📘 MATHEMATICS

1. **Problem Statement:** Plot the functions $y_1(x) = \sin(x)$ and $y_2(x) = \cos(x)$ on the same graph for $x$ ranging from $-2\pi$ to $2\pi$ with increments of $0.1$. 2. **Setti

1. Let's interpret the problem: We want to understand how to find the tally or count in math contexts. 2. Tally is typically a way to count occurrences or groupings, often by marki

1. Problem 1 part i: Statement: The roots of the equation $x^2+3x-7=0$ are $\alpha$ and $\beta$; find the new quadratic whose roots are $\alpha-2\beta$ and $\beta-2\alpha$. 2. Solu

1. Problem statement: We are given the quadratic $x^2+3x-7=0$ with roots $\alpha$ and $\beta$, and we must find a new quadratic whose roots are $\alpha-2\beta$ and $\beta-2\alpha$,

1. Stating the problem: Find the domain, range, and level curves for the given functions. 2. Domain: The domain of a function is the set of all input values $x$ for which the funct

1. For Question 1, we are asked to find the exact value of $$Q=\frac{(\sin 2x + b)(2 \sin x - 1)}{a^2 - 4 \tan x}$$ given $x=45^\circ$, $a=18$, and $b=\sqrt{2}$.

1. The dot on a number typically represents multiplication in mathematics. 2. For example, $3 \cdot 4$ means $3$ multiplied by $4$, which equals $12$.

1. The question seems to be about the meaning of the symbols \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\). 2. These are standard notations for different sets

1. Let's start by stating the problem: You want to understand common math symbols and what they mean. 2. Here are some common math symbols with their meanings:

1. **Easy:** Convert 4500 to scientific notation. Answer: $4.5 \times 10^{3}$

1. Write in words 1101001. 1101001 in words is "one million, one hundred and one thousand, and one".

1. **Understanding Prime Factorisation:** Prime factorisation means breaking a number into its prime factors. Among the options given, the correct one is D: breaking a number into

1. Let's clarify what DNE (Does Not Exist) means in mathematics. It indicates that a limit, value, or solution is not defined or does not have a finite or meaningful value. 2. To d

1. The problem asks to find the union of set A and its complement A'. The union of a set and its complement is the universal set, but the choices include A' and A U B etc. The corr

1. **Solve 6x² + 5x + 1 = 0 using quadratic formula** The quadratic formula is given by:

1. The problem is to understand and correctly enter the constant $\pi$ in STACK as an exact numerical constant. 2. The constant $\pi$ is approximately 3.14 but is an irrational num

1. We need to express numbers in scientific notation with 4 decimal places. 2. Scientific notation writes numbers as $a \times 10^b$, where $1 \leq |a| < 10$ and $b$ is an integer.

1. المشكلة: تمثل دالة زيتا لريمان \(\zeta(s)\) من خلال نموذج تصحيحي ديناميكي حسب المعادلة: $$\zeta(s) \approx \sum_{n=1}^{V} \frac{1}{n^s} + \beta(V, s) \cdot V^{1-s}$$

1. The question "Is the question a function?" is somewhat ambiguous, but generally, a function is a relation where each input has exactly one output. 2. To determine if a mathemati

1. The problem discusses how to enter common mathematical numbers and constants in STACK, a system for mathematical input. 2. Numbers such as 1, -23, 2.614, and fractions like 3/5

1. **Find the derivative dy/dx for each function:** (a) Given $$y = 3x^4 - x^3 + x^2 + 25x$$