🧮 algebra

1. Problem 27: Find the new coordinates of the point (6,7) after rotating the xy-axes through an angle of 45°. Step 1: The rotation formulas for coordinates when axes are rotated b

1. Problem 29: Find the value of $k$ such that the line $y = kx + 1$ is tangent to the hyperbola $3x^2 - 4y^2 = 12$. 2. Substitute $y = kx + 1$ into the hyperbola equation:

1. **State the problem:** Solve the system of equations using substitution and identify the error in the student's solution. Given system:

1. **State the problem:** Given the equation $ (\sin x + \cos x) y = \cos^2 x $ and the value $ x = \frac{\pi}{2} $, find the value of $ y $. 2. **Substitute the value of $ x $:**

1. The problem is to find the value of $w$ in the equation $$\sqrt{82^2 + 15^2 + 144^2} = \sqrt{82^2 + 15^2 + w^2}.$$\n\n2. First, calculate the squares of the known numbers:\n$$82

1. The problem is to simplify the expression $15c10$. 2. Assuming the expression means multiplication of three terms: $15 \times c \times 10$.

1. The problem is to simplify the expression $15c10$. 2. Assuming the expression means multiplication of constants and variables, it can be written as $15 \times c \times 10$.

1. **State the problem:** We analyze the function f represented by curve (C) and line (d) in the given graphs. 2. **Domain of definition of f:** From the graph, the curve (C) is de

1. The problem is to find the formula for the sum of the first $n$ natural numbers, i.e., $s = 1 + 2 + 3 + \cdots + n$. 2. We can use the formula for the sum of an arithmetic serie

1. The problem is to simplify the expression $2C + 7\sqrt{3}C$. 2. Notice that both terms have a common factor $C$.

1. The problem is to divide the mixed number $6\frac{3}{5}$ by the fraction $\frac{9}{10}$. 2. First, convert the mixed number $6\frac{3}{5}$ to an improper fraction.

1. **State the problem:** Simplify the expression $$\frac{1+\sqrt{8}}{3-\sqrt{2}}$$ and identify which of the options (a) 7 + \sqrt{2}, (b) 7 + 7\sqrt{2}, (c) 1 - 7\sqrt{2}, or (d)

1. **State the problem:** We are given matrices \(P = \begin{bmatrix}3 & 4 \\ 2 & x\end{bmatrix}\), \(Q = \begin{bmatrix}1 & 3 \\ -2 & 4\end{bmatrix}\), and \(R = \begin{bmatrix}-5

1. **State the problem:** We want to express the rational function $$\frac{x^2+x+4}{(1-x)(x^2+1)}$$ as a sum of partial fractions and check which of the given options (a), (b), (c)

1. Stating the problem: We are given the matrix $$P = \begin{bmatrix} y-2 & y-1 \\ y-4 & y+2 \end{bmatrix}$$ and its determinant $$|P| = -23$$. We need to find the value of $$y$$.

1. The problem asks which of the given numbers is not an integer. - An integer is a whole number without fractions or decimals.

1. **State the problem:** Find the range of values of $x$ for which $$x^2 + 4x + 5 < 3x^2 - x + 2.$$\n\n2. **Rewrite the inequality:** Move all terms to one side to compare with ze

1. The problem asks to simplify the product of three fractions: $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4}$. 2. Multiply the numerators together: $1 \times 1 \times 1 = 1$.

1. **State the problem:** We are given matrices \(P = \begin{bmatrix} 2 & 1 \\ 5 & -3 \end{bmatrix}\) and \(Q = \begin{bmatrix} 4 & -8 \\ 1 & -2 \end{bmatrix}\). We need to find \(

1. **State the problem:** Solve the inequality $$x^2 - 2x \geq 3$$ and analyze the given options. 2. **Rewrite the inequality:** Move all terms to one side to get zero on the other

1. The problem states that the roots of a quadratic equation are $-3$ and $1$. We need to find which of the given equations has these roots. 2. Recall that if a quadratic equation