Practice Test 1 — Detailed Solutions

GRE Math Subject · Axiom Academy

LESSON Practice Test 1 — Detailed Solutions Step-by-step solutions for 16 GRE Math Subject Test problems, organized by topic Recognize this as the Gamma function: ... Here n=2 , so the integral equals 2! = 2 . The radius of convergence of the power series ... is: Apply the ratio test. Let ... . Then: So the radius of convergence is R = e . By the Fundamental Theorem of Calculus combined with the chain rule (Leibniz rule): Find the area enclosed by r = 2cos(theta) in polar coordinates. The curve r=2cos(theta) is a circle of radius 1 centered at (1,0) . For what value of p does ... converge? For 1">p > 1 , integrate by parts with u = ln x , dv = x^ -p dx : For p = 1 : diverges . For p it also diverges since ln x / x^p doesn't decay fast enough. How many groups of order 6 are there, up to isomorphism? S_3 (symmetric group on 3 elements, non-abelian) 6 = 2*3 with gcd(2,3)=1, so there is exactly one abelian group. By Sylow's theorem, a group of order 6 either has a normal Sylow 3-subgroup (giving Z_6 ) or does not (giving S_3 ). Find the remainder when 2^100 is divided by 7. By Fermat's little theorem, 2^6 = 1 (mod 7) . Let T: R^3 -> R^3 be a linear transformation with characteristic polynomial (lambda-1)^2(lambda-3) . Which is necessarily true? The trace equals the sum of eigenvalues (with multiplicity): tr(T) = 1+1+3 = 5 . The determinant equals the product: det(T) = 3, not 1 .

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