C.B.S.E. 2012 Sample Papers for X Maths (5 Sets) (1)

Important Instructions for the School Principal (Not to be printed with the question paper) 1) This question paper is strictly meant for use in school based SA-I, September-2012 only. This question paper is not to be used for any other purpose except mentioned above under any circumstances. 2) The intellectual material contained in the question paper is the exclusive property of Central Board of Secondary Education and no one including the user school is allowed to publish, print or convey (by any means) to any person not authorised by the board in this regard. 3) The School Principal is responsible for the safe custody of the question paper or any other material sent by the Central Board of Secondary Education in connection with school based SA-I, September-2012, September-2012, in any form including the print-outs, compact-disc or any other electronic form. 4) Any violation of the terms and conditions mentioned above may result in the action criminal or civil under the applicable laws/byelaws against the offenders/defaulters. offenders/defaulters.

Note: Please ensure that these instructions are not printed with the question paper being administered to the examinees.

Page 1 of 9

I, 2012 SUMMATIVE ASSESSMENT – I, 2012

MA2-021

/ MATHEMATICS X / Class – X 3

90

Time allowed : 3 hours

Maximum Marks : 90

(i) (ii)

34

8

1

6

3

10

(iii)

1

2

10

4

8

(iv)

2 3

4

3

2

(v)

General Instructions: (i) (ii)

(iii) (iv)

(v)

All questions are compulsory. The question paper consists of 34 questions divided into four sections A, B, C and D. Section-A comprises of 8 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 10 questions of 4 marks each. Question numbers 1 to 8 in Section-A are multiple choice questions where you are required to select one correct option out of the given four. There is no overall choice. However, internal choices have been provided in 1 question of two marks, 3 questions of three marks each and 2 questions of four marks each. You have to attempt only one of the alternatives in all such questions. Use of calculator is not permitted.

Page 2 of 9

SECTION–A 1

8

1

Question numbers 1 to 8 carry one mark each. each. For each question, question, four alternative choices have been provided of which only one is correct. You have to select the correct correct choice. 1.

3 8

(A) 3 8

0.125

(B)

0.0125

(C)

0.0375

(D)

0.375

(B)

0.0125

(C)

0.0375

(D)

0.375

(C)

3, 4

(D)

3, 4

in decimal form is :

(A) 2.

0.125

p(x p(x)4x212x 12x9 (A)

3 2

,

3 2

(B)

3

3

2

2

 ,

The zeroes of the polynomial p(x p(x)4x212x 12x9 are : (A)

3 2

,

3 2

3.

(B)

3

3

2

2

 ,

(C)

3, 4

(D)

3, 4

(C)

2.75

(D)

3

(C)

2.75 cm

(D)

3 cm

(D)

2ab

(D)

2ab

ABC  PQR

x (A)

2.5

(B)

3.5

In the given figure if ABC  PQR

The value of x is : (A) 2.5 cm 4.

(B)

3.5 cm

xa cos, yb sin b2x2a2y2a2b2 (A) 1 (B) (C) 0 1 If xa cos, yb sin, then b2x2a2y2a2b2 is equal to : (A) 1 (B) (C) 0 1

Page 3 of 9

5. 111

(A)

125

127

(B)

8

19

(C)

3

5 2

2

9

(D)

455

A rational number which has non terminating decimal representation is : 111

(A)

6.

125

127

(B)

8

xa, yb

19

(C)

xy2

3

5 2

xy4

2

9

(D)

a

455

b

(A) 3, 5 (B) 5, 3 (C) 3, 1 (D) 1, 3 If xa, yb is the solution of the pair of equation xy2 and xy4, then the respective values of a and b are : (A) 3, 5 (B) 5, 3 (C) 3, 1 (D) 1, 3 7.

sin260sin230 (A)

1 4

1

(B)

2

3

(D)



(D)



(C)

17.5

(D)

15

(C)

17.5

(D)

15

(C)

4

1 2

The value of sin 260sin230 is : (A) 8.

1 4

1

(B)

2

3

(C)

4

1 2

10 – 25 (A) 17 (B) 18 The class mark of the class 10 – 25 is : (A) 17 (B) 18

/ SECTION-B 9

14

2

Question numbers 9 to 14 carry two marks each. 9.

255

867

Find the HCF of 255 and 867 by Euclid division algorithm. 10.

f (x)2x27x3

p, q

p2q2

If p, q are zeroes of polynomial f polynomial f (x)2x27x3, find the value of p2q2. 11.

QPR90,

PQR KR8

PQ24

QR26

PKR

PKR90,

PK

In the given triangle PQR,

QPR90,

PQ24 cm and QR26 cm and in PKR, Page 4 of 9

PKR90 and

12.

3

sinA If sinA

KR8 cm find PK.

2cot2A1

2 3

2

, find the value of 2cot2A1.

13.

2

2

2

Find the quadratic polynomial whose zeroes are

2

and

2

2

.

14.

0 – 6

6 – 12

12 – 18

18 – 24

24 – 30

7 5 10 12 6 Find the mean of the following frequency distribution : Class : 0 – 6 6 – 12 12 – 18 18 – 24 24 – 30 Frequency : 7 5 10 12 6 /OR 0 – 6

6 – 12

12 – 18

18 – 24

24 – 30

7 5 10 12 6 Find the mode of the following frequency distributions : Class : 0 – 6 6 – 12 12 – 18 18 – 24 24 – 30 Frequency : 7 5 10 12 6 SECTION-C 15

24

3

Question numbers 15 to 24 carry three marks each. 15.

Prove that the sum of squares on the sides of a rhombus is equal to sum of squares on its diagonals. 16.

Show that

1 2

1

3

2

2

and

3

2

4x24x3

are the zeroes of the polynomial 4x24x3 and verify the

relationship between zeroes and co-efficients of polynomial.

Page 5 of 9

17.

43 Prove that 43

2

2

is an irrational number. /OR a

0.3 178

b

Express the number

0.3 178

in the form of rational number

a b

.

18. cos50  2sin40 

4 cosec 59   tan 31  2



2

2

3tan 45



2 3

tan12  tan78.sin90

Find the value of the following without using trigonometric tables : cos50  2sin40 

19.

4 cosec 59   tan 31  2



2

2

3tan 45



3

tan12  tan78.sin90

2x39x2xb

(2x3)

b

2

Find the value of b for which (2x3) is a factor of 2x39x2xb 20.

3x5y20, 6x10y400 Using graph, find whether the pair of linear equations 3x5y20, 6x10y400 is consistence or inconsistent. Write its solution. /OR x

y 6



x 1

5 x 1

3

1

y 2 

1 y 2

2

,

x 1, y  2

Solve for x and y : 6



x 1

5 x 1

3

1

y 2 

1 y 2

2

, where x 1, y  2

21.

27 0 – 10

10 – 20

p 20 – 30

30 – 40

40 – 50

8 p 12 13 10 If the mean of the following distribution is 27, find the value of p : Class : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 Frequency : 8 p 12 13 10 22.

If the areas of two similar triangles are equal, then prove that they are congruent. /OR Page 6 of 9

ABC

DBC

PQBA

BC

PRBD

QRAD.

In the given figure, two triangles ABC and DBC lie on same side of BC such that PQ BA and PRBD. Prove that QRAD.

23.

sin3cos(6), 3 6  If sin3cos(6), where 3 and 6 are both acute angles, find the value of .

24.

0 – 10

10 – 20

20 – 30

30 – 40

8 16 36 34 Find mean, and median for the following data : Class : 0 – 10 10 – 20 20 – 30 30 – 40 Frequency : 8 16 36 34

40 – 50 6 40 – 50 6

/ SECTION-D 25

34

4

Question numbers 25 to 34 carry four marks each. 25.

3n

3n1

By Euclid division algorithm, show that square of any positive integer is of the form 3n or 3n1. 26.

k 3xy1 (2k1)x(k1)y2k1 For what value of k will the pair of equations have no solution ? 3xy1 (2k1)x(k1)y2k1 Page 7 of 9

27.

(secAtanA)2(1sinA)1sinA Prove that (secAtanA)2 (1sinA)1sinA

28.

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

70 – 80

8

10

14

12

4

2

Draw ‘less than’ and ‘more than’ ogives for the following distribution : Scores : 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 Frequency : 8 10 14 12 4 2 Hence find they median. Verify the result through calculations. 29.

p(x)8x414x32x28x12

4x23x2

p(x)

What must be subtracted or added to p(x)8x414x32x28x12 so that 4x23x2 is a factor of p(x) ? /OR x

y 133x 87y353 87x133y307

Solve for x and y 133x 87y353 and 87x133y307 30.

ABC

AB

AC

P

BC

Q

AD

PQ

BC

A

PQ

In ABC, P and Q are the points on the sides AB and AC respectively such that PQ is parallel to BC. Prove that median AD drawn from A to BC bisects PQ also. /OR ABC

3AB24AD2.

ADBC

In an equilateral ABC, ADBC. Prove that 3AB24AD2. 31.

sincosm

seccosecn,

n(m21)2m

If sincosm and seccosecn, then prove that n(m21)2m 32.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides. 33.

tan  sin tan  sin

Prove that :

tan  sin tan  sin





sec  1 sec  1

sec  1 sec  1

Page 8 of 9

34.

f 1

65 0 – 20

20 – 40

40 – 60

60 – 80

80 – 100

100 – 120

6

8

f 1

12

6

5

6, 8, f 1 12 Find the value of f 1 from the following data if its mode is 65 : Class 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 Frequency 6 8 f 1 12 6 where frequency 6, 8, f 1 and 12 are in ascending order.

100 – 120 5

-oOo-

Page 9 of 9

Important Instructions for the School Principal (Not to be printed with the question paper) 1) This question paper is strictly meant for use in school based SA-I, September-2012 only. This question paper is not to be used for any other purpose except mentioned above under any circumstances. 2) The intellectual material contained in the question paper is the exclusive property of Central Board of Secondary Education and no one including the user school is allowed to publish, print or convey (by any means) to any person not authorised by the board in this regard. 3) The School Principal is responsible for the safe custody of the question paper or any other material sent by the Central Board of Secondary Education in connection with school based SA-I, September-2012, in any form including the print-outs, compact-disc or any other electronic form. 4) Any violation of the terms and conditions mentioned above may result in the action criminal or civil under the applicable laws/byelaws against the offenders/defaulters.


Page 1 of 9


MA2-025


90


Maximum Marks : 90

(i) (ii)

34

8

1

6

3

10

(iii)

1

2

10

4

8

(iv)

2 3

4

3

2

(v)


(iii) (iv)

(v)


Page 2 of 9

SECTION–A 1

8

1

Question numbers 1 to 8 carry one mark each. For each question, four alternative choices have been provided of which only one is correct. You have to select the correct choice. 1.

189 125

(A)

1

(B)

2

(C)

3

(D)

4

189

The decimal expansion of (A) (C) 2.

125

will terminate after :

1 place of decimal 3 places of decimal

(B) (D)

2 places of decimal 4 places of decimal

3 (A)

(B)

(C)

(D)

The maximum number of zeroes that a polynomial of degree 3 can have is : (A) One (B) Two (C) Three (D) None 3.

ABC

PQR

60

PQ9

36

AB (A)

6

(B)

10

(C)

15

(D)

24

The perimeters of two similar triangles ABC and PQR are 60 cm and 36 cm respectively. If PQ9 cm, then AB equals : (A) 6 cm (B) 10 cm (C) 15 cm (D) 24 cm 4.

sin (A)

1 2

3

(B)

2

(C)

1

(D)

(C)

1

(D)

1 2

The maximum value of sin is : (A)

1 2

5.

3

(B)

2

20

3x2y6

2

24

(A) 240 (B) 480 (C) 120 The least positive integer divisible by 20 and 24 is : (A) 240 (B) 480 (C) 120 6.

1

(D)

960

(D)

960

y-

(A) (2, 0) (B) (0, 3) (C) (2, 0) (D) (0, 3) The point of intersection of the lines represented by 3x2y6 and the y-axis is : (A) (2, 0) (B) (0, 3) (C) (2, 0) (D) (0, 3) Page 3 of 9

7.

A, B

C

sin  

(A)

C 2

tan  

ABC  

(B)

cos  

C 2

 

(C)

AB 

cot  

 

2

C 2

  

(D)

tan  

C

tan  

C

2

 

AB  If A, B and C are interior angles of a ABC, then tan   equals :  2 

sin  

(A)

C 2

 

(B)

cos  

C 2

 

(C)

cot  

C 2

  

(D)

8.

2

 

(20.5, 15.5) (A) 36.0 (B) 20.5 (C) 15.5 (D) 5.5 If the „less than‟ type ogive and „more than‟ type ogive intersect each other at (20.5, 15.5), then the median of the given data is : (A) 36.0 (B) 20.5 (C) 15.5 (D) 5.5 / SECTION-B 9

14

2

Question numbers 9 to 14 carry two marks each. 9.

(867, 255) Find the HCF (867, 255) using Euclid‟s division lemma.

10.



4

1

5

3 4

1

5

3

Write the quadratic polynomial whose zeroes are  and 11.

ABCDEF ,

ABCD

. AE ED

BF 

FC

In the given figure, if ABCD is a trapezium in which ABCDEF, then prove that AE ED

12.

BF 

FC

.

2 sin2

3

cos2

Find the value of cos2 if 2 sin2

3

. Page 4 of 9

13.

4t25 Find the zeroes of the polynomial 4t25.

14.

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

1 3 5 9 7 3 Find the sum of lower limit of mediun class and the upper limit of model class : Classes : 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 Frequency : 1 3 5 9 7 3 /OR 50 – 55

55 – 60

60 – 65

65 – 70

70 – 75

2 8 12 24 38 Convert the following data into more than type distribution : Class : 50 – 55 55 – 60 60 – 65 65 – 70 70 – 75 Frequency : 2 8 12 24 38

75 – 80 16 75 – 80 16

SECTION-C 15

24

3


AB2CD2BD2AC2

ADBC

In the given figure, if ADBC, prove that AB2CD2BD2AC2

16.

x

2

6xa





a

2

If  and  are zeroes of the polynomial x26xa, find a if 2.

Page 5 of 9

17.

x, y

„ y‟

z

„z‟

'x'

Find the value of x, y, and z in the following factor tree. Can the value of 'x' be found without finding the value of „y‟ and „z‟, if yes, explain :

/OR 2

Prove that 18.

2

cosec If cosec

19.

is irrational.

13

2 sin  3 cos

12

4 sin  9 cos

13 12

, then evaluate ax25xc

2 sin  3 cos 4 sin  9 cos

.

10

„a‟

„c‟

If the sum and product of the zeroes of the polynomial a x25xc is equal to 10 each, find the value of „a‟ and „c‟. 20. x5y6

; 2x10y12 Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair : x5y6 ; 2x10y12 /OR





2x3y7 ; 2  x()y28 Find the value of  and  for which the following pair of linear equations has infinite number of solutions : 2x3y7 ; 2  x()y28 Page 6 of 9

21.

0 – 20

20 – 40

40 – 60

25 16 28 Compute the mode of the following data : Class : 0 – 20 20 – 40 40 – 60 Frequency : 25 16 28 22.

ABC

AB AD



AB

60 – 80

80 – 100

20

5

60 – 80 20

80 – 100 5

AC AE AC

D

E

BC

.

If a line segment intersects sides AB and AC of a ABC at D and E respectively and is parallel to BC, prove that

AD



AB

AE AC

. /OR

ABCD

AOB

ABDC

AB2CD

O

COD

The diagonals of a trapezium ABCD, in which AB DC intersect at O. If AB2CD, then find the ratio of areas of triangles AOB and COD. 23.

sinA, tanA cosecA secA Express sinA, tanA and cosecA in terms of secA.

24.

50 – 60

60 – 70

70 – 80

80 – 90

90 – 100

6 5 9 12 6 Draw the less than type ogive for the following data and hence find the median from it. Classes : 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100 Frequency : 6 5 9 12 6 / SECTION-D 25

34

4


n

n2n 2

Prove that n2n is divisible by 2 for every positive integer n. 26.

x

y

: 2(3xy)5xy ; 2(x3y)5xy

Solve for x and y : 2(3xy)5xy ; 2(x3y)5xy 27.

sec41.sin49cos49.cosec41 Evaluate : sec41.sin49cos49.cosec41

2 3

2 3

tan20tan60tan703(cos245sin290)

tan20tan60tan703(cos245sin290)

Page 7 of 9

28.

28.5 0 – 10

60

10 – 20

20 – 30

30 – 40

40 – 50

„p‟

„q‟

50 – 60

5 p 20 15 q 5 The median of the following frequency distribution is 28.5 and the sum of all the frequencies is 60. Find the values of „p‟ and „q‟ : Classes : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 Frequency : 5 p 20 15 q 5 29.

2x3x213x6

3

Show that 3 is a zero of the polynomial 2 x3x213x6. Hence find all the zeroes of this polynomial. /OR 5

3

9

3

2

67 The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and the breadth is increased by 3 units. The area is increased by 67 square units if length is increased by 3 units and breadth is increased by 2 units. Find the perimeter of the rectangle. 30.

Prove that “The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides”. /OR Prove that “In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides”. 31.

(sinAsecA)2(cosAcosecA)2(1secAcosecA)2 Prove that (sinAsecA)2(cosAcosecA)2(1secAcosecA)2

32.

If two sides and a median bisecting one of these sides of a triangle are respectively proportional to the two sides and the corresponding median of another triangle, then prove that the two triangles are similar. tanA

33.

secA  1

Prove that :

tanA secA  1





tanA

 2 cosecA

secA  1 tanA

 2 cosecA

secA  1

Page 8 of 9

34.

` `

< 100

< 200

< 300

< 400

< 500

12 28 34 41 50 Calculate the average daily income (in ` ) of the following data about men working in a company : Daily income (in ` ) < 100 < 200 < 300 < 400 < 500 Number of men 12 28 34 41 50 -oOo-

Page 9 of 9



Page 1 of 9


MA2-035


90


Maximum Marks : 90

(i) (ii)

34

8

1

6

3

10

(iii)

1

2

10

4

8

(iv)

2 3

4

3

2

(v)


(iii) (iv)

(v)


Page 2 of 9

SECTION–A 1

8

1

Question numbers 1 to 8 carry one mark each. For each question, four alternative choices have been provided of which only one is correct. You have to select the correct choice. 1.

23 3 2

2 5

(A)

(B)

(C)

(D)

Decimal expansion of (A) (C)

23 3 2

2 5

will be :

terminating non-terminating and repeating

2.

5

(B) (D)

non-terminating non-terminating and non-repeating

4

(A) x25x4 (B) x25x4 (C) x2x20 (D) x29x20 The polynomial whose zeroes are 5 and 4 is : (A) x25x4 (B) x25x4 (C) x2x20 (D) x29x20 3.

DEF ABC ABC 

(A)

DEF

99 

DE : AB2 : 3

(B)

120

DEF (C)

44

176 9

(D)

66

ABC ; If DE : AB2 : 3 and ar(DEF) is equal to 44 square units, then area (ABC)

in square units is : (A) 4.

5.

99

(B)

120

3sin2202tan2453sin270 (A) 0 (B) 1 3sin2202tan2453sin270 is equal to : (A) 0 (B) 1 2332

2233

176 9

(D)

66

(C)

2

(D)

1

(C)

2

(D)

1

(C)

2333

(D)

2232

(C)

2333

(D)

2232

(LCM)

(A) 23 (B) 33 L.C.M. of 2332 and 2233 is : (A) 23 (B) 33 6.

(C)

x2, y3 (A) 2x3y130 (C) 2x3y130 x2, y3 is a solution of the linear (A) 2x3y130 (C) 2x3y130

(B) 3x2y310 (D) 2x3y130 equation : (B) 3x2y310 (D) 2x3y130 Page 3 of 9

7.

sin

a

b

(A) a

2

b

2

Given that sin b

(A) a

2

b

tan

b

2

b

(B)

2

b a a b

2

a

(C) a

2

b

2

(D)

a 2

b a

2

, then tan is equal to : b

(B)

2

b a

2

a

(C) a

2

b

2

(D)

a 2

b a

2

8.

(A)

3

(B)

3

(C)

3

   3

2 2 2 2

(D) Relationship among mean, median and mode is : (A) 3 MedianMode2 Mean (B) 3 MeanMedian2 Mode (C) 3 ModeMean2 Median (D) Mode3 Mean2 Median / SECTION-B 9 14 2 Question numbers 9 to 14 carry two marks each. 9.

240 228 (HCF) Using Euclid’s algorithm, find the HCF of 240 and 228.

10.

2 3

2 3

Find a quadratic polynomial whose zeroes are 2 3 and 2 3 . 11.

ABDC

x

In the given figure, if ABDC, find the value of x.

Page 4 of 9

12.

3 sincos0 If

0 <  < 90



3 sincos0 and 0 <  < 90, find the value of .

13.

21

5

8

16

Find the quadratic polynomial whose sum and product of the zeroes are

21 8

and

5 16

respectively. 14.

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

4 8 10 12 10 Convert the following distribution to a ‘more than type’ cumulative frequency distribution : Class : 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 Frequency : 4 8 10 12 10 /OR 0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

3 8 9 10 3 Find the mode of the following frequency distribution : Class : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 Frequency : 3 8 9 10 3 SECTION-C 15

24

3

Question numbers 15 to 24 carry three marks each. 15. D, E

ABC

F

AB, BC

CA

DEF

ABC

D, E, F are respectively the mid-point of the sides AB, BC and CA of ABC. Find the ratios of the area of DEF and ABC. 16.

2x235x Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the co-efficients 2x235x.

17. 336

54



(LCM)

(HCF)



Find the LCM and HCF of 336 and 54 and verify that LCM HCFProduct of the two numbers. /OR Page 5 of 9

847, 2160 Using Euclids division algorithm, find whether the pair of numbers 847, 2160 are coprimes or not. 18.

Prove that : 19.

1  secA secA



sin 2A 1  cosA

1  secA secA



sin 2A 1  cosA

x311x223x35

1

5

Find all the zeroes of x311x223x35, if two of its zeros are 1 and 5. 20. k

3xy1; (2k1)x(k1)y2k1

For which value of k will the following pair of linear equations have no solution ? 3xy1, (2k1)x(k1)y2k1. /OR 7

2

The sum of digits of a two-digit numbers is 7. If the digits are reversed, the new number decreased by 2 equals twice the original number. Find the number. 21.

10

6

20

15

30

29

40

41

50

60

60

70 Calculate the median for the following distribution : Marks obtained Number of students Below 10 6 Below 20 15 Below 30 29 Below 40 41 Below 50 60 Below 60 70 22.

ABCD,

AOB

ABDC

O

AB2CD

COD

Diagonals of a trapezium ABCD with AB DC intersect each other at the point O. AB2CD, find the ratio of the area of triangles AOB and COD. /OR

If

Page 6 of 9

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

 1  tan 2A  cotA

23.

2

cosec A

Prove that :

 1  tan 2 A  cotA 2

cosec A

 tanA

 tanA

24.

10

14

20

22

30

37

40

58

50

67

60

75 Compute the arithmetic mean for the following data : Marks obtained No. of students Less than 10 14 Less than 20 22 Less than 30 37 Less than 40 58 Less than 50 67 Less than 60 75 / SECTION-D 25

34

4


3m

3m1

m Use Euclids Division Lemma to show that the square of any positive integer is either of the form 3m or 3m1 for some integer m. 26.

x3y6 ; 2x3y12 Check graphically, whether the pair of equations x3y6 ; 2x3y12 is consistent. If so, than solve them graphically.

27.

sec  tan  1 cos  tan  sec  1 1  sin sec  tan  1 cos Prove that :  tan  sec  1 1  sin

Page 7 of 9

28.

N100 0 – 10

10 – 20

32 20 – 30

f 1

30 – 40

40 – 50

f 2

50 – 60

10 f 1 25 30 f 2 10 100 Find the missing frequencies f 1 and f 2 in the following frequency distribution table, if N100 and median is 32. Class : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 Total Frequency : 10 f 1 25 30 f 2 10 100 29.

x46x316x225x10

(x22xk)

xa

k

a

If the polynomial x46x316x225x10 is divided by (x22xk) the remainder comes out to be xa, find k and a. /OR x

5

y

x  1

Sovle for x and y :

30.

5

1



x  1

ABC O



y 2

1 y 2 2 ;

2 ;

6 x  1

6 x  1 

3



3 y 2

y 2

1

1

DBC

BC

AD

BC

(ABC) AO  . DO (DBC)

In the given figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC ar(ABC) AO at O, show that .  ar( DBC) DO

/OR

If the area of two similar triangles are equal, prove that they are congruent. 31.

cot(90  ) sin(90  ) cot 40 2 2    cos 20  cos 70  sin tan 50 cot(90  ) sin(90  ) cot 40 2 2 Evaluate :    cos 20  cos 70  sin tan 50 Page 8 of 9

32.

Prove that the ratio of the area of two similar triangles is equal to the ratio of the squares of their corresponding sides. 33.

cos sin   (cos  sin ) 1  tan 1  cot cos sin Prove that :   (cos  sin ) 1  tan 1  cot

34.

200 – 250 – 300 – 350 – 400 – 450 – 500 – 550 – 250 300 350 400 450 500 550 600 30 15 45 20 25 40 10 15 For the following frequency distribution, draw a cumulative frequency curve of less than type. 200 – 250 – 300 – 350 – 400 – 450 – 500 – 550 – Class : 250 300 350 400 450 500 550 600 Frequency: 30 15 45 20 25 40 10 15 -oOo-

Page 9 of 9



Page 1 of 9


MA2-038


90


Maximum Marks : 90

(i) (ii)

34

8

1

6

3

10

(iii)

1

2

10

4

8

(iv)

2 3

4

3

2

(v)


(iii) (iv)

(v)


Page 2 of 9

SECTION – A

1

8

1

Question numbers 1 to 8 carry 1 mark each. In each question, select one correct option out of the given four. 1.

(HCF) (a, b)12

ab1800

(LCM )(a, b)

(A) 1800 (B) 900 (C) 150 If HCF (a, b) 12 and ab1800, then LCM (a, b) is : (A) 1800 (B) 900 (C) 150 2x25x1

2.

(A)

2

 (B)



(D)

90

(D)

90

(D)

3



1

(C)

1

If  and  are the zeroes of the polynomial 2x25x1, then the value of  is : (A) (B) (C) 1 (D) 3 2 1 3.

ABC

BDE

D

ABC

BC

BDE

(A) 2:1 (B) 1:2 (C) 4:1 (D) 1:4 ABC and BDE are two equilateral triangles such that D is the mid point of BC. Ratio of the areas of triangle ABC and BDE is : (A) 2:1 (B) 1:2 (C) 4:1 (D) 1:4 4.

1 , 0<  < 90 sec 

(A)

1

(B)

Maximum value of (A)

1

1 sec



(B)

2

(C)

1 2

(D)

(D)

1 2

, 0<  < 90 is : 2

(C)

1 2

1 2

5.

(A)

2, 3

5 (B)

2

3

(C)

3

5

(D)

2

5

A rational number can be expressed as a terminating decimal if the denominator has factors : (A) 2, 3 (or) 5 only (B) 2 (or) 3 only (C) 3 (or) 5 only (D) 2 (or) 5 only 6.

(A)

(B) Page 3 of 9

(C)

(D)

If a pair of linear equations is consistent, then the lines represented by these equations will be : (A) parallel (b) coincident always (C) intersecting (or) coincident (D) intersecting always 7.

2 tan 30 1  tan 2 30

(A)

sin60

The value of (A)

(B)

cos60

(C)

tan60

(D)

sin30

cos60

(C)

tan60

(D)

sin30

60 - 80

(D)

80 - 100

60 - 80

(D)

80 – 100

2 tan 30 1  tan 2 30

sin60

(B)

8.

20 – 40 40 – 60 60 – 80 80 – 100 10 (A)

20 - 40

12 (B)

20

22

40 - 60

(C)

The median class for the following data is : Class 20 – 40 40 – 60 60 – 80 80 – 100 Frequency 10 12 20 22 (A)

20 - 40

(B)

40 - 60

(C)

/ SECTION-B

9

14

2

Question numbers 9 to 14 carry 2 marks each. 9.

918 162 (HCF) Find the HCF of 918 and 162 using Euclid’s Division Algorithm.

10.

8 56 Form a quadratic polynomial whose one zero is 8 and the product of the zeroes is 56.

11.

X

Page 4 of 9

In the given figures, find the measure of X.

12.

ABCD APBAQD30

AP

AQ

(APAQ)

In the given figure, ABCD is a rectangle in which segment AP and AQ are drawn such that APBAQD30. Find the length of (APAQ).

13.

p(x)(a29) x245x6a

a

If one zero of the polynomial p(x)(a29) x245x6a is reciprocal of the other, find the value of a. 14.

0 – 20

20 – 40 40 – 60 60 – 80 80 – 100

6 21 24 5 4 Find the mode of the following data. Marks obtained : 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 No. of students 6 21 24 5 4 /OR 0 – 10

10 – 20 20 – 30 30 – 40 40 – 50

5 15 20 23 17 Write the following frequency distribution as “more than type” and “less than type” cumulative frequency distribution. Class : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 Frequency : 5 15 20 23 17 Page 5 of 9

SECTION-C 15

24

3


DEBC

 ADE

DE : BC3 : 5



In the given figure, DEBC. If DE : BC3 : 5, find

p(x)(a1)x2(2a3)x(3a4)

16.

BCED 

ar  ADE  ar  trap.BCED 

1

If the sum of the zeroes of the polynomial p(x)(a1)x2(2a3)x(3a4) is 1, then find the product of its zeroes. 17.

3 5

Prove that

3  5 is irrational. /OR

404

96



(HCF)

(LCM)



Find the HCF and LCM of 404 and 96 and verify HCFLCMProduct of two given numbers. 18.

3 cos 55 7 sin 35

Find the value of :



3 cos 55 7 sin 35

4 cos 70  . cosec 20  . cos 30 





7 tan 5 . tan 25 . tan 60  . tan 65 . tan 85 



4 cos 70 . cosec 20  . cos 30 





7 tan 5 . tan 25 . tan 60  . tan 65 . tan 85

 Page 6 of 9

19.

2x34x25x7

g(x)

2x

75x

g(x) On dividing the polynomial 2x34x25x7 by a polynomial g(x), the quotient and the remainder were 2x and 75x respectively. Find g(x). 20.

x y 3x2y9xy ; 9x4y21xy ; x, y ≠ 0. Solve the following pair of equations for x and y : 3x2y9xy ; 9x4y21xy ; x, y ≠ 0. /OR p q 4x5y2 ; (2p7q)x(p8q)y2qp1 For what values of p and q will the following pair of linear equations has infinitely many solutions ? 4x5y2 ; (2p7q)x(p8q)y2qp1

21.

200 100

200

300

400

500

40 82 154 184 200 Find the median for the following table which shows the daily wages drawn by 200 workers in a factory. Daily wages Less than Less than Less than Less than Less than (in Rs.) 100 200 300 400 500 No. of workers 40 82 154 184 200 22.

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. /OR

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. 23.

24.

cot cosec  1   2 sec cosec  1 cot cot cosec  1 Prove that :   2 sec cosec  1 cot

50

f 1

f 2

10 – 30 30 – 50 50 – 70 70 – 90 90 – 110 f 1 f 2 90 30 40 200 The mean of the data in the following table is 50. Find the missing frequencies f 1 and f 2. Class : 10 – 30 30 – 50 50 – 70 70 – 90 90 – 110 Total f 1 f 2 Frequency : 90 30 40 200 Page 7 of 9

/ SECTION-D 25

34

4


12, 72

120

(LCM)

(HCF)



≠ Find the LCM and HCF of 12, 72 and 120 using prime factorisation. Also show that HCFLCM ≠ Product of three given numbers. 26.

2x3y40 ; 2x3y80 x Solve the following pair of equations graphically : 2x3y40 ; 2x3y80 Also shade the region formed by the lines with the xaxis. 27.

cossinp

q(p21) 2p

seccosecq

If cossinp and seccosecq, prove that q(p21) 2p 28.

a, b, c, d, e

f

150–155 155–160 160–165 165–170 170–175 175–180 12

b

10

d

e

2

a

25

c

43

48

f

50

Find the unknown entries a, b, c, d, e and f in the following distribution and hence find their mode. Height 150–155 155–160 160–165 165–170 170–175 175–180 Total (in cm) : Frequency : 12 b 10 d e 2 50 Cumulative f a 25 c 43 48 frequency : x45x32x210x8

29.A



2

2

Find the other zeroes of the polynomial x45x32x210x8 if it is given that two of its zeroes are 

2

and

2.

/OR 29.B

11 : 7

9:5

400 The ratio of incomes of two persons is 11 : 7 and the ratio of their expenditures is 9 : 5. If each of them manages to save Rs. 400 per month, find their monthly incomes. Page 8 of 9

30.

Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. /OR

Prove that in a triangle, if the square on one side is equal to the sum of the squares on the other two sides, the angle opposite to the first side is a right angle. 31.

3  4 sin 2 A

17 sec A 8

4 cos 2 A3

3  4 sin 2 A 17 If sec A , verify that : 8 4 cos 2 A3

32.

ABC

XYAC





3  tan 2 A 1  3 tan 2 A

3  tan 2 A 1  3 tan 2 A

ABC

AX AB

In the given figure, XY AC in triangle ABC and it divides the triangle into two parts of equal area. Find

33.

AX . AB

15 tan24 sec223

(seccosec)2  sin2

If 15 tan24 sec223, then find the value of (seccosec)2  sin2. 34.

0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 7 10 23 51 6 3 Convert the following distribution into a “less than type” cumulative frequency distribution and draw its ogive. Also find the median from the ogive. Class : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 Frequency : 7 10 23 51 6 3 -o0oPage 9 of 9



Page 1 of 9

I, 2012

MA2-039

SUMMATIVE ASSESSMENT – I, 2012 / MATHEMATICS X / Class – X 3

90


Maximum Marks : 90

(i) (ii)

34

8

1

6

3

10

(iii)

1

2

10

4

8

(iv)

2 3

4

3

2

(v)


(iii) (iv)

(v)


Page 2 of 9

SECTION–A 1

1.

8

1

Question numbers 1 to 8 carry one mark each. For each question, four alternative choices have been provided of which only one is correct. You have to select the correct choice. a 3 q r a  3q  r, r

(a) 0 r < 3 (b) 1
3x2  kx  6 3 k (a) 9 (b) 3 (c) (d) 6 3 2 If the sum of the zeroes of the quadratic polynomial 3 x  k x 6 is 3, then the value of k is : (a) 9 (b) 3 (c) (d) 6 3

3.

ABC PQR 6 (a)

ABC

32

PQR

48

PR

AC 9

(b)

4

(c)

8

(d)

18

If ABC PQR, perimeter of ABC  32 cm, perimeter of PQR  48 cm PR  6 cm, then the length of AC is equal to : (a) 9 cm (b) 4 cm (c) 8 cm (d) 18 cm 4.

5.

and

tancot  5 tan2  cot2 (a) 23 (b) 25 (c) 27 (d) 15 If tancot  5, then the value of tan2  cot2 is : (a) 23 (b) 25 (c) 27 (d) 15 

22 7

(a)

(b)

(c)

(d)



(a) (c)

22 is : 7

a rational number a prime number

(b) (d)

6.

an irrational number an even number 4x3y  14

(a) (c)

3x4y  14 12x9y  42

(b) (d)

8x6y  28 12x9y Page 3 of 9

Two lines are given to be parallel. The equation of one of the lines is 4x3y  14. The equation of the second line can be : (a) 3x4y  14 (b) 8x6y  28 (c) 12x9y  42 (d) 12x9y 7.

sec2A  cosec(A  27) 2A A (a) 35 (b) 37 (c) 39 (d) 21 If sec2A  cosec(A  27) where 2A is an acute angle, then the measure of A is : (a) 35 (b) 37 (c) 39 (d) 21

8.

(a)

(b)

(c) (d) Mode is the value of the variable which has : (a) maximum frequency (b) (c) mean frequency (d)

minimum frequency middle most frequency

/ SECTION-B 9 9.

14

2

Question numbers 9 to 14 carry two marks each. HCF) (LCM) 9

459

27

HCF and LCM of two numbers is 9 and 459 respectively. If one of the number is 27, find the other number. 10.

3

2

3

2

Form a quadratic polynomial whose zeroes are 3  2 and 3  2 11.

ABC DF

AB

AC

P1

DEF

DE

( ABC ) ( DEF )

P2

The sides AB and AC and the perimeter P 1 of ABC are respectively three times the corresponding sides DE and DF and the perimeter P 2 of DEF. Are the two triangles ar ( ABC ) similar ? If yes, find ar ( DEF ) 12.

cos (AB)0

sin ( AB) 

If cos (AB)0 and sin ( AB) 

1 2 1 2

,

A, B

A

B

then find the value of A and B where A and B are

acute angles. 13.

x2

 px  q

2x2  5 x  3

p

q

Page 4 of 9

If the zeroes of the polynomial x2  px  q are double in value to the zeroes of 2x2  5x  3, find the value of p and q 14.

0 - 20

20 – 40

40 – 60

60 – 80

80 - 100

5

9

12

8

6

Convert the following frequency distribution to a more than type’ cumulative frequency distribution. Marks obtained 0 - 20 20 – 40 40 – 60 60 – 80 80 - 100 No. of Students 5 9 12 8 6 /OR

0 - 10

10 – 20

6

10

Find the mode of the following data. Height (in cms) 0 - 10 10 – 20 No. of students 6 10

20 – 30

30 – 40

12 20 – 30 12

32 30 – 40 32

40 - 50 20 40 - 50 20

SECTION-C 15

24

3


CDLA

DEAC

BE 4

EC2

CL

In the given figure, CD LA and DEAC. Find the length of CL if BE  4 cm and EC  2 cm.

16.





6y2  7y  2

1

1





If  and  are the zeros of the polynomial 6 y2  7y  2, find a quadratic polynomial whose Page 5 of 9

zeros are

1 

and

17.

1



.

5

3 5

5 is irrational and hence show that 3  5 is also irrational

Prove that

/OR

510

92

(HCF)



(LCM)



Find the HCF and LCM of 510 and 92. And verify that HCF  LCM  Product of two given numbers. 18. 2 (tan20 .tan60 .tan70 3 3(sin 2 31  sin 2 59 ) 2 Sec 41.sin49  cos29.cosec61  (tan20 .tan60 .tan70 3 Evaluate 3(sin 2 31  sin 2 59 ) Sec 41.sin49  cos29.cosec61 

19.

x22

x4x3x22x3

Check by division whether x22 is a factor of x4x3x22x3. 20.

x

a2



x

b2

0

;

y

a2 b



x

y

b2 a

 a b

,

x



0;

y



0

y

Solve the following pair of equations for x and y a2 x



b2 y

0

;

a2 b



x

b2 a

 a b

,

x



0;

y



0

y

/OR

7x  4y  49 ; 5x  6y  57 Find whether the following pair of linear equations has a If unique solution yes, find the solution. 7x  4y  49 ; 5x  6y  57 21.

20

40 15

60 37

Find the mean of the following data. Class less than 20 less than 40 Frequency 15 37

80 74

100 99

less than 60 74

120 less than 80 99

less than 100 120

Page 6 of 9

22.

BD 

AD BC

1 3

2AC2  2AB2  BC2

CD

In the given figure, AD BC and BD 

1 3

CD. Prove that 2AC2  2AB2  BC2

/OR

If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium. 23. sin  cos sin  cos



sin   cos sin  cos



sin   cos sin  cos



2 2sin

2

2sin

2

 1

Prove that : sin  cos sin  cos

24.



2  1

600 0 - 1000

1000 – 2000

2000 – 3000

3000 – 4000

4000 – 5000

5000  6000

250

190

100

40

15

5

Weekly income of 600 families is given below. Income in Rs 0 - 1000 1000 – 2000 2000 – 3000 3000 – 4000 4000 – 5000 5000  6000 250 190 100 40 15 5 No. of families Find the median. / SECTION-D 25 25.

34

4

Question numbers 25 to 34 carry four marks each. 4m 4 m  1

m

Show that square of any positive integer is of the form 4 m (or) 4 m  1, where m is any integer.

Page 7 of 9

26.

x y

 3y  6; 2x3y  12

–

Solve the following pair of linear equations graphically. x  3y  6 ; 2x3y  12 Also find the area of the triangle formed by the lines representing the given equations with y – axis. 27.

cosec  cot  p

cos 

If cosec  cot  p, then prove that cos  28.

31

x

p2  1 p2  1

p2  1 p2  1

y

0 - 10

10 - 20

20 – 30

30 – 40

40 – 50

50 – 60

5

x

6

y

6

5

40

50 – 60 5

Total 40

Find the values of x and y if the median for the following data is 31. Class 0 - 10 10 - 20 20 – 30 30 – 40 40 – 50 x y Frequency 5 6 6 29.

3

(x  5)

x  3

5 x2

 5 x  15

Given that x  5 is a factor of the polynomial zeroes of the polynomial.

5, 3

x 3

5

2

x  5 x  15

5, find, all the

/OR

20 The age of the father is twice the sum of the ages of his 2 children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father. 30.

Prove that if in a triangle, the square on one side is equal to the sum of the squares on the other two sides, then the angle opposite to the first side is a right angle. /OR

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. 31.

sec  1 sec  1



sec  1  2 cosec sec  1 Page 8 of 9

C.B.S.E. 2012 Sample Papers for X Maths (5 Sets) (1)

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