tktung/LC2024 · Datasets at Hugging Face

problem stringlengths 31 644	answer stringlengths 1 118 ⌀	question_id stringlengths 11 14
Solve the following equation for $n \in \mathbb{N}$: $n - 3 = \sqrt{3n + 1}$	8	Day 1 - Q1a
Write the following sentence as a single paragraph in terms of $t$: $\frac{4}{2t + 1} - \frac{7}{12t}$	$\frac{34t - 7}{(2t + 1)(12t)}$	Day 1 - Q1b
Solve the following parallel equations for $x, y, w \in \mathbb{Z}$: $\begin{cases} x + 2y = 143 \\ y + 3w = -74 \\ 4x + 5w = 4 \end{cases}$	51;46;-40	Day 1 - Q1c
In this question, $i^2 = -1$. Find both solutions to the equation in $z$ the following, where $z$ in number Complex.Each answer should be in the form $a + bi$, where $a, b \in \mathbb{R}$. $z^2 + 12z + 261 = 0$	-6+15i;-6-15i	Day 1 - Q2a
In this question, $i^2 = -1$. He used a theory of Moivre to $(1 - \sqrt{3}i)^9$ written in the form $a + bi$, where $a, b \in \mathbb{R}.$	-512	Day 1 - Q2b
Find the sum: $\int \cos6x\ dx$	\frac{1}{6}\sin(6x)+C	Day 1 - Q3a
The function is defined $f$ as follows for $x \in \mathbb{R}$: $f(x) = 2x^3 - 9x^2 + 5x - 11$ Find the equation of the circle with a graph.$f$ at the point at which the$x = 2$. You don't need to simplify your answer.	y − (−21) = −7(x − 2)	Day 1 - Q3bi
The function is defined $f$ as follows for $x \in \mathbb{R}$: $f(x) = 2x^3 - 9x^2 + 5x - 11$ Find the coordinates $x$ the rebound point of $f$.	\frac{3}{2}	Day 1 - Q3bii
The following function is distinguished from basic principles, in respect of $x$: $f(x) = x^2 - 7x - 10$	\frac{-238}{361}	Day 1 - Q4a
The function is defined $g(x)$ for the$x \in \mathbb{R}$ in accordance with: $g(x) = \frac{6x + 1}{x^4 + 3}$ Get the value $g'(-2)$, the following paragraph is added:$g(x)$ where $x = -2$.	-238/361	Day 1 - Q4b
It is a continuous operation.$h: \mathbb{R} \rightarrow \mathbb{R}$. The graph has a logarithmic minimum.$h(x)$ at point $(0, 5)$. Indicate whether the following statement is true or false: It must be$5$ is at least $h(x)$ for each real value of $x$. other Defend your answer.	False	Day 1 - Q4c
The first three terms of a concatenation sequence are as follows, where $p \in \mathbb{R}: \begin{cases} T_1 = 2p + 1 \\ T_2 = 5p - 3 \\ T_3 = 6p + 7 \end{cases}$ Get the value $p$.	7	Day 1 - Q5a
They are $G_7 = 6$ and $G_11 = \frac{3}{8}$ the 7th and 11th term of a polynomial sequence, respectively. Find the two possible values of $r$, The Commission shall be assisted by the European Parliament.$r \in \mathbb{R}$.	\frac{1}{2};\frac{-1}{2}	Day 1 - Q5b
A sequence of functions is defined.$F_0, F_1, F_2, \dots$ for $x \in \mathbb{R}, x \gt 0$: \The first one is the \The Commission shall:$F_0 = x^{2024}$ \Item for $n \geq 1$, The function is $F_n$ the following paragraph is added:$F_{n-1}$ , $x$. \end{title} Write $F_1$ and $F_2$ in terms of $x$.	F1(x) = 2024 x^2023; F2(x) = 2024 × 2023 x^2022	Day 1 - Q5ci
A sequence of functions is defined.$F_0, F_1, F_2, \dots$ for $x \in \mathbb{R}, x \gt 0$: \The first one is the \The Commission shall:$F_0 = x^{2024}$ \Item for $n \geq 1$, The function is $F_n$ the following paragraph is added:$F_{n-1}$ , $x$. \end{title} Find the first value of $n$ This leaves a$F_n=0$.	2025	Day 1 - Q5cii
$h(x) = x^2 + bx - 12$, where is$x \in \mathbb{R}$ and place it on a doorstep.$b$. Get the value $b$ This leaves a$x - 4$ in a factor of $h(x)$.	-1	Day 1 - Q6a
Two functions are defined, $f(x)$ and $g(x)$, for $x \in \mathbb{R}, x \gt 0$: \The first two terms of the equation are: Get the value $f(1.2)$. Please answer in the form $a \times 10^n$, where $a \in \mathbb{R}, 1 \leq a \lt 10, n \in \mathbb{N}$, and which shall be$a$ right to one decimal place.	4.9\times10^4	Day 1 - Q6bi
Two functions are defined, $f(x)$ and $g(x)$, for $x \in \mathbb{R}, x \gt 0$: \The first two terms of the equation are: Get the value $x$ This leaves a$g(x) = 3.5$. Please answer in the form $e^p$, where $p \in \mathbb{R}$.	x=e^7	Day 1 - Q6bii
Two functions are defined, $f(x)$ and $g(x)$, for $x \in \mathbb{R}, x \gt 0$: \The first two terms of the equation are: Write the function $g(f(x))$ in terms of $x$, in the simplest form.	4.5x	Day 1 - Q6biii
Fiadh has a total annual salary of €54000. She pays income tax at a rate of 20% on the first €40000 of her salary and at a rate of 40% on the balance sheet. Work out his annual net salary, assuming no other deductions.	42175	Day 1 - Q7a
The first monthly repayment is made directly after the mortgage is taken out, and the second monthly repayment is made directly after the mortgage is taken out. Write down the immediate value of each of the first three monthly repayments they make, at the time they take out the mortgage. Let all values be in pieces. Do not overwhelm any powers.	\frac{1647.75}{1.00279};\frac{1647.75}{1.00279^2};\frac{1647.75}{1.00279^3}	Day 1 - Q7bi
The first monthly repayment is made directly after the mortgage is taken out, and the second monthly repayment is made directly after the mortgage is taken out. Work out the amount of money that Wood and his partner borrowed for their mortgage. Correct your answer to the nearest euro.	334563	Day 1 - Q7bii
Fide puts money into a savings account, and she leaves it there for several years. This is the bottom line.$F(t)$, the amount of money in the account in euro after $t$ year, whereas$t \in \mathbb{R}, t \geq 0$: $F(t) = 5000 e^{0.04t}$ Use a differential and find the rate at which the amount of money in the account increases after 3.5 years.	230	Day 1 - Q7ci
Fide puts money into a savings account, and she leaves it there for several years. This is the bottom line.$F(t)$, the amount of money in the account in euro after $t$ year, whereas$t \in \mathbb{R}, t \geq 0$: $F(t) = 5000 e^{0.04t}$ Use an estimate and find the average amount of money in the account over the first 5 years.	5535	Day 1 - Q7cii
Fide puts money into a savings account, and she leaves it there for several years. This is the bottom line.$F(t)$, the amount of money in the account in euro after $t$ year, whereas$t \in \mathbb{R}, t \geq 0$: $F(t) = 5000 e^{0.04t}$ Calculate the annual interest rate (AER) for this account. So let's say that you have a percentage increase in the amount of money in the account over the course of a year, and your answer is right up to 2 decimal places.	4.08%	Day 1 - Q7ciii
It is time.$R$ unit by sphere. The part of the sphere that intersects a smooth surface is called a cap, and its volume is: $C=\frac{\pi k^2}{3}(3R - k)$ Here, it is.$C$ The volume of the pipe is $k$ The height of the pipe, where the$0 \lt k \leq R$. Get the value $C$, volume of the pipe, where:$R = 13$ and $k = 4$. # Please answer in terms of $\pi$.	\dfrac{560}{3} \pi	Day 1 - Q9a
It is time.$R$ unit by sphere. The part of the sphere that intersects a smooth surface is called a cap, and its volume is: $C=\frac{\pi k^2}{3}(3R - k)$ Here, it is.$C$ The volume of the pipe is $k$ The height of the pipe, where the$0 \lt k \leq R$. A sphere has 8 units and a tube has $y$ Unit at altitude, where$0 \lt y \leq 8$. The volume of these pipes is ($36\pi y$) Unit$^3$. Used that, and out of the blue for $C$ above, and demonstrate that: $\frac{y}{3}(24 - y) = 36$	null	Day 1 - Q9bi
It is time.$R$ unit by sphere. The part of the sphere that intersects a smooth surface is called a cap, and its volume is: $C=\frac{\pi k^2}{3}(3R - k)$ Here, it is.$C$ The volume of the pipe is $k$ The height of the pipe, where the$0 \lt k \leq R$. A sphere has 8 units and a tube has $y$ Unit at altitude, where$0 \lt y \leq 8$. The volume of these pipes is ($36\pi y$) Unit$^3$. The equation has widened.$\frac{y}{3}(24 - y) = 36$ and set it up to the top of the ladder. I'll get this.	6	Day 1 - Q9bii
The diameter of the$x$ cm at the hemisphere. The following is added:$V(x)$, the volume of that hemisphere in cm$^3$, in accordance with: $V(x) = \frac{\pi}{12} x^3$ Get the value $x$ when the volume of the hemisphere is more than 3 litres. Be sure your answer is correct to one decimal place.	22.5	Day 1 - Q9c
The diameter of the$x$ cm at the hemisphere. The following is added:$V(x)$, the volume of that hemisphere in cm$^3$, in accordance with: $V(x) = \frac{\pi}{12} x^3$ The volume of the hemisphere is increasing at a constant rate of 450 cm$^3$ in the second. Find the rate at which a diameter ($x$) The radius of the hemisphere increases with time, when the$x = 20$ cm. Your answer should be in cm per second, to the right of one decimal place.	1.4	Day 1 - Q9d
It is time.$r$ cm at the corner and it is$h$ cm in height. The maximum surface area of the curve of the cone,$S$, be written as: $S = \pi r \sqrt{r^2 + h^2}$ Re-allocate this to $h$ The Commission shall be assisted by the European Parliament.$S$, $r$, and $\pi$. Please answer in the form $\frac{\sqrt{S^2 - ar^n}}{br}$, where they were tendered$a$, $b$, and $n$.	\frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r}	Day 1 - Q9e
A plant body grows and sells. The function is defined $W(x)$ It can be used to simulate the height of a water spice plant, in mm, during the first 35 days after it starts growing. $W(x) = 0.667x + 1.5x^2 - 0.025x^3$ Here, it is equal to $x$ and the number of days after the plant has started growing, where $0 \leq x \leq 35, x \in \mathbb{R}$. Use of$W(x)$ to assess the height of water spionage plant after 15 days. Please correct your answer to the nearest mm.	263	Day 1 - Q10ai
A plant body grows and sells. The function is defined $W(x)$ It can be used to simulate the height of a water spice plant, in mm, during the first 35 days after it starts growing. $W(x) = 0.667x + 1.5x^2 - 0.025x^3$ Here, it is equal to $x$ and the number of days after the plant has started growing, where $0 \leq x \leq 35, x \in \mathbb{R}$. Write it down.$W'(x)$, the following paragraph is added:$W(x)$	0.667+3x-0.075x^2	Day 1 - Q10aii
A plant body grows and sells. The height of a plant can be simulated with the function $P(x)$, wherever it is.$x$ number of days after the plant starts growing. The derivative of this function is: $P'(x) = 1.1 + 2.73x - 0.078x^2$ Find the range of values of $x$ This leaves a$P'(x) \gt 24$. In your answer, each value should be rounded to the nearest integer.	14 <= x <= 21	Day 1 - Q10b
A plant body grows and sells. In this part, they are proposals.$p$ and $r$, by $p, r \in \mathbb{R}$ and $0 \lt r \lt 0.9p$. The body sells plant food bags. The normal price per item is €$p$. In the case of a retail outlet, the customer shall be able to choose between the following options: \The first one is the \Option 1: the normal price reduced by 10%, and a further reduction of €$r$ beyond that. \Optional item 2: the reduced normal price of €$r$, and that new price reduced by 10%. \end{title} Which option (1 or 2), if any, is cheaper? Write the price for each of the options (1 and 2) in terms of $p$ and $r$, to support your answer.	Rogha 1	Day 1 - Q10d
A group of 22 students were tested to find out how far, in metres, each of them could swim without stopping to take a break.$a$, $b$, $c$, and $d$ four entries have been replaced. \[\begin{array}{c\|l} 2 & b \quad 2 \quad 7 \\ 3 & a \quad 4 \quad 4 \quad 5 \quad 8 \\ 4 & 0 \quad 1 \quad d \quad 5 \quad 6 \quad 9 \\ 5 & 2 \quad 3 \quad 7 \quad 7 \quad 8 \\ 6 & 1 \quad 8 \quad c \\ \end{array} \] Key: $2 \mid 7 = 27$ of a kind used in motor vehicles The data mode is 34 meters.$a$ write it down.	4	Day 2 - Q1ai
A group of 22 students were tested to find out how far, in metres, each of them could swim without stopping to take a break.$a$, $b$, $c$, and $d$ four entries have been replaced. \[\begin{array}{c\|l} 2 & b \quad 2 \quad 7 \\ 3 & a \quad 4 \quad 4 \quad 5 \quad 8 \\ 4 & 0 \quad 1 \quad d \quad 5 \quad 6 \quad 9 \\ 5 & 2 \quad 3 \quad 7 \quad 7 \quad 8 \\ 6 & 1 \quad 8 \quad c \\ \end{array} \] Key: $2 \mid 7 = 27$ of a kind used in motor vehicles The range of the data is 49 meters.$b$ and value $c$ to receive.	b=0;c=9	Day 2 - Q1aii
A group of 22 students were tested to find out how far, in metres, each of them could swim without stopping to take a break.$a$, $b$, $c$, and $d$ four entries have been replaced. \[\begin{array}{c\|l} 2 & b \quad 2 \quad 7 \\ 3 & a \quad 4 \quad 4 \quad 5 \quad 8 \\ 4 & 0 \quad 1 \quad d \quad 5 \quad 6 \quad 9 \\ 5 & 2 \quad 3 \quad 7 \quad 7 \quad 8 \\ 6 & 1 \quad 8 \quad c \\ \end{array} \] Key: $2 \mid 7 = 27$ of a kind used in motor vehicles The average of the data is 43.5 metres.$d$ to receive.	2	Day 2 - Q1aiii
The table below shows the prizes, in euros, a player can win in a game, as well as the probability of winning each prize.$x \in \mathbb{R}$. \I'm not sure I'm ready to go. \centering \The first is the "Table of Contents" section. \Other \textbf{Prize (€) } & Does not exist & 2 & \( x - 10 \) & \(x\) \\ \Other \The probability of the event is 30% and 40% and 28% and 2% . \The following table shows the \The following table shows the It costs €10 to play the game once. The game is fair game i.e. the expected value of the money won is €0, and the costs taken into account. Work out the value $x$.	40	Day 2 - Q2a
They are collective coincidences.$A$ and $B$. $P(A) = 0.1$ and $P(B) = 0.4$. Describe the value of each of the following: $P(A \cap B)$ and $P(A \cup B)$.	P(A \cap B)=0; P(A \cup B)=0.5	Day 2 - Q2b
They are two cases.$C$ and $D$, with a universal set $U$. $P(C)=0.5$ and $P(D)=0.7$. Get the most value.$P[(C \cup D)']$. Note: This is $(C \cup D)'$ completion of the inventory $C \cup D$ in the set $U$.	Max $P[(C \cup D)']=0.3$	Day 2 - Q2c
It is a parallel.$ABCD$. $\vert AB \vert = 10 \text{cm}$, $\vert BC \vert = 13 \text{cm}$, and $\vert \angle ABC \vert = 110^\circ$. Get away from me.$ABCD$, right to the$\text{cm}^2$ is close.	$122$	Day 2 - Q3a
It's an angle.$X$, by $0^\circ \leq X \leq 360^\circ$, and $\cos(2x) = \frac{\sqrt{3}}{2}$ Work out all the possible values of $X$.	$X=15^\circ, 165^\circ, 195^\circ, 345^\circ$	Day 2 - Q3b
It's a triangle.$KLM$ where is$\vert MK \vert = 15\sqrt{3}$ cm, $\vert ML \vert = 45$ cm, and $\vert \angle KLM \vert = 25^\circ$. It is .$\theta$ the angle $\angle LKM$. Work out the two possible values of $\theta$, for the$0^\circ \lt \theta \lt 180^\circ$. Each answer should be correct to the nearest degree.	$\theta = 47^\circ, 133^\circ$	Day 2 - Q3c
The equator is the circle.$s$ than: $x^2 + y^2 + 4x - 6y + 5 = 0$. Write down a centre point and circle it.$s$.	Centre $= (-2, 3)$, $r = 2\sqrt{2}$	Day 2 - Q5ai
A circle has a midpoint on the perpendicular line through the point $(9, 0)$. The points are:$(7, 10)$ and $(12, 8)$ This is the first time that the Commission has proposed to the Council to adopt a directive on this matter. Find the equator of this circle. Note that your answer may contain non-integer values.	Centre $= (9, 5)$, $(x - 9)^2 + (y - \frac{31}{4})^2 = \frac{145}{16}$, or $x^2 + y^2 - 18x - \frac{31}{2}y + 132 = 0$	Day 2 - Q5b
It is a straight line.$[AB]$. The point is shared $C (6,11)$ the paragraph $[AB]$ internally in the ratio $1:3$. It is .$A$ point $(1, 13)$. Find the coordinates of a point $B$.	$(21, 5)$	Day 2 - Q6a
Find the vertical distance from a point $(5, −2)$ to the line: $y = \frac{4}{3}x - 11$.	$1.4$	Day 2 - Q6b
PK Hotels is a chain of hotels in Europe. The ages of those who stayed in PK hotels in The average life expectancy of a population is expected to be around 20 years, with a mean of 48.2 years and a standard deviation of 10.6 years. One person is chosen at random from those who stayed in a PK hotel in Find the probability that this person is less than 50 years old.	$56.75%$	Day 2 - Q7ai
PK Hotels is a chain of hotels in Europe. The ages of those who stayed in PK hotels in The average life expectancy of a population is expected to be around 20 years, with a mean of 48.2 years and a standard deviation of 10.6 years. Of those who stayed in PK hotels in 2023, just 10% are female.$A$ year Find the value $A$, to the nearest integer.	$62$	Day 2 - Q7aii
PK Hotels is a chain of hotels in Europe. During their most recent visit, the$\frac{1}{5}$ of PK Hotel customers use the swimming pool. 6 of the PK Hotels' customers are selected at random. Find the probability that 2 of them directly used the swimming pool.	$\frac{768}{3125}=0.24576$	Day 2 - Q7bi
PK Hotels is a chain of hotels in Europe. During their most recent visit, the$\frac{1}{5}$ of PK Hotel customers use the swimming pool. The following shall be chosen:$n$ of the customers of PK Austria randomly, where $n \in \mathbb{N}$. It is .$0.0047$, right to 4 random centres, the probability that none of them The use of the swimming pool.$n$.	$24$	Day 2 - Q7bii
PK Hotels is a chain of hotels in Europe. PK Hotels is testing a new reservation system. The old booking system is shown to 45% of people who log on to the PK Hotel website; the new booking system is shown to the other 55%. It is randomly determined which booking systems (old or new) people will see. Of those who see the old system bookings, one third book a room. Of those who see the new booking system, two-fifths book a room. One person is randomly selected from those who booked a room via the PK Hotel website. Your answer should be in percentages, right down to the nearest percent.	59\%	Day 2 - Q7c
Tommy makes ornaments out of metal and glass. It makes an open metal cylinder that is 15 cm high and 5 cm thick. The circuit is a loop. Find the dimensions of this cube. Your answers should be in cm, to the right of one decimal place, as appropriate.	$15, 31.4$	Day 2 - Q8a
Tommy makes ornaments out of metal and glass. Tommy makes another roller coaster that is 22 cm high and 12 cm in diameter. The dimensions of this circuit are just right so that it can be placed inside a glass sphere. Find the volume of the sphere, in cm$^3$, You used the Pythagorean theorem in your solution.	$8240.2$	Day 2 - Q8b

Solve the following equation for $n \in \mathbb{N}$: $n - 3 = \sqrt{3n + 1}$

8

Day 1 - Q1a

Write the following sentence as a single paragraph in terms of $t$: $\frac{4}{2t + 1} - \frac{7}{12t}$

$\frac{34t - 7}{(2t + 1)(12t)}$

Day 1 - Q1b

Solve the following parallel equations for $x, y, w \in \mathbb{Z}$: $\begin{cases} x + 2y = 143 \\ y + 3w = -74 \\ 4x + 5w = 4 \end{cases}$

51;46;-40

Day 1 - Q1c

In this question, $i^2 = -1$. Find both solutions to the equation in $z$ the following, where $z$ in number Complex.Each answer should be in the form $a + bi$, where $a, b \in \mathbb{R}$. $z^2 + 12z + 261 = 0$

-6+15i;-6-15i

Day 1 - Q2a

In this question, $i^2 = -1$. He used a theory of Moivre to $(1 - \sqrt{3}i)^9$ written in the form $a + bi$, where $a, b \in \mathbb{R}.$

-512

Day 1 - Q2b

Find the sum: $\int \cos6x\ dx$

\frac{1}{6}\sin(6x)+C

Day 1 - Q3a

The function is defined $f$ as follows for $x \in \mathbb{R}$: $f(x) = 2x^3 - 9x^2 + 5x - 11$ Find the equation of the circle with a graph.$f$ at the point at which the$x = 2$. You don't need to simplify your answer.

y − (−21) = −7(x − 2)

Day 1 - Q3bi

The function is defined $f$ as follows for $x \in \mathbb{R}$: $f(x) = 2x^3 - 9x^2 + 5x - 11$ Find the coordinates $x$ the rebound point of $f$.

\frac{3}{2}

Day 1 - Q3bii

The following function is distinguished from basic principles, in respect of $x$: $f(x) = x^2 - 7x - 10$

\frac{-238}{361}

Day 1 - Q4a

The function is defined $g(x)$ for the$x \in \mathbb{R}$ in accordance with: $g(x) = \frac{6x + 1}{x^4 + 3}$ Get the value $g'(-2)$, the following paragraph is added:$g(x)$ where $x = -2$.

-238/361

Day 1 - Q4b

It is a continuous operation.$h: \mathbb{R} \rightarrow \mathbb{R}$. The graph has a logarithmic minimum.$h(x)$ at point $(0, 5)$. Indicate whether the following statement is true or false: It must be$5$ is at least $h(x)$ for each real value of $x$. other Defend your answer.

False

Day 1 - Q4c

The first three terms of a concatenation sequence are as follows, where $p \in \mathbb{R}: \begin{cases} T_1 = 2p + 1 \\ T_2 = 5p - 3 \\ T_3 = 6p + 7 \end{cases}$ Get the value $p$.

7

Day 1 - Q5a

They are $G_7 = 6$ and $G_11 = \frac{3}{8}$ the 7th and 11th term of a polynomial sequence, respectively. Find the two possible values of $r$, The Commission shall be assisted by the European Parliament.$r \in \mathbb{R}$.

\frac{1}{2};\frac{-1}{2}

Day 1 - Q5b

A sequence of functions is defined.$F_0, F_1, F_2, \dots$ for $x \in \mathbb{R}, x \gt 0$: \The first one is the \The Commission shall:$F_0 = x^{2024}$ \Item for $n \geq 1$, The function is $F_n$ the following paragraph is added:$F_{n-1}$ , $x$. \end{title} Write $F_1$ and $F_2$ in terms of $x$.

F1(x) = 2024 x^2023; F2(x) = 2024 × 2023 x^2022

Day 1 - Q5ci

A sequence of functions is defined.$F_0, F_1, F_2, \dots$ for $x \in \mathbb{R}, x \gt 0$: \The first one is the \The Commission shall:$F_0 = x^{2024}$ \Item for $n \geq 1$, The function is $F_n$ the following paragraph is added:$F_{n-1}$ , $x$. \end{title} Find the first value of $n$ This leaves a$F_n=0$.

2025

Day 1 - Q5cii

$h(x) = x^2 + bx - 12$, where is$x \in \mathbb{R}$ and place it on a doorstep.$b$. Get the value $b$ This leaves a$x - 4$ in a factor of $h(x)$.

-1

Day 1 - Q6a

Two functions are defined, $f(x)$ and $g(x)$, for $x \in \mathbb{R}, x \gt 0$: \The first two terms of the equation are: Get the value $f(1.2)$. Please answer in the form $a \times 10^n$, where $a \in \mathbb{R}, 1 \leq a \lt 10, n \in \mathbb{N}$, and which shall be$a$ right to one decimal place.

4.9\times10^4

Day 1 - Q6bi

Two functions are defined, $f(x)$ and $g(x)$, for $x \in \mathbb{R}, x \gt 0$: \The first two terms of the equation are: Get the value $x$ This leaves a$g(x) = 3.5$. Please answer in the form $e^p$, where $p \in \mathbb{R}$.

x=e^7

Day 1 - Q6bii

Two functions are defined, $f(x)$ and $g(x)$, for $x \in \mathbb{R}, x \gt 0$: \The first two terms of the equation are: Write the function $g(f(x))$ in terms of $x$, in the simplest form.

4.5x

Day 1 - Q6biii

Fiadh has a total annual salary of €54000. She pays income tax at a rate of 20% on the first €40000 of her salary and at a rate of 40% on the balance sheet. Work out his annual net salary, assuming no other deductions.

42175

Day 1 - Q7a

The first monthly repayment is made directly after the mortgage is taken out, and the second monthly repayment is made directly after the mortgage is taken out. Write down the immediate value of each of the first three monthly repayments they make, at the time they take out the mortgage. Let all values be in pieces. Do not overwhelm any powers.

\frac{1647.75}{1.00279};\frac{1647.75}{1.00279^2};\frac{1647.75}{1.00279^3}

Day 1 - Q7bi

The first monthly repayment is made directly after the mortgage is taken out, and the second monthly repayment is made directly after the mortgage is taken out. Work out the amount of money that Wood and his partner borrowed for their mortgage. Correct your answer to the nearest euro.

334563

Day 1 - Q7bii

Fide puts money into a savings account, and she leaves it there for several years. This is the bottom line.$F(t)$, the amount of money in the account in euro after $t$ year, whereas$t \in \mathbb{R}, t \geq 0$: $F(t) = 5000 e^{0.04t}$ Use a differential and find the rate at which the amount of money in the account increases after 3.5 years.

230

Day 1 - Q7ci

Fide puts money into a savings account, and she leaves it there for several years. This is the bottom line.$F(t)$, the amount of money in the account in euro after $t$ year, whereas$t \in \mathbb{R}, t \geq 0$: $F(t) = 5000 e^{0.04t}$ Use an estimate and find the average amount of money in the account over the first 5 years.

5535

Day 1 - Q7cii

Fide puts money into a savings account, and she leaves it there for several years. This is the bottom line.$F(t)$, the amount of money in the account in euro after $t$ year, whereas$t \in \mathbb{R}, t \geq 0$: $F(t) = 5000 e^{0.04t}$ Calculate the annual interest rate (AER) for this account. So let's say that you have a percentage increase in the amount of money in the account over the course of a year, and your answer is right up to 2 decimal places.

4.08%

Day 1 - Q7ciii

It is time.$R$ unit by sphere. The part of the sphere that intersects a smooth surface is called a cap, and its volume is: $C=\frac{\pi k^2}{3}(3R - k)$ Here, it is.$C$ The volume of the pipe is $k$ The height of the pipe, where the$0 \lt k \leq R$. Get the value $C$, volume of the pipe, where:$R = 13$ and $k = 4$. # Please answer in terms of $\pi$.

\dfrac{560}{3} \pi

Day 1 - Q9a

It is time.$R$ unit by sphere. The part of the sphere that intersects a smooth surface is called a cap, and its volume is: $C=\frac{\pi k^2}{3}(3R - k)$ Here, it is.$C$ The volume of the pipe is $k$ The height of the pipe, where the$0 \lt k \leq R$. A sphere has 8 units and a tube has $y$ Unit at altitude, where$0 \lt y \leq 8$. The volume of these pipes is ($36\pi y$) Unit$^3$. Used that, and out of the blue for $C$ above, and demonstrate that: $\frac{y}{3}(24 - y) = 36$

null

Day 1 - Q9bi

It is time.$R$ unit by sphere. The part of the sphere that intersects a smooth surface is called a cap, and its volume is: $C=\frac{\pi k^2}{3}(3R - k)$ Here, it is.$C$ The volume of the pipe is $k$ The height of the pipe, where the$0 \lt k \leq R$. A sphere has 8 units and a tube has $y$ Unit at altitude, where$0 \lt y \leq 8$. The volume of these pipes is ($36\pi y$) Unit$^3$. The equation has widened.$\frac{y}{3}(24 - y) = 36$ and set it up to the top of the ladder. I'll get this.

6

Day 1 - Q9bii

The diameter of the$x$ cm at the hemisphere. The following is added:$V(x)$, the volume of that hemisphere in cm$^3$, in accordance with: $V(x) = \frac{\pi}{12} x^3$ Get the value $x$ when the volume of the hemisphere is more than 3 litres. Be sure your answer is correct to one decimal place.

22.5

Day 1 - Q9c

The diameter of the$x$ cm at the hemisphere. The following is added:$V(x)$, the volume of that hemisphere in cm$^3$, in accordance with: $V(x) = \frac{\pi}{12} x^3$ The volume of the hemisphere is increasing at a constant rate of 450 cm$^3$ in the second. Find the rate at which a diameter ($x$) The radius of the hemisphere increases with time, when the$x = 20$ cm. Your answer should be in cm per second, to the right of one decimal place.

1.4

Day 1 - Q9d

It is time.$r$ cm at the corner and it is$h$ cm in height. The maximum surface area of the curve of the cone,$S$, be written as: $S = \pi r \sqrt{r^2 + h^2}$ Re-allocate this to $h$ The Commission shall be assisted by the European Parliament.$S$, $r$, and $\pi$. Please answer in the form $\frac{\sqrt{S^2 - ar^n}}{br}$, where they were tendered$a$, $b$, and $n$.

\frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r}

Day 1 - Q9e

A plant body grows and sells. The function is defined $W(x)$ It can be used to simulate the height of a water spice plant, in mm, during the first 35 days after it starts growing. $W(x) = 0.667x + 1.5x^2 - 0.025x^3$ Here, it is equal to $x$ and the number of days after the plant has started growing, where $0 \leq x \leq 35, x \in \mathbb{R}$. Use of$W(x)$ to assess the height of water spionage plant after 15 days. Please correct your answer to the nearest mm.

263

Day 1 - Q10ai

A plant body grows and sells. The function is defined $W(x)$ It can be used to simulate the height of a water spice plant, in mm, during the first 35 days after it starts growing. $W(x) = 0.667x + 1.5x^2 - 0.025x^3$ Here, it is equal to $x$ and the number of days after the plant has started growing, where $0 \leq x \leq 35, x \in \mathbb{R}$. Write it down.$W'(x)$, the following paragraph is added:$W(x)$

0.667+3x-0.075x^2

Day 1 - Q10aii

A plant body grows and sells. The height of a plant can be simulated with the function $P(x)$, wherever it is.$x$ number of days after the plant starts growing. The derivative of this function is: $P'(x) = 1.1 + 2.73x - 0.078x^2$ Find the range of values of $x$ This leaves a$P'(x) \gt 24$. In your answer, each value should be rounded to the nearest integer.

14 <= x <= 21

Day 1 - Q10b

A plant body grows and sells. In this part, they are proposals.$p$ and $r$, by $p, r \in \mathbb{R}$ and $0 \lt r \lt 0.9p$. The body sells plant food bags. The normal price per item is €$p$. In the case of a retail outlet, the customer shall be able to choose between the following options: \The first one is the \Option 1: the normal price reduced by 10%, and a further reduction of €$r$ beyond that. \Optional item 2: the reduced normal price of €$r$, and that new price reduced by 10%. \end{title} Which option (1 or 2), if any, is cheaper? Write the price for each of the options (1 and 2) in terms of $p$ and $r$, to support your answer.

Rogha 1

Day 1 - Q10d

A group of 22 students were tested to find out how far, in metres, each of them could swim without stopping to take a break.$a$, $b$, $c$, and $d$ four entries have been replaced. \[\begin{array}{c|l} 2 & b \quad 2 \quad 7 \\ 3 & a \quad 4 \quad 4 \quad 5 \quad 8 \\ 4 & 0 \quad 1 \quad d \quad 5 \quad 6 \quad 9 \\ 5 & 2 \quad 3 \quad 7 \quad 7 \quad 8 \\ 6 & 1 \quad 8 \quad c \\ \end{array} \] Key: $2 \mid 7 = 27$ of a kind used in motor vehicles The data mode is 34 meters.$a$ write it down.

4

Day 2 - Q1ai

A group of 22 students were tested to find out how far, in metres, each of them could swim without stopping to take a break.$a$, $b$, $c$, and $d$ four entries have been replaced. \[\begin{array}{c|l} 2 & b \quad 2 \quad 7 \\ 3 & a \quad 4 \quad 4 \quad 5 \quad 8 \\ 4 & 0 \quad 1 \quad d \quad 5 \quad 6 \quad 9 \\ 5 & 2 \quad 3 \quad 7 \quad 7 \quad 8 \\ 6 & 1 \quad 8 \quad c \\ \end{array} \] Key: $2 \mid 7 = 27$ of a kind used in motor vehicles The range of the data is 49 meters.$b$ and value $c$ to receive.

b=0;c=9

Day 2 - Q1aii

A group of 22 students were tested to find out how far, in metres, each of them could swim without stopping to take a break.$a$, $b$, $c$, and $d$ four entries have been replaced. \[\begin{array}{c|l} 2 & b \quad 2 \quad 7 \\ 3 & a \quad 4 \quad 4 \quad 5 \quad 8 \\ 4 & 0 \quad 1 \quad d \quad 5 \quad 6 \quad 9 \\ 5 & 2 \quad 3 \quad 7 \quad 7 \quad 8 \\ 6 & 1 \quad 8 \quad c \\ \end{array} \] Key: $2 \mid 7 = 27$ of a kind used in motor vehicles The average of the data is 43.5 metres.$d$ to receive.

2

Day 2 - Q1aiii

The table below shows the prizes, in euros, a player can win in a game, as well as the probability of winning each prize.$x \in \mathbb{R}$. \I'm not sure I'm ready to go. \centering \The first is the "Table of Contents" section. \Other \textbf{Prize (€) } & Does not exist & 2 & $ x - 10 $ & $x$ \\ \Other \The probability of the event is 30% and 40% and 28% and 2% . \The following table shows the \The following table shows the It costs €10 to play the game once. The game is fair game i.e. the expected value of the money won is €0, and the costs taken into account. Work out the value $x$.

40

Day 2 - Q2a

They are collective coincidences.$A$ and $B$. $P(A) = 0.1$ and $P(B) = 0.4$. Describe the value of each of the following: $P(A \cap B)$ and $P(A \cup B)$.

P(A \cap B)=0; P(A \cup B)=0.5

Day 2 - Q2b

They are two cases.$C$ and $D$, with a universal set $U$. $P(C)=0.5$ and $P(D)=0.7$. Get the most value.$P[(C \cup D)']$. Note: This is $(C \cup D)'$ completion of the inventory $C \cup D$ in the set $U$.

Max $P[(C \cup D)']=0.3$

Day 2 - Q2c

It is a parallel.$ABCD$. $\vert AB \vert = 10 \text{cm}$, $\vert BC \vert = 13 \text{cm}$, and $\vert \angle ABC \vert = 110^\circ$. Get away from me.$ABCD$, right to the$\text{cm}^2$ is close.

$122$

Day 2 - Q3a

It's an angle.$X$, by $0^\circ \leq X \leq 360^\circ$, and $\cos(2x) = \frac{\sqrt{3}}{2}$ Work out all the possible values of $X$.

$X=15^\circ, 165^\circ, 195^\circ, 345^\circ$

Day 2 - Q3b

It's a triangle.$KLM$ where is$\vert MK \vert = 15\sqrt{3}$ cm, $\vert ML \vert = 45$ cm, and $\vert \angle KLM \vert = 25^\circ$. It is .$\theta$ the angle $\angle LKM$. Work out the two possible values of $\theta$, for the$0^\circ \lt \theta \lt 180^\circ$. Each answer should be correct to the nearest degree.

$\theta = 47^\circ, 133^\circ$

Day 2 - Q3c

The equator is the circle.$s$ than: $x^2 + y^2 + 4x - 6y + 5 = 0$. Write down a centre point and circle it.$s$.

Centre $= (-2, 3)$, $r = 2\sqrt{2}$

Day 2 - Q5ai

A circle has a midpoint on the perpendicular line through the point $(9, 0)$. The points are:$(7, 10)$ and $(12, 8)$ This is the first time that the Commission has proposed to the Council to adopt a directive on this matter. Find the equator of this circle. Note that your answer may contain non-integer values.

Centre $= (9, 5)$, $(x - 9)^2 + (y - \frac{31}{4})^2 = \frac{145}{16}$, or $x^2 + y^2 - 18x - \frac{31}{2}y + 132 = 0$

Day 2 - Q5b

It is a straight line.$[AB]$. The point is shared $C (6,11)$ the paragraph $[AB]$ internally in the ratio $1:3$. It is .$A$ point $(1, 13)$. Find the coordinates of a point $B$.

$(21, 5)$

Day 2 - Q6a

Find the vertical distance from a point $(5, −2)$ to the line: $y = \frac{4}{3}x - 11$.

$1.4$

Day 2 - Q6b

PK Hotels is a chain of hotels in Europe. The ages of those who stayed in PK hotels in The average life expectancy of a population is expected to be around 20 years, with a mean of 48.2 years and a standard deviation of 10.6 years. One person is chosen at random from those who stayed in a PK hotel in Find the probability that this person is less than 50 years old.

$56.75%$

Day 2 - Q7ai

PK Hotels is a chain of hotels in Europe. The ages of those who stayed in PK hotels in The average life expectancy of a population is expected to be around 20 years, with a mean of 48.2 years and a standard deviation of 10.6 years. Of those who stayed in PK hotels in 2023, just 10% are female.$A$ year Find the value $A$, to the nearest integer.

$62$

Day 2 - Q7aii

PK Hotels is a chain of hotels in Europe. During their most recent visit, the$\frac{1}{5}$ of PK Hotel customers use the swimming pool. 6 of the PK Hotels' customers are selected at random. Find the probability that 2 of them directly used the swimming pool.

$\frac{768}{3125}=0.24576$

Day 2 - Q7bi

PK Hotels is a chain of hotels in Europe. During their most recent visit, the$\frac{1}{5}$ of PK Hotel customers use the swimming pool. The following shall be chosen:$n$ of the customers of PK Austria randomly, where $n \in \mathbb{N}$. It is .$0.0047$, right to 4 random centres, the probability that none of them The use of the swimming pool.$n$.

$24$

Day 2 - Q7bii

PK Hotels is a chain of hotels in Europe. PK Hotels is testing a new reservation system. The old booking system is shown to 45% of people who log on to the PK Hotel website; the new booking system is shown to the other 55%. It is randomly determined which booking systems (old or new) people will see. Of those who see the old system bookings, one third book a room. Of those who see the new booking system, two-fifths book a room. One person is randomly selected from those who booked a room via the PK Hotel website. Your answer should be in percentages, right down to the nearest percent.

59\%

Day 2 - Q7c

Tommy makes ornaments out of metal and glass. It makes an open metal cylinder that is 15 cm high and 5 cm thick. The circuit is a loop. Find the dimensions of this cube. Your answers should be in cm, to the right of one decimal place, as appropriate.

$15, 31.4$

Day 2 - Q8a

Tommy makes ornaments out of metal and glass. Tommy makes another roller coaster that is 22 cm high and 12 cm in diameter. The dimensions of this circuit are just right so that it can be placed inside a glass sphere. Find the volume of the sphere, in cm$^3$, You used the Pythagorean theorem in your solution.

$8240.2$

Day 2 - Q8b