微臣GRE数学自编题目(最新版).pdf

GRE Quantitative Reasoning (Step 2) Bo Wang University of Virginia Aug 2015 Bo Wang (UVA)GRE MathAug 20151 / 106 Overview 1Arithmetic 2Algebra 3Geometry 4Data Analysis 5Problem-solving Techniques Bo Wang (UVA)GRE MathAug 20152 / 106 0 SN !%cG #GRE%:8B K.KE|!o( SK !GRE!rz#!) uGRE!%kr#I!) K8%$!) #!O 5%:KE|!o( KSKJ%(”u$puGRE!pJ) Bo Wang (UVA)GRE MathAug 20153 / 106 1 O)P$P%SN!: 2 veZuSKS 3 U!OwindowsgO%O 4 Xk7O5 o. 5 3(O Bo Wang (UVA)GRE MathAug 201514 / 106 1 K8w+AKn)!c 2 :# 3 KkKvw%M%(! 4 %! Bo Wang (UVA)GRE MathAug 201515 / 106 Arithmetic 1 Integers 2 Factors and multiples 3 quotients and reminders 4 odd and even number 5 prime and composite number 6 fractions 7 exponents and roots 8 decimals 9 real number 10 ratio 11 percent Bo Wang (UVA)GRE MathAug 201516 / 106 Integers) Integers)Positive Integers)Negative IntegersK) %:+=*=KKK+KKK=KKKKKK*KKK= Example 1 a,b,c are all integers, suppose a3b4c5is negative then which of the following CANNOT be true? A a + b 0 B b + c 0 C a + c 0 D ac 0 E abc 0 Bo Wang (UVA)GRE MathAug 201517 / 106 Factors and Multiplesf0 factors (divisors)f, multiple 0, least common multiple)0, greatest common divisor% %:)%)000 Example 2 Let S be the set of all positive integers n such that n2is a multiple of both 24 and 108. Which of the following integers are divisor of every integer n in S? Indicate all such integers A 12 B 24 C 36 D 72 Bo Wang (UVA)GRE MathAug 201518 / 106 Quotients and Remainders) a divided by b is q remainder r , a=bq+r where (0 r b) ab uqr quotient; remainder %: Example 3 An amazing fact is that the number 229 has 9 digits, and all of them are dierent. Find the remainder of 22 9 divided by 9. And fi nd the missing digit. Bo Wang (UVA)GRE MathAug 201519 / 106 odd and even number odd number, even number %:XXXJJJ)oookkk XXXJJJ)ooo000kkk Example 4 a,b,c are all integers, suppose a3b4c5is an odd number then which of the following is true? Indicate all such statements. A a + b is odd B b + c is odd C a + c is even D ac is odd E abc is even Bo Wang (UVA)GRE MathAug 201520 / 106 prime and composite number prime number , composite , prime factorization) %:555!XXX Example 5 a,b,c are dierent prime numbers and a 0Kk”!) ? = b2? 4ac = 0Kk) ? = b2? 4ac 0K*) %:)ggg Example 14 x2?3x+2 x4+3x2+4x?3 = 0 Quantity AQuantity B x2 Bo Wang (UVA)GRE MathAug 201536 / 106 Solving linear inequality) %:) Example 15 How many integers are there in the set ?4 x2+ 5x 1) cf (x)“rf (x)!p(shrink vertically)c0(0 c n are positive integers Column AColumn B m! ? n!(m ? n)! Exercise7 n = k+r s t v In the equation above, k,r,s,t,v represent positive numbers. Multiplying which one of these numbers by 2 will reduce the value of n to 1 2 of its present value? A k B r C s D t E v Bo Wang (UVA)GRE MathAug 201546 / 106 Exercise8 3 x 10 A x + (w ? 10)y + 10 B x + (w ? 10) + y C 10 x + 10(w ? y) D 10 x + y(w ? 10) E 10 x + wy Bo Wang (UVA)GRE MathAug 201547 / 106 Exercise10 T is no more than 20 percent greater than the integer x, and T is no smaller than a number 20 percent less than the integer x. If T = 60, which of the following indicates the smallest possible interval for x? A 40 x 80 B 48 x 75 C 48 x 72 D 50 x 78 E 50 x 75 Exercise11 0 /“k!/ sides3:vertices trianglen2/quadrilateralo/pentagon /,hexagon8 /,octagonl/ n!/S2180(n ? 2)o /regular polygon, perimeter(, area%:SSS222 Example 20 What is the maximum possible number of interior angles that are right angles of a decagon (10-side polygon)? Bo Wang (UVA)GRE MathAug 201552 / 106 polygons/ Example 21 What is a + b + c + d + e + f + g? Bo Wang (UVA)GRE MathAug 201553 / 106 trianglesn2/ n2/S2180o n2/equilateral triangle,+n2/isosceles triangle 2n2/right triangle2legs.hypotenuse ./*nPythagorean theorem.x.d*n Bo Wang (UVA)GRE MathAug 201554 / 106 A!n2/ n1 : 1 : p2 n1 : p3 : 2 Bo Wang (UVA)GRE MathAug 201555 / 106 n2/! Bo Wang (UVA)GRE MathAug 201556 / 106 n2/qn2/ n2/“/G)!,!n2/ qn2/“/G,6)*,!n2/ XABCDEF“qn2/!(AB CD = BC EF = AC DF %:kkk Bo Wang (UVA)GRE MathAug 201557 / 106 n2/ Example 22 In the fi gure above, what is the area of triangle BDE and ABE given that the area of triangle ACE is 4, the area of triangle CDE is 3? Bo Wang (UVA)GRE MathAug 201558 / 106 n2/ Example 23 suppose 2 + ? 100 then which of the following could be the value of ? Indicates all possible values A) 60B) 89C) 80D) 45E) 50F) 51 Bo Wang (UVA)GRE MathAug 201559 / 106 n2/ Example 24 What is the perimeter of ABCDE if both and ? are 150o, A,B are right angles and AC=AB=BD=1? Bo Wang (UVA)GRE MathAug 201560 / 106 quadrilateralso/ (/rectangle,/square,$1o/parallelogram,F/trapezoid $1o/.*p F/.+e.*p/2 Bo Wang (UVA)GRE MathAug 201561 / 106 circles, circle,center,%,radius,diameter chordu,arcl,circumference(C = 2r areaA = r2,sector/,concentric circles”%, tangent line, :point of tangency, ,S4/inscribed polygon, ,(4/circumscribed polygon Bo Wang (UVA)GRE MathAug 201562 / 106 , Example 25 suppose A(o) is the origin and the radius of the circle is 2. suppose CE=3, DEFG is a rectangle. Quantity AQuantity B area of DEFG7 Bo Wang (UVA)GRE MathAug 201563 / 106 , Example 26 Quantity AQuantity B area of ABCarea of region CD Bo Wang (UVA)GRE MathAug 201564 / 106 three-dimensional fi guresN/ rectangular solid(N cubeN,faces edges,vertex3: volumeNV = lwh surface areaL A = 2(lw + lh + wh) circular cylinder, axis/(PQ) volumeNV = r2h surface areaLA = 2r2+ 2pirh Bo Wang (UVA)GRE MathAug 201565 / 106 three-dimensional fi guresN/ %: LLL%NNN Example 27 An ice cube is 1cm long, if we slice it into two pieces. Quantity AQuantity B Resulting surface area of all the slices8 cm2 Bo Wang (UVA)GRE MathAug 201566 / 106 Exercise13 AC=40, BC=25, EF=10, and AD is parallel to EC. Quantity AQuantity B area of ABD90 Bo Wang (UVA)GRE MathAug 201567 / 106 Exercise14 triangle CDE is equilateral. Column AColumn B area of CDEtotal area of the other parts of the circle Bo Wang (UVA)GRE MathAug 201568 / 106 Exercise15 In the fi gure above AB AC = AC BC What is AB BC? Bo Wang (UVA)GRE MathAug 201569 / 106 Exercise16 AB=3, AC=6 Column AColumn B area of ABC8 Bo Wang (UVA)GRE MathAug 201570 / 106 Exercise17 In the above circle CG = 10, FG = 8, CF = 9 and DG = 12. What is the length of DE? A) 10B) 11.5C) 12D) 13.5E) 15 Bo Wang (UVA)GRE MathAug 201571 / 106 Data Analysis 1 Graphical Methods for Describing Data 2 Numerical Methods for Describing Data 3 Counting Methods 4 Probability 5 Distributions of Data, Random Variables, and Probability Distributions 6 Data Interpretation Bo Wang (UVA)GRE MathAug 201572 / 106 Graphical Methods for Describing Data/ 1 Frequency Distribution 2 Bar Graphs/ 3 Circle GraphsG 4 Histograms 5 Scatterplots: 6 Time Plotsm: K8!L=*wpI%/ L! Bo Wang (UVA)GRE MathAug 201573 / 106 measure of central tendency!% arithmetic mean(average)$ X* 6,4,7,10,74$6+4+7+10+7+4 6 = 38 6 median l),Xo2k“m! Xk“m!$ X*2,4,4,5,7,7,7,7,7,7,8,8,9,9,9,9 216o7 mode yg!X2,4,4,5,7,7,7,7,7,7,8,8,9,9,9,97“ 8k Bo Wang (UVA)GRE MathAug 201574 / 106 measure of position! 8k!A“)1414 1n4 14Q1L%k1 4!)14 Q2L%k2 4!)1n4Q3L%k 3 4 !) “rl),N31 4!i“Q1 da ,(kzP1,P2, ,P99OLk%1%,2%, ,99%! )“l),N31% !i“P1da 5median,Q2P50L!g” Bo Wang (UVA)GRE MathAug 201575 / 106 measure of dispersion! range4(0) outliersl+(“l-!) interquartile range4(Q3? Q1) boxplots(box-and-whisker plots) XLL)GLQ114M Q314G Bo Wang (UVA)GRE MathAug 201576 / 106 standard deviationIO X* 0,7,8,10,10. $0+7+8+10+10 5 = 7 (0?7) 2+(7?7)2+(8?7)2+(10?7)2+(10?7)2 5 = 13.6 IOp13.6 3.7 5!3$!m3IOS Bo Wang (UVA)GRE MathAug 201577 / 106 %: mean, median, mode Example 28 Find which of the ABC is the mean, median and mode. Bo Wang (UVA)GRE MathAug 201578 / 106 %: statistics Example 29 If the average GRE score of Bo, Uncle Qi, Jialing, Glee, Jimmy is 330. Jialing has a score of 333, Glee is 3 points higher than Jimmy. Bo is 10 points lower than Uncle Qi. Then what is the mean of Bo and Jimmy? Bo Wang (UVA)GRE MathAug 201579 / 106 %: mean, median Example 30 Suppose here is the chart describing the annual snow depth of Mathsville ski resort in 40 years. 1. What is the size of the IQR (interquartile range)? 2. How many of years are there that has a snow depth higher than 100? 3. Suppose in 2014 the snow depth is 90; this is the 60th percentile value on this chart. How many years has a snow depth higher than 50 and lower than 90? Bo Wang (UVA)GRE MathAug 201580 / 106 %: Percentiles and Quartiles Example 31 suppose all the 10 teachers of Weichen took the GRE test and Bo got a very good score of 335. Unluckily Uncle Qi (still one of the 10 teachers) had an even better score of 339. And everybody else is lower than Bo. Suppose now Weichen hires 10 more teachers who have GRE scores more than 335, than what is the percentile of Bo now? Bo Wang (UVA)GRE MathAug 201581 / 106 %: Percentiles and Quartiles Example 32 Bo took a test that is graded on a scale from 0 to 100. Quantity A Quantity B Bo.s scorethe percentile of Bo.s score Bo Wang (UVA)GRE MathAug 201582 / 106 sets and lists set8(*E)!o8p!zelement) k08k!fi nitek0*!infi nite 8;empty set!8nonempty setf88! X2,8 0,2,4,6,8!f8 list,L$8,L,LdE X1,1,1,1,1,2setlist 8!intersection/82k!)/!f88 !union0u,8!)!8X0,11,2!/ 1, 0,1,2 e8!/8disjoint(mutually exclusive)/ Bo Wang (UVA)GRE MathAug 201583 / 106 Venn diagram* *5L8m!X u!8ST|S T| = |S| + |T| ? |S T| Bo Wang (UVA)GRE MathAug 201584 / 106 Theorem Theorem (*n) !kNa1akM11#a kM2,1NakMNo! kM1+ + MN Theorem (*n) !IN1kM1” 1#kM2”,1NkMN”o! !kM1 MN” Bo Wang (UVA)GRE MathAug 201585 / 106 permutations and combinations,| ln”)?*m)U*!S,