Tkgs 2026 Sec 4 T2 WA Paper Solutions with feedback

Mark scheme for Sec 4 Math WA2 2026 1 (a) Show that   3 4 8 3 6433 x y x y   . [1] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria     3 484 48 36 3 4 3 3 3 xy xy xy      For “show” questions, working must show use of Laws of Indices explicitly. Apply Law & Definition of Indices: •     mm n mn na a a  • () n n nab a b  (b) Hence, simplify   1 63 2484 8 813 xxy y             and leave your answer in positive index form. [2]   1 63 2484 8 3 36 4 10 813 93 27 xxy y xxy y y                          Apply Law & Definition of Indices: • n n n aa bb          • 1 ,0 n naa a   • m n m na a a   • m n m naaa   • 0 1, 0 aa  Overall Item Demand Content Complexity Context Response A01 Low Medium Low Low 2 Mandy deposited $150 000 in a 5 -year investment plan. For the first two years, it offers a simple interest rate of 4% per annum. After two years, the accumulated sum will enjoy a compound interest of 5% per annum, compounded quarterly for 3 years. Show that the amount of interest she will receive at the end of 5 years is about $38 000 (rounded to the nearest thousand). [4] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria Simple interest earned after 2 years $150000 4 2 100 $12000 I = = Accumulated sum after 2 years = $162 000 Total amount at the end of 5 years Apply simple interest formula 100 PRTI  with correct P, R and T to find simple interest. Adjust r and n according to frequency of calculating compound interest.

2 34 54$162000 1 100 $188042.2319  =+  = Total interest earned at the end of 5 years $188042.2319 $150 000 $38042.23 $38000 (nearest thousand)    Alternatively, Total interest earned at the end of 5 years ($188042.2319 $162000) $12000 $38042.23 $38000 (nearest thousand)    For “show” question, students should show value $38042 before rounding off to required accuracy. Apply compound interest formula 1 100 n rAP          to find total amount with compound interest. Find total interest by taking the difference of total amount and initial principal. Overall Item Demand Content Complexity Context Response A02 Medium Medium Medium Somewhat unfamiliar 3 The diagram shows the positions of the points X, P and Q where 4 2PQ =  is represented in the grid below. (a) Y is the point such that PQXY is a parallelogram. (i) Mark and label the point Y on the grid above. [1] (ii) Express PY  as a column vector. [1] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria (i) Mark Y using a cross. (ii) Since PQXY is a parallelogram, PY QX=  . 2 3PY −=  . Apply concept that vectors on opposite sides of a parallelogram are equal. Write down a vector in column vector notation. Draw point Y correctly to form parallelogram PQXY. Overall Item Demand Content Complexity Context Response x Y Q X P

A01 Low Low Low Low (c) Find QX  . Leave your answer in 2 significant figures. [1] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria 22( 2) 3 13 3.6 units (2 s.f.) QX = − + = =  Take note of accuracy required is 2 s.f. Find magnitude of a vector using 22x xyy  =+ Overall Item Demand Content Complexity Context Response A01 Low Low Low Low 4 (a) Two fair dice with sides numbered 1 to 6 are thrown at the same time. Find the probability that both dice show the same number. [1] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria P(both dice show same number) 6 1 1 1= or 6 6 6 6 6 1 6    = or P(both dice show same number) No. of combined outcomes with same no.= Total no. of combined outcomes 6 66 1 6 =  = Or use Possibility Diagram to visualize how to get answer. Die 1 1 2 3 4 5 6 Die 2 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 Can visualize using Tree Diagram but no need to show Tree Diagram No need to show Possibility Diagram for 1 mark question Apply Multiplication Law of Independent Events in Probability: P((1,1))= 11 66 Apply Addition Law of Mutually Exclusive Events in Probability: P((1,1) or (2,2) or (3,3) or…(6,6)) = P((1,1))+P((2,2))+P((3,3))+P((6,6)) = 11 6 66 or Since each combined outcome is equally likely (has same probability) to occur, we can use (favourable outcomes in )() (Sample space) nEPE n  or use Possibility Diagram Overall Item Demand Content Complexity Context Response

4 A02 Low Medium Low Medium 4 (b) The table below shows the results of a survey in which 60 students were asked whether they played video games every day during the school holidays and whether they had trouble sleeping at night. Played video games Daily Not daily Total Trouble sleeping 46 2 48 No trouble sleeping 8 4 12 Total 54 6 60 Two students were selected at random from the whole group of students. Find the probability that (i) at least one student had trouble sleeping at night. [2] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria (at least one student had trouble sleeping) 1 (none had trouble sleeping) 12 111 60 59 284 295 (at least one student had trouble sleeping) 12 48 48 12 48 47 60 59 60 59 60 59 284 295 Accept 0.963 (3 s.f.) P P OR P =− = −  = =  +  +  = for Real - World - Context question. Note there are 3 possible cases here: P(S’S) +P(SS’) +P(SS) where S rep the event that the student has trouble sleeping Apply ( ) 1 ( ')P E P E  and concepts of multiplying probabilities of dependent events (similar to multiplying probabilities along branches of Tree Diagram): ( ) ( ) ( given )P AB P A P A B  Overall Item Demand Content Complexity Context Response A02 Medium Medium Medium Medium (ii) neither of them played video games daily nor had trouble sleeping. [1] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria Let A and B rep event that a student played games daily and have trouble sleeping respectively. Apply ( ) ( ) ( given )P AB P A P A B 

. both r (neit her pl ' ayed games dail ) y nor had trouble sleeping) 43 60 59 1 295 ) f om ( o ' Accept 0.00339 (3 s.f. f r Real - World - Context question P AB= =  Overall Item Demand Content Complexity Context Response A02 Medium Medium Medium Taught 5 A little girl has 9 coins in her pocket. She has one 10-cent coin, two 50-cent coins and six $1 coins. To pay for a drink which costs $1.50, she randomly takes out a coin from her pocket. If the coin is a 10-cent coin, it is put back into the pocket. Then, she takes out the next coin. (a) Complete the Tree Diagram. [1] Solutions/Alternative Methods Remarks Skills/Concept/Success Criteria Compute probability and fill up Tree Diagram Overall Item Demand Content Complexity Context Response A02 Low Low Medium Low 1st coin 2nd coin 10-cent 10-cent 50-cent 50-cent 10-cent $1 1 8 5 8 ( 1 9 ) 62 93= 1 8 $1 1 9 2 9 62 93= ( 2 9 ) 50-cent 50-cent 10-cent $1 $1 1 8 63 84= 21 84=

6 (b) Given that the first coin is a 50-