Roll No.

Serial No. of Q. C. A. B.

≤¡ÈJ …ÂX‡Rπ› ‚¢zW : 58 ]
Total No. of Questions : 58 ]

[

≤¡ÈJ ÆÂÈÈåX∆ …ÂÏ¡π› ‚¢zW : 40

[ Total No. of Printed Pages : 40

81-E Code No. : 81-E
‚¢xË∆ ‚¢zW :

ê·ÂŒÈ : πÂä∆Â
Subject : MATHEMATICS
(ߢÇYË·˜Ô ü˘Ê·Ê¢∆¬ / English Version )
[ Date : 08 04. 2013 [ Time : 9-30 A.M. to 12-45 P.M. [ Max. Marks : 100

å»Ê¢∑ : 08. 04. 2013 ] ‚ÂÆÂÈŒÈ : üúπB 9-30 ࢫ ÆÂÈ«˘ÊW„ÂR 12-45 ¬ÂÕ¬π ] …¬ÂÆÂ⁄ÕÂå˘ •¢∑π›ÂÈ : 100 ]

FOR OFFICE USE ONLY
Q. No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Marks Q. No. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. Marks Q. No. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. Marks Q. No. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. Marks Q. No. 53. 54. 55. 56. 57. 58. Marks

× × × × × × ×

Total Marks
Total Marks in words 1. 2. Grand Total

✓ ✓
Signature of Evaluators Registration No.

✓
Signature of the Deputy Chief

✓
Signature of the Room Invigilator

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81-E General Instructions : i) ii)

2

The Question-cum-Answer Booklet consists of objective and subjective types of questions having 58 questions. Space has been provided against each objective type question. You have to choose the correct choice and write the complete answer along with its alphabet in the space provided. For subjective type questions enough space for each question has been provided. You have to answer the questions in the space. Follow the instructions given against both the objective and subjective types of questions. Candidate should not write the answer with pencil. Answers written in pencil will not be evaluated. ( Except Graphs, Diagrams & Maps ) In case of Multiple Choice, Fill in the blanks and Matching questions, scratching / rewriting / marking is not permitted, thereby rendering to disqualification for evaluation. Candidates have extra 15 minutes for reading the question paper.

iii) iv) v) vi)

vii)

viii) Space for Rough Work has been printed and provided at the bottom of each page. I. Four alternatives are given for each of the following questions / incomplete statements. Only one of them is correct or most appropriate. Choose the correct alternative and write the complete answer along with its alphabet in the space provided against each question. 1. 20 × 1 = 20

If A, B and C are non-empty sets then the ‘Intersection of sets is distributive over union of sets’ is represented as (A) (B) (C) (D) A I (B U C)=(A I B) U (A I C) A I (B I C)=(A I B) I (A I C) (A U B) U C=(A I C) U (B U C) (A I B) U C=(A U C) I (B U C)

Ans. :
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3 2.

81-E

If 5 and 2 are the Arithmetic Mean and Harmonic Mean of two distinct numbers, then their Geometric Mean is (A) (C) 3 10 (B) (D) 7 10.

Ans. : 3.

 2 If A + B =   4
(A)

3 

 1 2   and A =    0 3  then matrix B is 5 
(B)

 1 1     4 2   2 4     1 1 

 1 4     1 2   4 2     1 1 

(C)

(D)

Ans. : 4. If n C 8 = n C 5 , then the value of n is (A) (C) Ans. : 5. The H.C.F. of 5x 2 y 3 and 10x 3 y 2 is (A) (C) 10x 3 y 3 5xy (B) (D) 5x 2 y 2 5x 3 y 3 . 2 1 (B) (D) 3 13.

Ans. :
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81-E 6. The expansion of

4

∑ p‚q‚r p 2 is

(A) (C) Ans. : 7.

p2 q2 r2 p2

(B) (D)

pq r p2 +q2 +r2.

The value of

∑ a‚b‚c a ( b – c ) is

(A) (C) Ans. : 8.

2 ( ab + bc + ca ) 0

(B) (D)

ab + bc + ca a + b + c.

If one factor of a 3 + b 3 is ( a + b ), then the other factor is (A) (C) a 3 + b 3 + ab a 2 + b 2 – ab (B) (D) a – b + ab a 2 + b 2 + ab

Ans. : 9. If x y = 80 , then the value of y is (A) (C) 5 4 (B) (D) 16 20.

Ans. :
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5 10. The simplified form of 10

81-E

3

x

–8

3

x

is

(A)

18

3

x x

(B)

2 x

(C) Ans. :

2

3

(D)

18 x

81 11. If 4x = x , then the value of x is (A) (C) Ans. : 12. The quadratic equation having the roots (A) (C) Ans. : 13. If 3 ⊕ y ≡ 2 ( mod 6 ), then the value of y is (A) (C) Ans. : 14. Out of the following sets, Z 4 is (A) (C) Ans. :
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– 4·5 4·5

(B) (D)

± 4·5 ± 0·45.

(2+

3

)

and

(2–

3

)

is

x 2 – 4x + 1 = 0 x 2 – 4x – 1 = 0

(B) (D)

x 2 + 4x – 1 = 0 x 2 + 4x + 1 = 0

2 5

(B) (D)

4 6.

{ 0, 1, 2 } { 0, 1, 2, 3, 4 }

(B) (D)

{ 0, 1, 2, 3 } { 1, 2, 3, 4 }

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81-E

6 15. In ∆ ABC, D and E are the mid-points of AB and AC respectively, then the area of ∆ ADE is (A) (C) Ans. : AX 16. In the given figure, XY || BC, then BX =
A

4 ∆ ABC 2 ∆ ABC

(B) (D)

1 4 ∆ ABC 1 2 ∆ ABC.

X

Y

B

C

(A) (C) Ans. :

AY AC AX AB

(B) (D)

YC AY AY CY .

17. In ∆ ABC, ∠ ABC = 90°. If AC = ( x + y ) and BC = ( x – y ), then the length of AB is (A) (C) Ans. :
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x2 –y2 2 xy

(B) (D)

2xy x2 +y2

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7 respectively. If CB = 5 cm, and EF = 3 cm, then the length of CE is

81-E

18. In the given figure, AC, CE and EH are tangents drawn to the circle at B, D and F

(A) (C) Ans. :

2 cm 3 cm

(B) (D)

5 cm 8 cm.

19. The formula to find the coefficient of variation is (A) σ X × 100 (B) X σ × 100

(C)

X 100 × σ

(D)

σ 100 × X

Ans. : 20. If the circumference of the base of a cylinder is 44 cm and height 20 cm, then its lateral surface area is (A) (C) Ans. :
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440 sq. cm 88 sq.cm

(B) (D)

880 sq.cm 44 sq.cm.

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81-E II.

8 Fill in the blanks with suitable answers : 10 × 1 = 10

21. If A and B are the subsets of the universal set U then

( A U B ) l = ……………………… .

Ans. :

22. If A is a matrix of order ( m × n ) and B is a matrix of order ( n × p ) then order of AB is …………………………… .

Ans. :

23. The value of n P 0 is ………………………… .

Ans. :

24. Rationalising factor of

(

x+y

)

is …………………………… .

Ans. :

25. The standard form of the quadratic equation is ……………………… .

Ans. :

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9

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26. If the value of the discriminant of the quadratic equation ax 2 + bx + c = 0 is less than 0 then the nature of the roots is ……………………… .

Ans. :

27. If R and r are the radii of two circles having their centres d cm apart, then the length of the transverse common tangent t is ……………………… .

Ans. :

28. If the square on one side of a triangle is equal to the sum of the squares on the other two sides, then those two sides contain ………………………… .

Ans. :

29. The formula to find volume of a right circular cylinder is …………………… .

Ans. :

30. Shape of each face of Dodecahedron is ………………………… .

Ans. :
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81-E III.

10 31. Which term of the Geometric Progression 2, 2 2 , 4, …… is 64 ? 2

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11 32. Find the sum of the series 1 + 2 + 4 + ……… up to 9 terms. [ using the formula ]

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2

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81-E

12 33. Three numbers are in harmonic progression. The harmonic mean between first and third numbers is 20. If the 1st number is twice the third number, find the three terms of the progression. 2

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13 34. What is meant by transposing of a matrix ? Give an example.

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81-E 35. (a) (b)

14 What is fundamental counting principle ? What is the meaning of n P r ? 2

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15

81-E

36. There are 3 white and 5 red roses in a basket. In how many ways can 4 flowers be removed from the basket so that they contain 2 red flowers ? 2

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81-E

16 37. The H.C.F. and L.C.M. of two expressions are ( a – 7 ) and

( a 3 – 10a 2 + 11a + 70 ) ( a 2 – 12a + 35 )

respectively. If one of the expressions is 2

, find the other.

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17 38. Rationalise the denominator and simplify : 5+ 2 . 5– 2

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2

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81-E

18 39. By selling an article for Rs. 18·75, a dealer loses as much per cent as its cost price. Find the cost price of the article. 2

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19 40. Solve the equation by using the fomula x 2 – 8x + 1 = 0.

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81-E

20 41. What is a pure quadratic equation ? Give an example. 2

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21 42. For what value of k the equation kx 2 + 6x + 1 = 0 has equal roots ?

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81-E

22 43. Construct two tangents to a circle of radius 3·5 cm from a point 4·5 cm away from the circle. 2

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23 44. ABCD is a rhombus. Prove that AC 2 + BD 2 = 4 AB 2 .

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2

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81-E

24 45. In the given figure, TP and TQ are tangents drawn to a circle with centre O. Prove that ∠ PTQ = 2 ∠ OPQ. 2

P

O

T

Q

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25 46. Draw a plan for the recordings from the Surveyor’s field book given below : [ Scale : 20 m = 1 cm ] Metres to D 160 120 100 60 From A 60 to C 40 to B

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To E 80

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81-E

26 47. Draw a network for the following matrix : 2

0 1 2

1 2 1

2

 1  0

.

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27 48. Verify Euler’s formula for the given solid.

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81-E IV.

28 49. In an examination 82% of the candidates passed in Maths, 72% passed in Science and 55% passed in both. Find the percentage of students failed in both. ( Draw Venn diagram to verify ) 3

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29 50. Calculate the Mean and Standard Deviation for the following distribution : Class-interval 0–4 5–9 10 – 14 15 – 19 20 – 24 Frequency 2 3 10 3 2

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81-E

30 51. Find the L.C.M. of x 3 – 2x 2 – 13x – 10 and x 3 – x 2 – 10x – 8. 3

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31

81-E

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81-E 52. If a + b + c = abc, show that a

32

( b 2c 2 – 1 ) bc + 1

+

b

( c 2a 2 – 1 ) ca + 1

+

c

( a 2b 2 – 1 ) ab + 1

= 2abc.

3

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33

81-E

53. If two circles touch each other externally, prove that their point of contact and their centres are collinear. 3

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81-E

34 54. Find the total surface area of a sphere whose volume is equal to the volume of the cone having the radius 12 cm and height 6 cm. 3

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35 V.

81-E

55. In an Arithmetic progression the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that the 20th term is equal to – 112. 4

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81-E

36 56. Two circles of radii 4 cm and 2 cm, have their centres 10 cm apart. Draw two direct common tangents and measure their length and write. 4

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37

81-E

57. If two triangles are equiangular, prove that their corresponding sides are proportional. 4

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81-E

38 58. Draw the graph of y = x 2 and y = 3 – 2x and hence solve the equation x 2 + 2x – 3 = 0. 4

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39

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81-E

40

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