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SIMOC, Secondary 3 Contest SIMOC SECONDARY 3 MOCK TEST Section A 1.



Find the next term of the following sequence: 1, 2, 5, 8, 15, 28, 51, … (a) (b) (c) (d) (e)



2.



79 83 87 91 None of the above



The following histogram shows the average amount of money spent by students in the school canteen every day. No. of students



Money Spent



200 100 0 0



1



2



3



4



Money (in dollars)



Which one of the following statement is true? (a) (b) (c) (d) (e) 3.



The median is $1. The students spent more money in the canteen during the middle of the day. If the school decided to increase the price of the food in the canteen, then the heights of the columns would get higher. The mode is 200 students. None of the above



The cost price of a pair of shoes was $25. Mr Lee sold the pair of shoes to a customer for $40. The latter gave Mr Lee a $50 note, but Mr Lee did not have enough change. So Mr Lee went to Mr Tan in the shop next door and exchanged the $50 note into smaller notes. Then Mr Lee gave the customer a $10 change. After the customer had left, Mr Tan discovered that the $50 note was a counterfeit. So Mr Lee compensated Mr Tan $50. How much money did Mr Lee lose as compared to if the $50 note was real? (a) (b) (c) (d) (e)



$10 $20 $35 $50 $65



1



SIMOC, Secondary 3 Contest 4.



The area of a right-angled triangle is 150√3 cm2. If one of the angles of the triangle is 30, find the perimeter of the triangle. (a) (b) (c) (d) (e)



5.



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 20 cm lying on the same plane and tangential to the first two circles. (a) (b) (c) (d) (e)



6.



20(√2  1) cm 20(√2 + 1) cm 30(√3  1) cm 30(√3 + 1) cm None of the above



2 3 4 5 None of the above



The roots,  and , of a quadratic equation satisfy the following inequality: 2 + 2 < 0. What can you conclude about the nature of the roots of the quadratic equation? (a) (b) (c) (d) (e)



7.



The two roots are real and distinct The two roots are real and equal The two roots are non-real No conclusion can be drawn None of the above



The diagram shows a point E inside a rectangle ABCD such that AE = 16 cm, DE = 20 cm and CE = 13 cm. Find the length of BE. A



B 16 20



13



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



E



C



4 cm 5 cm 6 cm 7 cm 8 cm



2



SIMOC, Secondary 3 Contest 8.



The decimal numeral system uses ten symbols 0 to 9 to form numbers, while the binary numeral system uses only two symbols 0 and 1 to form numbers. The following table shows the binary numbers for the decimal numbers 0 to 5. Decimal Number Binary Number



0 0



1 1



2 10



3 11



4 100



5 101



What is the decimal number for the binary number 110 100? (a) (b) (c) (d) (e) 9.



If the five-digit number 32N83 is divisible by 7, find N. (a) (b) (c) (d) (e)



10.



28 52 100 2015 None of the above



7 8 9 0 None of the above



The diagram shows parts of a city where the lines are roads. Charles walks from point X to point Y along the roads without ‘backtracking’ (i.e. he can only walk downwards or to the right in the diagram below). How many ways can he walk from X to Y? X



Y (a) (b) (c) (d) (e) 11.



54 55 56 57 None of the above



Find the smallest whole number n for which (a) (b) (c) (d) (e)



28 42 56 84 None of the above



3



336 is a factor of 360. n



SIMOC, Secondary 3 Contest 12.



The diagram shows a circle with centre O. ABC and EDC are straight lines. Given that AOE =  and BOD = , find ACE in terms of  and/or .



A B 



O







C



D



E (a) (b) (c) (d) (e) 13.



 𝛼+𝛽 2 𝛼−𝛽 2 𝛼+𝛽 3



None of the above



Alice throws a ball into the air. The path of the ball can be modelled by the equation h = t 2 + 4t + 1, where t, in seconds, is the time from the moment the ball is thrown, and h, in metres, is the height of the ball above the ground. Find the difference in height between the ball at its highest point and at the point from which it is thrown. (a) (b) (c) (d) (e)



14.



2 metres 3 metres 4 metres 5 metres None of the above



Given that x2 + y2 = 7 and x + y  xy = 2 + √3 , where x and y are real numbers, find the value of |x + y  1|. (a) (b) (c) (d) (e)



1  √3 1 + √3 2  √3 2 + √3 None of the above



4



SIMOC, Secondary 3 Contest 15.



Find the remainder when 22015 is divided by 5. (a) (b) (c) (d) (e)



0 1 2 3 4



5



SIMOC, Secondary 3 Contest Section B 16.



17.



18.



In the Fibonacci sequence, T1 = T2 = 1 and Tn = Tn1 + Tn2 for all positive integers n  3. Each of the first 2015 Fibonacci terms is written on an identical card. All the 2015 cards are then placed inside a box. Philip randomly picks up a card from the box. What is the probability that he will pick an odd number? Find the value of (8) + (8) + (8) + ⋯ + (8), where each (8) is a Binomial coefficient. 1



2



3



8



𝑟



The diagram shows a room in the shape of a cuboid of dimensions 6 metres by 4 metres by 3 metres. A spider is at the ceiling corner X. It crawls along the ceiling / wall / floor to reach the opposite floor corner Y. It so happens that the spider has moved along the shortest possible path. Find the length of this path, leaving your answer in surd form if necessary.



X 3m



Y 4m 6m



19.



The following is a conversation between Gabriel and Heather. Gabriel: Heather: Gabriel: Heather:



I thought of two distinct one-digit numbers. Can you guess the sum of these two numbers? No. Can you give me a clue? The last digit of the product of the two numbers is your house number. Now I know the sum of the two numbers.



So what is the sum of the two numbers?



6



SIMOC, Secondary 3 Contest 20.



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 )



, where h is the height



6



and r is the radius of the cap. r h



A cylindrical hole 12 cm long has been drilled straight through the centre of a solid sphere. What is the volume of the remaining solid? Leave your answer in terms of . (Note: Volume of cylinder =   radius2  height) 21.



In a chess tournament, each player has to play one game with every other player. 2n players, where n is a positive integer, took part in the tournament. So far, Player 1 has played 2n  1 games, Player 2 has played 2n  2 games, Player 3 has played 2n  3 games, and so on until Player 2n  1 has played 1 game. How many games has Player 2n played?



22.



Find the value of



23.



In the following cryptarithm, all the different letters stand for different digits. Find the 5-digit number SIXTY. F



O



+ S



24.



I



1 1



+



1 1+2



+



R



T



Y



T



E



N



T



E



N



X



T



Y



1 1+2+3



+. . . +



1



.



1+2+3+ ... + 2015



The following shows the top view and the front view of a three-dimensional structure. The structure does not have any curved surface and it does not have any painted lines on it. Notice also that there are no dotted (or hidden) lines. Draw the left or the right side view. (It does not matter which one you draw. Both answers will be accepted.)



Top View



Front View



7



SIMOC, Secondary 3 Contest 25.



Find the missing number in the grid below. (Hint: Four-digit perfect squares.) 9



9



8



1



0



4



4



3



3



2



6



3



5



9



1



8