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... | \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 incr... | 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 ... | 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... | 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 $k = 4$.
# P... | \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$0 \lt ... | null | Day 1 - Q9bi |
It is time.$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$0 \lt ... | 6 | Day 1 - Q9bii |
The diameter of the$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 hemisp... | 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 $\frac{\sqrt{S^2 - ar^n}}{br}$,... | \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 \le... | 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 \le... | 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 sh... | 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
\Opti... | 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 & ... | 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 & ... | 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 | 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 prob... | 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 probabili... | $56.75%$ | 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 $... | $62$ | 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 ... | $24$ | 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 t... | 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 |
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