SIMOC, Secondary 2 Contest [PDF]

Citation preview

SIMOC, Secondary 2 Contest SIMOC SECONDARY 2 MOCK TEST Section A 1.



The LCM of 4, 9 and n is 108. Find the least value of n. (a) (b) (c) (d) (e)



2.



3 9 27 36 81



The diagram shows a triangle ACE. The lines AD, BE and CF intersect at the point G. How many triangles are there in the diagram altogether? E F A (a) (b) (c) (d) (e)



3.



B



D



C



14 15 16 17 None of the above



A particular month has 5 Saturdays. The first and the last day of the month are not Saturdays. What day is the first day of the month? (a) (b) (c) (d) (e)



4.



G



Thursday Friday Saturday Sunday Monday



Three men, Albert, Ben and Charles, and three women, Denise, Evelyn and Fiona, met at a restaurant. Each of the men is married to exactly one of the women, and vice versa. Albert’s wife and Denise’s husband do not know each other. Evelyn’s husband and Ben’s wife do not know each other. Ben knows everybody. Who is Fiona’s husband? (a) (b) (c) (d) (e)



Albert Ben Charles It cannot be determined from the given information None of the above 1



SIMOC, Secondary 2 Contest 5.



The diagram shows a circle with centre O. ABC and EDC are straight lines. Given that AOE = 80 and BOD = 20, find ACE.



A B 80



O



20



C



D



E (a) (b) (c) (d) (e) 6.



If the four-digit number 66N2 is divisible by 28, find N. (a) (b) (c) (d) (e)



7.



2 3 5 7 9



A television set has a 30-inch screen size, i.e. the length of the diagonal of the screen is 30 inches. The aspect ratio of the screen is 4 : 3. Find the height of the screen. Leave your answer to the nearest whole number if necessary. (Note: The aspect ratio of the screen refers to the ratio of the width of the screen to its height.) (a) (b) (c) (d) (e)



8.



20 25 30 35 None of the above



13 inches 18 inches 21 inches 24 inches None of the above



Given that x2  x + 1 = 0, find the value of x 3  (a) (b) (c) (d) (e)



1 . x3



3 1 2 3 Undefined [common mistake because x is undefined]



2



SIMOC, Secondary 2 Contest 9.



The diagram shows a diagonal passing through 6 squares of a 4-by-3 rectangle. Find the number of squares passed through by a diagonal for a 18-by-12 rectangle.



(a) (b) (c) (d) (e) 10.



There are two circles, each of radius 8 cm, lying on a plane and tangential to each other (i.e. the two circles just touch each other at one point). Find the number of circles of radius 16 cm lying on the same plane and tangential to the first two circles. (a) (b) (c) (d) (e)



11.



12 18 24 30 None of the above



2 3 4 5 None of the above



The diagram shows a point E inside a rectangle ABCD such that AE = 14 cm, DE = 16 cm and CE = 8 cm. Find the length of BE. A



B 14 16



8



D (a) (b) (c) (d) (e) 12.



E



C



2 cm 3 cm 4 cm 5 cm 6 cm



A list of whole numbers from 1 to 2015 is written on a sheet of paper. All the multiples of 5 are then struck off from the list. What is the last digit of the product of the remaining numbers? (a) (b) (c) (d) (e)



3 4 5 6 None of the above 3



SIMOC, Secondary 2 Contest 13.



The diagram shows a spherical cap, which is obtained by making a straight cut across a sphere. The volume of a spherical cap is given by



𝜋ℎ(3𝑟 2 +ℎ2 ) 6



, where h is the height



and r is the radius of the cap. r h A cylindrical hole 6 cm long has been drilled straight through the centre of a solid sphere. What is the volume of the remaining solid? (Note: Volume of cylinder =   radius2  height) (a) (b) (c) (d) (e) 14.



27 cm3 36 cm3 45 cm3 54 cm3 None of the above



Five rectangular sheets of paper can be put up on a bulletin board by using a minimum of 11 thumbtacks if the corners are overlapped as shown in the diagram, where the black dots represent thumbtacks and the dotted lines represent the edges of the sheets of paper that are covered by other sheets of paper.



What is the minimum number of thumbtacks required to put up 18 rectangular sheets of paper on a very large bulletin board in the same way as described above? (a) (b) (c) (d) (e) 15.



27 28 29 30 None of the above



Find the next term of the following sequence: 2, 12, 1112, 3112, 132112, … (a) (b) (c) (d) (e)



332112 422112 432112 1113122112 None of the above 4



SIMOC, Secondary 2 Contest Section B 16.



Fill in the missing number in the box.   31  37 + 41  431 = 2015



17.



Find the smallest prime factor of 9991.



18.



What is the largest product that can be formed from using the digits 2, 3, 4 and 5, and one multiplication sign? You are only allowed to combine the digits to form two numbers, e.g. 2  345, but you are not allowed to use indices, e.g. 23  45 is not allowed.



19.



Find the last digit of 1 + 2 + 22 + 23 + … + 22015.



20.



The following is a conversation between Esther and Frank who met on a bus. Esther:



Frank: Esther: Frank:



I have three children. Assuming that their ages are whole numbers, the product of their ages is 72, and the sum of their ages is the bus number of this bus that we are on. Of course I know the bus number, but I still don’t know their ages. Oh, I forgot to tell you that my two youngest children have the same age. Oh, now I know their ages.



So what are the ages of the three children?



21.



A data set contains 10 whole numbers. The smallest number is 2 and the largest number is 7. The mean, median and mode of the data set are all equal to 3. List the 10 numbers in ascending order (i.e. from the smallest to the largest).



22.



The maximum number of parts that can be obtained from cutting a circle using 3 straight cuts is 7. What is the maximum number of parts that can be obtained from cutting a circle using n straight cuts? Express your answer in the form of (Note: Do not count the parts outside the circle.)



5



𝑎𝑛2 +𝑏𝑛+𝑐 𝑑



.



SIMOC, Secondary 2 Contest 23.



Two towns, A and B, lie to the north of a straight road running in the East-West direction (as shown in the diagram below). Unfortunately they are neither connected to the road nor to each other. The people living in the two towns decide to build two new roads, one from each town, to the existing road, such that the route connecting the two towns via the existing road will be the shortest. Given that AC = 10 km, BD = 20 km and CD = 40 km, find the length of the shortest route. Town B Town A



20 km



10 km Road C



24.



40 km



D



The diagram shows 12 identical match sticks, each of length one unit, forming a square with an area of 9 square units.



Use the same 12 match sticks to form a figure with an area of 4 square units. The entire length of each match stick must be used.



25.



In the following exact long division, the letter X can stand for different digits. Find the eight-digit dividend. XX8XX XXXXXXXXXXX  XXX XXXX  XXX XXXX XXXX 0



6