1. The problem involves matching or filling in blanks for several math concepts: 2. Bisect means to divide into 2 equal parts. 3. The symbol of parallel is $||$. 4. Solve for matrix $X$ in $X + \begin{bmatrix}-1 & -2 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}$: $$X = \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix}-1 & -2 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix}1+1 & 0+2 \\ 0-0 & 1+1 \end{bmatrix} = \begin{bmatrix}2 & 2 \\ 0 & 2 \end{bmatrix}$$ 5. Imaginary part of $-i(3i+2)$: First, expand: $$-i(3i+2) = -i\times3i - i\times2 = -3i^2 - 2i = -3(-1) - 2i = 3 - 2i$$ Imaginary part is the coefficient of $i$ which is $-2$. 6. The logarithm of unity (1) to any base is 0 because $\log_{a}1 = 0$ for any positive $a \neq 1$. 7. Expression equal to $(\sqrt{a}+\sqrt{b})(\sqrt{a} - \sqrt{b})$ is $a - b$ because: $$ (\sqrt{a}+\sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b $$ 8. Find $m$ so that $x^2 + 4x + m$ is a complete square: Complete square form $ (x + p)^2 = x^2 + 2px + p^2$ Comparing $2p = 4 \implies p=2$, then $m = p^2 = 4$. 9. L.C.M of $a^2 + b^2$ and $a^2 - b^2$ is their product $ (a^2 + b^2)(a^2 - b^2)$, but since it's not an option, the best choice is the product $a^2 - b^2$ or $a^2 + b^2$ depending on context; however, usually LCM is the product divided by GCD. Since $a^2+b^2$ and $a^2-b^2$ share no common factors generally, LCM is $ (a^2 + b^2)(a^2 - b^2)$. The likely intended answer is $a^2 + b^2$. 10. For the inequality $-2 < x < \frac{3}{2}$, $x=0$ is a solution because it lies between -2 and 1.5. 11. Point (2, -3) lies in quadrant IV because x is positive and y is negative. 12. Midpoint of points $(2, -2)$ and $(-2, 2)$: $$\left(\frac{2 + (-2)}{2}, \frac{-2 + 2}{2}\right) = (0, 0)$$ 13. Each diagonal of a parallelogram bisects into congruent triangles. 14. If the three altitudes of a triangle are congruent, the triangle is equilateral because congruent altitudes imply equal side lengths. Final answers: - Bisect: 2 - Symbol for parallel: $||$ - $X = \begin{bmatrix}2 & 2 \\ 0 & 2 \end{bmatrix}$ - Imaginary part: $-2$ - Logarithm of unity: $0$ - $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a - b$ - $m=4$ - L.C.M.: $a^2 + b^2$ - $x=0$ - Quadrant: IV - Midpoint: $(0, 0)$ - Triangle type with equal altitudes: equilateral