But she finished. And the solution bank said “Correct.” Her heart beat a little faster.
She set down her pen. The screen glowed with the green checkmark of the official answer. Seven out of seven. A perfect paper.
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. ib math aa hl exam questionbank
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: But she finished
By the fourth question—a probability distribution with a hidden binomial and a condition that required Bayes’ theorem—she wasn't just solving. She was reading . She saw the trap before she stepped in it. The questionbank had trained her. She knew that when they said “at least two,” they meant “1 minus the probability of zero and one.” She knew that when they gave a complex number in polar form and asked for the least positive integer n such that z^n was real, they were really asking about the argument modulo π.
Prove by mathematical induction that for all n ∈ ℤ⁺, Σ_{k=1}^n (k * k!) = (n+1)! – 1. The screen glowed with the green checkmark of
“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 .