70 minutes, closed notes, 100 points. A basic scientific calculator is allowed. Show setup, reasoning and units or domain checks. Questions 1, 2, 4 and 5 carry 15 points each: 5 for setup, 5 for results and 5 for interpretation. Questions 3 and 6 carry 20 points each: 5 for setup, 10 for results and 5 for interpretation. Afterward, allow 30 minutes for correction.
For f(u)=√u and g(x)=x²−4, find both compositions and their real domains.
Solve sin θ=−1/2 on 0≤θ<2π. Evaluate arcsin(sin(7π/6)) and explain the difference between a principal value and a full solution set.
A display uses d(x)=3x+10 on 0≤x≤50, but caps at 100 counts. Write the piecewise model, recover x from 70 counts, and state all x compatible with 100 counts.
A 9 V source feeds R1=1 kΩ and R2=2 kΩ. Find the unloaded output across R2. Add a 2 kΩ load across R2 and find the new output, source current and both lower branch currents. Check KVL.
A node connects to 10 V through 2 kΩ, to 4 V through 4 kΩ and to ground through 4 kΩ. Write a signed KCL equation, solve the node voltage and interpret each branch current.
Ideal 15 V and 6 V sources oppose each other in a loop with 3 kΩ and 6 kΩ series resistors. Choose a current reference, write KVL and find current and both drops. Explain what reverses when you reverse the reference arrow.