Maths AA HL practice questions

At a skiing competition the mean time of the first three skiers is 34.1 seconds. The time for the fourth skier is then recorded and the mean time of the first four skiers is 35.0 seconds. Find the time achieved by the fourth skier.[4]

The $n^{\text{th}}$ term of an arithmetic sequence is given by $u_n = 15 - 3n$.

(a) State the value of the first term, $u_1$.[1]

(b) Given that the $n^{\text{th}}$ term of this sequence is $-33$, find the value of $n$.[2]

(c) Find the common difference, $d$.[2]

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

(a) One of the players is chosen at random. Find the probability that this player's score was 5 or more.[2]

(b) Calculate the mean score.[2]

The function $f$ is defined by $f(x) = \frac{1}{x}$, $x \neq 0$.

The graph of the function $y = g(x)$ is obtained by applying the following transformations to the graph of $y = f(x)$:

a translation by the vector $\begin{pmatrix} -3 \\ 0 \end{pmatrix}$;

a translation by the vector $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$.

(a) Find an expression for $g(x)$.[2]

(b) State the equations of the asymptotes of the graph of $g$.[2]

A mathematics test is given to a class of 20 students. One student scores 0, but all the other students score 10.

(a) Find the mean score for the class.[2]

(b) Write down the median score.[1]

(c) Write down the number of students who scored

(i) above the mean score;[2]

(ii) below the median score.

Consider two consecutive positive integers, $n$ and $n+1$.

Show that the difference of their squares is equal to the sum of the two integers.[4]

Events $A$ and $B$ are such that P$(A) = 0.65$, P$(B) = 0.75$ and P$(A \cap B) = 0.6$.

(a) Find P$(A \cup B)$.[2]

(b) Hence, or otherwise, find P$(A' \cap B')$.[2]

The marks obtained by a group of students in a class test are shown below.

Given the mean of the marks is 6.5, find the value of $k$.[4]

Let $A$ and $B$ be events such that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.6$. Find $P(A \mid B)$.[5]

Consider the complex numbers $z_1 = \cos \frac{11\pi}{12} + i\sin \frac{11\pi}{12}$ and $z_2 = \cos \frac{\pi}{6} + i\sin \frac{\pi}{6}$.

(a)(i) Find $\frac{z_1}{z_2}$[1]

(ii) Find $\frac{z_2}{z_1}$[3]

(b) $0, \frac{z_1}{z_2}$ and $\frac{z_2}{z_1}$ are represented by three points O, A and B respectively on an Argand diagram. Determine the area of the triangle OAB.[2]

It is believed that the lifespans of Manx cats are normally distributed with a mean of 13.5 years and a variance of 9.5 years$^2$.

(a) Calculate the range of lifespans of Manx cats whose lifespans are within one standard deviation of the mean.[2]

(b) Estimate the number of Manx cats in a population of 10 000 that will have a lifespan of less than 10 years. Give your answer to the nearest whole number.[3]

A set of 15 observations has mean 11.5 and variance 9.3. One observation of 22.1 is considered unreliable and is removed. Find the mean and variance of the remaining 14 observations.[6]

The number of hours spent exercising each week by a group of students is shown in the following table.

The median is 4.5 hours.

(a) Find the value of $x$.[2]

(b) Find the standard deviation.[2]

The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean $0.2$.

(a) Find the probability that Lucca eats at least one banana in a particular day.[2]

(b) Find the expected number of weeks in the year in which Lucca eats no bananas.[4]

Consider the arithmetic sequence $8, 26, 44, \ldots$

(a) Find an expression for the $n^{\text{th}}$ term.[1]

(b) Write down the sum of the first $n$ terms using sigma notation.[1]

(c) Calculate the sum of the first 15 terms.[2]