LIVE: FINAL EXAM PREPARATION PAPER 1 30 OCTOBER 2014

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1 LIVE: FINAL EXAM PREPARATION PAPER 0 OCTOBER 04 Lesson Descrition In this lesson we: Work through uestions from various Paer aers. Challenge Question Phili is designing large fish troughs in the shae of rectangular risms with an oen to as shown in the diagram below. The table shows some standard sized troughs and their surface areas. Oen at the to Fish Trough Surface Area [ + ()] + [ + 5()] + ( + )( + 5) 44 m [ + ()] + [ + 5 ()] + ( + )( + 5) 8 m [ + ()] + [ + 5()] + ( + )( + 5) m 4 [4 + (4)] + [4 + 5(4)] + (4 + )(4 + 5) 9 m 5 [5 + (5)] + [5 + 5(5)] + (5 + )(5 + 5) 60 m m A + B + C D m Find the value of A + B + C in terms of m. (b) (Place your answer into simlest form.) (5) Phili's fish troughs are all built using the surface area formula of A + B + C. What is the surface area of a trough so that the trough can hold 5 m of water if filled right to the to? (5) P a g e

2 Question Exam Questions Give the next two terms in the seuence, assuming that it remains consistent:,, 5 5, 7 (b) Solve for x:... () x x (x ) (correct to one decimal digit) (5) () () 8 x 5 0 x (4) () x () (c) (d) Check whether x is a root of the euation: x 7 x x + () Simlify: () (e) Determine the 60 th term of the geometric seuence, leaving your answer in exonential form , 5,,,... () 4 (f) Find n so that k 0 n k (g) Given: f(x) x x x + 40 Question 808 (7) () Use the Factor Theorem to fully factorise f(x). (5) () Determine f ' (). () Refer to the figure showing stacking of congruent triangles. Fig. Fig. Fig. Fig.4 P a g e

3 Qestion () Comlete the table: Figure 4 5 No. of sides 8 0 () Determine a formula for the number of sides in the n th figure. (7) () Hence determine which figure has 900 sides of triangles. (4) Refer to the figure. The largest triangle has an area of one suare unit. The biggest grey triangle has area 4 s. units () and each subseuent triangle's area is 4 the size of the triangle bigger than it. These triangles continue indefinitely. Determine the area of the unshaded art of the triangle Question 4 Determine dy x 6x 5x 4 for y dx x (b) Given: f(x) x 4x, find f ' () Question 5 If a + b 5 and c, what is the value of a + (b + c)? () (b) Given: f(x) 9 and g(x) x () State the range of each function. () () Determine the value(s) of x for which f(x) > g(x). () (4) P a g e

4 Question 6 Refer to the figure below showing the grahs of f(x) x and g(x) f (x). Calculate the average gradient of the curve of f between the oints A (where x ) and B (where x ). () (b) Give the values of x for which f ' (x) > 0. () (c) Give the euation of g in the form g(x) () Question 7 Given: f(x) x Draw the grahs of f and g. (b) Question 8 (b) and g(x) x 7 All the intercets with the axes and asymtotes should be clearly shown. (7) Use your grahs to determine the value(s) of x for which: () f(x) g(x) () () f(x) > g(x) () Clare needed R500 urgently. A 'loan shark' agreed to give it to her for one month but she would have to return R600 to him. () Determine the interest rate that he is charging for this one month loan. () () Show that if this monthly rate is comounded for months then it is euivalent to an effective annual ercentage rate of nearly 800%. (4) The owner of a small business decides that in one month's time he must start deositing R 000 er month into a sinking fund earning 0,5%.a. comounded monthly in order to be able to relace his ower generator. It is exected to cost R Calculate how many months it will take before he has sufficient funds. (8) P a g e 4

5 Question 9 There are 5 eole in a grou. The Venn-diagram below shows the number of eole who enjoy listening to radio (R), enjoy gardening (G) and/or enjoy cooking (C). There are eole who enjoy all three activities. There areeole who do not enjoy any of the activities. 9. If there are 8 eole who enjoy gardening, calculate the value of x () 9. Hence determine the value of y () Question 0 The 9 letters in the word CELLPHONE are each written on a card and rearranged. 0. How many different arrangements can be made if the reeated letters (E and L) are considered as different? () 0. Determine the robability that the two E s will be laced next to each other if the reeated letters are considered as different. () 0. How many different arrangements that start with the letter P can be made if the reeated letters are considered as the same? () P a g e 5

6 Answers Exam Questions Question 7 ; 9 9 A (b) () x x (x ) x M x 5x + 0 A x M 5 7 A 4,6 or 0,4 CA () 8 x 5 0 x x x M A x x x 65 x 5 M A () x x 8 x 5 x 5 M A (c) LHS ( ) 7 ( ) RHS ( ) A A x is NOT a root. CA P a g e 6

7 (d) 4 A M A (e) 0, 5, 5 5,, 4 G.P. a 0, r A T 60 ar A n (f) k 0 k n 0 M n A.P. a 7, d, S n n n 4 n 606 A 808 M n 7n A 7 n 76 or M A i.e. n 76 A P a g e 7

8 (g) f(x) x x x + 40 () f() +40 M x is a factor A f(x) (x )(x + x 0) M A (x )(x + 5)(x 4) A Question () () f(x) x x x + 40 f'(x) x x A f'() Figure 4 5 No. of sides M A () st diff.: A A nd diff.: M A Formula of form: T n an + bn + c a a T 0 0 T n n c 0 bn A A Question T Shaded area forms G.S. a r S 4 4 M + b 4 A A M 4 4 A P a g e 8

9 CA Unshaded art is s. units Question 4 a) y x 6x 5x 4 x x + 6x 5 + 4x M A dy x + 6 4x dx x x M A (b) f(x) x 4x Question 5 f'(x) x 4 A f' 4 M A () a + b + c 5 + M A (b) () f: Range: y 9 A g: Range: y 0 A Question 6 () f(x) > g(x) () Ave. Grad. 9 > x A < x < A 4 f () f ( ) M A P a g e 9

10 () x ε R A () g(x) log x M A Question 7 a.) (4) A'( ; ) B'(0; ) A A (5; 0) (0; -,5) y - x (0; -7) b.) () (; -4) and (; ) Question 8 () x < and < x < () Interest R00 00 M % er month A () + i (, ) M A 8,96 A i 7,96 79,6%.a. (nearly 800%.a.!) CA P a g e 0

11 (b) 0,05, A A Question 9 n x i i 9. x 8 9. y Question A n A M A (A) n 0,4508 M (A)n,45 n log A,45 i.e. 6 months A M 5, A CA 0. 9! n(e s next to each other) 8 x 7! X P ((E s next to each other) n(start with the letter P, reeated letters the same) 8!!.! P a g e