Stúdentspróf (Iceland) · Mathematics

Calculus, algebra, series and probability

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

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All 92 questions

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 $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.125$ . What is $P (A ∣ B)$ ?
If $f (x) = (2 x + 1)^{2}$ , find $f^{'} (2)$ .
Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 12 x$ .
What are the coordinates of the vertex of the parabola $y = - 2 (x + 2)^{2} + 5$ ?
The function $f$ is defined by $f (x) = 5 x + 1$ . Find $f^{- 1} (2)$ .
A variable is normally distributed with mean 60 and standard deviation 15. Approximately what percentage of values lie between 45 and 75?
A set of scores has a mean of 60 and a standard deviation of 12. What is the $z$ -score of a score of 78?
A random variable $X$ follows a binomial distribution with $n = 20$ and $p = 0.3$ . What is $Var (X)$ ?
What is the sum of the first 7 terms of the geometric sequence with first term 1 and common ratio 2?
An amount of 2000 dollars is invested at 10% per annum compounded annually. What is its value after 2 years, in dollars?
How many real roots does the equation $1 x^{2} + 3 x - 4 = 0$ 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 $y = 3 x^{2}$ , the $x$ -axis and the line $x = 3$ (in square units)?
Differentiate $y = e^{4 x}$ with respect to $x$ .
Let $f (x) = x^{2} (1 x + 2)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 4 x^{3} + 3 x^{2} - 5$ , what is $f^{'} (1)$ ?
What is the value of $lo g_{3} (2 7^{3})$ ?
What is the sum to infinity of the geometric series with first term 2 and common ratio $\frac{2}{3}$ ?
If $α$ and $β$ are the roots of $x^{2} - 5 x + 6 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $cos 12 0^{\circ}$ ?
What is the sum of the first 25 terms of an arithmetic progression with first term 3 and common difference 6?
A line is given by $y = - 3 x + 6$ . What is the slope of a line perpendicular to it?
In how many ways can 4 books be chosen from 9 different books?
Find $lim_{x \to 5} \frac{x ^{2} - 25}{x - 5}$ .
A rectangle has a fixed perimeter of 32 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?
For events $A$ and $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.375$ . What is $P (A ∣ B)$ ?
The function $f$ is defined by $f (x) = 5 x - 2$ . Find $f^{- 1} (0)$ .
A variable is normally distributed with mean 170 and standard deviation 15. Approximately what percentage of values lie between 155 and 185?
An amount of 2000 dollars is invested at 20% 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 $1 x^{2} + 0 x + 6 = 0$ have?
What is the area enclosed by $y = 3 x^{2}$ , the $x$ -axis and the line $x = 4$ (in square units)?
Differentiate $y = e^{6 x}$ with respect to $x$ .
Let $f (x) = x^{2} (2 x + 3)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 4 x^{3} + 4 x^{2} - 2$ , what is $f^{'} (1)$ ?
What is the value of $lo g_{5} (12 5^{4})$ ?
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 $x^{2} + 9 x + 20 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $cos 6 0^{\circ}$ ?
What is the sum of the first 17 terms of an arithmetic progression with first term 4 and common difference 5?
A line is given by $y = 3 x + 3$ . What is the slope of a line perpendicular to it?
In how many ways can 2 books be chosen from 8 different books?
Find $lim_{x \to 6} \frac{x ^{2} - 36}{x - 6}$ .
A rectangle has a fixed perimeter of 20 units. What is the greatest possible area of the rectangle?
If $f (x) = (3 x - 3)^{4}$ , find $f^{'} (0)$ .
What are the coordinates of the vertex of the parabola $y = (x - 4)^{2} - 1$ ?
The function $f$ is defined by $f (x) = 2 x + 4$ . Find $f^{- 1} (0)$ .
A variable is normally distributed with mean 170 and standard deviation 15. Approximately what percentage of values lie between 140 and 200?
A set of scores has a mean of 60 and a standard deviation of 10. What is the $z$ -score of a score of 40?
A random variable $X$ follows a binomial distribution with $n = 40$ and $p = 0.3$ . What is $E (X)$ ?
What is the sum of the first 7 terms of the geometric sequence with first term 5 and common ratio 2?
Solve $2^{x + 1} = 4$ for $x$ .
How many real roots does the equation $3 x^{2} + 1 x - 1 = 0$ have?
Find the gradient of the tangent to $y = x^{2} - 4 x$ at the point where $x = - 2$ .
What is the area enclosed by $y = 2 x^{2}$ , the $x$ -axis and the line $x = 3$ (in square units)?
Let $f (x) = x^{2} (2 x + 1)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = x^{3} + 3 x^{2} - 1$ , what is $f^{'} (2)$ ?
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 $x^{2} - 11 x + 30 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 6 0^{\circ}$ ?
What is the sum of the first 20 terms of an arithmetic progression with first term 5 and common difference 2?
A line is given by $y = 2 x + 4$ . What is the slope of a line perpendicular to it?
In how many ways can 4 books be chosen from 6 different books?
Find $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .
A radioactive substance has a half-life of 2 years. How much of a 800 g sample remains after 6 years?
For events $A$ and $B$ , $P (B) = 0.4$ and $P (A \cap B) = 0.2$ . What is $P (A ∣ B)$ ?
If $f (x) = (3 x - 1)^{5}$ , find $f^{'} (2)$ .
Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 75 x$ .
What are the coordinates of the vertex of the parabola $y = 3 (x + 5)^{2} + 8$ ?
The function $f$ is defined by $f (x) = 5 x + 7$ . Find $f^{- 1} (- 2)$ .
A variable is normally distributed with mean 50 and standard deviation 10. Approximately what percentage of values lie between 30 and 70?
A set of scores has a mean of 50 and a standard deviation of 12. What is the $z$ -score of a score of 68?
A random variable $X$ follows a binomial distribution with $n = 20$ and $p = 0.4$ . What is $Var (X)$ ?
What is the sum of the first 6 terms of the geometric sequence with first term 3 and common ratio 2?
An amount of 1000 dollars is invested at 20% per annum compounded annually. What is its value after 2 years, in dollars?
Solve $3^{x + 2} = 27$ for $x$ .
How many real roots does the equation $4 x^{2} + 6 x + 1 = 0$ have?
What is the area enclosed by $y = x^{2}$ , the $x$ -axis and the line $x = 3$ (in square units)?
Differentiate $y = e^{2 x}$ with respect to $x$ .
Let $f (x) = x^{2} (4 x + 3)$ . Use the product rule to find $f^{'} (1)$ .
If $f (x) = 3 x^{3} + 4 x^{2} - 4$ , what is $f^{'} (2)$ ?
What is the value of $lo g_{5} (12 5^{3})$ ?
If $α$ and $β$ are the roots of $x^{2} - 2 x - 8 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $cos 15 0^{\circ}$ ?
What is the sum of the first 20 terms of an arithmetic progression with first term 9 and common difference 6?
A line is given by $y = 3 x - 1$ . What is the slope of a line perpendicular to it?
In how many ways can 3 books be chosen from 8 different books?
Find $lim_{x \to 7} \frac{x ^{2} - 49}{x - 7}$ .
A rectangle has a fixed perimeter of 48 units. What is the greatest possible area of the rectangle?

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