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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,0.06505809724330902,0.06505809724330902,0.10726267844438553,0.3303923010826111 |
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,0,1,2,3,D,False,0.08172506093978882,0.15268249809741974,0.17301194369792938,0.15268249809741974 |
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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,0.07468792796134949,0.03997756540775299,0.058167021721601486,0.15811434388160706 |
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The shortest distance from the curve xy = 8 to the origin is,4,8,16,2sqrt(2),A,False,0.0946265459060669,0.12150289118289948,0.12150289118289948,0.2003243863582611 |
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,2/69,1/30,2/23,12/125,C,True,0.11355126649141312,0.12867043912410736,0.18721438944339752,0.14580270648002625 |
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,,,,,D,True,0.10234677046537399,0.13141584396362305,0.13141584396362305,0.27820733189582825 |
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,12*sqrt(2),6*sqrt(2),4*sqrt(2),1/(2*sqrt(2)),C,True,0.1014937162399292,0.1303205043077469,0.16733483970165253,0.14767247438430786 |
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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,True,0.09937309473752975,0.14458699524402618,0.18565337359905243,0.14458699524402618 |
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,,I only,II only,III only,C,False,0.11349324136972427,0.08838862925767899,0.0780026838183403,0.11349324136972427 |
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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,0.053172893822193146,0.0993400439620018,0.08766727894544601,0.18559162318706512 |
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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,0.250966340303421,0.1522187739610672,0.10461833328008652,0.13433261215686798 |
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,1/64,15/64,1/4,3/8,B,False,0.13554799556732178,0.13554799556732178,0.11962068825960159,0.22348086535930634 |
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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,0.09876146167516708,0.14369706809520721,0.18451069295406342,0.18451069295406342 |
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,4,7,10,13,B,False,0.15959025919437408,0.12428902089595795,0.10968467593193054,0.1408379077911377 |
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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,0.12469074875116348,0.16010607779026031,0.12469074875116348,0.20558027923107147 |
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,I only,II only,III only,I and II only,D,True,0.08504480123519897,0.07505177706480026,0.09636839479207993,0.26195642352104187 |
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,I only,II only,III only,I and II only,B,False,0.1300923377275467,0.07890499383211136,0.10131601244211197,0.18928322196006775 |
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,-1/4,-1/(4sqrt(2)),0,sqrt(2),C,False,0.0744767114520073,0.1227913349866867,0.13914082944393158,0.29456111788749695 |
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,I and II only,III only,I and III only,,D,True,0.07181868702173233,0.04356024041771889,0.09221702814102173,0.36472541093826294 |
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,3,e,sqrt(5),0,A,False,0.11146429926156998,0.14312298595905304,0.20824271440505981,0.20824271440505981 |
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,I only,II only,III only,,D,True,0.045410942286252975,0.05830880254507065,0.0961349681019783,0.43084701895713806 |
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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,0.12140799313783646,0.1766475886106491,0.12140799313783646,0.15589094161987305 |
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,I only,II only,III only,I and II only,B,False,0.07687362283468246,0.08710923790931702,0.09870769083499908,0.23678743839263916 |
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,2/81,1/18,2/27,1/9,A,False,0.15081892907619476,0.11745790392160416,0.13309724628925323,0.1936553418636322 |
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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,0.31993743777275085,0.11769841611385345,0.055596787482500076,0.11769841611385345 |
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,I and II only,I and III only,II and III only,III only,D,True,0.07844959944486618,0.0888950526714325,0.07844959944486618,0.24164177477359772 |
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,9/2 days,5 days,41/8 days,36/7 days,D,True,0.07219954580068588,0.0818127989768982,0.1348865032196045,0.15284642577171326 |
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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,0.13552388548851013,0.11959940940141678,0.15356867015361786,0.1740161031484604 |
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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,0.13131605088710785,0.13131605088710785,0.11588601022958755,0.2165035605430603 |
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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,0.07535804063081741,0.12424440681934357,0.15953296422958374,0.14078736305236816 |
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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,0.06311143189668655,0.06311143189668655,0.1336069107055664,0.3205063045024872 |
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,,,,,B,False,0.07282558083534241,0.07282558083534241,0.0825222060084343,0.17469951510429382 |
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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,0.10105333477258682,0.08917925506830215,0.06945286691188812,0.1666087806224823 |
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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,0.13516901433467865,0.15316654741764069,0.11928623169660568,0.19666974246501923 |
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,I only,II only,III only,,D,True,0.040090642869472504,0.05833154171705246,0.09617245942354202,0.4310150444507599 |
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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,0.08256540447473526,0.10601608455181122,0.13612735271453857,0.19806411862373352 |
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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,0.08505567163228989,0.12375526875257492,0.0750613734126091,0.20403794944286346 |
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,2,3,4,5,A,False,0.12934426963329315,0.16608130931854248,0.18819478154182434,0.10073341429233551 |
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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,0.25991708040237427,0.12277614325284958,0.05799534171819687,0.10834955424070358 |
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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,True,0.084920234978199,0.140009805560112,0.1797761470079422,0.1797761470079422 |
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,the entire real axis,a point,a ray,an open finite interval,B,False,0.10305369645357132,0.0909445658326149,0.13232356309890747,0.21816466748714447 |
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,finite,cyclic,of order two,abelian,D,True,0.15941104292869568,0.0966876819729805,0.12414944171905518,0.20468781888484955 |
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,3,4,5,8,D,True,0.11634717136621475,0.1318386197090149,0.1918240487575531,0.21736513078212738 |
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,II only,III only,I and II only,II and III only,A,False,0.09273736923933029,0.10508520156145096,0.17325620353221893,0.17325620353221893 |
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What is the units digit in the standard decimal expansion of the number 7^25?,1,3,5,7,D,False,0.0929904356598854,0.11940208077430725,0.19686074554920197,0.17372900247573853 |
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,,,,,A,True,0.2185264229774475,0.13254298269748688,0.06260886788368225,0.11696877330541611 |
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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,0.10619531571865082,0.12033505737781525,0.12033505737781525,0.2248154729604721 |
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,3,pi,3pi,3/2,B,False,0.14955733716487885,0.13198387622833252,0.14955733716487885,0.2176046371459961 |
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,Only (II),Only (IV ),(I) and (IV ),(II) and (III),C,False,0.05455903708934784,0.05455903708934784,0.07938292622566223,0.11550149321556091 |
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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,0.10305587947368622,0.1499454230070114,0.0909465029835701,0.24721820652484894 |
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,the open first quadrant,a closed curve,a ray in the open first quadrant,a single point,A,True,0.19790871441364288,0.12003770470619202,0.13602054119110107,0.13602054119110107 |
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For which p is the norm ||x||_p expressable as an inner product?,1,2,p >= 1,infinity,B,False,0.1014551967382431,0.1014551967382431,0.1476164311170578,0.16727133095264435 |
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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,0.14762288331985474,0.14762288331985474,0.16727864742279053,0.18955154716968536 |
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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,0.11355623602867126,0.08843769133090973,0.18722258508205414,0.16522334516048431 |
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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,0.09639284759759903,0.15892493724822998,0.12377086281776428,0.15892493724822998 |
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,|x| ≥ 0,|x · y| ≤ |x||y|,|x| = 0 if and only if x = 0,|x + y| = |x| + |y|,D,True,0.07848520576953888,0.14662966132164001,0.14662966132164001,0.24175144731998444 |
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,4,5,6,7,B,False,0.12194033712148666,0.12194033712148666,0.20104563236236572,0.10761197656393051 |
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,1 only,2 only,0 and 1 only,,D,True,0.09778342396020889,0.07615380734205246,0.12555640935897827,0.2345702350139618 |
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,2,π,4,3π/2,C,True,0.10005367547273636,0.1284714639186859,0.18692487478256226,0.18692487478256226 |
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,9,8,0,-5,A,False,0.12064781039953232,0.12064781039953232,0.17554152011871338,0.22539977729320526 |
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How many real roots does the polynomial 2x^5 + 8x - 7 have?,,One,Two,Three,B,True,0.06283261626958847,0.21930737793445587,0.17079675197601318,0.11738678067922592 |
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,20!/2^(10),10!/2^9,10^(20) - 2^(10),10^(20) - 100,A,False,0.188126802444458,0.11410468071699142,0.11410468071699142,0.21317559480667114 |
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Which of the following are true:,,,Neither (a) nor (b),Both (a) and (b).,A,False,0.045835696160793304,0.1099541187286377,0.05193864926695824,0.15998229384422302 |
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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,False,0.11860089749097824,0.10466492176055908,0.19553981721401215,0.13439242541790009 |
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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,0.099668487906456,0.099668487906456,0.12797687947750092,0.23909227550029755 |
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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,True,0.12736660242080688,0.14432525634765625,0.12736660242080688,0.18531730771064758 |
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,0,-2,-sqrt(34),-sqrt(35),C,False,0.08546636253595352,0.15967218577861786,0.15967218577861786,0.2050231397151947 |
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,II only,III only,I and II only,I and III only,D,False,0.1561223715543747,0.09469300508499146,0.1561223715543747,0.13777750730514526 |
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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,0.13002508878707886,0.13002508878707886,0.13002508878707886,0.18918538093566895 |
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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,0.1077786386013031,0.20135699212551117,0.1077786386013031,0.1568169891834259 |
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,−48/π cm/min,−4/(3π) cm/min,−8/(3π) cm/min,−16/(3π) cm/min,D,True,0.07820626348257065,0.11378943920135498,0.11378943920135498,0.2729666829109192 |
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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,0.09188585728406906,0.133693128824234,0.10412031412124634,0.3207131624221802 |
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,f(c),f(e^c),f(1/c),f(log c),D,True,0.10912572592496872,0.10912572592496872,0.14012020826339722,0.2617790102958679 |
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,I only,II only,III only,I and II only,D,True,0.08894374221563339,0.08894374221563339,0.11420603096485138,0.2739660143852234 |
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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,0.12038049846887589,0.10623541474342346,0.15457162261009216,0.2548454999923706 |
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,H = 50,T >= 60,51 <= H <= 55,H >= 48 and T >= 48,D,True,0.050313420593738556,0.07320559024810791,0.08295280486345291,0.42126911878585815 |
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,,,,,D,False,0.2861796021461487,0.11929740011692047,0.056352101266384125,0.1052796021103859 |
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,2,1,0,-1,C,True,0.1307608187198639,0.1679002195596695,0.21558813750743866,0.1679002195596695 |
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,x^2/9,x^2/8,x^2/4,x^2,B,False,0.12098343670368195,0.07338016480207443,0.12098343670368195,0.12098343670368195 |
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,,,,,B,False,0.25029703974723816,0.1518128365278244,0.06328500807285309,0.1339743584394455 |
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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,0.17565546929836273,0.12072612345218658,0.12072612345218658,0.19904373586177826 |
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,24,30,32,36,B,False,0.12319764494895935,0.12319764494895935,0.15818890929222107,0.20311857759952545 |
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How many structurally distinct Abelian groups have order 72?,4,6,8,9,B,False,0.11703653633594513,0.11703653633594513,0.11703653633594513,0.13261976838111877 |
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,27,28,35,37,C,False,0.11684474349021912,0.15003161132335663,0.1324024349451065,0.1700080931186676 |
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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,0.14770174026489258,0.11503023654222488,0.11503023654222488,0.24351899325847626 |
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,,,,,B,False,0.16656875610351562,0.12972387671470642,0.11448092758655548,0.21387851238250732 |
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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,0.05967827886343002,0.08683138340711594,0.14316076040267944,0.2674594819545746 |
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How many positive numbers x satisfy the equation cos(97x) = x?,1,15,31,49,C,False,0.13555440306663513,0.13555440306663513,0.15360327064990997,0.1740553081035614 |
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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,0.10616354644298553,0.08268025517463684,0.09368901699781418,0.2546731233596802 |
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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,0.16027051210403442,0.12481880187988281,0.12481880187988281,0.2057914137840271 |
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,-2,-1,0,2,B,False,0.0649658665060997,0.07361596822738647,0.1765955239534378,0.2911567986011505 |
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,I only,II only,I and II only,I and III only,C,False,0.1341446340084076,0.09219617396593094,0.15200577676296234,0.1722451150417328 |
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,1/2,7/12,5/8,2/3,C,False,0.16662010550498962,0.1297638714313507,0.1297638714313507,0.14704172313213348 |
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,0,1/4,sqrt(2)/4,1/sqrt(2),B,False,0.09357088804244995,0.1748134195804596,0.1748134195804596,0.19808955490589142 |
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(1+i)^10 =,1,i,32,32i,D,False,0.0472836047410965,0.06071334704756737,0.1870102882385254,0.1870102882385254 |
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,0,1,0 or 1,,D,True,0.11854927241802216,0.10461936146020889,0.13433392345905304,0.17248816788196564 |
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,closed,open,connected,totally disconnected,C,False,0.160764679312706,0.160764679312706,0.14187432825565338,0.1252036690711975 |
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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,0.12160956114530563,0.12160956114530563,0.13780169188976288,0.17694087326526642 |
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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,0.11095160990953445,0.14246468245983124,0.18292827904224396,0.14246468245983124 |
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,,,,,A,False,0.1022428572177887,0.1022428572177887,0.16856996715068817,0.19101479649543762 |
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