Practice Paper 1 (Cambridge International A-Level Mathematics)

A charity records the weight, in grams, of each rescued otter when it first arrives. The weights are represented by the box-and-whisker plot.

Find the median and the interquartile range of the weights on arrival.

The otters are weighed again after one month. The weights, in grams, are summarised in the table.

On the grid, draw a box-and-whisker plot to represent the weights after one month.

A random sample of 50 students was asked how long they spent revising for their Mathematics examination in the 24 hours before the examination. The results are shown in the table below.

Write down the modal class for these data.

Using $x$ to represent the mid-point of each class, $Σ x f = 8340$ and $Σ x^{2} f = 1636200$.

Calculate estimates of the mean and the standard deviation of the time spent revising.

Hugo records the number of days that each of the 23 employees in the IT department of a company was absent last quarter. The box-and-whisker plot summarises his results.

(i) State the median number of days absent.

(ii) State, with a reason, whether the mean or the median is the more suitable measure of central tendency for these data.

For the 23 employees, $\sum x = 286$ and $\sum x^{2} = 4238$, where $x$ is the number of days absent.

The employee who was absent for 29 days had a long-term illness, and Hugo decides to remove this value from the data.

Calculate the mean and the standard deviation of the number of days absent for the remaining 22 employees.

Praewa plays one sport every day after school: football, tennis or basketball. On any day, the probability that she chooses football is 4 times the probability that she chooses tennis, and the probability that she chooses basketball is 3 times the probability that she chooses tennis. Her choices on different days are independent.

Show that the probability that Praewa chooses tennis on a particular day is $\frac{1}{8}$.

Find the probability that Praewa chooses three different sports on the next three days.

For a 5-day week chosen at random, find the probability that Praewa chooses football every day.

Dennis and Minnie play a game using a box containing black counters and red counters. The number of red counters in the box is 3 more than the number of black counters.

In each turn of the game, a player selects two counters at random from the box at the same time. The first player to obtain two counters of the same colour wins the game.

Dennis takes the first turn. The probability that Dennis wins on his first turn is $\frac{17}{35}$.

Find the two possible values for the number of black counters in the box.

Dennis does not win on his first turn, and he does not replace his counters in the box. Minnie then selects two counters at random from the counters left in the box.

Given that Minnie wins on her first turn, the probability that she selects two red counters is $\frac{14}{19}$.

Find the original number of black counters in the box.

Twelve friends travel home from a party in three taxis: Taxi A, Taxi B and Taxi C. Four of the friends travel in each taxi.

Find the number of different ways in which the twelve friends can be allocated to the three taxis.

To join a superhero club, Bucky must pass a superhero examination. Bucky takes the examination repeatedly until he passes. The probability that Bucky passes the examination on any attempt is 0.16, independently of all previous attempts.

Find the probability that

(i) Bucky passes on his 5th attempt,

(ii) Bucky takes at least 5 attempts to pass,

(iii) Bucky passes within his first 10 attempts.

Find the expected number of attempts Bucky needs to pass the examination.

State one criticism of this model, in context.

The random variable X \sim \text{N}(23, \; 4^{2}).

Find

(i) $\text{P} ( X < 20 )$

(ii) $\text{P} ( X \geq 29 )$

(iii) $\text{P} ( 20 \leq X < 29 )$

The random variable Y \sim \text{N}(100, \; 225).

Find

(i) $\text{P} ( Y \leq 90 )$

(ii) $\text{P} ( Y > 140 )$

(iii) $\text{P} ( 85 \leq Y \leq 115 )$

Leonardo has a biased spinner with six sectors labelled 0, 1, 1, 2, 3 and 5. The probability of the spinner landing on each sector is shown in the following table.

Find the value of $p$.

Leonardo spins the spinner once. The random variable $X$ denotes the number on the sector where the spinner lands.

Find the probability that

(i) $X$ is no more than 1,

(ii) $X$ is at least 3.

Find the mean and the standard deviation of $X$.

Leonardo spins the spinner twice and adds the two scores together.

Find the probability that his total score is 5.