Regions with projected low support ratios by 2050 will primarily face challenges in: a. Increasing agricultural productivity. b. Attracting international migration. c. Reducing fertility rates. d. Sustaining elderly care systems.

12. Find a standard basis vector for \( R^{3} \) that can be added to the set \( \left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\} \) to produce a basis for \( R^{3} \). (a) \( \mathbf{v}_{1}=(-1,2,3), \mathbf{v}_{2}=(1,-2,-2) \) (b) \( \mathbf{v}_{1}=(1,-1,0), \mathbf{v}_{2}=(3,1,-2) \)

A historian is exploring the demographic patterns in West Africa's high forest zone. Which historical factor best explains the hig population density in this area? a. Absence of significant slave raids b. Consistent low temperatures c. Frequent and devastating wars d. Lack of fertile soils for agriculture

(ix) The equation \( \frac{1}{r r} \frac{\partial}{r}\left(r \frac{\partial u}{r r}\right)=\frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}} \) is one dimensional wave equation in: (a) Cylindrical coordinate (b) Polar coordinate (c) Spherical coordinate (d) None of these

4. ¿Cuál de estas igualdades es correcta? (A) \( \log _{3} 8=2 \) (B) \( \log _{3} 2=8 \) (C) \( \log _{2} 8=3 \) (D) \( \log _{2} 3=8 \) (D)

5. A psychologist is interested in exploring the effect oftutorial support on student academic performance and assign students into two groups. Students in group on the tutorial support and those in groupltwo do not. In this example, what is the a) Dependent variable b) Independent variable c) Control group d) Experimental group