Mathematics I Two Hours

G.C.E.(O.L.) Support Seminar - 2015

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Part A Answer all questions on this question paper itself.

1. If 250 rupees is paid as rates for a quarter for a certain house, find the value of the annual rates charged.

2. The circumference of a circle is 44 cm. Find the arc length of a semicircular portion of this circle.

3. Simplify :1 y

+ 2 y

4. The triangle ABC in the figure is an equilateral triangle. Find the value of x.

x°

A

B C

5. Simplify(√ 3 × √ 3

6. What is the actual length represented by 5 cm in a scale diagram drawn to the scale 1 : 100?

7. The first, second and third quartiles of a frequency distribution are respectively 6, 8 and 15. Find the inter-quartile range of this frequency distribution.

8. Write down the shaded region in the figure using set notation.

A B

ε

9. Simplify(5 a × a

10. In how many days can three men complete a task which can be completed in 18 man days?

[See page 2

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11. Solve(2(x + 5) = 6

12. BDE in the figure is a straight line. Find the magnitude of ACB,^ based on the information in the

figure. E

D

C

BA

112°

13. 1728 = 26 ×33. Find the value of (1728)13 .

14. The histogram represents information on the marks obtained by a group of students at an examination. Find the total number of students who faced the examination.

1

20 0 40 60 80 100

2

3

4

5

marks

num

ber

of s

tude

nts

15. It takes 20 minutes to fill a tank completely with water at a rate of 50 litres per minute. What is the capacity of the tank in cubic metres?

16. The figure denotes a circle with centre O. Find the values of a and b based on the information

in the figure.

a°

2b°

b°

S

O

R

Q

P

17. Write down a quadratic equation with roots +2 and −5.

[See page 3

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18. ABCD in the figure is a parallelogram. AE = EF. Based on the information in the figure, name

two triangles with area equal to exactly half the area of the parallelogram ABCD.

D C F

E

BA

19. Determine the value ofx :3 loga x + log

a 5 = log

a40

20. Show that the total outer surface area of a right cylindrical tin without a lid, of radius r units and height 2 r units is 5 πr 2 square units.

21. The figure depicts two circles with a common centre O. The chord AB of the larger circle of radius 17 cm, touches the smaller circle at C. If AB = 30 cm , find the radius of the smaller circle.

B

CO

A

22. Simplify1 ÷ 1.25 and express the answer as a decimal number.

23. In the figure, ABCD is a parallelogram and DXCY is a rhombus. If ABC =CXD^= 130°, find the

magnitude of ADY .^

130°

130°

B

C

Y

D

A

X

24. Fill in the blanks in the following expression.

(a + 2)3 = a3 + 6 .... + .... a + 8

[See page 4

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25. The value of an imported toy car is 350 rupees. After paying import duty its value is 420 rupees. Find the percentage charged as duty.

26. AD is the tangent drawn at A to the circle with centre O depicted in the figure. Find the value of x based on the information in the figure.

75°

40°

C

B

A

D

x°

O

27. Make r the subject of the formula V = 1 3 π r2 h.

28. Solve the inequality2 x + 3 > 8 and find the smallest integral value that x can take.

29. It is required to fix a post, 5 m from the boundary AB and 10 m from the corner A, within the plot of land ABCD depicted in the figure. Using the knowledge on loci, sketch how the location

where the post should be fixed is found.

A

D

B

C

10 m

20 m

30. The least common multiple of 6, 15 and x is 90. Find two values that x can take which are odd numbers.

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[See page 5

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Part BAnswer all questions on this question paper itself. Each question carries 10 marks.

1. In a garment factory,1— 3

of the employees serve in the production section, 1— 4

in the packing

section and 2— 5

of the remaining employees in the quality control section. The rest of the employees

serve in other sections.

(i) What fraction of the total number of employees serves in either the production section or the packing section?

(ii) What fraction of the total number of employees serves in the quality control section?

(iii) The management spends Rs. 300 000 in total to give a bonus of Rs. 5 000 to each employee in the production section. Find the total number of employees in the garment factory.

(iv) If a total of Rs. 207 000 is required to give a bonus to each employee in the packing section, show that the amount received as bonus by an employee in the packing section is less than the amount received as bonus by an employee in the production section.

2. The figure depicts a square shaped park of side length 28 m. A semicircular shaped pond is located within the park at one end. (Take π = 22—

7) 28 m

28 m

D

A B

C

(i) Find the perimeter of the portion of the park without the pond.

(ii) Find the area of the portion of the land in which the pond is located.

(iii) If it costs Rs. 60 to turf an area of 1 m2, show that Rs. 28 600 is sufficient to turf the area of the park apart from the land on which the pond is located.

(iv) The manager of the park plans to construct a rectangular shaped playground outside the park, of area equal to that of the turfed lawn, with AB as one boundary of the playground. Sketch this on the given figure together with the correct measurements.

[See page 6

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3. A mixture of plaster contains cement, lime and sand in the ratio 1 : 2 : 3.

(i) In how many pans of this mixture are there 24 pans of lime?

(ii) How many pans of sand are there in 48 pans of this mixture?

(iii) If a new mixture is made by adding 4 pans of cement and 8 pans of sand to a quantity of 48 pans of the above mixture, find the ratio of sand to lime to cement in the new mixture.

(iv) If four pans of this new mixture are required to plaster an area of 1 square metre of a wall, how many square metres of the wall can be plastered with the total mixture prepared in (iii) above.

4. A box contains 5 identical cards, each of which has exactly one of the numbers 1, 2, 3, 4, 5 written on it. A student randomly draws out one card from the box.

(i) Draw a tree diagram with the probabilities indicated on it, to show the events of drawing an odd numbered card or an even numbered card.

(ii) Without replacing the first card, another card is drawn randomly from the box. Extend the tree diagram to represent the events of the number on the second card being odd or even. (iii) Find the probability of one of the drawn cards being even and the other one being odd.

(iv) Represent the sample space of the above random

1

1 2 3 4 5

2

3

4

5

First Draw

Seco

nd D

raw

experiment of drawing the cards out from the box, on the given grid.

(v) If a two digit number is formed using the two cards that are drawn, find the probability of this number being one which is divisible by three.

[See page 7

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5. The following table provides information on the marks obtained by 60 students at an examination.

(i) Complete the table by filling in the blanks.

Marks Frequency Cumulative Frequency

0 − 10

10 − 20

20 − 30

30 − 40

40 − 50

50 − 60

3

.....

11

14

.....

7

3

9

.....

34

53

.....

(ii) Draw the cumulative frequency curve on the coordinate plane given below by selecting a

suitable scale based on the information in the table.

(iii) Find the median of the marks by using the cumulative frequency curve.

(iv) If 40% of the students failed this examination, using the cumulative frequency curve, determine the mark above which students were passed.

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Mathematics II Two Hours and Thirty Minutes

G.C.E.(O.L.) Support Seminar - 2015

Part AAnswer five questions only.

1. An owner of a certain building rents it out for Rs. 10 000 per month. He takes an advance of twelve months rent initially. He uses 25% of the advance money for maintenance of the building and deposits the remaining amount for a year in a financial institute such that he receives an annual interest of 12%. (a)(i) How much does the owner of the building receive as the advance? (ii) How much does he spend for maintenance of the building? (iii) Calculate the interest he receives at the end of the year by depositing the rest of the advance moneyinthefinancialinstitute.

(b) Instead of depositing the rest of the advance money in the financial institute, if the owner invests it to buy Rs. 20 shares at Rs. 18 per share in a company that pays an annual dividend of 12%, explain with reasons, which of these two investments would be the more profitable one.

2. An incomplete table of values suitable to draw the graph of the function y = x2 + 2x − 3 is given below. x -4 -3 -2 -1 0 1 2

y 5 0 -3 ... -3 0 5

(a) (i) Find the value of y when x = − 1. (ii) By taking 10 small divisions along the x axis and along the y axis to represent one unit as scale, draw the graph of the given function on a graph paper, using the values in the above table. (b)By considering the graph, (i) find the minimum value of the function. (ii) write down the interval of values of x for which the function takes values which are less than 2. (iii) Rearrange the function to take the form y = (x + a)2 + b and thereby deduce the equation of the axis of symmetry and the minimum value of the function y = (x - 3)2 + 2.

3. (a) Simplify( 1 - 2y x - y x2- y2

(b)The figure represents a lamina in the shape of a rhombus of side length (x + 3) cm. The perpendicular distance from D to AB is (x - 1) cm.

(x - 1) cm

(x + 3) cm A

D

B

C (i)Obtain an expression in terms of x for the area of the rhombus.

(ii) If the area of the rhombus is 9 cm2, show that x satisfies the equation x2 + 2x - 12 = 0

(iii)By completing the square or by some other method, solve the above equation and find the length of a side of the rhombus. (Take √13 =3.61)

[See page 2

✽ Answer ten questions by selecting five questions from part A and five questions from part B.✽Each question carries 10 marks.

✽ The volume of a right circular cone of base radius r and height h is 1 3

π r2h.

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11th Proof

[See page 3

4.(a) If x 2 0 4 1 3

A = 2 -1 5 6 3 y

, B = 2A - B = 4 5 -5 2 -1 9

and , find the values of x and y.

(b) The perimeter of a rectangle of length x + 1 units and breadth y units is 20 units. When the length of the rectangle is increased by x units while the breadth is kept unchanged, the perimeter of the new rectangle is 30 units. (i) Construct a pair of simultaneous equations to represent the above information. (ii) Find the values of x and y by solving the above constructed simultaneous equations. (iii) Calculate the area of the new rectangle.

5. (a) In the given figure, CD represents a vertical post fixed to

A C

D

B

a horizontal ground AB. The angle of elevation of D when observed from A is 35°. The distance from A to the foot of the post is 50 m.

(i) Copy the figure, onto your answer script and mark the given information on it.

(ii) Find the height of the post CD to the nearest metre. (iii) The length of a supporting wire connecting B to a point E on the post, 5m below D is 40 m. Find the magnitude of CBE^

'

(b) At the commencement of a race, the announcer observes the person who gives the starting signal at a distance of 80 m from him on a bearing of 030°, and he observes the judge at a

distance of 60 m from him on a bearing of 120°. (i) Draw a sketch to represent the above information. (ii) Using the sketch, find the distance between the person who gives the starting signal and the judge.

6. The following table provides information on the number of minutes that each employee of a certain organizationusedthemobilephoneduringthe8officehoursofaworkday.

Time spent on calls (Minutes)

Number of employees

(f)

0 - 6 6 - 1212 - 1818- 2424 - 3030 - 3636 - 4242 - 48

2261612741

(i) To which time interval does the amount of time spent on the mobile phone by the most number of employees fall?(ii) Find to the nearest minute, the mean amount of time that an employee spent on the mobile phone during office hours that day. (iii) Accordingly, how many hours could be expected to be spent on the mobile phone during office hours by all the employees of this organization during a month consisting of 22 working days?

(iv) The daily wage of an employee is Rs. 800. The Head of the Organization states that due to the use of mobile phones during office hours, the organization bears of a loss of more than Rs. 42 000 in a month which has 22 working days. Explain with reasons whether this

statement could be true.

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11th Proof

[See page 4

(i) f,dayoKafvysm˙udjπ, aiylweiqfrkaolajkak'

(ii) f,dayf.da,30ysm˙udjπ iy aweiqfrkaolajkak'

(iii) f,dayoKafvys†."f.da,hlwrhfuka10.=Khlanjby;m%;sM,weiqfrkafmkajkak'

Part B Answer five questions only.

7.(a) Thefigureshowsaportionofasquareshapedcushioncover.Apatternconsistingofsquaresof side length 3 cm, 4 cm, 5 cm etc., of increasing magnitude, has been drawn on this cushion cover. Amali plans to sew lace around each of these squares.

3cm

4cm5cm

(i) By taking that a strip of 12 cm of lace is required for the smallest square, write down in order, the length of each strip of lace required for the first three squares. (ii) Accordingly, find the length of the strip of lace required for the 8th square. (iii) For which square in this pattern is a strip of lace of length 56 cm required? (iv) The 18th square in this pattern is the largest square on the cushion cover. Amali says that a roll of 8.5 m of lace is sufficient for all the squares on the cushion cover. Explain with reasons whether you agree with this.

(b) Show that the 9th term of the geometric progression 25, 125, 625, . . . is510.

8. Use only a straight edge with a cm/mm scale and a pair of compasses to do the following constructions.

(i) Draw a straight line segment PQ such that PQ = 9 cm. (ii) Mark the point S such that QPS ^ = 60°and PS = 5.2 cm. (iii) Mark the point R which is at an equal distance from the straight lines PQ and PS and is 7 cm

from S, and complete the quadrilateral PQRS. (iv) Construct the circle that passes through the point Q and touches the line SR at R. Measure

and write down the radius of this circle.

9. The midpoint of the side BC of the triangle ABCinthefigureisD. The straight line drawn through D parallel to AC, meets AB at E. A

CDB

E

(i) Copy the figure, onto your answer script and mark the given information. (ii) Write down the theorem by which we can conclude that AE = EB.

(iii) Show that Δ BDE = 14 Δ ABC.

Suppose ED is produced to X such that ED = DX. (iv) Show that Δ BED ≡ Δ DXC'

(v) Give reasons why BECX is a parallelogram.

10. In the figure,ABCD is a cyclic quadrilateral. The tangent drawn to the circle at A, meets CD produced at E. E A

B

C

D

(i) Name an angle equal in magnitude to EAD .^

(ii) Show that EAC = ADE .^

(iii) If EAD = 45° and AD = DC, show that AC is a diameter of the circle.

(iv) Show that EA2= EC. ED'

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11th Proof

11. The side length of a solid cubic metal block is 2a cm. This metal block is heated and a solid right circular cone of base radius a cmand height 3a cmis made.

(i) Find the volume of the metal block in terms of a. (ii) Find the volume of the solid cone in terms of a. (iii) Show that the volume of metal that goes waste in the construction of the cone is a3 (8 - π) cm3 .

(iv) By taking a = 2.32 cm and (8 - π) = 4.858 , and using the table of logarithms, find the volume of the metal that goes waste.

12. Information on the results obtained by 80 employees of a certain private organization for Mathematics and English at the G.C.E. (O.L.) Examination is given in the following incomplete Venn diagram, distinguishing those who obtained ordinary passes and those who obtained a credit pass or above.

37

2

21

Passed EnglishPassed Mathematics

Obtained a credit pass or above

ω

5

(i) How many employees obtained a credit pass or above for English only? (ii) How many employees passed Mathematics but failed English?(iii) Describe the set represented by the shaded region in the Venn diagram. (iv) Those who have obtained a credit pass or above in both Mathematics and English are eligible

to follow a computer training course. The Head of the Organization claims that 75% of the employees are eligible. Providing reasons show whether this claim is true of false.

(v) If all those who obtained credit passes or above for English also obtained credit passes or above for Mathematics, show by drawing a new Venn diagram how the above Venn diagram should be changed to represent this information.

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