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0,1,2,3,4,5,correct,choiceA_probs,choiceB_probs,choiceC_probs,choiceD_probs |
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,k = 0 and n = 1,k = 1 and n = 0,k = n = 1,k > 1,B,False,0.04647026211023331,0.08681788295507431,0.14313848316669464,0.34337136149406433 |
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,0,1,2,3,D,False,0.05466615781188011,0.24499671161174774,0.1683836132287979,0.14859801530838013 |
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,n = 1 and r = 6,n = 1 and r = 7,n = 2 and r = 5,n = 2 and r = 6,D,True,0.09239182621240616,0.04945378005504608,0.05603847652673721,0.15232837200164795 |
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The shortest distance from the curve xy = 8 to the origin is,4,8,16,2sqrt(2),A,False,0.07425286620855331,0.15719333291053772,0.12242229282855988,0.2287149429321289 |
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,2/69,1/30,2/23,12/125,C,True,0.08924098312854767,0.14713351428508759,0.16672411561012268,0.14713351428508759 |
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,,,,,D,True,0.07434363663196564,0.13889218866825104,0.12257194519042969,0.3331850469112396 |
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,12*sqrt(2),6*sqrt(2),4*sqrt(2),1/(2*sqrt(2)),C,False,0.09316645562648773,0.15360553562641144,0.1055714413523674,0.11962810158729553 |
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Consider a segment of length 10. Points A and B are chosen randomly such that A and B divide the segment into three smaller segments. What is the probability that the three smaller segments could form the sides of a triangle?,0,10%,25%,50%,C,False,0.056219831109046936,0.2223537415266037,0.17316927015781403,0.17316927015781403 |
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,,I only,II only,III only,C,False,0.11179883778095245,0.12668468058109283,0.07683814316987991,0.14355254173278809 |
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Which of the following is true?,Every compact space is complete,Every complete space is compact,Neither (a) nor (b)., Both (a) and (b).,A,False,0.02711123414337635,0.0946274995803833,0.08350849151611328,0.22699958086013794 |
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A discrete graph is complete if there is an edge connecting any pair of vertices. How many edges does a complete graph with 10 vertices have?,10,20,25,45,D,False,0.2166232317686081,0.19116932153701782,0.10232557356357574,0.14888282120227814 |
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,1/64,15/64,1/4,3/8,B,False,0.09348514676094055,0.17465323209762573,0.10593254119157791,0.254118949174881 |
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Let V and W be 4-dimensional subspaces of a 7-dimensional vector space X. Which of the following CANNOT be the dimension of the subspace V intersect W?,0,1,2,3,A,False,0.062066033482551575,0.19117683172225952,0.16871295869350433,0.2166317254304886 |
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,4,7,10,13,B,True,0.10376735776662827,0.19386295974254608,0.09157438576221466,0.1509806215763092 |
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Sofia and Tess will each randomly choose one of the 10 integers from 1 to 10. What is the probability that neither integer chosen will be the square of the other?,0.64,0.72,0.81,0.95,D,True,0.1018776148557663,0.19033244252204895,0.11544246971607208,0.24439169466495514 |
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,I only,II only,III only,I and II only,D,True,0.055611979216337204,0.09168864786624908,0.09168864786624908,0.36263564229011536 |
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,I only,II only,III only,I and II only,B,False,0.07625021040439606,0.09790721535682678,0.09790721535682678,0.2661393880844116 |
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,-1/4,-1/(4sqrt(2)),0,sqrt(2),C,False,0.057558391243219376,0.1218511164188385,0.09489775449037552,0.2579587996006012 |
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,I and II only,III only,I and III only,,D,True,0.03620356321334839,0.052675873041152954,0.08684782683849335,0.4410496950149536 |
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,3,e,sqrt(5),0,A,False,0.08560388535261154,0.18122340738773346,0.2053530216217041,0.23269546031951904 |
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,I only,II only,III only,,D,True,0.019215775653719902,0.05223389342427254,0.0670696496963501,0.5615673065185547 |
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For how many positive integers k does the ordinary decimal representation of the integer k! end in exactly 99 zeros?,,One,Four,Five,D,False,0.09903807193040848,0.3050587475299835,0.07713092863559723,0.14409954845905304 |
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,I only,II only,III only,I and II only,B,False,0.051337338984012604,0.10868114233016968,0.09591077268123627,0.33476150035858154 |
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,2/81,1/18,2/27,1/9,A,False,0.13614492118358612,0.1542724072933197,0.1201474741101265,0.19808968901634216 |
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Statement 1 | Suppose {s_n} and {t_n} are sequences of complex numbers such that lim_n→∞ s_n = s and lim_n→∞ tn = t. lim_n→∞ s_nt_n = st. Statement 2 | There exists a bounded sequence of real numbers with no convergent subsequence.,,,,,B,False,0.20356465876102448,0.1399078071117401,0.04008425399661064,0.15853631496429443 |
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,I and II only,I and III only,II and III only,III only,D,True,0.04782857745885849,0.10125311464071274,0.06141311675310135,0.27523449063301086 |
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,9/2 days,5 days,41/8 days,36/7 days,D,True,0.03816533461213112,0.10374413430690765,0.10374413430690765,0.1938195675611496 |
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What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?,1/2,1,sqrt(2),pi,A,False,0.10965634882450104,0.1242569163441658,0.1408015340566635,0.18079274892807007 |
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,f has all of its relative extrema on the line x = y.,f has all of its relative extrema on the parabola x = y^2.,,,A,False,0.14646019041538239,0.1292506605386734,0.07839449495077133,0.1659611463546753 |
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What is the largest order of an element in the group of permutations of 5 objects?,5,6,12,15,B,False,0.048175737261772156,0.1483917087316513,0.1483917087316513,0.16814984381198883 |
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,S(n0 + 1) is not true.,S(n0 - 1) is not true.,S(n) is not true for any n <= n0,S(n) is not true for any n >= n0,C,False,0.044300492852926254,0.0937841385602951,0.15462391078472137,0.3709234893321991 |
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,,,,,B,False,0.06122586503624916,0.08908310532569885,0.06937798112630844,0.16642914712429047 |
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For which integers n such that 3 <= n <= 11 is there only one group of order n (up to isomorphism)?,For no such integer n,,,,B,False,0.06294815242290497,0.11760261654853821,0.0808270201086998,0.17111079394817352 |
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A tree is a connected graph with no cycles. How many nonisomorphic trees with 5 vertices exist?,1,2,3,4,C,False,0.08441634476184845,0.22946739196777344,0.10839272290468216,0.22946739196777344 |
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,I only,II only,III only,,D,True,0.011117910966277122,0.0439721941947937,0.06397915631532669,0.6070173978805542 |
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,x^2 + y^2 + z^2 + 8 = 6y,(x^2 + y^2 + z^2)^2 = 8 + 36(x^2 + z^2),(x^2 + y^2 + z^2 + 8)^2 = 36(x^2 + z^2),(x^2 + y^2 + z^2 + 8)^2 = 36(x^2 + y^2),D,True,0.07539255172014236,0.12430128455162048,0.10969550162553787,0.20493817329406738 |
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Which of the following is an equation of the line tangent to the graph of y = x + e^x at x = 0 ?,y = 2x + 1,y = 2x,y = x + 2,y = x + 1,A,False,0.08842140436172485,0.18718810379505157,0.06077107787132263,0.18718810379505157 |
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,2,3,4,5,A,False,0.10483156144618988,0.2219284027814865,0.1728380173444748,0.11878972500562668 |
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Statement 1 | Suppose ∑|a_i| diverges and ∑ a_i = 2. There is a rearrangement a_i_k of the terms such that ∑ a_i_k = 4. Statement 2 | There exists metric spaces X and Y with X closed and bounded and a continuous mapping f : X → Y such that f(X) is NOT “closed and bounded”.,,,,,A,True,0.14990654587745667,0.14990654587745667,0.04294894263148308,0.14990654587745667 |
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The maximum number of acute angles in a convex 10-gon in the Euclidean plane is,1,2,3,4,C,False,0.046365756541490555,0.16183240711688995,0.16183240711688995,0.23546475172042847 |
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,the entire real axis,a point,a ray,an open finite interval,B,False,0.07120823860168457,0.10360736399888992,0.11740253120660782,0.24854116141796112 |
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,finite,cyclic,of order two,abelian,D,True,0.11345069110393524,0.10011989623308182,0.10011989623308182,0.2721540629863739 |
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,3,4,5,8,D,True,0.07215297967195511,0.1730860024690628,0.1730860024690628,0.2853705585002899 |
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,II only,III only,I and II only,II and III only,A,False,0.054037973284721375,0.11439839005470276,0.18861106038093567,0.21372433006763458 |
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What is the units digit in the standard decimal expansion of the number 7^25?,1,3,5,7,D,True,0.03584182634949684,0.14175717532634735,0.2062554657459259,0.23371806740760803 |
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,,,,,A,False,0.14564965665340424,0.1650426834821701,0.053581517189741135,0.18701787292957306 |
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The region bounded by the curves y = x and y = x^2 in the first quadrant of the xy-plane is rotated about the y-axis. The volume of the resulting solid of revolution is,π/12,π/6,π/3,2π/3,B,False,0.07216938585042953,0.13483016192913055,0.11898720264434814,0.2518959045410156 |
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,3,pi,3pi,3/2,B,False,0.13399295508861542,0.1720503717660904,0.13399295508861542,0.22091704607009888 |
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,Only (II),Only (IV ),(I) and (IV ),(II) and (III),C,False,0.04002378508448601,0.06598807126283646,0.07477426528930664,0.12328194081783295 |
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,f is Riemann integrable.,There does not exist a countably infinite E of R such that f is nondifferentiable at every x \in E.,There does not exist an uncountable subset E of R such that f is nondifferentiable at every x \in E.,None of the above.,D,True,0.07195381820201874,0.15232624113559723,0.07195381820201874,0.28458285331726074 |
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,the open first quadrant,a closed curve,a ray in the open first quadrant,a single point,A,True,0.1714811623096466,0.13354966044425964,0.11785716563463211,0.1714811623096466 |
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For which p is the norm ||x||_p expressable as an inner product?,1,2,p >= 1,infinity,B,False,0.09912046790122986,0.14421942830085754,0.12727320194244385,0.1851814091205597 |
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,f is a homeomorphism.,f is continuous and one-to-one,f is continuous.,f is one-to-one,C,False,0.11369667202234268,0.18745410442352295,0.12883520126342773,0.21241334080696106 |
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What is the area of an equilateral triangle whose inscribed circle has radius 2?,12,16,12*sqrt(3),16*sqrt(3),C,True,0.0921667218208313,0.11834441125392914,0.1721900999546051,0.1721900999546051 |
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For which value of n are there exactly two abelian groups of order n up to isomorphism?,4,7,8,12,A,False,0.07229721546173096,0.2226911038160324,0.10519183427095413,0.17343200743198395 |
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,|x| ≥ 0,|x · y| ≤ |x||y|,|x| = 0 if and only if x = 0,|x + y| = |x| + |y|,D,True,0.05670657381415367,0.1746685653924942,0.10594183206558228,0.32632380723953247 |
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,4,5,6,7,B,False,0.09283565729856491,0.13507509231567383,0.19653309881687164,0.13507509231567383 |
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,1 only,2 only,0 and 1 only,,D,True,0.07137975096702576,0.08088385313749313,0.11768531799316406,0.24913980066776276 |
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,2,π,4,3π/2,C,False,0.07731477916240692,0.18546847999095917,0.18546847999095917,0.21016332507133484 |
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,9,8,0,-5,A,False,0.08930127322673798,0.16683675348758698,0.16683675348758698,0.2750672996044159 |
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How many real roots does the polynomial 2x^5 + 8x - 7 have?,,One,Two,Three,B,True,0.035950228571891785,0.3410862982273102,0.12547864019870758,0.08624012023210526 |
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,20!/2^(10),10!/2^9,10^(20) - 2^(10),10^(20) - 100,A,False,0.15288536250591278,0.1190672367811203,0.08183364570140839,0.22244687378406525 |
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Which of the following are true:,,,Neither (a) nor (b),Both (a) and (b).,A,False,0.029686234891414642,0.10361514240503311,0.048944324254989624,0.15075914561748505 |
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In a game two players take turns tossing a fair coin; the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is,1/4,1/3,1/2,2/3,D,True,0.0956386998295784,0.10837284475564957,0.15768155455589294,0.17867662012577057 |
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Which of the following is NOT a group?,The integers under addition,The nonzero integers under multiplication,The nonzero real numbers under multiplication,The complex numbers under addition,B,False,0.059204280376434326,0.11060815304517746,0.11060815304517746,0.3860603868961334 |
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Let x and y be positive integers such that 3x + 7y is divisible by 11. Which of the following must also be divisible by 11?,4x + 6y,x + y + 5,9x + 4y,4x - 9y,D,False,0.10943610221147537,0.1804296374320984,0.10943610221147537,0.1804296374320984 |
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,0,-2,-sqrt(34),-sqrt(35),C,False,0.053995199501514435,0.1884617656469345,0.14677417278289795,0.1884617656469345 |
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,II only,III only,I and II only,I and III only,D,True,0.13146938383579254,0.10238846391439438,0.14897432923316956,0.16881002485752106 |
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The region bounded by the x-axis and the function f(x) = x/(1 + x^3) is rotated about the x-axis. What is the volume of the solid generated?,π/3,π/4,π,2π,A,False,0.0900922566652298,0.14853700995445251,0.1156807392835617,0.21612006425857544 |
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Suppose today is Wednesday. What day of the week will it be 10^(10^(10)) days from now?,Sunday,Monday,Tuesday,Wednesday,A,False,0.05032822862267494,0.2555878460407257,0.1065448671579361,0.19905202090740204 |
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,−48/π cm/min,−4/(3π) cm/min,−8/(3π) cm/min,−16/(3π) cm/min,D,True,0.0411323718726635,0.08707723021507263,0.06781581044197083,0.20888741314411163 |
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,The system is consistent.,The system has infinitely many solutions.,The sum of any two solutions is a solution.,,D,True,0.04654688760638237,0.1265275627374649,0.09853976964950562,0.44162461161613464 |
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,f(c),f(e^c),f(1/c),f(log c),D,True,0.09239797294139862,0.13443826138973236,0.11864134669303894,0.28460580110549927 |
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,I only,II only,III only,I and II only,D,True,0.05203952267765999,0.09722262620925903,0.11016767472028732,0.384522944688797 |
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Let y = f(x) be a solution of the differential equation x dy + (y - xe^x) dx = 0 such that y = 0 when x = 1. What is the value of f(2)?,1/(2e),1/e,e^2/2,2e,C,False,0.10858216136693954,0.157986119389534,0.13942226767539978,0.2604750692844391 |
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,H = 50,T >= 60,51 <= H <= 55,H >= 48 and T >= 48,D,True,0.03378530219197273,0.10406605899333954,0.08104672282934189,0.41158926486968994 |
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,,,,,D,False,0.19769565761089325,0.1539655327796936,0.034354351460933685,0.13587410748004913 |
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,2,1,0,-1,C,False,0.1021609827876091,0.24507147073745728,0.1684349924325943,0.21627481281757355 |
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,x^2/9,x^2/8,x^2/4,x^2,B,False,0.10784720629453659,0.09517481923103333,0.09517481923103333,0.12220689654350281 |
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,,,,,B,False,0.17082728445529938,0.19357267022132874,0.05545949935913086,0.21934658288955688 |
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Which of the following sets has the greatest cardinality?,R,The set of all functions from Z to Z,,The set of all finite subsets of R,C,False,0.12416394054889679,0.140696182847023,0.09669896960258484,0.23196877539157867 |
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,24,30,32,36,B,False,0.09894828498363495,0.16313812136650085,0.14396889507770538,0.2689692974090576 |
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How many structurally distinct Abelian groups have order 72?,4,6,8,9,B,True,0.1076744943857193,0.15666545927524567,0.1076744943857193,0.15666545927524567 |
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,27,28,35,37,C,False,0.09648287296295166,0.1590733677148819,0.12388645857572556,0.20425423979759216 |
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,local maximum at P and at Q,saddle point at P and at Q,local maximum at P and a saddle point at Q,local minimum at P and a saddle point at Q,C,False,0.15406565368175507,0.13596245646476746,0.0934455394744873,0.2241642028093338 |
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,,,,,B,False,0.1288992166519165,0.1460619419813156,0.10038681328296661,0.2125188708305359 |
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Let M be a 5 x 5 real matrix. Exactly four of the following five conditions on M are equivalent to each other. Which of the five conditions is equivalent to NONE of the other four?,,The homogeneous system Mx = 0 has only the trivial solution.,The system of equations Mx = b has a unique solution for each real 5 x 1 column vector b.,The determinant of M is nonzero.,A,False,0.03649432584643364,0.09920185804367065,0.14433786273002625,0.3055632412433624 |
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How many positive numbers x satisfy the equation cos(97x) = x?,1,15,31,49,C,False,0.13081227242946625,0.19033074378967285,0.13081227242946625,0.19033074378967285 |
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For which of the following rings is it possible for the product of two nonzero elements to be zero?,The ring of complex numbers,The ring of integers modulo 11,,The ring {a + b * sqrt(2) : a and b are rational numbers},C,False,0.0781787633895874,0.1137494295835495,0.0781787633895874,0.2728706896305084 |
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The map x -> axa^2 of a group G into itself is a homomorphism if and only if,G is abelian,G = {e},a^3 = e,a^2 = a,C,False,0.12387708574533463,0.15906132757663727,0.12387708574533463,0.23143285512924194 |
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,-2,-1,0,2,B,False,0.029414603486657143,0.10266705602407455,0.13182710111141205,0.31623679399490356 |
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,I only,II only,I and II only,I and III only,C,False,0.09373053908348083,0.10621059685945511,0.17511168122291565,0.2547859847545624 |
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,1/2,7/12,5/8,2/3,C,False,0.15157462656497955,0.1717565506696701,0.09193466603755951,0.1717565506696701 |
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,0,1/4,sqrt(2)/4,1/sqrt(2),B,True,0.0695117637515068,0.24261988699436188,0.14715640246868134,0.18895256519317627 |
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(1+i)^10 =,1,i,32,32i,D,True,0.023301806300878525,0.06334087997674942,0.19510364532470703,0.22108139097690582 |
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,0,1,0 or 1,,D,True,0.08636501431465149,0.14239183068275452,0.12566035985946655,0.20717889070510864 |
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,closed,open,connected,totally disconnected,C,False,0.13616955280303955,0.1748451590538025,0.12016920745372772,0.15430031716823578 |
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Which of the following circles has the greatest number of points of intersection with the parabola x^2 = y + 4 ?,x^2 + y^2 = 1,x^2 + y^2 = 2,x^2 + y^2 = 9,x^2 + y^2 = 16,C,False,0.10422664880752563,0.1516488790512085,0.11810426414012909,0.17184068262577057 |
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X is a finite set with the following property: The number of subsets of X that contain exactly 3 elements is 14 more than the number of subsets of X that contain exactly 2 elements. How many subsets of X contain exactly 4 elements?,24,25,28,35,D,False,0.05375128611922264,0.21259047091007233,0.165565624833107,0.1876104176044464 |
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,,,,,A,False,0.0795116201043129,0.13109248876571655,0.1485472470521927,0.2161349654197693 |
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