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SUMMATIVE ASSESSMENT- I 2015-16 Class – XMATHEMATICS

Time allowed: 3 hours Maximum Marks: 90

General Instructions: 1. All questions are compulsory. 2. The question paper consists of 31 questions divided in to four sections A,B,C and D. Section-A comprises of 4 questions of 1 mark each; section-B comprise of 6 questions of 2 marks each; sections-C comprise of 10 questions of 3 marks each and Section –D comprises of 11 questions of 4 marks each. 3.There is no overall choice in this question paper. 4. Use of calculator is not permitted.

Section-A

Question number 1 to 4 carry one mark each

1. In PQR� ,S and T are points in the sides PQ and PR respectively such that ST QR . If PS =4 cm,

PQ=9 cm and PR = 4.5 cm, then find PT.

2. Find the valor of cos +sec ,when it is given that cos =1

2

3. If 3 sin ,find the valor of sin .tan .(1 cot

sin cos

.

4. Find the sum of upper limit and lower limit of the class interval in which the 20th observation of

the following data lies:

Class

interval

0-100 100-200 200-300 300-400 400-500 500-600 600-700

Frequency 5 7 6 3 20 4 8

SECTION-B

Question number 5 to 10 carry two marks each.

5. Find the prime factorization of the denominator of the rational number equivalent to 8.39. 2

6. Show that 5 6 is an irrational number 2

7. Find the quadratic polynomial whose zeroes are 2 +3 and 2 -3. 2

8. State which of the two triangles given in the figure are similar. Alsip state the similarity

criterion used.


9. Prove that :1

tan cot 21 1

2sec A1 sinA 1 sinA

10. Determine missing frequency x, from the following data, when Mode is 67.

Class 40-50 50-60 60-70 70-80 80-90

Frequency 5 X 15 12 7

Questions number 11 to 20 carry three marks each.

11. Use Euclid division lemma to show that square of any positive integer cannot be of the from

5m+2 or 5m +3 for some integer m.

12. A man has certain note of denomination 20 and 5 which amount to 380 . If the number of notes

of each kind are interchanged ,they amount to 60 less than before . Find the number of notes of

each denomination.

13. Divide the polynomial 4 3 23x 5x 4x +10x-2 by the polynomial 3x -2x and verify the division

algorithm.

14. Show graphically the following pair of linear equations if inconsistent: 3

2x-2y-2=0

3x-3y+5=0

15. ABC� and EBC� are in the same base BC.If AE priduced intersects BC at D then ,prove that

ar( ABC) AD

ar( EBC) ED

�

�

16. In a ABC� , AD is perperdicular to BC and 2AD =BD xCD,Prove that ABC is a right angled

Triangle.

17. 0 0 2 0 2 0

0 0 0 0. 0

sec sosec(90 tan ,sot(90 ) sin 55 sin 35

tan10 .tan 20 .tan 60 , tan 70 tan80

18. Prove that:

2

sin A cosecA + 2

cosA secA =7+ 2 2tan A cot A

19. The following data gives the information on the observed life times (in hours) of 150 electrical

components:

Life time (in

hours)

0-20 20-40 40-60 60-80 80-100

Frequency 15 10 35 50 40


Find the mode of the distribution.

20. The weekly pocket money of the students of class ix of a school are given in the following table:

Pocket

money (in)

0-40 40-80 80-120 120-160 160-200 200-240

Number of

students

5 7 15 10 5 8

Find the median for the above data.

Section-D

Question numbers 21 to 31 carry four marks each.

21. Can the number n6 ,n being a natural number ,end with the digit 5? Give reasons.

22. Draw the graph of the following pair of linear equations:

X+3y=6 and 2x-3y=12

Find the ratio of the areas of the two triangles formed by first line, x=0,y=0 and second line

x=0,y=0

23. Obtain all other zeroes of the polynomial 4 3x 3 2x + 23x 3 2 -x if two of its zeroes are 2 and

2 2 .

24. Mr. Sharma and Mr. Arora are family friends and they decided to go for a trip with family .

For the trio they reserved their rail tickets . Mr. Arora has not taken a half ticket for his child

who is 6 years old where as Mr, Sharma has taken half tickets for his two children who are 65

years and 8 years old . A railway half ticket costs half of the full fare but the reservation charges

are the same as in a full ticket . Mr. and Mrs. Arora paid 1700 ,while Mr. and Mrs. Sharma paid

2700. Find the full fare of one ticket and the reservation charges per ticket what difference you

find in their behavior and which one you will choose for youself?

25. In the given figure ,ABC is a triangle and GHED is a rectangle. BC=12 cm, HE =6cm, FC=BF and

altitude AF= 24 cm. Find the area of the rectangle.

26. “In a triangle if square of one side is equal to the sum of the squres of the other two jsides, then

the angle opposite the first side is a right angle’. Prove it.

27. If = 030 , verify the following:

i) Cos 3 =4 3cos -3 cos ,

ii) Sin 3 =3 sin -4 3sin


28. Prove that:

2(sec tan =cosec 1

cosec 1

29. (cose -sin )((cose -sin )=sin cos =1

tan cot

30. The daily income of 150 families if given below . Calculate the arithmetic mean.

Income No. of families

More than75

More than85

More than95

More than105

More than115

More than125

More than135

More than145

150

140

115

95

70

60

40

25

31. The following table gives the daily income of 50 workers of a factory .draw both types(“less than

type’ and’ greater than type’)ogives

Daily income(in) 100-120 120-140 140-160 160-180 180-200

Number of workers 12 14 8 6 10s


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GYAN SAGAR PUBLI SCHOOL SUMMATIVE ASSESSMENT-I, 2015-16

CLASS-X, MATHERMATYICS V9Y3QA1 Time allowed: 3 hours Maximum Marks: 90

General Instructions: 1. All questions are compulsory. 2. The question paper consists of 31 questions divided into four sections A, B, C and D. Section-A comprises of 4 question of 1 mark each; Section-B comprises of 6 question of 2 marks each; Section-C comprises of 3 marks each and Section- D comprises of 11 question of 4 marks each. 3. There is no overall choice in this question paper. 4. Use of calculator is not permitted.

Section A

Question number 1 to 4 carry one mark each

Q.1 In DEW, AB||EW. If AD=4 cm, DE=12 cm and DW=24 cm, then find the value of DB.

Q.2 In a ABC, write tan A B

2

in terms of angle C.

Q.3 If 3 sin =cos , find the value of 23cos 2cos

3cos 2

Q.4 If the mode of the data: 3, 5, 8, 9, 8, 12, 7, 12 and x is 8, find the value of x.

Section B

Question number 5 to 10 carry two mark each

Q.5 Prove that 5 2 is an irrational number

Q.6 Use Euclid division algorithm to find if the following pair of numbers is co-prime : 121, 573

Q.7 On dividing 2 2x 3x x 3, by a polynomial g(x), the quotient and the remainder were 2x x 1 and 2x 5 respectively. Find g(x).

Q.8 R and S are points on the sides DE and EF respectively of a DEF such that ER=5 cm, RD=2.5

cm, SD=1.5 cm and FS=3.5 cm. Find whether RS||DF or not.

Q.9 Express sinA and cosA in terms of cotA

Q.10 Given below is a frequency distribution table showing daily income of 50 workers of a

factory:



Daily income of

Workers (in rs)

200-205 250-300 300-350 350-400 400-450

Number of

workers

60 10 12 08 14

Change this tabel to a ‘less than type’ cumulative frequency table.

Section C

Question number 11 to 20 carry three mark each

Q.11 During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32

each. If you want full packs of both both and the same number of pencils and crayons, how

many of each would you need to buy?

Q.12 Solve the following pair of linear equations by the cross multiplication method:

x 2y 2

x 3y 7

Q.13 Find the zeros of the polynomial 3x 7x 6 .

Q.14 Check graphically whether the following pair of linear equations is consistent. If yes, solve it

graphically:

2x 5y 0 , x y 0

Q.15 Prove that area of the equilateral triangle described on the side of a square is half of the area

of the equilateral triangle described on its diagonal.

Q.16 In the figure of ABC, D divides CA in the ration 4 : 3 If DE||BC, then find ar (BCDE) : ar (

ABC )

Q.17 If b cos a , then prove that b a

cosec cotb a

Q.18 Prove the identity:2 2

22

cos tan 1tan

sin

Q.19 In a study on asthmatic patients, the following frequency distribution was obtained. Find the

average (mean) age at the detection.

Age at detection (in

years)

0-9 10-19 20-29 30-39 40-49

Number of patients 12 25 13 10 5



Q.20 Find the mean and median for the following data:

Class 0-4 4-8 8-12 12-16 16-20

Frequency 3 5 9 5 3

Section D

Question number 21 to 31 carry four mark each

Q.21 Show that 2n 1 is divisible by 8, if n is an odd positive integer.

Q.22 A boat goes 30 km upstream and 20km downstream in 7 hours. In 6 hours, it can go 18 km

upstream and 30 km downstream. Determine the speed of the stream and that of the boat in

still water.

Q.23 Find the values of a and b so that 4 3 2x x 8x ax b is divisible by

2x 1 .

Q.24 The ratio of income of two persons A and B are in the ration 3:4 and the ratio of their

expenditures is 5:7 If their saving are Rs.15000 annually, find their annual incomes. What

value will be promoted if expenditure is under control?

Q.25 In ABC, from A and B altitudes AD and BE are drawn. Prove that ADC BEC. Is and ADB ADC ?

Q.26 The sides of triangle are 30 cm, 70 cm and 80 cm. If an altitude is dropped upon the side of

length 80 cm, then find the length of larger segment cut off on this side.

Q.27 If cos(A+B)=0 and cot(A-B)= 3 , then evaluate :

(i) cosA. cosB – sinA. sinB

(ii) cos tB cot A

cot A cotB+1

Q.28 If m = cosA – sinA and n = cosA + sinA, show that

2 2

2 2

m 1

2m

n

- n

secA. cosecA =

(cos tA tanA)

2

Q.29 If seca

msec

and seca

ncosec

, show that 2 2 2 2m n n cos ec .

Q.30 Find the median and mode of the following data and then find the mean from the empirical

relationship between them :

Class interval Frequency



0-20

20-40

40-60

60-80

80-100

100-120

120-140

6

8

10

12

6

5

3

Q.31 Following distribution give the marks obtained, out of 200, by the students of Class IX in their

class test:

Find the mean and mode of the data.

marks 0-25 25-50 50-75 75-100 100-125 125-150 150-175 175-200

Number of

students

10 15 22 30 28 27 12 6



SUMMATIVE-ASSESSMENT-1 2015-16 SUBJECT –MATHEMATICS

CLASS-X zzdr-130 Time allowed: 3 hours Maximum Marks: 90

General Instructions: 1. All questions are compulsory 2. The question paper consists of 31 questions divided in to four sections A,B,C and D 3. section –A comprises of 4 questions of 1 mark each, Section -B comprises of 4 questions of 2 mark each. Section -c comprises of 4 questions of 3 mark each Section -D comprises of 4 questions of 4 mark each 4.use of calculator is not permitted. 5. An additional 15 minuts time has been allotted to read this question paper only.

SECTION A

Directions: 4 question of 1 mark each

1. Find a quadratic polynomial having zeroes as 3

and

2

2. Write the formula for the mid-point of a class interval .

3. If Sin A=3

45 ,calculate cos A

4. Given an example of a pair of similar fugures.

SECTION-B

Directions: 6 questions of 2marks each.

5. Find the zeroes of 2t -15 and verify the relationship between the zeroes and Coefficients

6. Determine whether the following system of linear equation has a unique solution, no

solution or infinitely many solution.

4x-5y=3

8x-10y=6

7. In the given figure DE//BC. Find Ec



8. Sin 2A=2 sin A is true when A=?

(a) 00

(b) 030

(c) 045

(d) 060

9. Express sin 067 +cos 075 in terms of trigonometric rations of angles between 00 and 045

10. Find mode of the given distribution

Family size 1-3 3-5 5-7 7-9 9-11

No. of families 7 8 2 2 1

Or

The following table gives the literacy rates (in percentage) of 35 cities. Find the mean literacy

rate.

Literacy rate (in %) number of cities

45-55 3

55-65 10

65-75 11

75-85 8

85-95 3

SECTION –C

Directions: 10 QUESTIONS OF 3MARKS EACH

11. Prove that 5 is an irrational number.

12. If a and are the zeroes of the polynomial 2x -5x+k and a - =-1.

Find the value of k.

13. Determine a and b for which the following system of linear eqations has infinitely amny

solutions

2x-(a-4)y=2b+1

4x-(a-1)y+5b-1

14. If the areas of two similar triangles are equal, Prove that they are congruent.

15. PQR is a right angels triangle right angled at p.m is a point on QR such that PM QR.show

that 2PM =QM.MR

16. Prove that:

sec0 1

sec0 1

+sec0 1

sec0 1

=2cosec 0

17. If tan A=cotB,

Prove that A+B=090



Or

If A,B,C are interior angles of a ABC, show that

2sec B C

2

-1= 2cotA

2

18. Evaluate :

5 25cos 060 + 04sec 030 - 0tan 045

_______________________________________ 2sin 030 + 2cos 030

19. The length of 40 leaves of a plant are measured correct to nearest millimeter and the data

obtained is represented in the table below:

Find the median length of the leaves

Length (in mm) no. of leaves

118-126

127-135

136-144

145-153

154-162

163-171

172-180

3

5

9

12

5

4

2

20. The following distribution gives the daily income of 50workers of a factory

Daily income (in Rs) No. of workers

100-120

120-140

140-160

160-180

180-200

12

14

8

6

10

Convert the distribution above a less than type cumulative frequency distribution and draw its

ogive.

SECTION –D

Directions: 11 question of 4 marks each

21. (a) Find the HCF of 1305, 1365 by using Euclid’s division algorithim.

(b) Also deduce the LCM of 1305 and 1365.



22. Prove that 2 3 5 is anirrational number.

23. Solve graphically the following system of equations:

X+2y=5

2x3y=4

24. Yash scored 40 marks in a test ,getting 3 marks for each right answer and losing 1 mark for

each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks for

each incorrect answer, then Yash would have scored 40marks. How many question were

there in the test?

25. Solve the following by substitution method:

3x+4y=10

2x-2y=2

26. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sided

in distinct points, then the other two sides are divided in the same ration.

27. In the given figure ,

OA.OB=OC.OD

Show that = c and B=D

Or

If AD and PM are medians of triangles ABC and PQR, respectively whereABCPQR,

Prove that AB

PQ=AD

PM

28. If sec +tan =p Show that 2

2

p 1

p 1

=sin

29. If tan +sin =m, and Tan -sin =n. Show that 2m - 2n =4 mn

30. The annual profits earned by 30 shops of shopping complex in a locality give rise to the

following distribution:

Profit in lakhs (RS) no. of shops (frequency)

More than or equal to5

More than or equal to 10





30

28

16

14

10

7



More than or equal to 35 3

Draw both ogives for the above data and hence obtain the media profit.

31. If the median of the distribution given below is 28.5, find the value of x and y

Class Interval Frequency

010

10-20

20-30

30-40

40-50

50-60

5

x

20

15

y

5

Total 60



Mathematics

Set-3

Time: 3 hrs M.M: 90

General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections – A, B, C and D. (iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks

each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.

(iv) Use of calculated is not permitted.

Section A

Q1 Find the 9th term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185. Q2 Cards marked with number 3, 4, 5, ...., 50 are placed in a box and mixed thoroughly. A

card is drawn at random form the box. Find the probability that the selected card bears a perfect square number.

Q3 From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB.

Q4 In Fig. 1, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder. (use3√=1.73)

Q5 Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–

1, –4). Also find the coordinates of the point of division.

Q6 If 2

33

x and x= = − are roots of the quadratic equation 2 7 0ax x b+ + = , find the values

of a and b. Q7 The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5)

and R(–3, 6), find the coordinates of P. Q8 In Fig. 2, a circle is inscribed in a ABC∆ , such that it touches the sides AB, BC and CA at

points D, E and F respectively. If the lengths of sides AB, BC and CA and 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF.



Q10 In Fig. 3, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and

60 .APB∠ = ° Find the length of chord AB.

Q10 How many terms of the A.P. 65, 60, 55, .... be taken so that their sum is zero? Q11 A well of diameter 4 m is dug 21 m deep. The earth taken out of it has been spread

evenly all around it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment.

Q12 If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

Q13 The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal

distance between the tower and the building. ( ) 3 1.73 use =

Q14 In Fig. 4, ABCD is a square of side 14 cm. Semi-circles are drawn with each side of square

as diameter. Find the area of the shaded region. (22

7

useπ = )

Q15 In Fig. 5, is a decorative block, made up two solids – a cube and a hemisphere. The base

of the block is a cube of side 6 cm and the hemisphere fixed on the top has diameter of

3.5 cm. Find the total surface area of the bock. (22

7

useπ = )



Q16 In Fig. 6, ABC is a triangle coordinates of whose vertex A are (0, −1). D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of D .ABC and EF∆ ∆

Q17 In Fig. 7, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and

radius OP while arc PBQ is a semi-circle drawn on PQ ad diameter with centre M. If OP =

PQ = 10 cm show that area of shaded region is 225 36

cmπ

−

Q18 A box consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major

defects. Ramesh, a shopkeeper will buy only those shirts which are good but 'Kewal' another shopkeeper will not buy shirts with major defects. A shirt is taken out of the box at random. What is the probability that (i) Ramesh will buy the selected shirt? (ii) 'Kewal' will buy the selected shirt?

Q19 Solve the following quadratic equation for x: 2 1 0 a a b

x xa b a ++ + +

=+

Q20 A toy is in the form of a cone of base radius 3.5 cm mounted on a hemisphere of base diameter 7 cm. If the total height of the toy is 15.5 cm, find the total surface area of the

toy. 22

7

useπ =

Q21 A passenger, while boarding the plane, slipped form the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane. What value is depicted in this question?

Q22 In Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.



Q23 A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of

elevation of the bird is 45°. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30°. Find the speed of flying of the bird.

( ) 3 1.732 . Take =

Q24 A game of chance consists of spinning an arrow on a circular board, divided into 8 equal parts, which comes to rest pointing at one of the numbers 1, 2, 3, ..., 8 (Fig. 9), which are equally likely outcomes. What is the probability that the arrow will point at (i) an odd number (ii) a number greater than 3 (iii) a number less than 9.

Q25 Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is

independent of t. Q26 A bucket open at the top is in the form of a frustum of a cone with a capacity of

312308.8 cm . The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. ( ) 3.14 use π =

Q27 An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one

point C on the belt, the elastic belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from the point O. Find the length of the belt that is still in contact

with the pulley. Also find the shaded area. ( ) 3.14 3 1.73use andπ = =

Q28 The sum of three numbers in A.P. is 12 and sum of their cubes is 288. Find the numbers. Q29 Prove that the lengths of tangents drawn from an external point to a circle are equal.



Q30 The time taken by a person to cover 150 km was 1

22

hours more than the time taken in

the return journey. If he returned at a speed of 10 km/hour more than the speed while going, find the speed per hour in each direction.

Q31 Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are times the corresponding sides of ΔABC.



Mathematics

Set-2

Time: 3 hrs M.M: 90

General Instructions:

(i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections – A, B, C and D. (iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks

each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.


Section A

Q1 Cards marked with number 3, 4, 5, ...., 50 are placed in a box and mixed thoroughly. A card is drawn at random form the box. Find the probability that the selected card bears a perfect square number.

Q2 In Fig. 1, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder. (use3√=1.73)

Q3 Find the 9th term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185. Q4 From an external point P, tangents PA and PB are drawn to a circle with centre O. If

50 , . PAB then find AOB∠ = ° ∠ Section B

Q5 The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and R(–3, 6), find the coordinates of P.

Q6 In Fig. 2, a circle is inscribed in a ΔABC, such that it touches the sides AB, BC and CA at points D, E and F respectively. If the lengths of sides AB, BC and CA and 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF.

Q7 In Fig. 3, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and ∠APB =

60°. Find the length of chord AB.



Q8 If 2

33

x and x= = − are roots of the quadratic equation 2 7 0ax x b+ + = , find the values of

a and b. Q9 Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–

1, –4). Also find the coordinates of the point of division. Q10 How many terms of the A.P. 27, 24, 21, .... should be taken so that their sum is zero? Q11 If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum

of first n terms of the A.P. Q12 A well of diameter 4 m is dug 21 m deep. The earth taken out of it has been spread evenly

all around it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment.

Q13 In Fig. 4, ABCD is a square of side 14 cm. Semi-circles are drawn with each side of square

as diameter. Find the area of the shaded region. (22

7

useπ = )

Q14 In Fig. 5, is a decorative block, made up two solids – a cube and a hemisphere. The base of

the block is a cube of side 6 cm and the hemisphere fixed on the top has diameter of 3.5

cm. Find the total surface area of the bock. (22

7

useπ = )

Q15 In Fig. ABC is a triangle coordinates of whose vertex A are (0, −1). D and E respectively

are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of .ABC and DEF∆ ∆



Q16 In Fig. 7, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and

radius OP while arc PBQ is a semi-circle drawn on PQ ad diameter with centre M. If OP =

PQ = 10 cm show that area of shaded region is 225 36

cmπ

− .

Q17 The angles of depression of the top and bottom of a 50 m high building from the top of a

tower are 45° and 60° respectively. Find the height of the tower and the horizontal

distance between the tower and the building. ( ) 3 1.73 use =

Q18 Solve for x : 1 2 2 3

4 1 2 2

x x x

x x x

+ − ++ = −− + −

Q19 Two different dice are thrown together. Find the probability of: (i) getting a number greater than 3 on each die (ii) getting a total of 6 or 7 of the numbers on two dice

Q20 A right circular cone of radius 3 cm, has a curved surface area of 247.1 cm . Find the volume of the cone. ( ) 3.14 . use π

Section D

Q21 A passenger, while boarding the plane, slipped form the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane. What value is depicted in this question?

Q22 In Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.

Q23 Prove that the lengths of tangents drawn from an external point to a circle are equal.



Q24 Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is independent of t.

Q25 A game of chance consists of spinning an arrow on a circular board, divided into 8 equal parts, which comes to rest pointing at one of the numbers 1, 2, 3, ..., 8 (Fig. 9), which are equally likely outcomes. What is the probability that the arrow will point at (i) an odd number (ii) a number greater than 3 (iii) a number less than 9.

Q26 An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one

point C on the belt, the elastic belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from the point O. Find the length of the belt that is still in contact

with the pulley. Also find the shaded area. ( ) 3.14 3 1.73use andπ = =

Q27 A bucket open at the top is in the form of a frustum of a cone with a capacity of

312308.8 cm . The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. ( ) 3.14 use π =

Q28 The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m

from the base of the tower and in the same straight line with it are 60° and 30° respectively. Find the height of the tower.

Q29 Construct a triangle ABC in which BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct

another triangle whose sides are3

4 times the corresponding sides of . ABC∆

Q30 The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle.

Q31 A thief, after committing a theft, runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5 m/minute every succeeding minute. After how many minutes, the policeman will catch the thief?


Mathematics

Set-1

Time: 3 hrs M.M: 90

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.

(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks

each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of

4 marks each.


Section A

Q1 In Fig. 1, AOB is a diameter of a circle with centre O and AC is a tangent to the circle at A.

If 130BOC∠ = ° v, the find .ACO∠

Q2 An observer, 1.7 m tall, is 20 3 m away from a tower. The angle of elevation from the of

observer to the top of toweris 30°. Find the height of tower.

Q3 For what value of k will the consecutive terms 2k + 1, 3k + 3 and 5k − 1 form an A.P. ?

Q4 20 tickets, on which numbers 1 to 20 are written, are mixed thoroughly and then a ticket

is drawn at random out of them. Find the probability that the number on the drawn ticket

is a multiple of 3 or 7.

Section B

Q5 A two digit number is four times the sum of the digits. It is also equal to 3 times the

product of digits. Find the number.

Q6 Find the ratio in which the point (−3, k) divides the line-segment joining the points (−5,

−4) and (−2, 3). Also 6ind the value of k.

Q7 In Fig. 2, from a point P, two tangents PT and PS are drawn to a circle with centre O such

that ∠SPT = 120°, Prove that OP = 2PS.

Q8 Prove that the points (2, −2), (−2, 1) and (5, 2) are the vertices of a right angled triangle.

Also find the area of this triangle.


Q9 If the ratio of sum of the first m and n terms of an AP is m2 : n2, show that the ratio of its

th thm and n terms is (2m − 1) : (2n − 1).

Q10 In fig. 3 are two concentric circles of radii 6 cm and 4 cm with centre O. If AP is a tangent

to the larger circle and BP to the smaller circle and length of AP is 8 cm, find the length of

BP.

Section C

Q11 Find the area of shaded region in Fig. 4, where a circle of radius 6 cm has been drawn

with vertex O of an equilateral triangle OAB of side 12 cm. ( ) 3.14 3 1.73Use andπ = =

The rate of 347 litre per second. How much time will it take to make the tank half empty?

22

7Useπ

=

Q13 If the point C (–1, 2) divides internally the line-segment joining the points A (2, 5) and B

(x, y) in the ratio 3 : `4, find the value of x2 + y2.

Q14 In fig. 5 is a chord AB of a circle, with centre O and radius 10 cm, that subtends a right

angle at the centre of the circle. Find the area of the minor segment AQBP. Hence find the

area of major segment ALBQA ( ). 3.14use π =

Q15 Divide 56 in four parts in AP such that the ratio of the product of their extremes (1st and

4th) to the product of means (2nd and 3rd) is 5 : 6.


Q16 Solve the given quadratic equation for ( ) ( )2 2 2: 9 – 9 2 5 2 0. x x a b x a ab b+ + + + =

Q17 A cylindrical tub, whose diameter is 12 cm and height 15 cm is full of ice-cream. The

whole ice-cream is to be divided into 10 children in equal ice-cream cones, with conical

base surmounted by hemispherical top. If the height of conical portion is twice the

diameter of base, find the diameter of conical part of ice-cream cone.

Q18 A metal container, open from the top, is in the shape of a frustum of a cone of height 21

cm with radii of its lower and upper circular ends as 8 cm and 20 cm respectively. Find

the cost of milk which can completely fill the container at the rate of Rs 35 per litre.

22

7Useπ

=

Q19 Two men on either side of a 75 m high building and in line with base of building observe

the angles of elevation of the top of the building as 30° and 60°. Find the distance

between the two men. ( ) 3 1.73 Use =

Q20 A game consist of tossing a one-rupee coin 3 times and noting the outcome each time.

Ramesh will win the game if all the tosses show the same result, (i.e. either all thee heads

or all three tails) and loses the game otherwise. Find the probability that Ramesh will lose

the game.

Section D

Q21 A pole has to be erected at a point on the boundary of a circular park of diameter 17 m in

such a way that the differences of its distances from two diametrically opposite fixed

gates A and B on the boundary is 7 metres. Find the distances from the two gates where

the pole is to be erected.

Q22 Prove that the lengths of tangents drawn from an external point to a circle are equal.

Q23 Draw a ABC∆ in which AB = 4 cm, BC = 5 cm and AC = 6 cm. Then construct another

triangle whose sides are 35 of the corresponding sides of ∆ABC.

Q24 In fig. 6, AB is a chord of a circle, with centre O, such that AB = 16 cm and radius of circle

is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA.

Q25 Find the positive value(s) of k for which quadratic equations

2 2 64 0 – 8 0x kx and x x k+ + = + = both will have real roots.

Q26 A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 5

m. From a point on the ground the angles of elevation of the top and bottom of the

flagstaff are 60° and 30° respectively. Find the height of the tower and the distance of the

point from the tower. ( ) 3 1.732 take =

Q27 Reshma wanted to save at least Rs 6,500 for sending her daughter to school next year

(after 12 month.) She saved Rs 450 in the first month and raised her savings by Rs 20

every next month. How much will she be able to save in next 12 months? Will she be able

to send her daughter to the school next year? What value is reflected in this question.


Q28 The co-ordinates of the points A, B and C are (6, 3), (−3, 5) and (4, −2) respectively. P(x, y)

is any point in the plane. Show that ( )

( )2

7

ar PBC x y

ar ABC

+ −=∆

∣ ∣

Q29 In fig. 7 is shown a disc on which a player spins an arrow twice. The fraction a

b is formed,

where 'a' is the number of sector on which arrow stops on the first spin and 'b' is the

number of the sector in which the arrow stops on second spin. On each spin, each sector

has equal chance of selection by the arrow. Find the probability that the fraction 1.a

b>

Q30 Find the area of the shaded region in Fig. 8, where � � � �, , APD AQB BRC and CSD are semi-circles

of diameter 14 cm, 3.5 cm, 7 cm and 3.5 cm respectively. 22

7

Useπ =

Q31 In fig. 9 is shown a right circular cone of height 30 cm. A small cone is cut off from the top

by a plane parallel to the base. If the volume of the small cone is 1

27of the volume of cone,

find at what height above the base is the section made.

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