Mock Eam Mock Eam Section A. Reference: HKDSE Math M 0 Q (a) ( + ) ( - ) n nn ( ) ( + + ) n + + Coefficient of - n - n -7 n (b) Coefficient of nn ( - ) - n + (- ) - () + (). Reference: HKDSE Math M PP Q6 d + h ( ) lim d h 0 h lim h 0 + h + h + h + h + + h lim h 0 h ( + h + ) lim h 0 + h + (). Reference: HKDSE Math M PP Q (a) + cot θ cot θ + cos θ sin θ cos θ + sin θ cos θ sin θ cosθ (b) ( + )( ) ( ) + + Since is real, we let cot θ for some θ. ( + )( ) cos θ (by (a)) + - cos θ - cos θ ( + )( ) The least value of + is -. () Hong Kong Educational Publishing Company
Mathematics: Mock Eam Papers Module (Etended Part) Second Edition Solution Guide. Reference: HKDSE Math M 0 Q (a) f () k(e k ) + ()e k ke k + e k f () k(ke k + e k ) + ke k k e k + ke k (b) f () - kf () + f () k e k + ke k k(ke k + e k ) + e k (by (a)) ( - k )e k Since f () - kf () + f () 0 for all real, we have - k 0 k ±. Reference: HKDSE Math M 0 Q (a) Let tan θ. Then d sec θ dθ. d sec θ dθ + 9 9sec θ dθ θ + C + C tan, where C is a constant () (b) y d + 9 + 9 9 d + 9 9 + d 9 9 + C a tan (by( )) tan + C Since Γ passes through the point (0, 6), we have C 6. The equation of Γ is y tan + 6. (7) 6. Reference: HKCEE A. Math 006 Q (a) From the graph, we have a < < 0 < < b b b < < c f () 0 + 0 The -coordinate of the maimum point is b. The -coordinate of the minimum point is 0. Hong Kong Educational Publishing Company
Mock Eam b (b) (i) f ( ) d Area of R 0 f( b) f (0) 6 f( b) 6 f( b) 7 (ii) Area of R Area of R 0 f ( ) d f ( ) d a c f( a) f(0) f( b) f( c) f( a) 7 f( a) f( a) b (7) 7. Reference: HKDSE Math M PP Q The homogeneous system of linear equations has non-trivial solutions if and only if k k 0 + 9 ()(-k)(-9) + ()(-)() + (k)()() - ()(-k)(k) - ()(-)() - (-9)()() 0 9k - 6 + k + k + 0 + 7 0 k + k + 0 (k + )(k + ) 0 k - or k - (). Reference: HKDSE Math M 0 Q6 (a) Note that A T A and -A (-) A - A. From A T -A, we have A T -A. Therefore, A - A. So, we have A 0. A 0 For n n matri A and constant k, we have ka k n A. (b) (i) I - M 0 y 0 z y z 0 (I - M) T -(I - M) By (a), we have I - M 0. (ii) Note that I - M - -M - (I - M). I - M - -M - I - M -M - (0) (by (b)(i)) 0 I - M - is a singular matri. The claim is agreed. (6) Hong Kong Educational Publishing Company
Mathematics: Mock Eam Papers Module (Etended Part) Second Edition Solution Guide 9. (a) Since R divides PQ in the ratio :, OP + OQ OR + i + j + (i j) 0i j OR 0 + ( ) 0 0i j The required unit vector 0 i 6 6 j (b) + + ( i j) i j ()cos ROP 6 6 0 7 cos ROP 6 6 9 cos ROP 6 7 ROP ( cor. to the nearest degree) () Section B 0. Reference: HKDSE Math M SP Q (a) (i) AB 0 0 0 BA 0 0 0 (ii) From (a)(i), AB BA -I, where I is the identity matri. A B B A I A - B 0 () Hong Kong Educational Publishing Company
Mock Eam (b) (i) ACA 6 0 0 0 0 (ii) det (ACA - ) ()()() det A det C det A - det C det (AA - ) det C 0 C is invertible. (iii) By (b)(i), D 0 0. D 0 0 0 0 0 0 0 0 D - is a diagonal matri. 06 ( D ) 0 0 0 0 T 06 Hong Kong Educational Publishing Company
Mathematics: Mock Eam Papers Module (Etended Part) Second Edition Solution Guide Note that D - (ACA - ) - (A - ) - (AC) - AC - A -. (D - ) 06 A(C - ) 06 A - 06 (C - ) 06 A ( D ) A 0 06 06 0 06 ( ) 06 06 06 06 06 06 0 06 06 06 (7). Reference: HKDSE Math M PP Q (a) (i) OP ( roa ) + rob ( r)( i k) + r(i + j + k) (r ) i + rj + (r ) k OQ ( soc ) + sod ( s)( i + j k) + s( i + j + k) (s ) i + ( ) s j + (s ) k PQ OQ OP (s r) i + ( s r) j + (s r) k (ii) AB OB OA i + j + k PQ AB 0 [(s - r)i + ( - s - r)j + (s - r)k] (i + j + k) 0 (s - 6r) + (6 - s - r) + (s - r) 0 6 + s - r 0 s r -... () CD OD OC i j + k PQ CD 0 [(s - r)i + ( - s - r)j + (s - r)k] (i - j + k) 0 (s - r) + (-6 + s + r) + (6s - r) 0 s - r - 6 0 s - r - 0... () 6 Hong Kong Educational Publishing Company
Mock Eam Substituting () into (), (r - ) - r - 0 6r - 0 r s PQ (s r) i + ( s r) j + (s r) k i + j + k The shortest distance between the straight lines AB and CD PQ ( ) + + () (b) (i) AC OC OA j AB AC (i + j + k) () j 6i + k (ii) Since RD ( AB AC), let RD tab ( AC), where t is a constant. RD 6ti + tk AD OD OA i + j + k AR AD + DR (i + j + k) ( 6ti + tk) ( + 6) t i + j + ( t) k AR ( AB AC) 0 [( + 6 t) i + j + ( ) t k] ( 6i + k) 0 ( 6) t + ( t) 0 t OR OD + DR 6 ( i + j + k) + i k i + j + k Hence the coordinates of R are,, () Since AR lies on the plane ABC, AR ( AB AC). We have AR ( AB AC) 0. 7 Hong Kong Educational Publishing Company
Mathematics: Mock Eam Papers Module (Etended Part) Second Edition Solution Guide. (a) When 0, y 0. When y 0, 0. The -intercept is 0 and the y-intercept is 0. () (b) f ( ) ( + )( ) ( ) 6 + ( + ) ( + ) ( + )(6 + ) ( + ) ( + )( ) f ( ) ( + ) 6 6 6 ( + )[( + )( + ) ( + )] ( + ) 6 ( + )( + 6 + ) ( ) ( + ) 6 6 ( + ) (c) (i) When f () 0, 6 + 0 ( + ) 0 0 or - 0 or - () Note that f (0) 0, the second derivative test cannot be applied. < - - < < < < 0 0 > 0 f () + 0 - - 0 + f (0) 0 f (-) - The maimum point is,, the minimum point is (0, 0). When f () 0, ( - ) 0 0 or < < 0 < < > f () + 0 + 0 - (ii) f ( ) The point of infleion is,. The denominator + is 0 when -. The vertical asymptote is -. Hong Kong Educational Publishing Company
Mock Eam f( ) + + + + Note that lim 0 + The oblique asymptote is y. (iii) y y +,, for the shape of the graph for the asymptotes for all correct (). Reference: HKDSE Math M PP Q (a) The volume of the solid of revolution π + h π y ydy + h π + h [( ) ] π ( h + h + 6 6) π h + h ( ) () (b) (i) By (a), V V + π h + h ( ) for 0 h, where V cm is the capacity of the frustum and h H - for H. Differentiating both sides with respect to t, dv π + h dh dh When the depth of milk is 7 cm, i.e. h, π dh + () π dh () dh 7 π dh The rate of increase of the depth of milk is 7π cm s -. 9 Hong Kong Educational Publishing Company
Mathematics: Mock Eam Papers Module (Etended Part) Second Edition Solution Guide (ii) Let r cm and l cm be the lengths as shown in the figure. By similar triangles, l l + l l + l By similar triangles, r + H H r V π l + π r l + H ( )( ) ( ) π π H ( )() π π H () ( ) π π H () () π H (iii) The volume of the upper portion π [ + ()] cm 6π cm ( H ) The volume of the milk that leaks out after 0 seconds π 0 cm 6π cm > 6π cm The depth of the remaining milk is less than cm. 0 Hong Kong Educational Publishing Company
Mock Eam 6π+ V π π π H 6 6π+ π H π π H 6 π 09π 09 H 9 π H H H 9 Differentiating both sides with respect to t, dv π H 0 H 6π dh dh After 0 seconds, π π 9 6 dh dh 9 0. ( cor. to sig. fig. ) The rate of decrease of the depth of milk is 0. cm s -. () Hong Kong Educational Publishing Company