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Additional math 1999

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MUSAKILAVE

This document contains a 14 question mathematics exam with questions covering a range of topics including: - Matrix algebra and inverses - Integration using substitution - Graphing functions - Probability - Arithmetic and geometric progressions - Complex numbers - Linear transformations - Logical statements The exam consists of two sections - Section A contains 8 short answer questions, and Section B contains 6 multi-part questions where students must show working for partial or full marks. Overall the exam covers a wide breadth of mathematical concepts and techniques.

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042 SECTION A (56 marks) Answer ALL questions in this section. ALL working for each question must be shown clearly. ·""· 1 . (a) Evaluate f nh sin ( rt - x ) dx by substitution method. 0 2 (4 marks) (b) Find the equation of a curve which passes through the point (1 ,0) whose gradient function is 3x2 - 1 ;;2 2. (a) Find the inverse of the matrix A= [1 2 J2 - 1 (3 marks) (3 marks) (b) Solve the following system of equations by using the inverse of matrix A obtained in (a) above 2y + X= 10 2x-y 5 3. (a) Write 3 + i in standard form 4- 3i (b) Give that z = i (3 + 4i), fmd I z 1. (c) Express 1 + i in polar fomi (2 marks) (3 marks) (2 marks) 4. (a) If n(A) = 60, n(B) = 50, n(C) = 1 8, n (AnB)= 30, n(BnC) = 10, n(Anq= 6, n(An BnC)= 4 Find n(Au BUC) (3 marks) (b) In a class of 34 girls, 21 play tennis and 1 8 play netball. If all the girls play at least one of these games, how many play both netball and tennis? :-,.. {i (a) If y= 1 + sin 8 , show that _1_ cos 8 y (b) If s = sin 8 and c = cos 8 simplify: (i) (ii) - 6. (a) Given that f(x)= s {-2 for x > 1 1 for x = 1 3 for x < 1 ·f-- 1- sin 8 Cos 8 2 (4 marks) (3 marks) (4 marks) Find this and other free resources at: http://maktaba.tetea.org
042 (i) (ii) Sketch the graph of this function . Evaluate f(- 5) (b) Solve for t, 13-2tl < 5 (4 marks) (3marks) 7. (a) Find the values of a and b if the expression x4 +ax3 + bx2- 4 is exactly divisible by x2-9. (4 marks) (b) Determine the factors of the expression x2- 2x- 3. (3 marks) 8. (a) The probability that Halima will pass Mathematics is 0.4 and that Perpetua will do the same is 0.7. Find the probability that: (i) both will pass Mathematics (ii) Halima or Perpetua will pass Mathematics. (4 marks) (b) Find x in (x] = 3when (n J= · _n_! _ 2 r (n-r)!r! SECTION B (44 marks) (3marks) Answer ANY FOUR (4) questions from this section. ALL workings for each question answered must be shown clearly. v 9. (a) The 4th, 6th and 9th terms of an arithmetic progression forms the first three terms of a geometric progression. Determine the common ratio of the geometric progression. (7 marks) (b) Determine the equation of the tangent to the circle x2 + y2-4x + 6y- 77 = 0 when the tangent passes through the point of tangency at (5,6). (4 marks IO (a) Show that z3 + 2z- 3 = (z-.1) (� + 2 + 3)and hence fmd the zeros of f(z) = z3 + 2z = 3(5 marks) (b) The complex number z has modulus 4 and argument rr_. Find the modulus and argument of 4 (i) (ii) (6 marks) II. (a) Show that the equation of the normal at the point P(p, L) on the curve xy = I is 3 _ I 4py-p X- -p 3 p (5 marks) Find this and other free resources at: http://maktaba.tetea.org
042 .2X.L(, tlfX ---_Ct-! &�·t-c�qK . ( -:· 9,(_ ��I (b) The diagram above shows the normal specified in (a), intersects the line y = x at Q (i) Find in a simplified form interms of p, the coordinates of Q (ii) Find the equation of the circle with centre P and radius OP, where 0 is the origin. (iii) Verify that the circle obtained in (iii) passes through Q (6 marks) 12. Carpentry section of one of the technical schools in Tanzania make tables and chairs. The section has two departments. Assembly and Finishing departments. For a particular order from a customer only 48 hours of work are available in the Assembly and 36 hours in the Finishing departments. To manufacture one table it requires 3 hours in the Assembly and 3 hours in the Finishing department; while a chair requires 4 hours in Assembly and 2 hours in Finishing department. If chair fetches Tshs.400 as profit and a table Tshs. 400/=, determine the best combination of tables and chairs to produce so as to get maximum profit. ...../ 13. (a) Solve for x (4 x+J) (16x) = 83x (11 marks) (4 marks) (b) A linear transformation T sends any point p(x, y) into (x', y1) such that X 1 = 2X + 2y y1 = 2x + 2y (i) Find the matrix which represents the transformation T. (ii)Find the image of the point (3,- 2) under linear transformation T. (4 marks) (o) Givrn that B � [�+1 y - 0 i.a 'ingulmnatrix,fmd po,iblo v•lu" of y. (3 muks) v 14. (a) Show that (pA q) --+ r :: (p--+ r) V(q -H). (5 inarks) (b) Let p denote "It is cold" and let q denote "It rains". Write the following statements in symbolic form. (i) It rains only if it is cold. (ii) It is not cold or it rains. (iii) It never rains when it is cold. (3 marks) (c) Sketch an electrical circuit to represent the following statement. (p A q) V (r As) ).J( +"Lt;j ·= J6 L-0 JL �Jx �� ?._-. (_ �.:' '<-6-3� � <_)?( -r 'J ( J G -j(') ,__ "-{d: "--2� 3 K t- 2} J C � 3rrJ� 'I �- 4 (3 marks) -(Lp- -- )- 7 �� Lj � ]£-W (__ ('1. ....i_ L Find this and other free resources at: http://maktaba.tetea.org

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Additional math 1999

  • 1. 042 SECTION A (56 marks) Answer ALL questions in this section. ALL working for each question must be shown clearly. ·""· 1 . (a) Evaluate f nh sin ( rt - x ) dx by substitution method. 0 2 (4 marks) (b) Find the equation of a curve which passes through the point (1 ,0) whose gradient function is 3x2 - 1 ;;2 2. (a) Find the inverse of the matrix A= [1 2 J2 - 1 (3 marks) (3 marks) (b) Solve the following system of equations by using the inverse of matrix A obtained in (a) above 2y + X= 10 2x-y 5 3. (a) Write 3 + i in standard form 4- 3i (b) Give that z = i (3 + 4i), fmd I z 1. (c) Express 1 + i in polar fomi (2 marks) (3 marks) (2 marks) 4. (a) If n(A) = 60, n(B) = 50, n(C) = 1 8, n (AnB)= 30, n(BnC) = 10, n(Anq= 6, n(An BnC)= 4 Find n(Au BUC) (3 marks) (b) In a class of 34 girls, 21 play tennis and 1 8 play netball. If all the girls play at least one of these games, how many play both netball and tennis? :-,.. {i (a) If y= 1 + sin 8 , show that _1_ cos 8 y (b) If s = sin 8 and c = cos 8 simplify: (i) (ii) - 6. (a) Given that f(x)= s {-2 for x > 1 1 for x = 1 3 for x < 1 ·f-- 1- sin 8 Cos 8 2 (4 marks) (3 marks) (4 marks) Find this and other free resources at: http://maktaba.tetea.org
  • 2. 042 (i) (ii) Sketch the graph of this function . Evaluate f(- 5) (b) Solve for t, 13-2tl < 5 (4 marks) (3marks) 7. (a) Find the values of a and b if the expression x4 +ax3 + bx2- 4 is exactly divisible by x2-9. (4 marks) (b) Determine the factors of the expression x2- 2x- 3. (3 marks) 8. (a) The probability that Halima will pass Mathematics is 0.4 and that Perpetua will do the same is 0.7. Find the probability that: (i) both will pass Mathematics (ii) Halima or Perpetua will pass Mathematics. (4 marks) (b) Find x in (x] = 3when (n J= · _n_! _ 2 r (n-r)!r! SECTION B (44 marks) (3marks) Answer ANY FOUR (4) questions from this section. ALL workings for each question answered must be shown clearly. v 9. (a) The 4th, 6th and 9th terms of an arithmetic progression forms the first three terms of a geometric progression. Determine the common ratio of the geometric progression. (7 marks) (b) Determine the equation of the tangent to the circle x2 + y2-4x + 6y- 77 = 0 when the tangent passes through the point of tangency at (5,6). (4 marks IO (a) Show that z3 + 2z- 3 = (z-.1) (� + 2 + 3)and hence fmd the zeros of f(z) = z3 + 2z = 3(5 marks) (b) The complex number z has modulus 4 and argument rr_. Find the modulus and argument of 4 (i) (ii) (6 marks) II. (a) Show that the equation of the normal at the point P(p, L) on the curve xy = I is 3 _ I 4py-p X- -p 3 p (5 marks) Find this and other free resources at: http://maktaba.tetea.org
  • 3. 042 .2X.L(, tlfX ---_Ct-! &�·t-c�qK . ( -:· 9,(_ ��I (b) The diagram above shows the normal specified in (a), intersects the line y = x at Q (i) Find in a simplified form interms of p, the coordinates of Q (ii) Find the equation of the circle with centre P and radius OP, where 0 is the origin. (iii) Verify that the circle obtained in (iii) passes through Q (6 marks) 12. Carpentry section of one of the technical schools in Tanzania make tables and chairs. The section has two departments. Assembly and Finishing departments. For a particular order from a customer only 48 hours of work are available in the Assembly and 36 hours in the Finishing departments. To manufacture one table it requires 3 hours in the Assembly and 3 hours in the Finishing department; while a chair requires 4 hours in Assembly and 2 hours in Finishing department. If chair fetches Tshs.400 as profit and a table Tshs. 400/=, determine the best combination of tables and chairs to produce so as to get maximum profit. ...../ 13. (a) Solve for x (4 x+J) (16x) = 83x (11 marks) (4 marks) (b) A linear transformation T sends any point p(x, y) into (x', y1) such that X 1 = 2X + 2y y1 = 2x + 2y (i) Find the matrix which represents the transformation T. (ii)Find the image of the point (3,- 2) under linear transformation T. (4 marks) (o) Givrn that B � [�+1 y - 0 i.a 'ingulmnatrix,fmd po,iblo v•lu" of y. (3 muks) v 14. (a) Show that (pA q) --+ r :: (p--+ r) V(q -H). (5 inarks) (b) Let p denote "It is cold" and let q denote "It rains". Write the following statements in symbolic form. (i) It rains only if it is cold. (ii) It is not cold or it rains. (iii) It never rains when it is cold. (3 marks) (c) Sketch an electrical circuit to represent the following statement. (p A q) V (r As) ).J( +"Lt;j ·= J6 L-0 JL �Jx �� ?._-. (_ �.:' '<-6-3� � <_)?( -r 'J ( J G -j(') ,__ "-{d: "--2� 3 K t- 2} J C � 3rrJ� 'I �- 4 (3 marks) -(Lp- -- )- 7 �� Lj � ]£-W (__ ('1. ....i_ L Find this and other free resources at: http://maktaba.tetea.org