The second question was a nightmare dressed in vectors. Line L1 passes through (1,2,3) with direction (2, -1, 2). L2 is given by (x-3)/2 = (y+1)/1 = (z-4)/-2. Find the shortest distance between L1 and L2. Maya groaned. This was the kind of problem that separated the 6s from the 7s. She sketched the cross product of the direction vectors, found a vector connecting the two lines, and then did the scalar projection. Her arithmetic was shaky—she forgot a negative sign halfway through, had to erase four lines, and nearly threw her pencil across the room.
But she finished. And the solution bank said “Correct.” Her heart beat a little faster.
Maya laughed. It was almost elegant. The base case: n=1, 1 1! = 1, and (2)! – 1 = 1. True. The inductive step: Assume true for n. Then add (n+1) (n+1)! to both sides. Left becomes sum to n+1. Right becomes (n+1)! – 1 + (n+1)*(n+1)! = (n+1)!(1 + n + 1) – 1 = (n+2)! – 1. Done.
Maya stared at the blinking cursor on her laptop. Around her, the dormitory was silent, save for the hum of an old refrigerator and the distant, rhythmic thump of a bass guitar from three floors down. On her screen, a single tab glowed: ib math aa hl exam questionbank
“Okay,” she whispered, pulling out a fresh sheet of paper. “Integration by parts. Twice. Then a trick.” Her pen flew, sketching the cyclic dance of derivatives. sin(x) becomes cos(x) becomes -sin(x) . e^x stays e^x . She wrote the lines, the u and dv, the careful subtraction. Ten minutes later, she had an answer: (e^π + 1)/2 .
She set down her pen. The screen glowed with the green checkmark of the official answer. Seven out of seven. A perfect paper.
She clicked “Generate Random Paper.” The second question was a nightmare dressed in vectors
Outside, a bird started singing. The deep blue of the night sky was bleeding into a pale, anxious gray. Maya saved her work, closed the laptop, and lay back on her pillow. The questionbank was merciless—a cold, infinite engine of suffering. But tonight, for a few quiet hours, she had been its master.
She checked the solution bank. Correct. A tiny, fragile smile.
She closed her eyes and dreamed of limits that didn't diverge. Find the shortest distance between L1 and L2
Prove by mathematical induction that for all n ∈ ℤ⁺, Σ_{k=1}^n (k * k!) = (n+1)! – 1.
The first question appeared. It was a beast: Find the area bounded by the curve y = e^x sin(x), the x-axis, and the lines x = 0 and x = π.