MOHD NAZAN BIN KAMARUL ZAMAN SMK. KOTA KLIAS, BEAUFORT TEKNIK MENJAWAB MATEMATIK SPM 2010 SETS 3 marks 1. The Venn diagram in the answer space shows sets, P, Q and set R such that the universal set On the diagrams in the answer space, shade R Q P = R Q) (P' ) Q P a) bP R Q P R Q Q P a) P R Q Firstly label each part at the diagram with numbers or letters 2 5 3 I 4 P = 1, 2, 3, 4 Q = 3, 4, 5 P Q = 3, 4 R Q) (P' ) bP R Q Firstly mark all the area at the diagram with numbers or letters 2 5 3 I 4 P = 1, 2, 3, 4 P= 5 Q = 3, 4, 5 P Q = 5 R = 2, 3 5 3, 2, R Q) (P' = Elimination method SIMULTANEOUS LINEAR EQUATIONS Substitution method Matrix method 4 MARKS 2 Calculate the value of d and of e that satisfy the following simultaneous linear equations: 8d 9e = 5 2d 3e = 1 (i) (ii) (ii) x 4 2d(4) 3e(4) = -1(4) 8d 12e = - 4 (iii) 8d = 32 0 3e = -9 (iii) (i) 339= =eeSubstitute e = 3 to (i) 8d 9(3) = 5 8d 27 = 5 8d = 5 + 27 8d 9e = 5 (i) 4832==dd1 mark 1 mark 2 marks 3 e and 4 d = = Using matrices 8d 9e = 5 2d 3e = 1 ||.|

\|=||.|

\|||.|

\|1 -5 ed 3 - 29 - 83 4158 29 361158 29 3) 9 ( 2 ) 3 ( 81 1= = ||.|

\|||.|

\|=||.|

\|||.|

\|||.|

\| =||.|

\|== e and dededC A B C B A1 mark 2 marks 1 mark 2 Calculate the value of d and of e that satisfy the following simultaneous linear equations: 27 2 3331= += e de dUsing matrices ||.|

\|=||.|

\||||.|

\| 2732 3311ed6 52731 3312312731 3312) 3 (31) 2 ( 11 1= = ||.|

\||||.|

\|=||.|

\|||.|

\||||.|

\||.|

\| =||.|

\|== e and dededC A B C B A1 mark 2 marks 1 mark QUADRATIC EQUATIONS - General Form - Factorisation 4 Marks 3. Solve the quadratic equation x73 2x2=+Change to general form ( )0 3 x 7 2xx 7 3 x 27 x 3 2xx73 2x2222= + = + = +=+21 x and 3 x = =1 2x 0 1 - 2x 0 3 - x = = =( ) 0 1 - 2x ) 3 - (x =1 mark 2 marks 1 mark MATRICES NOTES 1. When the matrix has no inverse 2. MATRIX FORM 3. Formula of the inverse matrix 4. State the value of x and of y 4. The inverse matrix of ||.|

\|||.|

\|2 m3 - 7 k1 is7 43 2a) Find the value of m and of k b) Write the following simultaneous linear equations as matrix equation : 2x + 3y = - 1 4x + 7y = 5 Hence, using matrix method, calculate the value of x and of y ||.|

\| =||.|

\| ||.|

\| =||.|

\| =||.|

\| 7 43 22123 217 43 212 1417 43 2) 4 ( 3 ) 7 ( 21 1 a)m ka c b dbc adk = 2 and m = - 4 b) Write the following simultaneous linear equations as matrix equation : 2x + 3y = - 1 4x + 7y = 5 Hence, using matrix method, calculate the value of x and of y 7 11711512 43 721517 43 2 1= = ||.|

\| =||.|

\|||.|

\|||.|

\| =||.|

\|== ||.|

\|=||.|

\|||.|

\|y and xyxyxC A B C B A yx1 mark 2 marks 1 mark THE STRAIGHT LINE 6 MARKS REMEMBER : 1. Gradient 2. Equation of a line 2 12 1x x y ym=c mx y + =1 = +byax3. Parallel lines , same gradient 4. Perpendicular lines , the product of their gradients = - 1 2 1 m m =12 1 = m my R(4,12) Q P(3, -6) 0 Diagram 3 5. In Diagram 3,OPQR is parallelogram and O is the origin. Find (a) the equation of the straight line PQ, (b) the y-intercept of the straight line QR x a) mPQ = mRO mRO = 34120 40 12x xy y1 21 2===mPQ = mRO = 3 m = 3 and P(3, -6) y = mx + c -6 = 3(3) + c -6 = 9 + c -6 9 = c - 15 = c m = 3 and c = -15 y = mx + c y = 3x - 15 b) mQR = mOP = 2360 30 6x xy y1 21 2 == =m = - 2 and R(4, 12) y = mx + c 12 = - 2(4) + c 12 = - 8 + c 12 + 8 = c 20 = c y-intercept of the straight line QR = 20 GRADIENT AND AREA UNDER A GRAPH 6. In the diagram, OPQ is the distance-time graph of a car traveling from town A to town B. The straight line RPS represents the distance-time graph of a van traveling from town B to town A 0 t 5 6 144 250 Distance from A (km) Time(hrs) P Q R S Calculate the a) average speed, in km h-1 , of the car from town A to B b) value of t if the van travelled at uniform speed. a) Average speed = timedistance total4240== 60 km h-1 b) t144 80t = 144 t = 1.2 = 80 LINES AND PLANES IN 3 DIMENSION a) Line and Plane b) Plane and plane 7. Diagram 10 shows a right prism. Right angled triangle SUT is the uniform cross-section of the prism U Q S T P R 5 cm 12 cm 20 cm Identify and calculate the angle between the plane PSR and the plane PUTR. Using open & close method S P U T R 5 cm i. Identify the plane PSR and the plane PUTR. ii. open the plane PSR and the plane PUTR. P R T U S Identify three points when we joint together become a straight line. The straight line is SPU or UPS , so the angle between the plane PRS and the plane PUTR is Z SPU or Z UPS S U P 12 20 Z SPU ' 0 057 30 @ 96 . 302012 tan== u D F E G 7228. The diagram shows a solid formed by combining a right pyramid with a half cylinder on the rectangular plane DEFG. DE = 7 cm, EF = 10 cm and the height of the pyramid is 9 cm. Clculate the volume, in cm3, of the solid. [ using t = Volume of pyramid + volume of half cylinder 210 9 x 10 x 7 x 31 height x base of Area x 31 pyramid of volume===192.5 10 x 27 x 722 x 21 height x r x 21 cylinder half of volume22=|.|

\|== tVolume of the combine solid = 402.5 CIRCLES : Perimeter and Area 1. Use the correct formulae 2. Substitute with the correct values. 9. In Diagram, BC and AD are archs of two different circle which have the same cente O It is given that a) The perimeter, in cm of the whole diagram b) The area, in cm of the shaded region calculate ,722 Using14cm OB , 30 OBC and 90 OAED0 0== = Z = ZtO C B A E D 7cm 32427317 7317 147 77222360607 1472223603014)=+ + + + =+|.|

\| ++|.|

\| + =+ + + + = DO ED CE BC OB perimeter a61646512315132257722360301472236030772236060)22 2= + =|.|

\| |.|

\| +|.|

\| = + = OAE OBC ODE area bPROBABILITY ) () () (S n A nA P =10. Diagram 9 shows two boxes , P and Q . Box P contains four cards labeled with letters and box Q contains three cards labeled with numbers. T S E B 7 6 4 Two cards are picked at random, a card from box P and another card from box Q . a) List the sample space and the outcomes of the events . b) Hence , find the probability that (i) a card labeled with letter E and a card labelled with an even number are picked (ii) a card lebelled with letter E or a card labelled with an even number are picked P Q a) {(B, 4), (B, 6), (B, 7), (E, 4), (E, 6), (E, 7), (S, 4), (S, 6), (S, 7), (T, 4), (T, 6), (T, 7)} Notes : 1. Accept 8 correct listings for 1 mark b) i) {(E, 4), (E, 6)} ii) {(E, 4), (E, 6), (E, 7), (B, 4), (B, 6), (S, 4), (S, 6), (T, 4), (T, 6)} 61@12243@1291(m) 1(m) 2(m) 1(m) 1(m) MATHEMATICAL REASORNING a) State whether each of the following statement is true or false 12 > 5 and 14 72=It is a false statement b) Write down Premise 2 to complete the following argument Premise 1 : If x is greater than zero, then x is a positive number Premise 2 : __________________________________________ Conclusion : 6 is a positive number 6 is greater than 0 c) Make a general conclusion by induction for the sequence of number 7, 14, 27,.which follows the following pattern. ... .......... ....3 ) 2 ( 3 272 ) 2 ( 3 141 ) 2 ( 3 7321= + = + = + =3 (2) n+ n n = 1, 2, 3,