SUMMATIVE ASSESSMENT-I, 2015-16 CLASS-X ...icsk-kw.com/pdf/pqp/10/mat/2.pdfQ.21 What is the HCF and LCM of two prime numbers a and b? Three alarm clocks ring at intervals of 6, 9 and

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SUMMATIVE ASSESSMENT-I, 2015-16 CLASS-X, MATHEMATICS LYSVYI3 Time allowed: 3 hours Maximum Marks: 90

General Instructions: 1. This question are compulsory. 2. The question paper is divided into four sections Section A: 4 question (1mark each) Section B: 6 question (2 marks each) Section C: 10 questions (3 mark each) Section D: 11 questions (4 mark each). 3. There is no overall choice in this question paper. 4. Use of calculators is not permitted.

Section A

Question number 1 to 4 carry one mark each

Q. 1 If the corresponding medians of two similar triangles are in the ratio 5:7, then find the ratio

of their corresponding sides.

Q. 2 If 24 cotA – 7 , Find the value of sinA

Q.3 Simplify : 2

2

cosec A

1 cot A

Q.4 If the point of intersection of two gives is (18, 54), then find the value of median.

Section B

Question number 5 to 10 carry two mark each

Q.5 Show that 5 6 is an irrational number

Q.6 Express 5050 as product of its prime factors. Is it unique?

Q.7 The taxi charges in a city consists of a fixed change together with the charge for the distance

covered. For a distance of 6 km, the charges paid are Rs 58 while for a journey of 10 km, the

charges paid are Rs. 90. Find the charge per km and the fixed charge.

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

criterion used.

Q.9 Prove that: 2 2sec 1 1 cosec 1



Q.10 Show that the mode of the series obtained by combining the two series 1S and 2S give below

is different from that of 1S and 2S taken separately:

1S : 3,5,8,8,9,12,13,9,9

2S : 7,4,7,8,7,8,13

Section C

Question number 11 to 20 carry three mark each

Q.11 Pens are sold in pack of 8 and notepads are sold in pack of 12. Find the least number of pack

of each type that one should buy so that there are equal number of pen and notepads.

Q.12 Solve using cross multiplication method :

2u 7v 1

4u 3v 15

Q.13 If 2p x x 5x 2 , what is the value of p 3 q 2 ?

Q.14 For what value of K, will the following system of equations have no solution?

23k 1 x 3y 2; k 1 x k 2 y 5

Q.15 In ABC,X is middle point of AC. If XY||AB then prove that Y is middle point of AB

Q.16 In the figure, ABCD is a rectangle. If in ADE and ABE , E F , then prove that

AD AB

AE AF .

Q.17 If 2 27sin A_3cos A 4 , show that 1

tanA3

.

Q.18 Prove that: 3

3

sin 2sintan

2cos cos

Q.19 In a class test, marks scored by students are given in the following frequency distribution :



Marks 0-6 6-12 12-18 18-24 24-30

Number of students 1 4 9 3 3

Find the mean and median of the data.

Q.20 Some surnames were picked up from a local telephone directory and the frequentation

distribution of the number of letters of the English alphabets was obtained as follows:

Number of letters 1-4 4-7 7-10 10-13 13-16 16-19

Number of surnames 10 25 35 x 12 8

If it is given that mode of the distribution is 8, then find the missing frequency (x).

Section D

Question number 21 to 31 carry four mark each

Q.21 What is the HCF and LCM of two prime numbers a and b?

Three alarm clocks ring at intervals of 6, 9 and 15 minutes respectively. If they start ringing

together, after what time will they next ring together.

Q.22 Draw the graph of the following pair of linear equation: 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

Q.23 If the polynomial 4 3 2x 2x 8 12x 18 is divided by another polynomial 2x 5 , the

remainder comes out to be px q , find the values of p and q.

Q.24 Ram’s mother has given him money to buy some boxes from the market at the rate of 24x 3x 2 . The total amount of money is represented by 4 3 28x 14x 2x 7x 8 . Out of this

money he donated some amount to a child who was studying in the light of street Iamp. Find

how much amount of money he donated and purchased how many boxes from the market?

Why Ram did so?

Q.25 In a right angled triangle ABC, oA 90 and AD BC . Prove that:

(i) 2AB BD BC

(ii) 2AD BD DC



(iii) 2AC BC CD

Q.26 Find the length of the diagonal of the rectangle BCDE, If BCE DCF , AC=6 m and

CF=12m.

Q.27 Evaluate o o o otan1 tan 2 tan3 ......tan89

Q.28 Prove that 2 2 2 2 2 2b x a y a b , if :

(i) x a sec , y b tan

(ii) x a coec , y bcot

Q.29 Prove that: cot cosec 1 . cot cosec 1 .tan 2

Q.30 Cost of Living Index fox some period is given in the following frequency distribution :

Index 1500-

1600

1600-

1700

1700-

1800

1800-

1900

1900-

2000

2000-

2100

2100-

2200

Number

of weeks

3 11 12 7 9 8 2

Q.31 Following is the ages of asthmatic patients admitted during a year in a hospital. Find the

mean age of the patients.

Age (in

years)

0-8 8-16 16-24 24-32 32-40 40-48 48-56 56-64

Number of

weeks

6 25 12 13 11 14 11 8



SUMMATIVE ASSESSMENT-I, 2015-16 CLASS-X, MATHEMATICS GG-RO-103 Time allowed: 3 hours Maximum Marks: 80

General Instructions: 1. All questions are compulsory. 2. Question paper contains 31 questions divided into 4 section A, B, C & D. 3. Section-A comprises of 4 questions carrying 1 mark each, Section-B comprises of 6 questions carrying 2 marks each, Section-C comprises of 10 questions carrying 3 marks each, Section-D comprises of 11 questions carrying 4 marks each, 4. Al questions in section-A are very short answer questions. 5. There are no overall choices in the question paper. 6. Use of calculator is not permitted. 7. If required Graph papers will be provided.

Section A

Question numbers 1 to 4 carry 1 mark each.

Q. 1 Find the value of o

o

tan30

cot 60

Q. 2 What is the altitude of an equilateral triangle of each side 6 cm.

Q. 3 If the sum of zeroes of quadratic polynomial 23x kx 6 is 3, then fine the value of k.

Q. 4 Find the class marks of class 35-55.

Section B

Question numbers 5 to 10 carry 2 marks each.

Q. 5 In a right isosceles triangle ABC right angled at C prove that 2 2AB 2AC

Q. 6 Use Euclid’s division algorithm to find the HCF of 870 and 255.

Q. 7 Find the mode of the following distribution of marks obtained by 80 students:

Marks obtained No. of students

0-10

10-20

20-30

30-40

40-50

06

10

12

32

20



Q. 8 Find the value of k. If the pair of linear equation :

3x 4y k

9x 12y 6

Has infinitely many solutions.

Q. 9 If the LCM of ‘a’ and 18 is 36 and the HCF of ‘a’ and 18 is 2, then find the value of ‘a’.

Q. 10 Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2 respectively.

Section C


Q. 11 Use Euclid’s division Lemma, to show that the square of any positive integer is either of the

form 3m or 3m+1 for some integer m.

Q. 12 Solve the following pairs of linear equations by Cross-multiplication method :

4x y 14

5x 6y 27

Q. 13 If the areas of two similar triangles are equal, then prove that they are congruent.

Q. 14 1 sinA

secA tanA1 sin A

Q. 15 Find the mean of the following distribution, using step deviation method :

Class Frequency

0-10

10-20

20-30

30-40

40-50

07

12

13

10

08

Q. 16 Find the zeroes of the quadratic polynomial 26x 3 7x and verify the relationship between

the zeroes and the coefficients.

Q. 17 In triangle PQR right angled at Q, PR+QR=25cm and PQ=5cm determine the value of sin P, cos

P and tan P.

Q. 18 In figure PQ||CD and PR||CB,

Prove that AQ AR

QD RB



Q. 19 Find the median of the following frequency distribution table:

marks No. of students

0-10

10-20

20-30

30-40

40-50

50-60

05

08

06

10

06

05

Q. 20 If 1 1

sin A B cos A B2 2

, 0 00 A B 90 , A B, Find A and B.

Section D


Q. 21 Show that: 25 3 is an irrational number

Q. 22 Find all the zeroes of the polynomials 4 3 2x x 9x 3x 18 . If it is given that two of its zeroes

are 3 and 3

OR

Divide 2 33x x 3x 5 by 2x 1 x and verify the division algorithm.

Q. 23 Prove that in a Right Angled Triangle, the square of the Hypotenuse is equal to the sum of the

square of the others two sides’

OR

Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their

corresponding medians.

Q. 24 Prove that:tan cot

1_ sec cosec1 cot 1 tan

OR



3

3

sin 2sintan

2cos cos

Q. 25 Solve the following system of Linear Equation Graphically:

X – Y = 1, 2 X + Y = 8.

Shade the area bounded by these two lines and Y-axis.

Q. 26 The median of the following data is 50. Find the values of p and q if the sum of all the

frequencies is 90.

marks No. of students

20-30

30-40

40-50

50-60

60-70

70-80

80-90

P

15

25

20

Q

08

10

Q. 27 If 1

sin2

Show that 3 3cosB 4Cos B 0

Q. 28 During the medical check-up of the 35 students of a class their weights were recorded as

follows:

Weight in KG No. of students

38-40

40-42

42-44

44-46

46-48

48-50

50-52

03

02

04

05

14

04

03

Draw a less than type give for the above data.

Q. 29 Solve the pair of linear equation:1 1 3

3x y 3x y 4

,

1 1 1

2 3x y 2 3x y 8



OR

The Sum of the digits of a two digit number is 9, Also nine times this number is twice the –

number obtain by reversing the order the digits. Find the number.

Q. 30 Evaluate:o o o

o o o o o o

7cos70 3 cos55 cosec35

2sin 20 2tan5 tan45 tan85 tan65 tan25

Q. 31 Mr. Balwant Singh has a triangular field ABC. He has three sons. He wants to divide the field

into four equal and identical parts, so that he may give three parts to his three sons and

retain the fourth part with him.

i) Is it possible to divide the field into four parts which are equal and identical?

ii) If yes, explain the method of division.

iii) By doing so, which values is depicted by Mr. Balwant Singh.



Summative Assessment-1 2014-2015

Mathematics

Class – X

Time allowed: 3:00 hours Maximum Marks: 90

General Instructions:

a) All questions are compulsory.

b) Question paper contains 31 questions divide into 4 sections A, B, C and D.

c) Question No. 1 to 4 are very short type questions, carrying 1 mark each. Question No. 5 to

10 are of short answer type questions, carrying 2 marks each. Question No. 11 to 20 carry

3 marks each. Question No. 21 to 31 carry 4 marks each.

d) There are no overall choices in the question paper.

e) Use of calculator is not permitted.

Section A

Question numbers 1 to 4 carry 1 mark each.

1. In XYZ∆ , A and B are points on the sides XY and XZ respectively such that AB YZ� . If

AY=2.2cm, XB=3.3cm and XZ=6.6cm, then find AX.

2. If tan cot 2θ θ+ = , then find the value of 2 2tan cotθ θ+ .

3. If 45θ = ° , then find the value of 2 22sin 3cosecθ θ+ ?

4. Life time of electric bulbs are given in the following frequency distribution:

Life time

(in hours)

250-300 300-350 350-400 400-450 450-500

Number of

bulbs

5 14 21 12 10

Find the class mark of the modal class interval.

Section B

Question numbers 5 to 10 are two marks each.

5. Find whether decimal expansion of 13

64is a terminating or non-terminating decimal. If it

terminates, find the number of decimal places its decimal expansion has.

6. Write the decimal expansion of 27

1250without actual division.

7. Ifα and β are the zeroes of a polynomial 29 12 4y y+ + , then find the value of α β αβ+ + .

8. Are the given figures similar? Give reason.



9. Simplify: (1-sin A)(tan A + sec A)

10. The following distribution shows the daily pocket allowance of children of a locality:

Daily pocket

allowance (in Rs.)

12 15 20 25 30

Number of

children

8 7 15 6 4

Find the median of the data.

Section C

Question numbers 11 to 20 carry three marks each.

11. Prove that 3 5+ is an irrational number.

12. Solve for x and y:

x+4y=27xy

x+2y=21xy

13. Determine graphically whether the following pair of linear equations

2x-3y=8

4x-6y=16 has

a) A unique solution,

b) Infinitely many solution or

c) No solution

14. If 4 3 24 7 4 7x x x x k+ − − + is completely divisible by 3x x− , then find the value of k.

15. As shown in the figure, a 26m long ladder is placed at A. if it is placed along wall PQ, it

reaches a height of 24m whereas it reaches a height of 10m if it is placed against wall RS.

Find the distance between the walls.

16. If in ABC∆ , AD is median and AM BC⊥ , then prove that 2 2 2 212

2AB AC AD BC+ = +

17. Prove that: 2 2

2 22 2

cossec cos 2

cos sin

sin A AA ec A

A A+ = ⋅ −

18. In ABC∆ , right angled at C, if1

tan3

A = , show that sin A. cos B + cos A. sin B=1



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

20. For the following distribution, draw a ‘less than type’ ogive and from the curve, find the

median.

Marks

obtained

Less

than

20

Less

than

30

Less

than

40

Less

than

50

Less

than

60

Less

than

70

Less

than

80

Less

than

90

Less

than

100

Number

of

students

2 7 17 40 60 82 85 90 100

Section D

Question numbers 21 to 31 carry four marks each.

21. Dhudnath has two vessels containing 720 ml and 405 ml of milk respectively. Milk from

these containers is poured into glasses of equal capacity to their brim. Find the minimum

number of glasses that can be filled.

22. The ratio of incomes of two persons A and B is 9:7and the ratio of their expenditure is 4:3. If

their savings are Rs. 200 per month, find their monthly incomes.

Why is it necessary to save money?

23. Find all the zeroes of 4 3 25 15 12x x x x− + + − , if it is given that two of its zeroes are 1 and 4.

24. A boat goes 30 km upstream and 20 km 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.

25. In ABC∆ , AD BC⊥ and D lies on BC such that 4DB=CD, then proves that 2 2 25 5 3AB AC BC= −

26. ABC is an isosceles triangle in which 90B∠ = ° and 3 2AC m= . Two equilateral triangles ACP

and ABQ are drawn on the sides AC and AB. Find the ratio of area ( )ABQ∆ and area ( )ACP∆ .

27. In the adjoining figure, ABCD is a rectangle with breadth BC=7cm and 30CAB∠ = ° . Find the

length of side AB of the rectangle and length of diagonal AC. If the 60CAB∠ = ° , then what is

the size of the side AB of the rectangle (use 3 1.73= and 2 1.41= , if required)

28. If cos sina b cθ θ− = , then prove that 2 2 2sin cosa b a b cθ θ+ = ± + −



29. Given that sin( ) sin cos cos sinA B A B A B− = ⋅ − ⋅ . Find the value ofsin15° in two ways.

a) Taking 60 , 45A B= ° = ° , and

b) Taking 45 , 30A B= ° = °

30. A class test in mathematics was conducted for class VI of a school. Following distribution

gives marks (out of 60) of students:

Marks 0-10 10-20 20-30 30-40 40-50 50-60

Number of

students

8 22 12 10 5 3

Find the mean of the marks obtained.

31. In an examination, 150 students appeared, and their marks (out of 200) are given in the

following distribution. Find the missing frequencies x and y, when it is given that mean

marks is 103.

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

Number

of

students

2 10 x 30 y 15 12 4

30/2 1 P.T.O.

narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð >na Adí` {bIo§ & Candidates must write the Code on the

title page of the answer-book.

Series RLH H$moS> Z§. 30/2

Code No.

amob Z§.

Roll No.

g§H${bV narjm – II

SUMMATIVE ASSESSMENT – II

J{UV

MATHEMATICS

{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 90

Time allowed : 3 hours Maximum Marks : 90

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ &

àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na {bI| &

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >31 àíZ h¢ &

H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU nydm©•

_| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &

Please check that this question paper contains 11 printed pages.

Code number given on the right hand side of the question paper should be

written on the title page of the answer-book by the candidate.

Please check that this question paper contains 31 questions.

Please write down the Serial Number of the question before

attempting it.

15 minute time has been allotted to read this question paper. The question

paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the

students will read the question paper only and will not write any answer on

the answer-book during this period.

SET-2

30/2 2

gm_mÝ` {ZX}e :

(i) g^r àíZ A{Zdm`© h¢ & (ii) Bg àíZ-nÌ _| 31 àíZ h¢ Omo Mma IÊS>m| A, ~, g Am¡a X _| {d^m{OV h¢ & (iii) IÊS> A _| EH$-EH$ A§H$ dmbo 4 àíZ h¢ & IÊS> ~ _| 6 àíZ h¢ {OZ_| go àËòH$ 2 A§H$

H$m h¡ & IÊS> g _| 10 àíZ VrZ-VrZ A§H$m| Ho$ h¢ & IÊS> X _| 11 àíZ h¢ {OZ_| go àËòH$ 4 A§H$ H$m h¡ &

(iv) H¡$bHw$boQ>a H$m à`moJ d{O©V h¡ &

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 calculators is not permitted.

IÊS> A

SECTION A

àíZ g§»`m 1 go 4 VH$ àË oH$ àíZ 1 A§H$ H$m h¡ & Question numbers 1 to 4 carry 1 mark each.

1. AmH¥${V 1 _|, O H|$Ð dmbo d¥Îm H$s PQ EH$ Ordm h¡ VWm PT EH$ ñne© aoIm h¡ & `{X QPT = 60 h¡, Vmo PRQ kmV H$s{OE &

AmH¥${V 1

30/2 3 P.T.O.

In Figure 1, PQ is a chord of a circle with centre O and PT is a tangent. If

QPT = 60, find PRQ.

Figure 1

2. `{X {ÛKmV g_rH$aU px2 – 2 5 px + 15 = 0 Ho$ Xmo g_mZ _yb hm|, Vmo p H$m _mZ kmV H$s{OE &

If the quadratic equation px2 – 2 5 px + 15 = 0 has two equal roots,

then find the value of p.

3. AmH¥${V 2 _|, EH$ _rZma AB H$s D±$MmB© 20 _rQ>a h¡ Am¡a BgH$s ^y{_ na naN>mB© BC H$s bå~mB© 20 3 _rQ>a h¡ & gy`© H$m CÞVm§e kmV H$s{OE &

AmH¥${V 2

In Figure 2, a tower AB is 20 m high and BC, its shadow on the ground,

is 20 3 m long. Find the Sun’s altitude.

Figure 2

30/2 4

4. Xmo {^Þ nmgm| H$mo EH $gmW CN>mbm J`m & XmoZm| nmgm| Ho$ D$nar Vbm| na AmB© g§»`mAm| H$m JwUZ\$b 6 AmZo H$s àm{`H$Vm kmV H$s{OE & Two different dice are tossed together. Find the probability that the

product of the two numbers on the top of the dice is 6.

IÊS> ~ SECTION B

àíZ g§»`m 5 go 10 VH$ àËòH$ àíZ 2 A§H H$m h¡ & Question numbers 5 to 10 carry 2 marks each.

5. `{X {~ÝXþ A(x, y), B(– 5, 7) VWm C(– 4, 5) ñ§maoIr` hm|, Vmo x VWm y _| gå~ÝY kmV H$s{OE & Find the relation between x and y if the points A(x, y), B(– 5, 7) and

C(– 4, 5) are collinear.

6. EH$ g_m§Va lo‹T>r Ho$ àW_ n nXm| Ho$ `moJ\$b H$mo Sn Ûmam Xem©`m OmVm h¡ & Bg lo‹T>r _| `{X

S5 + S

7 = 167 VWm S

10 = 235 h¡, Vmo g_m§Va lo‹T>r kmV H$s{OE &

In an AP, if S5 + S

7 = 167 and S

10 = 235, then find the AP, where S

n

denotes the sum of its first n terms.

7. AmH¥${V 3 _|, Xmo ñne© aoImE± RQ VWm RP d¥Îm Ho$ ~mø {~ÝXþ R go ItMr JB© h¢ & d¥Îm H$m

Ho$ÝÐ O h¡ & `{X PRQ = 120 h¡, Vmo {gÕ H$s{OE {H$ OR = PR + RQ.

AmH¥${V 3

30/2 5 P.T.O.

In Figure 3, two tangents RQ and RP are drawn from an external point R

to the circle with centre O. If PRQ = 120, then prove that

OR = PR + RQ.

Figure 3

8. AmH¥${V 4 _|, 3 go_r {ÌÁ`m dmbo EH$ d¥Îm Ho$ n[aJV EH$ {Ì^wO ABC Bg àH$ma ItMm J`m h¡ {H$ aoImIÊS> BD VWm DC H$s b§~mB`m± H«$_e… 6 go_r VWm 9 go_r h¡§ & `{X ABC H$m joÌ\$b 54 dJ© go_r h¡, Vmo ^wOmAm| AB VWm AC H$s bå~mB`m± kmV H$s{OE &

AmH¥${V 4

In Figure 4, a triangle ABC is drawn to circumscribe a circle of radius

3 cm, such that the segments BD and DC are respectively of lengths 6 cm

and 9 cm. If the area of ABC is 54 cm2, then find the lengths of sides

AB and AC.

Figure 4

30/2 6

9. {ZåZ {ÛKmV g_rH$aU H$mo x Ho$ {bE hb H$s{OE : 4x2 + 4bx – (a2 – b2) = 0

Solve the following quadratic equation for x :

4x2 + 4bx – (a2 – b2) = 0

10. `{X A(4, 3), B(–1, y) VWm C(3, 4) EH$ g_H$moU {Ì^wO ABC Ho$ erf© h¢, {Og_| A na g_H$moU h¡, Vmo y H$m _mZ kmV H$s{OE & If A(4, 3), B(–1, y) and C(3, 4) are the vertices of a right triangle ABC,

right-angled at A, then find the value of y.

IÊS> g SECTION C

àíZ g§»`m 11 go 20 VH$ àËòH$ àíZ 3 A§H$ H$m h¡ & Question numbers 11 to 20 carry 3 marks each.

11. AMmZH$ ~m‹T> AmZo na, Hw$N> H$ë`mUH$mar g§ñWmAm| Zo {_b H$a gaH$ma H$mo Cgr g_`

100 Q>|Q> bJdmZo Ho$ {bE H$hm VWm Bg na AmZo dmbo IM© H$m 50% XoZo H$s noeH$e H$s &

`{X àËòH$ Q>|Q> H$m {ZMbm ^mJ ~obZmH$ma h¡ {OgH$m ì`mg 4.2 _r. h¡ VWm D±$MmB© 4 _r.

h¡ VWm D$nar ^mJ Cgr ì`mg H$m e§Hw$ h¡ {OgH$s D±$MmB© 2.8 _r. h¡, Am¡a Bg na bJZo

dmbo H¡$Zdg H$s bmJV < 100 à{V dJ© _r. h¡, Vmo kmV H$s{OE {H$ BZ g§ñWmAm| H$mo

{H$VZr am{e XoZr hmoJr >& BZ g§ñWmAm| Ûmam {H$Z _yë`m| H$m àXe©Z {H$`m J`m ?

[ = 7

22 br{OE ]

Due to sudden floods, some welfare associations jointly requested the

government to get 100 tents fixed immediately and offered to contribute

50% of the cost. If the lower part of each tent is of the form of a cylinder

of diameter 4.2 m and height 4 m with the conical upper part of

same diameter but of height 2.8 m, and the canvas to be used costs < 100

per sq. m, find the amount, the associations will have to pay. What values

are shown by these associations ? [Use = 7

22]

30/2 7 P.T.O.

12. YamVb Ho$ EH$ {~ÝXþ A go EH$ hdmB© OhmµO H$m CÞ`Z H$moU 60 h¡ & 15 goH$ÊS H$s C‹S>mZ Ho$ níMmV², CÞ`Z H$moU 30 H$m hmo OmVm h¡ & `{X hdmB© OhmµO EH$ {ZpíMV D±$MmB© 1500 3 _rQ>a na C‹S> ahm hmo, Vmo hdmB© OhmµO H$s J{V {H$bmo_rQ>a/K§Q>m _| kmV H$s{OE &

The angle of elevation of an aeroplane from a point A on the ground is

60. After a flight of 15 seconds, the angle of elevation changes to 30. If

the aeroplane is flying at a constant height of 1500 3 m, find the speed

of the plane in km/hr.

13. EH$ AÕ©Jmobr` ~V©Z H$m AmÝV[aH$ ì`mg 36 go_r h¡ & `h Vab nXmW© go âm h¡ & Bg Vab H$mo 72 ~obZmH$ma ~moVbm| _| S>mbm J`m h¡ & `{X EH$ ~obZmH$ma ~moVb H$m ì`mg 6 go_r hmo, Vmo àËòH$ ~moVb H$s D±$MmB© kmV H$s{OE, O~{H$ Bg {H«$`m _| 10% Vab {Ja OmVm h¡ & A hemispherical bowl of internal diameter 36 cm contains liquid. This

liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height

of the each bottle, if 10% liquid is wasted in this transfer.

14. EH$ Oma _| Ho$db bmb, Zrbr VWm Zma§Jr a§J H$s J|X| h¢ & `mÑÀN>`m EH$ bmb a§J H$s J|X

Ho$ {ZH$mbZo H$s àm{`H$Vm 4

1 h¡ & Bgr àH$ma Cgr Oma go `mÑÀN>`m EH$ Zrbr J|X Ho$

{ZH$mbZo H$s àm{`H$Vm 3

1 h¡ & `{X Zma§Jr a§J H$s Hw$b J|X| 10 h¢, Vmo ~VmBE {H$ Oma _|

Hw$b {H$VZr J|X| h¢ &

The probability of selecting a red ball at random from a jar that contains

only red, blue and orange balls is 4

1. The probability of selecting a blue

ball at random from the same jar is 3

1. If the jar contains 10 orange balls,

find the total number of balls in the jar.

15. 10 go_r ^wOm dmbo EH$ KZmH$ma ãbm°H$ Ho$ D$na EH$ AY©Jmobm aIm hþAm h¡ & AY©Jmobo H$m A{YH$V_ ì`mg Š`m hmo gH$Vm h¡ ? Bg àH$ma ~Zo R>mog Ho$ g§nyU© n¥ð>r` joÌ H$mo n|Q> H$admZo H$m < 5 à{V 100 dJ© go_r H$s Xa go ì`` kmV H$s{OE & [ = 3.14 br{OE ]

A cubical block of side 10 cm is surmounted by a hemisphere. What is the

largest diameter that the hemisphere can have ? Find the cost of

painting the total surface area of the solid so formed, at the rate of < 5

per 100 sq. cm. [ Use = 3.14 ]

30/2 8

16. `{X (– 2, – 2) VWm (2, – 4) H«$_e… {~ÝXþ A VWm B Ho$ {ZX}em§H$ h¢, Vmo {~ÝXþ P Ho$

{ZX}em§H$ kmV H$s{OE O~{H$ P aoImIÊS> AB na h¡ VWm AP = 7

3AB.

If the coordinates of points A and B are (– 2, – 2) and (2, – 4) respectively,

find the coordinates of P such that AP = 7

3AB, where P lies on the line

segment AB.

17. 3.5 go_r ì`mg VWm 3 go_r D±$Mo 504 e§Hw$Am| H$mo {nKbmH$a EH$ YmpËdH$ Jmobm ~Zm`m J`m & Jmobo H$m ì`mg kmV H$s{OE & AV… BgH$m n¥ð>r` joÌ\$b kmV H$s{OE &

[ = 7

22 br{OE ]

504 cones, each of diameter 3.5 cm and height 3 cm, are melted and

recast into a metallic sphere. Find the diameter of the sphere and hence

find its surface area. [Use = 7

22]

18. EH$ g_MVw^w©O Ho$ g^r erf© EH$ d¥Îm na pñWV h¢ & `{X Bg d¥Îm H$m joÌ\$b 1256 dJ© go_r h¡, Vmo g_MVw^w©O H$m joÌ\$b kmV H$s{OE & [ = 3.14 br{OE ] All the vertices of a rhombus lie on a circle. Find the area of the rhombus,

if the area of the circle is 1256 cm2. [ Use = 3.14 ]

19. x Ho$ {bE hb H$s{OE :

2x2 + 6 3 x – 60 = 0

Solve for x :

2x2 + 6 3 x – 60 = 0

20. EH$ g_mÝVa lo‹T>r H$m 16dm± nX BgHo$ Vrgao nX H$m nm±M JwZm h¡ & `{X BgH$m 10dm± nX 41 h¡, Vmo BgHo$ àW_ 15 nXm| H$m `moJ\$b kmV H$s{OE &

The 16th term of an AP is five times its third term. If its 10th term is 41,

then find the sum of its first fifteen terms.

30/2 9 P.T.O.

IÊS> X

SECTION D

àíZ g§»`m 21 go 31 VH$ àËòH$ àíZ 4 A§H$ H$m h¡ & Question numbers 21 to 31 carry 4 marks each.

21. {gÕ H$s{OE {H$ d¥Îm H$s {H$gr Mmn Ho$ _Ü`-{~ÝXþ na ItMr JB© ñne© aoIm, Mmn Ho$ A§Ë` {~ÝXþþAm| H$mo {_bmZo dmbr Ordm Ho$ g_m§Va hmoVr h¡ &

Prove that the tangent drawn at the mid-point of an arc of a circle is

parallel to the chord joining the end points of the arc.

22. EH$ Prb _| nmZr Ho$ Vb go 20 _rQ>a D±$Mo {~ÝXþ A go, EH$ ~mXb H$m CÞ`Z H$moU 30 h¡ & Prb _| ~mXb Ho$ à{V{~å~ H$m A go AdZ_Z H$moU 60 h¡ & A go ~mXb H$s Xÿar kmV H$s{OE &

At a point A, 20 metres above the level of water in a lake, the angle of

elevation of a cloud is 30. The angle of depression of the reflection of the

cloud in the lake, at A is 60. Find the distance of the cloud from A.

23. AÀN>r Vah go \|$Q>r JB© EH$ Vme H$s JÈ>r go EH$ nÎmm `mÑÀN>`m {ZH$mbm J`m & àm{`H$Vm kmV H$s{OE {H$ {ZH$mbm J`m nÎmm

(i) hþHw$_ H$m nÎmm h¡ `m EH$ B¸$m h¡ &

(ii) EH$ H$mbo a§J H$m ~mXemh h¡ &

(iii) Z Vmo Jwbm_ h¡ VWm Z hr ~mXemh h¡ &

(iv) `m Vmo ~mXemh h¡ `m ~oJ_ h¡ &

A card is drawn at random from a well-shuffled deck of playing cards.

Find the probability that the card drawn is

(i) a card of spade or an ace.

(ii) a black king.

(iii) neither a jack nor a king.

(iv) either a king or a queen.

30/2 10

24. AmH¥${V 5 _|, PQRS EH$ dJm©H$ma bm°Z h¡ {OgH$s ^wOm PQ = 42 _rQ>a h¡ & Xmo d¥ÎmmH$ma \y$bm| H$s Š`m[a`m± ^wOm PS VWm QR na h¢ {OZH$m Ho$ÝÐ Bg dJ© Ho$ {dH$Um] H$m à{VÀN>oXZ

{~ÝXþ O h¡ & XmoZm| \y$bm| H$s Š`m[a`m| (N>m`m§{H$V ^mJ) H$m Hw$b joÌ\$b kmV H$s{OE &

AmH¥${V 5

In Figure 5, PQRS is a square lawn with side PQ = 42 metres. Two

circular flower beds are there on the sides PS and QR with centre at O,

the intersection of its diagonals. Find the total area of the two flower beds

(shaded parts).

Figure 5

25. EH$ R>mog YmVw Ho$ ~obZ Ho$ XmoZmo| {H$Zmam| go Cgr ì`mg Ho$ AÕ©Jmobo Ho$ ê$n _| YmVw {ZH$mbr JB© & ~obZ H$s D±$MmB© 10 go_r VWm BgHo$ AmYma H$s {ÌÁ`m 4.2 go_r h¡ & eof ~obZ H$mo {nKbmH$a 1.4 go_r _moQ>r ~obZmH$ma Vma ~ZmB© JB© & Vma H$s bå~mB© kmV H$s{OE &

[ = 7

22 br{OE ]

From each end of a solid metal cylinder, metal was scooped out in

hemispherical form of same diameter. The height of the cylinder is 10 cm

and its base is of radius 4.2 cm. The rest of the cylinder is melted and

converted into a cylindrical wire of 1.4 cm thickness. Find the length of

the wire. [Use = 7

22]

30/2 11 P.T.O.

26. EH$ Am`VmH$ma IoV H$m {dH$U© BgH$s N>moQ>r ^wOm go 16 _rQ>a A{YH$ h¡ & `{X BgH$s ~‹S>r ^wOm N>moQ>r ^wOm go 14 _rQ>a A{YH$ h¡, Vmo IoV H$s ^wOmAm| H$s bå~mB`m± kmV H$s{OE &

The diagonal of a rectangular field is 16 metres more than the shorter

side. If the longer side is 14 metres more than the shorter side, then find

the lengths of the sides of the field.

27. g_m§Va lo‹T>r 8, 10, 12, ... H$m 60dm± nX kmV H$s{OE, `{X Cg_| Hw$b 60 nX h¢ & AV… Bg lo‹T>r Ho$ A§{V_ 10 nXm| H$m `moJ\$b kmV H$s{OE &

Find the 60th term of the AP 8, 10, 12, ..., if it has a total of 60 terms and

hence find the sum of its last 10 terms.

28. EH$ ~g nhbo 75 {H$bmo_rQ>a H$s Xÿar {H$gr Am¡gV Mmb go MbVr h¡ VWm CgHo$ ~mX H$s 90 {H$bmo_rQ>a H$s Xÿar nhbo go 10 {H$bmo_rQ>a à{V K§Q>m A{YH$ H$s Am¡gV Mmb go MbVr h¡ & `{X Hw$b Xÿar 3 K§Q>o _| nyar hmoVr h¡, Vmo ~g H$s nhbr Mmb kmV H$s{OE &

A bus travels at a certain average speed for a distance of 75 km and then

travels a distance of 90 km at an average speed of 10 km/h more than the

first speed. If it takes 3 hours to complete the total journey, find its first

speed.

29. {gÕ H$s{OE {H$ d¥Îm Ho$ {H$gr {~ÝXþ na ñne© aoIm ñne© {~ÝXþ go OmZo dmbr {ÌÁ`m na b§~ hmoVr h¡ &

Prove that the tangent at any point of a circle is perpendicular to the

radius through the point of contact.

30. EH$ g_H$moU {Ì^wO ABC H$s aMZm H$s{OE, {Og_| AB = 6 go_r, BC = 8 go_r VWm B = 90 h¡ & B go AC na b§~ BD It{ME & {~ÝXþAm| B, C VWm D go hmoH$a OmZo dmbm EH$ d¥Îm It{ME VWm A go Bg d¥Îm na ñne© aoImAm| H$s aMZm H$s{OE &

Construct a right triangle ABC with AB = 6 cm, BC = 8 cm and B = 90.

Draw BD, the perpendicular from B on AC. Draw the circle through B, C

and D and construct the tangents from A to this circle.

31. k Ho$ _mZ kmV H$s{OE {OZgo (k+1, 1), (4, – 3) VWm (7, – k) erfm] dmbo {Ì^wO H$m joÌ\$b 6 dJ© BH$mB© hmo &$

Find the values of k so that the area of the triangle with vertices (k+1, 1),

(4, – 3) and (7, – k) is 6 sq. units.

30/2/1 1 P.T.O.

narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð >na Adí` {bIo§ & Candidates must write the Code on the

title page of the answer-book.

Series RLH/2 H$moS> Z§. 30/2/1

Code No.

amob Z§.

Roll No.

g§H${bV narjm – II

SUMMATIVE ASSESSMENT – II

J{UV

MATHEMATICS

{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 90

Time allowed : 3 hours Maximum Marks : 90

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 12 h¢ &

àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na {bI| &

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >31 àíZ h¢ &

H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| &

Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU nydm©• _| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &

Please check that this question paper contains 12 printed pages.

Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate.

Please check that this question paper contains 31 questions.

Please write down the Serial Number of the question before attempting it.

15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period.

SET-1

30/2/1 2

gm_mÝ` {ZX}e :

(i) g^r àíZ A{Zdm`© h¢ &

(ii) Bg àíZ-nÌ _| 31 àíZ h¢ Omo Mma IÊS>m| A, ~, g Am¡a X _| {d^m{OV h¢ &

(iii) IÊS> A _| EH$-EH$ A§H$ dmbo 4 àíZ h¢ & IÊS> ~ _| 6 àíZ h¢ {OZ_| go àËòH$ 2 A§H$m| H$m h¡ & IÊS> g _| 10 àíZ VrZ-VrZ A§H$m| Ho$ h¢ Am¡a IÊS> X _| 11 àíZ h¢ {OZ_| go àËòH$ 4 A§H$m| H$m h¡ &

(iv) H¡$bHw$boQ>a H$m à`moJ d{O©V h¡ &

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 calculators is not permitted.

IÊS> A

SECTION A

àíZ g§»`m 1 go 4 VH$ àË oH$ àíZ 1 A§H$ H$m h¡ & Question numbers 1 to 4 carry 1 mark each.

1. g_m§Va lo‹T>r – 5, 2

5 , 0, 2

5 , ... H$m 25dm± nX kmV H$s{OE &

Find the 25th term of the A.P. 5, 2

5, 0,

2

5, ...

30/2/1 3 P.T.O.

2. O~ gy`© H$m CÞ`Z H$moU 60° h¡, Vmo EH$ Iåô H$s ^y{_ na N>m`m H$s b§~mB© 2 3 _rQ>a h¡ & Iåô H$s D±$MmB© kmV H$s{OE &

A pole casts a shadow of length 2 3 m on the ground, when the sun’s

elevation is 60°. Find the height of the pole.

3. g§`moJ Ho$ EH$ Iob _| EH$ Vra H$mo Kw_m`m OmVm h¡, Omo éH$Zo na g§»`mAm| 1, 2, 3, 4, 5,

6, 7, 8 _| go {H$gr EH$ g§»`m H$mo B§{JV H$aVm h¡ & `{X `h g^r n[aUm_ g_àm{`H$ hm|, Vmo Vra Ho$ 8 Ho$ {H$gr EH$ JwUZIÊS> na éH$Zo H$s àm{`H$Vm kmV H$s{OE &

A game of chance consists of spinning an arrow which comes to rest

pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally

likely outcomes. Find the probability that the arrow will point at any

factor of 8.

4. {ÌÁ`mE± a VWm b (a > b) Ho$ Xmo g§Ho$ÝÐr` d¥Îm {XE JE h¢ & ~‹S>>o d¥Îm H$s Ordm, Omo N>moQ>o

d¥Îm H$s ñne© aoIm h¡, H$s bå~mB© kmV H$s{OE &

Two concentric circles of radii a and b (a > b) are given. Find the length of

the chord of the larger circle which touches the smaller circle.

IÊS> ~

SECTION B

àíZ g§»`m 5 go 10 VH$ àËòH$ àíZ Ho$ 2 A§H$ h¢ & Question numbers 5 to 10 carry 2 marks each.

5. AmH¥${V 1 _|, d¥Îm H$m Ho$ÝÐ O h¡ & PT VWm PQ Bg d¥Îm na ~mø {~ÝXþ P go Xmo

ñne©-aoImE± h¢ & `{X TPQ = 70° h¡, Vmo TRQ kmV H$s{OE &

AmH¥${V 1

30/2/1 4

In Figure 1, O is the centre of a circle. PT and PQ are tangents to the

circle from an external point P. If TPQ = 70°, find TRQ.

Figure 1

6. AmH¥${V 2 _|, 5 go_r {ÌÁ`m dmbo d¥Îm _| Ordm PQ H$s bå~mB© 8 go_r h¡ & P VWm Q na

ñne©-aoImE± nañna {~ÝXþ T na {_bVr h¢ & TP VWm TQ H$s bå~mB`m± kmV H$s{OE &

AmH¥${V 2

In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The

tangents at P and Q intersect at a point T. Find the lengths of TP and

TQ.

Figure 2

30/2/1 5 P.T.O.


x2 – ( 3 +1) x + 3 = 0

Solve for x :

x2 – ( 3 +1) x + 3 = 0

8. EH$ g_m§Va lo‹T>r H$m Mm¡Wm nX 11 h¡ & Bg g_m§Va lo‹T>r Ho$ nm±Md| VWm gmVd| nXm| H$m

`moJ\$b 34 h¡ & BgH$m gmd© AÝVa kmV H$s{OE &

The fourth term of an A.P. is 11. The sum of the fifth and seventh terms

of the A.P. is 34. Find its common difference.

9. {gÕ H$s{OE {H$ {~ÝXþ (a, a), (– a, – a) VWm (– 3 a, 3 a) EH$ g_~mhþ {Ì^wO Ho$ erf©

{~ÝXþ h¢ &

Show that the points (a, a), (– a, – a) and (– 3 a, 3 a) are the vertices of

an equilateral triangle.

10. k Ho$ {H$Z _mZm| Ho$ {bE {~ÝXþ (8, 1), (3, – 2k) VWm (k, – 5) g§aoIr` h¢ ?

For what values of k are the points (8, 1), (3, – 2k) and (k, – 5) collinear ?

IÊS> g

SECTION C

àíZ g§»`m 11 go 20 VH$ àËòH$ àíZ 3 A§H$m| H$m h¢ & Question numbers 11 to 20 carry 3 marks each.

11. {~ÝXþ A, {~ÝXþAm| P(6, – 6) VWm Q(– 4, –1) H$mo {_bmZo dmbo aoImIÊS> PQ na Bg àH$ma

pñWV h¡ {H$ 5

2

PQ

PA & `{X {~ÝXþ P aoIm 3x + k (y + 1) = 0 na ^r pñWV hmo, Vmo k H$m

_mZ kmV H$s{OE &

Point A lies on the line segment PQ joining P(6, – 6) and Q(– 4, –1) in

such a way that 5

2

PQ

PA . If point P also lies on the line 3x + k (y + 1) = 0,

find the value of k.

30/2/1 6


x2 + 5x – (a2 + a – 6) = 0

Solve for x :

x2 + 5x – (a2 + a – 6) = 0

13. `{X EH$ g_m§Va lo‹T>r H$m 12dm± nX –13 h¡ VWm BgHo$ àW_ Mma nXm| H$m `moJ\$b 24 h¡,

Vmo BgHo$ àW_ Xg nXm| H$m `moJ\$b kmV H$s{OE &

In an A.P., if the 12th term is –13 and the sum of its first four terms is 24,

find the sum of its first ten terms.

14. EH$ W¡bo _| 18 J|X§o h¢ {OZ_| x bmb J|X| h¢ &

(i) `{X W¡bo _| go EH$ J|X `mÑÀN>`m {ZH$mbr OmE, Vmo BgHo$ bmb J|X Ho$ Z hmoZo H$s

àm{`H$Vm Š`m h¡ ?

(ii) `{X W¡bo _| 2 bmb J|X| Am¡a S>mb Xr OmE±, Vmo bmb J|X Ho$ AmZo H$s àm{`H$Vm,

nhbr AdñWm _| bmb J|X Ho$ AmZo H$s àm{`H$Vm H$s 8

9 JwZm h¡ & x H$m _mZ kmV

H$s{OE &

A bag contains 18 balls out of which x balls are red.

(i) If one ball is drawn at random from the bag, what is the

probability that it is not red ?

(ii) If 2 more red balls are put in the bag, the probability of drawing a

red ball will be 8

9 times the probability of drawing a red ball in the

first case. Find the value of x.

30/2/1 7 P.T.O.

15. 50 _rQ>a D±$Mo Q>mda Ho$ {eIa go EH$ Iåô Ho$ erf© VWm nmX Ho$ AdZ_Z H$moU H«$_e: 30°

VWm 45° h¢ & kmV H$s{OE

(i) Q>mda Ho$ nmX go Iåô Ho$ nmX H$s Xÿar,

(ii) Iåô H$s D±$MmB© & ( 3 = 1·732 H$m à`moJ H$s{OE)

From the top of a tower of height 50 m, the angles of depression of the top

and bottom of a pole are 30° and 45° respectively. Find

(i) how far the pole is from the bottom of a tower,

(ii) the height of the pole. (Use 3 = 1·732)

16. EH$ K‹S>r H$s ~‹S>r gwB© VWm N>moQ>r gwB© H«$_e: 6 go_r VWm 4 go_r bå~r h¢ & gwB© m| H$s ZmoH$m|

Ûmam 24 K§Q>m| _| V` Xÿ[a`m| H$m `moJ\$b kmV H$s{OE & ( = 3.14 H$m à`moJ H$s{OE)

The long and short hands of a clock are 6 cm and 4 cm long respectively.

Find the sum of the distances travelled by their tips in 24 hours.

(Use = 3.14)

17. EH$ hr YmVw Ho$ Xmo Jmobm| H$m ^ma 1 {H$bmoJ«m_ VWm 7 {H$bmoJ«m_ h¡ & N>moQ>o Jmobo H$s {ÌÁ`m

3 go_r h¡ & XmoZmo Jmobm| H$mo {nKbm H$a EH$ ~‹S>m Jmobm ~Zm`m J`m & ZE Jmobo H$m ì`mg

kmV H$s{OE &

Two spheres of same metal weigh 1 kg and 7 kg. The radius of the

smaller sphere is 3 cm. The two spheres are melted to form a single big

sphere. Find the diameter of the new sphere.

18. YmVw H$o EH$ ~obZ H$s {ÌÁ`m 3 go_r VWm D±$MmB© 5 go_r h¡ & Bg H$m ^ma H$_ H$aZo Ho$

{bE ~obZ _| EH$ e§ŠdmH$ma N>oX {H$`m J`m & Bg e §ŠdmH$ma N>oX H$s {ÌÁ`m 2

3 go_r VWm

JhamB© 9

8 go_r h¡ & eof ~Mo ~obZ H$s YmVw Ho$ Am`VZ H$m e§ŠdmH$ma N>oX H$aZo hoVw

{ZH$mbr JB© YmVw Ho$ Am`VZ go AZwnmV kmV H$s{OE &

A metallic cylinder has radius 3 cm and height 5 cm. To reduce its

weight, a conical hole is drilled in the cylinder. The conical hole has a

radius of 2

3cm and its depth is

9

8cm. Calculate the ratio of the volume of

metal left in the cylinder to the volume of metal taken out in conical

shape.

30/2/1 8

19. AmH¥${V 3 _|, ABCD EH$ g_b§~ h¡ {Og_| AB | | DC h¡, AB = 18 go_r, DC = 32 go_r Am¡a AB VWm DC Ho$ ~rM H$s Xÿar 14 go_r h¡ & `{X A, B, C VWm D àËòH$ H$mo H|$Ð _mZ H$a g_mZ {ÌÁ`m 7 go_r H$s Mmn| {ZH$mbr JB© h¢, Vmo N>m`m§{H$V ^mJ H$m joÌ\$b kmV H$s{OE &

AmH¥${V 3

In Figure 3, ABCD is a trapezium with AB | | DC, AB = 18 cm,

DC = 32 cm and the distance between AB and DC is 14 cm. If arcs of

equal radii 7 cm have been drawn, with centres A, B, C and D, then find

the area of the shaded region.

Figure 3

20. nmZr go nyam âo 60 go_r {ÌÁ`m VWm 180 go_r D±$MmB© dmbo EH$ b§~d¥Îmr` ~obZ _|,

60 go_r D±$MmB© VWm 30 go_r {ÌÁ`m dmbm EH$ R>mog b§~d¥Îmr` e§Hw$ S>mbm J`m & ~obZ _|

~Mo nmZr H$m Am`VZ KZ _rQ>am| _| kmV H$s{OE & [ = 7

22 H$m à`moJ H$s{OE ]

A solid right-circular cone of height 60 cm and radius 30 cm is dropped in

a right-circular cylinder full of water of height 180 cm and radius 60 cm.

Find the volume of water left in the cylinder, in cubic metres.

[Use = 7

22]

30/2/1 9 P.T.O.

IÊS> X

SECTION D

àíZ g§»`m 21 go 31 VH$ àËòH$ àíZ 4 A§H$m| H$m h¢ & Question numbers 21 to 31 carry 4 marks each.

21. `{X x = – 2, g_rH$aU 3x2 + 7x + p = 0 H$m EH$ _yb h¡, Vmo k Ho$ dh _mZ kmV

H$s{OE, {H$ g_rH$aU x2 + k (4x + k – 1) + p = 0 Ho$ _yb g_mZ hm| &

If x = – 2 is a root of the equation 3x2 + 7x + p = 0, find the values of k so

that the roots of the equation x2 + k (4x + k – 1) + p = 0 are equal.

22. VrZ-A§H$m| dmbr CZ g^r g§»`mAm|, {OZH$mo 4 go ^mJ H$aZo na 3 eof AmVm h¡, go ~Zr lo ‹T>r H$m _Ü` nX kmV H$s{OE & _Ü` nX Ho$ XmoZm| Amoa AmZo dmbr g^r g§»`mAm| H$m AbJ-AbJ `moJ\$b ^r kmV H$s{OE &

Find the middle term of the sequence formed by all three-digit numbers

which leave a remainder 3, when divided by 4. Also find the sum of all

numbers on both sides of the middle term separately.

23. EH$ H$n ‹S>>o H$s Hw$N> b§~mB© H$s Hw$b bmJV <$ 200 h¡ & `{X H$n ‹S>m 5 _rQ>a A{YH$ bå~m hmo VWm àËòH$ _rQ>a H$s bmJV < 2 H$_ hmo, Vmo H$n ‹S>o H$s bmJV _| H$moB© n[adV©Z Zht hmoJm & H$n ‹S>o H$m dmñV{dH$ à{V _rQ>a _yë` kmV H$s{OE VWm H$n ‹S>o H$s bå~mB© ^r kmV H$s{OE & The total cost of a certain length of a piece of cloth is < 200. If the piece

was 5 m longer and each metre of cloth costs < 2 less, the cost of the piece

would have remained unchanged. How long is the piece and what is its

original rate per metre ?

24. {gÕ H$s{OE {H$ d¥Îm Ho$ {H$gr {~ÝXþ na ItMr JB© ñne©-aoIm Cg {~ÝXþ go JwµOaZo dmbr {ÌÁ`m na bå~ hmoVr h¡ &

Prove that the tangent at any point of a circle is perpendicular to the

radius through the point of contact.

30/2/1 10

25. AmH¥${V 4 _|, O Ho$ÝÐ dmbo d¥Îm Ho$ ~mø {~ÝXþ T go TP EH$ ñne©-aoIm h¡ & `{X

PBT = 30° h¡, Vmo {gÕ H$s{OE {H$ BA : AT = 2 : 1.

AmH¥${V 4

In Figure 4, O is the centre of the circle and TP is the tangent to the

circle from an external point T. If PBT = 30°, prove that

BA : AT = 2 : 1.

Figure 4

26. 3 go_r {ÌÁ`m H$m d¥Îm It{ME & Ho$ÝÐ go 7 go_r Xÿar na {~ÝXþ P go d¥Îm na Xmo ñne©-aoImE±

It{ME & BZ XmoZm| ñne©-aoImAm| H$s bå~mB© _m{nE &

Draw a circle of radius 3 cm. From a point P, 7 cm away from its centre

draw two tangents to the circle. Measure the length of each tangent.

27. g_mZ D±$MmB© Ho$ Xmo Iåô 80 _rQ>a Mm¡ ‹S>r g‹S>H$ Ho$ XmoZm| Amoa EH$-Xÿgao Ho$ gå_wI h¢ & BZ

XmoZm| Iå^m| Ho$ ~rM g ‹S>H$ Ho$ {H$gr {~ÝXþ P na EH$ Iåô Ho$ erf© H$m CÞ`Z H$moU 60° h¡

VWm Xÿgao Iåô Ho$ erf© go {~ÝXþ P H$m AdZ_Z H$moU 30° h¡ & Iå^m| H$s D±$MmB`m± VWm

{~ÝXþ P H$s Iå^m| go Xÿ[a`m± kmV H$s{OE &

30/2/1 11 P.T.O.

Two poles of equal heights are standing opposite to each other on either

side of the road which is 80 m wide. From a point P between them on the

road, the angle of elevation of the top of a pole is 60° and the angle of

depression from the top of another pole at point P is 30°. Find the heights

of the poles and the distances of the point P from the poles.

28. EH$ ~m°Šg _| g§»`m 6 go 70 VH$ H$s {JZVr Ho$ H$mS>© h¢ & `{X EH$ H$mS>© `mÑÀN>`m ~m°Šg go

ItMm OmE, Vmo àm{`H$Vm kmV H$s{OE {H$ ItMo JE H$mS>© na

(i) EH$ A§H$ H$s g§»`m h¡ &

(ii) 5 go nyU© {d^m{OV hmoZo dmbr g§»`m h¡ &

(iii) 30 go H$_ EH$ {df_ g§»`m h¡ &

(iv) 50 go 70 Ho$ _Ü` H$s EH$ ^mÁ` g§»`m h¡ &

A box contains cards bearing numbers from 6 to 70. If one card is drawn

at random from the box , find the probability that it bears

(i) a one digit number.

(ii) a number divisible by 5.

(iii) an odd number less than 30.

(iv) a composite number between 50 and 70.

29. EH$ g_~mhþ {Ì^wO ABC H$m AmYma BC, y-Aj na pñWV h¡ & {~ÝXþ C Ho$ {ZX}em§H$

(0, –3) h¢ & _yb {~ÝXþ AmYma H$m _Ü`-{~ÝXþ h¡ & {~ÝXþAm| A VWm B Ho$ {ZX}em§H$ kmV

H$s{OE & AV: EH$ AÝ` q~Xþ D Ho$ {ZX}em§H$ kmV H$s{OE {Oggo BACD EH$ g_MVw^w©O

hmo &

The base BC of an equilateral triangle ABC lies on y-axis. The

coordinates of point C are (0, –3). The origin is the mid-point of the base.

Find the coordinates of the points A and B. Also find the coordinates of

another point D such that BACD is a rhombus.

30/2/1 12

30. nmZr go âm EH$ ~V©Z CëQ>o e§Hw$ Ho$ AmH$ma H$m h¡ & Bg ~V©Z H$s D±$MmB© 8 go_r h¡ & ~V©Z

D$na go Iwbm h¡ {OgH$s {ÌÁ`m 5 go_r h¡ & Bg_| 100 Jmobr` Jmo{b`m± S>mbr JBª {Oggo

~V©Z H$m EH$-Mm¡WmB© nmZr ~mha Am J`m & EH$ Jmobr H$s {ÌÁ`m kmV H$s{OE &

A vessel full of water is in the form of an inverted cone of height 8 cm and

the radius of its top, which is open, is 5 cm. 100 spherical lead balls are

dropped into the vessel. One-fourth of the water flows out of the vessel.

Find the radius of a spherical ball.

31. EH$ XÿY dmbo ~V©Z, {OgH$s D±$MmB© 30 go_r h¡, EH$ e§Hw$ Ho$ {N>ÞH$ Ho$ AmH$ma H$m h¡,

{OgHo$ {ZMbo VWm D$nar d¥Îmr` {gam§o H$s {ÌÁ`mE± H«$_e: 20 go_r VWm 40 go_r h¢, _| âm

XÿY ~m ‹T>> nr{ ‹S>Vm| Ho$ {bE H¢$n _| {dV[aV {H$`m OmZm h¡ & `{X `h XÿY < 35 à{V brQ>a Ho$

^md go CnbãY h¡ VWm EH$ H¢$n Ho$ {bE H$_-go-H$_ 880 brQ>a XÿY à{V {XZ Mm{hE, Vmo

kmV H$s{OE {H$ Eogo {H$VZo ~V©Zmo§ H$m XÿY à{V {XZ H¢$n Ho$ {bE Mm{hE VWm XmVm EO|gr

H$mo à{V {XZ H¢$n Ho$ {bE Š`m ì`` H$aZm n‹S>oJm & Cnamoº$ go XmVm EO|gr Ûmam H$m¡Z-gm

_yë` àX{e©V {H$`m J`m h¡ ?

Milk in a container, which is in the form of a frustum of a cone of height

30 cm and the radii of whose lower and upper circular ends are 20 cm and

40 cm respectively, is to be distributed in a camp for flood victims. If this

milk is available at the rate of < 35 per litre and 880 litres of milk is

needed daily for a camp, find how many such containers of milk are

needed for a camp and what cost will it put on the donor agency for this.

What value is indicated through this by the donor agency ?



SUMMATIVE ASSESSMENT – II – 2014-2015

Mathematics

Class – X

Time allowed: 3:00 hours Maximum Marks: 90

General Instructions:

a. This question paper contains four parts A , B , C and D.

b. All questions are compulsory for all.

Section-A comprises of 4 questions of 1 mark each, Section-B comprises of 6 questions of 2

marks each, Section-C comprises of 10 questions of 3 marks each and Section-D comprises

of 11 questions of 4 marks each.

c. There is no overall choice

d. Use of calculator is not permitted

Section – A

1. Find sum of 10 terms of following A.P. :3 7

, 5 , ,...............5 5

2. A tower stands near an airport. The angle of elevation θ of the tower from a point on the

ground is such that its tangent is 5

,12

find the height of the lower, if the distance of the

observer from the tower is 120 meters.

3. A die is thrown once. Find the probability of getting “at most 2.”

4. A( 1, 1), B(6,1), C(8, 8), D(x, y)− − are the four vertices of a rhombus taken in order. Find the

co-ordinates of point D.

Section – B

5. Ram Prasad saved `10 in the first week of a year and then increased his weekly savings by

`2.75. If in the nth week, his savings become `59.50, find n.

6. Find the roots of the quadratic equation 25x 2 10x 2 0.− + =

7. Two tangents PA and PB are drawn to the circle with centre O such that 0APB 120∠ = Prove

that OA= 3 AP.

8. Draw a circle of radius 3.6 cm. Take a point P outside the circle and construct a pair of

tangents to the circle from that point.

9. In two concentric circles, a chord of length 24 cm of larger circle becomes a tangent to the

smaller circle whose radius is 5 cm. Find the radius of the larger circle.

10. In the given figure, OAPB is a sector of a circle of radius 3.5 cm with the centre at O. If

0AOB 120∠ = , then find the length of OAPBO. (use 22

7π = )



Section - C

11. Find the sum of n terms of the sequence na< > where na 5 6n= − and n is a natural

number.

12. The sum of a number and its reciprocal is 10

3 , find the number.

13. Draw a circle of radius 3 cm. Construct two tangents at the extreme ties of a diameter of this

circle.

14. A man observes the angle of elevation of a bird to be 030 . He then walks 100 m towards the

birds which is stationary and finds that the angle of elevation is060 . Find the height at which

the bird is sitting.

15. From a well shuffled pack of 52 cards, two black queens and two kings are removed. From

the remaining cards, a card is drawn at random. What is the probability that drawn card is :

a) a face card

b) an ace

16. Show that the line-segments joining the points (4, 2) and (-6,4) and (-10, 5) and (8, 1) bisect

each other.

17. The coordinates of the vertices of ABC△ are A( 7, 2), B (9,10) and C(1, 4). If E and F are the

mid points of AB and AC respectively, prove that 1

EF BC2

= .

18. In a cylinder of base radius 10 cm, liquid is filled to the height of 9 cm. A metal cube of

diagonal 8 3 cm is immersed completely in the liquid. Find the height by which the water

will rise in the cylinder.

19. The wheel of a motor cycle is of radius 21 cm. How many revolutions per minute must the

wheel make so as to keep a speed of 77 km/h?

20. A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller solid

cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.

SECTION – D

21. If th th thp ,q and r term of an A. P. are a, b and c respectively, then show that:

a(q r) (r p) c(p q) 0− + − + − =

22. Solve: 2y 2y 5 25

; y 3, 4y 4 y 3 3

−+ = ≠− −

23. If the equation ( )2 2 2 21 m x 2mcx (c a ) 0+ + + − = has equal roots, prove that 2 2 2c a (1 m )= +



24. In the figure FG is a tangent to the circle with centre A. If 0DCB 15∠ = and CE=DE, find

GCE and BCE∠ ∠ .

25. Draw ABC∆ such that BC=5 cm, 0 0ABC 60 and ACB 30∠ = ∠ = , now construct

ABC` ABC with A`B ; AB 3: 2.∆ ∆ =∼

26. Two pillars of equal heights stand on either side of a road, which is 200 m wide. The angles of

elevation of the top of the pillars are 0 060 and 30 at a point on the road between the pillars.

Find the position of the point between the pillars and height of each pillar.

27. 17 cards numbered 1, 2, 3 ………………….. , 16, 17 are put a box and mixed thoroughly . One

person draws a card from the box. Find the probability that the number on the card is

(a) odd (b) a prime

(c) divisible by 3. (d) divisible by 3 and 2 both.

28. Prove that the points A(0, 0), B( 0, 2 ) C (2, 0) are the vertices of an isosceles right triangle.

Also, find its area.

29. If h, C and V respectively represent the height, curved surface area and volume of a cone,

prove that 3 2

22

3 Vh 9VC

h

π +=

30. The area of equilateral triangle is 2196 3m . Three circles are drawn at the vertices of the

triangle with radius equal to the half of side of triangle. Find the area of the triangle not

included in the sectors.

31. A school thought to collect the rainwater from the roof of the building, whose dimensions are

22m 20m× by draining into a cylindrical vessel having diameter 7 m and height 4.2 m. If the

vessel is just full, find the rainfall recorded in cm.

Why it is necessary to conserve water by doing, these type of activities?

Documents

SUMMATIVE ASSESSMENT-I, 2015-16 CLASS-X ...icsk-kw.com/pdf/pqp/10/mat/2.pdfQ.21 What is the HCF and LCM of two prime numbers a and b? Three alarm clocks ring at intervals of 6, 9 and