7/ Modul Peningkatan Prestasi Matematik Tambahan (Kertas ) SPM 6 Zon B Kuching Sarawak y y P Substitute () into () * * y * y y ( ) ( ) ()( ) () ( ) ( ) ()( ) y ().,. y.66,.66 Note:. If the solutions of and y are matched wrongly, then SS- from full marks. 5 (a) Let w f ( ) Then f ( w) w w w f ( ) (Shown) 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
7/ (i) gf ( ) g 5.5 (ii) gh( ) f ( ) h( ) h( ) h( ) 8 6 (a) π R () 56π R a 6 r S 6 56 85 85. 6 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
7/ (a) Equation of the curve y 6 d c () () c c y At turning point, dy d 6, When, When, y () () y () () Second derivative, d d y 6 6 When, When, d y 6() 6 d d y 6() 6 d 6 6,, is the maimum point. is the minimum point. 8 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
7/ 5 (a) tan A tan B sin A sin B cos A cos B sin Acos B cos Asin B cos Acos B sin( A B) cos Acos B (i) Shape of sine graph P Amplitude = P.5 cycle f π P (ii) y equivalent π P Crect gradient crect y -intercept Number of solutions = 8 y y sin O π π π π 5π π y π 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
5 7/ 6 (a) cos 6 cos 8.9 8.9 96.8 96.8. 8.68 rad Area of maj sect POR 6. *.68 Area of triangle POR 6 6 sin*96.8 Area of immersed cross-sectional region 6. *.68 66sin* 96.8 7.5 cm 7 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
6 7/ 7 (a) T (5) () () ( ),, (shown) Area of QOR 5 8 9 unit (c) m ( ) 5 m m y y (d) ( ) ( y ) 8 6 y y 9 y 8 y 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
7 7/ 8 (a) A B,, Area under the curve from to * A ( ) d * d * (*) (*) 6 Area of the region required by the question * 6 * * (c) Volume generated π y d π d π 8 6 d π 6 5 5 5 5 () 6() () 6() π () () 5 5 56 π 5 7 π 5 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
8 7/ 9 (a) Range 5 5.5 5.5 (i) Midpoints 5.5, 5.5, 5.5, 5.5, 5.5 P Mean, (5.5) 8(5.5) (5.5) (5.5) 7(5.5) 8 7 8.5 Variance, (5.5) 8(5.5) (5.5) (5.5) 7(5.5) 8 7 8.5 8 8.5 (ii) New variance * P (c) The height of the bars are proptional to respective frequencies P Boundaries midpoints class intervals are crectly labeled (bars are of the same width) P Crect way of finding mode Mode.5 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
9 7/ f 8 6.5.5.5.5.5 5.5 6.5 mode.5 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
7/ (a) n p 5 q 5 P (i) (ii) P( X 8) P( X 8) P( X 9) P( X ) 8 9 C8 C9 C 5 5 5 5 5 5.668.5786.857.9 P( X 5) P Z 5 6 8 P Z.5 P Z.5.9 P Z.5.668 Number of students.9 7 P( X a). P Z a 6 8. a 6.8 8 a.8 8 6 The minimum marks to get grade A, a 7.5 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
7/ (a) 5 6 7 log y.79.8.68.98.8.58.75.75.675.975.75.575 Note:. In the table, the decimal number being rounded off must not be less precise than the nearest.5 f Y.. The N mark is given f all crect X and Y.. If the value(s) of X and Y are not shown through a table, then the N mark is given f all the crect points on the graph. At least one *point crect, using crect aes log y against in crect direction and ientation, and with unifm scales All 6 *points plotted accdingly Line of best fit (c) log y log a log b ( equivalent) P (i) log a log y -intercept *.75 *.75 a.985 Note:.985 a.6 (ii) log b m.7.6 7.... b.995 Notes:. Graph can only be read with precision until half of the smallest grid division.. Do not accept readings obtained by calculation from the wrong graph.. SS- from full marks if not using the given scale. 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
7/ log y.5. 7.,.7.5..5. c.75.5.,.6 5 6 7 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
7/ (a) a dv dt t At maimum velocity, t t a. vma 6 9 8 6 8 6.5 At initial acceleration, t. (c) a () When stopping instantaneously, v. t t 6 t t t, t t t Distance from point O v dt 6 t t t () () 6() 6 8 8.667 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
7/ (a) (i) PR 7 8 (7)(8)cos75 PR 7.5 (ii) *7.5 8.5 sin sin QPR QPR sin 8.5sin 7.5 7.9 75' Note: 7.8 QPR 7.9 QRP 8 *7.9.9 Area of Area of PQR PRS (8.5)(7.5)sin.9 (7)(8)sin75 Area of quadrilateral PQRS (8.5)(7.5)sin.9 (7)(8)sin 75 5.758679 6.8576.6 Note:.6 Area.6 (c) Let the shtest distance from point Q sin.9 h 8.5 OR h 8.5sin.9 to the line PR be h. (7.5) h * (8.5)(7.5)sin.9 5.787 Note: 5.787 h 5.788 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
5 7/ (a) I: y II: y III: y y At least one straight line crect from the *inequalities involving and y Draw all three lines crectly from the *inequalities Shade the crect region (c) (i) When, 7.5 y ymin 8 Note: N if answer ymin 7.5 (ii) Objective function 5 5y k Optimum point, kma 5 5 RM 78 Notes:. F (c)(i) and (c)(ii), accept the codinates only if they are integers.. SS- if in (a), symbol "<", ">" is used instead of " ", " " me than three inequalities are given (ecept y ) OR in, not using the given scale aes are reversed not using graph paper 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah
6 7/ y y 5 y 5 5 R 5 5y k ma y, 5 55y 875 5 6 7 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak
7 7/ 5 (a) 7.5 6. 5 y 5. y 6. 7.7 z z 5.5 I 8(5) (5) 6(5) () 8 6 8 K.8 (c) (i) Food.8 6. Food RM 79.68 (ii).85 I 6, equivalent I6, 5.7 7/ (C) 6 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah