . Find $f^{- 1} (0)$ .
A variable is normally distributed with mean 50 and standard deviation 15. Approximately what percentage of values lie between 35 and 65?
A set of scores has a mean of 70 and a standard deviation of 5. What is the $z$ -score of a score of 60?
What is the sum of the first 5 terms of the geometric sequence with first term 2 and common ratio 2?
An amount of 5000 dollars is invested at 20% per annum compounded annually. What is its value after 3 years, in dollars?
Solve $2^{x + 2} = 32$ for $x$ .
How many real roots does the equation $3 x^{2} - 8 x - 3 = 0$ have?
What is the area enclosed by $y = x^{2}$ , the $x$ -axis and the line $x = 2$ (in square units)?
Differentiate $y = e^{4 x}$ with respect to $x$ .
Let $f (x) = x^{2} (1 x + 1)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 4 x^{3} + x^{2}$ , what is $f^{'} (3)$ ?
What is the value of $lo g_{5} (62 5^{2})$ ?
What is the sum to infinity of the geometric series with first term 12 and common ratio $\frac{2}{3}$ ?
If $α$ and $β$ are the roots of $x^{2} + 3 x - 10 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 21 0^{\circ}$ ?
What is the sum of the first 21 terms of an arithmetic progression with first term 6 and common difference 7?
A line is given by $y = 2 x - 3$ . What is the slope of a line perpendicular to it?
In how many ways can 2 books be chosen from 6 different books?
Find $lim_{x \to 7} \frac{x ^{2} - 49}{x - 7}$ .
What are the coordinates of the vertex of the parabola $y = - (x - 1)^{2} - 2$ ?
The function $f$ is defined by $f (x) = 4 x - 4$ . Find $f^{- 1} (7)$ .
A variable is normally distributed with mean 100 and standard deviation 5. Approximately what percentage of values lie between 90 and 110?
A set of scores has a mean of 50 and a standard deviation of 8. What is the $z$ -score of a score of 62?
A random variable $X$ follows a binomial distribution with $n = 20$ and $p = 0.25$ . What is $E (X)$ ?
What is the sum of the first 6 terms of the geometric sequence with first term 3 and common ratio 2?
Solve $3^{x + 1} = 243$ for $x$ .
How many real roots does the equation $1 x^{2} - 2 x + 4 = 0$ have?
Find the gradient of the tangent to $y = 3 x^{2} - 2 x$ 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 $y = e^{5 x}$ with respect to $x$ .
Let $f (x) = x^{2} (2 x + 3)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 2 x^{3} + 2 x^{2} + 6$ , what is $f^{'} (3)$ ?
What is the value of $lo g_{5} (312 5^{4})$ ?
What is the sum to infinity of the geometric series with first term 6 and common ratio $\frac{1}{4}$ ?
If $α$ and $β$ are the roots of $x^{2} + 7 x + 12 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 30 0^{\circ}$ ?
What is the sum of the first 14 terms of an arithmetic progression with first term 2 and common difference 2?
A line is given by $y = 3 x - 4$ . What is the slope of a line perpendicular to it?
In how many ways can 4 books be chosen from 9 different books?
Find $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .
If $f (x) = (5 x + 2)^{5}$ , 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 $y = (x - 3)^{2} + 7$ ?
The function $f$ is defined by $f (x) = 2 x + 4$ . Find $f^{- 1} (4)$ .
A set of scores has a mean of 70 and a standard deviation of 8. What is the $z$ -score of a score of 86?
What is the sum of the first 6 terms of the geometric sequence with first term 2 and common ratio 2?
An amount of 5000 dollars is invested at 20% per annum compounded annually. What is its value after 2 years, in dollars?
How many real roots does the equation $1 x^{2} + 2 x + 2 = 0$ have?
Find the gradient of the tangent to $y = x^{2} - 4 x$ at the point where $x = 0$ .
What is the area enclosed by $y = 2 x^{2}$ , the $x$ -axis and the line $x = 2$ (in square units)?
Differentiate $y = e^{2 x}$ with respect to $x$ .
Let $f (x) = x^{2} (2 x + 1)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = x^{3} + 4 x^{2} - 4$ , what is $f^{'} (2)$ ?
What is the value of $lo g_{2} (3 2^{4})$ ?
What is the sum to infinity of the geometric series with first term 3 and common ratio $\frac{1}{4}$ ?
If $α$ and $β$ are the roots of $x^{2} - 5 x + 6 = 0$ , what is $α^{2} + β^{2}$ ?

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Calculus, functions and statistics

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