maths p1... · web view©2017 nairobi school f4 april holiday assignment paper 1 and paper 2 name:...

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Name: ....................................................................................... AdmNo: ...................Class: ...............................

121/1MATHEMATICS ALT APaper 12½ hours

Nairobi SchoolFORM FOUR APRIL HOLIDAY ASSIGNMENT

Kenya Certificate of Secondary Education (K.C.S.E.)MATHEMATICS ALT A

Paper 12½ hours

Instructions to candidates(a) Write your name and index number in the spaces provided above.(b) Sign in the spaces provided above.(c) This paper consists of TWO sections: Section I and Section II.(d) Answer ALL the questions in Section I in Section II.(e) All answers and working must be written on the question paper in the spaces provided below each

question.(f) Show all the steps in your calculations, giving your answers at each stage in the spaces below each

question.(g) Marks may be given for correct working even if the answer is wrong.(h) Non – programmable silent electronic calculators and KNEC Mathematical tables may be used except

where stated otherwise.(i) This paper consists of 16 printed pages.(j) Candidates should check the question papers to ascertain that all the pages are printed as indicated and

that no questions are missing.

For Examiner’s Use Only

Section I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Total

Section II

17 18 19 20 21 22 23 24 Total

©2017 NAIROBI SCHOOL F4 APRIL HOLIDAY ASSIGNMENT PAPER 1 AND PAPER 2

Grand

Total

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SECTION I (50 marks)

Answer all the questions in this section in the spaces provided.

1. Use tables of reciprocal only to evaluate

10 .325 , hence evaluate ;

3√0.0001250.325 (3 marks)

2. Solve the equation 3 x2+4 x=2giving the roots correct to two decimal places. (4 marks)

3. The straight line through the points D (6, 3) and E (3, -2) meets the y-axis at the point F. Determine the coordinates of F.

(3 marks)


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4. Using the grid provided below, draw and shade the unwanted regions to show the region satisfied by R given the following inequalities; y+x<5 , y−x≤1 and x+5 y>5

(3 marks)

5. Given that a=−2 , b=−1and c=3 , evaluate

2 ( a+c )2−(a−b ) (b−c )−2c3 (a+b )−2 (b−c ) (3

marks)


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6. A key cutting factory starts cutting keys at the rate of 500 per hour. The rate of production reduces by 10% every 2 hours. Calculate the numbers of keys cut in the first 6 hours.

(2 marks)

7. Two boys and a girl shared some money .The elder boy got 49 of it, the younger boy got

25 of the

reminder and the girl got the rest. Find the percentage share of the younger boy to the girl’s share.

(3 marks)

8. Annette has some money in two denominations only. Fifty shilling notes and twenty shilling coins. She has three times as many fifty shilling notes as twenty shilling coins. If altogether she has sh. 3400, find the number of fifty shilling notes and 20 shilling coins.

(3 marks)


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9. The figure below shows a rhombus PQRS with PQ = 9cm and ∠SPQ=600. SXQ is a circular arc; centre P.

Calculate the area of the shaded region correct to two decimal places. (Takeπ= 227 ) (4 marks)

10. A particle accelerates uniformly from rest and attains a maximum velocity of 30m/s after 16 seconds. It travels at this constant velocity for the next 20 seconds before decelerating to rest after another 8 seconds. Calculate the total distance covered by the car.

(3 marks)


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11. The figure below shows a right angled triangle with AB = 12cm. ED is parallel to BA, CE = 6.3cm and ED = 8cm.

Given that the area of EBAD = 31.5cm2, find the length of BC (4 marks)

12. Find the value of x in the equation 5

x4 = 1

25 (3 marks)

13. Given thattan x=2. 4 , evaluate without use of tables and calculators, sin x−cos x in the form of ab where a and b are integers. (4 marks)


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14. The difference between the interior and exterior angles at each vertex of a regular polygon is1620

. Find the number of sides of the polygon.(2 marks)

15. A number n is such that when it is divided by 27 and 30 or 45, the remainder is always 3. Find the smallest value of n. (2 marks)


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16. The figure below shows a regular tetrahedron PQRS of edges 4cm. Draw its net and measure the length of the straight path of PS through the midpoint T over the edge QR. (3 marks)


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SECTION II (50marks)

Answer only five questions in this section in the spaces provided.

17. Three business partners, Bela, Joan and Trinity contributed Kshs. 112, 000, Kshs. 128, 000 and Kshs. 210, 000 respectively to start a business. They agreed to share their profits as follows:

30% to be shared equally30% to be shared in the ratio of their contributions40% to be retained for the running of the business.

If at the end of the year, the business realised a profit of Kshs. 1. 35millionCalculate:(a) the amount of money retained for running the business at the end of the year. (1mark)

(b) the difference between the amounts received by Trinity and Bela. (6marks)

(c) Express Joan’s share as percentage of the total amount of money shared between the three partners. (3 marks)


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18. In the figure below, O1 and O2 are the centres of the circles whose radii are 4 cm and 7 cm respectively. The circles intersect at A and B and angle AO1O2 = 60˚

Find by calculation; take π = 3.142

(a) The angle AO2O1 (1 marks

(b) The area of the quadrilateral AO1BO2 (4 marks)

(c) The shaded area (5 marks)


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19. The figure below shows a plan of a roof with a rectangular base ABCD. AB = 20cm and BC = 12cm. the ridge PQ = 8cm and is centrally placed. The faces ADP and BCQ are equilateral triangles. N is the midpoint of AD

Calculate:

(a) The length of PN (2 marks)

(b) The altitude of P above the base. (3 marks)

(c) The angle between the planes ABQP and ABCD. (2 marks)

(d) The obtuse angle between the lines PQ and DB (3 marks)


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20. Complete the table below for the function y=x3+6 x2+8 x for −5≤x≤1 (3 marks)

x −5 −4 −3 −2 −1 0 1

x3 −125 −64 −1 0 1

6 x2 54 6 08 x −40 −24 −16 0 8y 0 3 0 15

(a) Draw the graph of the function y = x3 + 6 x2 + 8x for −5≤ × ≤ 1 (3 marks)(Use a scale of 1cm to represent 1 unit on the x axis. 1 cm to represent 5 units on the y-axis)

(b) Hence use your graph to estimate the roots of the equation

x3 + 5 x2 + 4 x = −x2 − 3 x − 1 (4 marks)


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21. Three islands P, Q, R and S are on at ocean such that island Q is 400kmon a bearing of 0300 from island P. Island R is 520kmand on a bearing of 1200from island Q. A port S is sighted 750km due south of island Q.

(a) Taking a scale of 1cm to represent 100km, give a scale drawing showing the relative positions of P, Q, R and S. (4 marks)

Use the scale drawing to(b) Find the bearing of:

(i) Island R from islandP (1 mark)

(ii) Port S from island R (1 mark)

(c) Find the distance between island P and R (2 marks)

(d) A warship T is such that it is equidistant from the islands P, S and R. By construction locate the position of T (2 marks)


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22. Two vertical columns A and B of height h and 2h respectively stand on level ground and are 100m apart. Two points P and Q are d metres apart, the elevation of the top of A and B from point P are 350 and 280 respectively and the elevation of top of B from point Q is 750

Calculate

(a) The vertical heights of the two columns in metres (7 marks)

(b) The distance PQ in metres (3 marks)


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23. In the figure below, E is the midpoint of AB, OD : DB = 2 : 3 and F is the point of intersection of OE and AD

Given that OA = a and OB = b,(a) Express in terms of a and b

(i) AD (1 mark)

(ii) OE (2 marks)

(b) Given further that AF = sAD and OF = tOE, find the values of s and t (5 marks)

(c) Show that E, F and O are collinear (2 marks)


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24. A swimming pool is 20m by 12m and it slopes gently from a depth of 1m at the shallow end to a depth of 3m at the deep end.

(a) Calculate the volume of water in the swimming pool (in m3) when it is full. (3marks)

(b) If the swimming pool is to be drained by a pump which removes water at the rate of 2.5m3 per minute, how long will it take this pump to drain the swimming pool if it was full? (3marks)

(c) If the sides of the swimming pool and its floor are to be covered with square tiles of side 20cm, find to the nearest 100 the number of tiles required. (4marks)


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Name: ....................................................................................... AdmNo: ...................Class: ...............................

121/1MATHEMATICS ALT APaper 22½ hours

Nairobi SchoolFORM FOUR APRIL HOLIDAY ASSIGNMENT

Kenya Certificate of Secondary Education (K.C.S.E.)MATHEMATICS ALT A

Paper 22½ hours

Instructions to candidates(k) Write your name and index number in the spaces provided above.(l) Sign in the spaces provided above.(m) This paper consists of TWO sections: Section I and Section II.(n) Answer ALL the questions in Section I in Section II.(o) All answers and working must be written on the question paper in the spaces provided below each

question.(p) Show all the steps in your calculations, giving your answers at each stage in the spaces below each

question.(q) Marks may be given for correct working even if the answer is wrong.(r) Non – programmable silent electronic calculators and KNEC Mathematical tables may be used except

where stated otherwise.(s) This paper consists of 16 printed pages.(t) Candidates should check the question papers to ascertain that all the pages are printed as indicated and

that no questions are missing.

For Examiner’s Use Only

Section I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Total

Section II

17 18 19 20 21 22 23 24 Total


Grand

Total

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SECTION I(50 Marks)Answer all the questions in this section in the spaces provided

1. Use logarithms to evaluate: (3 marks)

(36 . 15×0 . 27531. 9382 )

2. Given thata2 x2+6 ax+k is a perfect square, findk . (3 marks)

3. Makeh the subject of the formula. (3 marks)

E=1−∏ √ h−0 .51−h


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4. Given that P varies directly as V and inversely as the cube of R and that P = 12 when V = 3 and R = 2, (i) Find an equation connecting P, V and R. (3 marks)

(ii) Find the value of V when P = 10 and R = 1.5 (1 mark)

5. The figure below shows a toy which consists of a conical top and a hemispherical base.

The hemispherical base has a radius of 5cm and the total height of the toy is 17cm. calculate the volume of the toy. (Take π = 3.142) (3

marks)


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6. a. Find the inverse of the matrix

(3−4 ¿ )¿¿

¿¿(1 mark)

b. Hence solve the following simultaneous equations using matrix method (3 marks)3 x−4 y=23 x+5 y=13

7. The first term of an arithmetic sequence is (2 x+1 ) and the common difference is( x+1 ) . If the product of the first and the second terms is zero, find the first three terms of the two possible sequences. (4 marks)


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8. Solve for x in the equation log 5−2+ log (2x+10 )=log ( x−4 ) (3 marks)

9. (a) Expand (1+ 15 x )4 (1

marks)

(b) Use the first three terms of the expansion in (a) to find the approximate value of (0.98)4

(2 marks)

10. Draw a line DF=4.6cm.Construct the locus of point K above DF such that angle DKF = 700.(3 marks)


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11. Machine A can complete a piece of work in 6 hours while machine B can complete the same work in 10 hours. If both machines start working together and machine A breaks down after 2 hours, how long will it take machine B to complete the rest of the work? (3 marks)

12. Evaluate ∫−1

2(2 x2−3 x−14 ) dx (3mks)

13. The base and perpendicular height of a triangle measured to the nearest centimeter are 6cm and 4cm respectively. Find(a) The absolute error in calculating the area of the triangle. (2 marks)

(b) The percentage error in the area giving the answer to 1 decimal place. (1 mark)


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14. Given that

xx+2 y

=38 , find the ratio x:y (2 marks)

15. Complete the table below for the function y=3 x2−8 x+10

x 0 2 4 6 8 10y 10 6 26 138

Hence estimate the area bounded by the curve y=3 x2−8 x+10and the linesy=0 , x=0 and x=10 using trapezoidal rule with 5 strips. (3marks)

16. If

13−√5

−2+2√53+√5

=a+b√c , find the value of a, b and c (3

marks)


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SECTION II (50 MARKS)Answer only FIVE questions in this section in the spaces provided.

17. The table below shows the Kenya tax rates in a year

Income (Ksh per annum) Tax rate (per £)1 – 116,160 10%116,161 – 225,600 15%225,601 – 335,040 20%335,041 – 444,480 25%Over 444,481 30%

In that year, Ushuru earned a basic salary of Ksh 30000 per month. In addition, he was entitled to a medical allowance of Ksh 2,800 per month and a traveling allowance of Ksh 1800 per month. He is housed by the employer and pays a nominal rent of 2000. He also claimed a monthly family relief of Ksh 1056. Other monthly deductions were union dues Ksh 445, WCPS Ksh 490, NHIF Ksh 320, COOP shares Ksh 1000 and risk fund Ksh 100Calculate:(a) Ushuru’s annual taxable income. (2marks)

(b) The tax paid by Ushuru in that year (5marks)

(c) Ushuru’s net income in that year (3marks)


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18. (a)Complete the table for the function y=12 Sin2 x , where00≤x≤3600

(2 marks)x 00 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600

2 x 00 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200

Sin2x 00 0 .866 00 0 .866 −0 .866y= 1

2 Sin2 x 00 0 . 433 00

(b) On the grid provided, draw the graph of the function y= 12 Sin2 x for 0

0≤x≤3600using the

scale 1cm for 300 on the horizontal axis and 4cm for 1 unit of y axis. (3 marks)

(c) Use your graph to determine the amplitude and period of the functiony= 12 Sin2 x (2

marks)

(b) Use the graph to solve

(i)12 Sin2 x0=0 (1 mark)

(ii)12 Sin2 x0−0 . 5=0 (2 marks)


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19. The following are marks out of 100 scored by 40 learners in a Mathematics contest.Marks 40 – 49 50 – 59 60 – 69 70 – 79 80 - 89 90 – 99No. of learners 4 6 8 12 8 2

(a) (i) Using an assumed mean of 64.5, calculate the standard deviation of the data. (5marks)

b. On the grid provided, draw a cumulative frequency curve. (3 marks)

From your graph, determine;

(i) The median (1mark)

(ii) The interquartile range (1mark)


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20. A triangle ABC with vertices at A (1,-1) B (3,-1) and C (1, 3) is mapped onto triangle A1B1C1 by a

transformation whose matrix is

(−10 ¿ ) ¿¿

¿¿

Triangle A1B1C1 is then mapped onto A11B11C11 with vertices at A11 (2, 2) B11 (6, 2) and C11 (2,-6) by a second transformation.(i) Find the coordinates of A1B1C1 (3 marks)

(ii) Find the matrix which maps A1B1C1 onto A11B11C11. (2 marks)

(iii) Determine the ratio of the area of triangle A1B1C1 to triangle A11B11C11. (3 marks)

(iv) Find the transformation matrix which maps A11B11C11 onto ABC (2 marks)


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21. In a form 2 class 23 are boys and the rest are girls.

45 of the boys and

910 of the girls are right handed;

the rest are left handed. The probability that a right handed student will answer a question correctly

is 110 and the corresponding probability for a left handed student is

310 irrespective of the sex.

By use of tree diagram; Determine(a) The probability that a student chosen at random from the class is left handed. (5 marks)

(b) Given that getting a boy or a girl at any stage in a family of three children is equally likely; (i) Use the letters B and G to show the possibility space for all families with three children

(1mark)

(ii) Using the possibility space calculate the probability that a family of three children has at least one girl. (2marks)

(iii) The oldest and the youngest are of the same sex. (2marks)


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22. In the figure below O is the centre of the circle. DEF is a straight line. FCX is a tangent at C.∠DCX=600 ,∠AFD=50 and∠ABC=850

. FCX is the tangent to the circle and ∠BAF=100

B A

D O E G F

C X

a) Find the sizes of the following angles giving reasons.(i) ∠DFC (3

marks)

(ii) ∠DAF (2 marks)

(iii) ∠OCB(2 marks)

b) If GF is 10 cm and the radius of the circle is 7 cm.Calculate GF (3 marks)


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23. An aeroplane that moves at a constant speed of 600 knots flies from town A (140N, 300W)

southwards to town B (X0S, 300W) taking 312 hrs. It then changes direction and flies along latitude

to town C (X0S, 600E). Given π=3 . 142 and radius of the earth R= 6370 km(a) Calculate

(i) The value of X (3 marks)

(ii) The distance between town B and town C along the parallel of latitude in km. (2 marks)

(b) D is an airport situated at (50N, 1200W), calculate(i) The time the aeroplane would take to fly from C to D following a great circle through the

South Pole. (3 marks)

(ii) The local time at D when the local time at A is 12.20 p.m (2 marks)


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24. A businessman wants to buy machines that make plastic chairs. There are two types of machines that can make these chairs, type A and type B. Type A makes 120 chairs a day, occupies 20 m2of space and is operated by 5 men. Type B makes 80 chairs a day, occupies 24 m2of space and is operated by 3men.The businessman has 200m2 of space and 40 men.(a) List all inequalities representing the above information given that the business man

buys x machines of type A and y machines of type B. (3marks)

(b) Represent the inequalities above on the grid provided. (3marks)

(c) Using your graph find the number of machines of type A and those of type B that the business

man should buy to maximize the daily chair production. (2marks)


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(d) Given that the price of a chair isKsh.250, determine the maximum daily sales the businessman

can make. (2marks)


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