🧮 algebra

1. **State the problem:** We are given a geometric sequence with the first three terms positive and their sum equal to 42. The common ratio is given as $\frac{1}{2}$. We need to fi

1. The problem asks which ordered pair is NOT graphed on the coordinate grid. 2. The points graphed are described as:

1. The problem asks us to determine the location of the point $(2,-6)$ on the coordinate plane. 2. Recall the four quadrants of the coordinate plane:

1. **Problem a(i):** Produce a sequence for the first four palings where the first paling is 1760 mm and each successive paling is 14 mm higher. The first term $a_1 = 1760$ mm.

1. **State the problem:** Solve the equation $$X + \sqrt{X^2 + 1} = 2X + \frac{1}{X} + \sqrt{X^2 + 1}$$ for $X$. 2. **Simplify the equation:** Notice that $\sqrt{X^2 + 1}$ appears

1. The problem is to evaluate the 9th root of 9063, which is written as $\sqrt[9]{9063}$. 2. The 9th root of a number $x$ is the number $y$ such that $y^9 = x$.

1. The problem is to simplify the fraction $\frac{9063}{9}$. 2. To simplify, divide the numerator by the denominator: $9063 \div 9$.

1. Problem 33: Find the equation of the line perpendicular to $2y = x + 2$ passing through $(4,3)$. Step 1: Rewrite the given line in slope-intercept form.

1. Problem: Find the x-coordinate of the line with slope $-2$ passing through $(3,7)$ when $y=17$. Step 1: Use point-slope form: $y - y_1 = m(x - x_1)$.

1. **State the problem:** We have two points \(A(a, 3a + 9)\) and \(B(2a + 4, 4a - 1)\), and the distance between them \(AB = 260\). We need to find the possible values of \(a\). 2

1. **Problem statement:** (a)(i) Find the angle from line $l_1: y = -5x + 7$ to line $l_2: y = 3x - 8$.

1. The problem is to solve the quadratic equation $x^2 - 6x - 3 = 0$. 2. We use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=-6$, and $c=-3$.

1. **State the problem:** Factor the quadratic expression $x^2 - 4$. 2. **Recognize the form:** The expression is a difference of squares since $x^2$ and $4$ are both perfect squar

1. **Statement of the problem:** We study the functions \(\psi(x) = -x^6 + 6x^2 - 2\) and \(\varphi(x) = x^4 - 4x + 6\) defined on \(\mathbb{R}\). We analyze their limits, monotoni

1. **Stating the problem:** We are given a lattice $(L, \leq)$ and elements $a, b \in L$. We want to understand and prove the equivalence: $$a \leq b \iff a \wedge b = a \iff a \ve

1. **State the problem:** We are given the total monthly profit $P = 250000$, and the monthly revenue $R$ is twice the monthly cost $C$ minus 60000. We need to find $R$ and $C$.

1. **Problem 1.1**: Given the quadratic function $$f(x) = -x^2 - 2x - 1$$ 1. Find the x- and y-intercepts.

1. Problem: An alloy contains 120 g of copper and 150 g of zinc. (i) Find the ratio of copper to zinc.

1. **State the problem:** Simplify the expression $$(2x^2 - 3x - 6) - (-x^3 + 4x^2 - 5)$$. 2. **Remove parentheses carefully:** When subtracting a polynomial, change the signs of e

1. The problem is to compare the two fractions $\frac{4}{17}$ and $\frac{7}{10}$. 2. To compare fractions, we can find a common denominator or convert them to decimals.

1. The problem is to evaluate the expression $3 \div 2 \times \left( \frac{1}{6} \times 3.14 \times 2^3 \right)$.\n\n2. First, calculate the exponent: $2^3 = 8$.\n\n3. Next, multip