Sample Paper Maths

5/13/2018 Sample Paper Maths

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Design of Sample Question PaperMathematics, SA~.IClass XType of Question Marks per question Total No. of Questions Total MarksM.e.Q. 1 1 0 1 0SA-I 2 8 1 6SA-II 3 1 0 3 0LA 4 6 2 4TOTAL 34 80

Blue PrintSample Question PaperMathematics, SA~IClass XTopic I Unit MeQ SA(I) SA(II) LA TotalNumberSystem 2 ( 2 ) 1 ( 2 ) 2 ( 6 ) - 5 ( 1 0 )Algebra 2 ( 2 ) 2 ( 4 ) 2 ( 6 ) 2 ( 8 ) 8 ( 2 0 )Geometry 1 ( 1 ) 2 ( 4 ) 2 ( 6 ) 1 ( 4 ) 6 ( 1 5 )Trigonometry 4 ( 4 ) 1 ( 2 ) 2 ( 6 ) 2 ( 8 ) 9 ( 2 0 )Statistics 1 ( 1 ) 2 ( 4 ) 2 ( 6 ) 1 ( 4 ) 6 ( 1 5 )TOTAL 10(10) 8(16) 10(30) 6(24) 34(80)Note: Marks are within brackets.


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Sample Question PaperMathematicsClass X (SA-I)

Time: 3 to 3% hours M..M.: 80General Instructionsi) All questions are compulsory.ii) The questions paper consists of 34 questions divided into four sections A, B, C and D.

Section A comprises of 1 0 questions of 1 mark each, Section B comprises of 8 questionsof 2 marks each, Section C comprises of 10 questions of 3 marks each and Section Dcomprises of6 questions of 4 marks each.

iii) Question numbers 1 to 10 in Section A are multiple choice questions where you are toselect one correct option out of the given four.

iv) There is no overall choice. However, internal choice has been provided in 1 question oftwo marks, 3questions of three marks each and 2questions offour marks each. You haveto attempt only one of the alternatives in all such questions.

v) Use of calculators is not permitted.Section-A

Question numbers 1 to 10 are of one mark each.1. Euclid's Division Lemma states that for any two postive integers a and b, there exist

uniqueintegres q and r such that a=bq+r, where r must satisfy.(A) I


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4. If sin 38 = cos (8-6), where (38) and (8_6) are both acute angles, thenthe value of 8 is(A) 180

5. 1 cosec28-see28Given that tan8= h the value of is"\13 cosec'+sec'B

12 (0) 12A) -1 (8) (C)6. In Fig. 3, AO=4 em, 80= 3 em and

C8 = 12 em, then cotS equals A3443

(A) (8) 512125C ) (0)

1477. The decimal expansion of 120 will terminate after how many places of decimal?(A) (8) 2 (C) 3 (0) wi I I not terminate

8 . The pair of linear equations 3x+2y=5; 2x-3y=7 have(A) One solution (8) Two solutions(C) Many Solutions (0) No solution

159. If sec A = cosec B = 7' thenA+8 is equal to(A) Zero (8) 900 (C) 900

10. For a given data wi.th 70 observations the 'less then og.ive' and the 'more than oqive'intersect at (20.5, 35). The median of the data is(A) 20 (8) 35 (C) 70 (0) 20.5

SECTION-BQuestion numbers 11 to 18 carry 2 marks each.11. Is 7x5x3x2+3 a composite number? Justify your answer.12. Can (x-2) be the remainder on division of a polynomial p(x) by (2x+3)? Justify your

answer. x+yD .----------.----------------.--------- .. C13. In Fig. 4, ABeD is a rectangle.Find the values ofx and y.

. , .;!x-y,;i~L_ ~A .- ..- 1 2 - - B

Fig. 43

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14. If 7sin28+3cos28 = 4, show that tan8= ~

OR_15 (2+2sin8)(1-sin8)If cot8--, evaluate ( )( )8 1+cos8 2-2cos8

A15. In Fig. 5, DEIIAC and DFIIAE. Prove thatFE ECBF BE

A

16. In Fig. 6, AD..l BC and BD=~CD.Fig.5

Prove that 2CN=2AB2+BC' cFig. 6

17. The following distribution gives the daily income of 50 workers ofa factory:Daily income 100-120 120-140 140-160 160-180 180-200(in rupees)Number of 12 14 8 6 10WorkersWrite the above distribution as less than type cumulative frequency distribution.

18. Find the mode of the following distribution of marks obtained by 80 students:Marks obtained 0-10 10-20 20-30 30-40 40-50Number of students 6 10 12 32 20

SECTION CQuestion numbers 19-28 carry 3 marks each.19. Show that any positive odd integer is of the form 4q+1 or 4q+3 where q is a positive

integer.

20. P h 2.J3.. . Irove t at5S irrationa .OR

Prove that ( 5 - . J 2 ) is irrational.21. A person can row a boat at the rate of 5km/hour in still water. He takes thrice as much time

in going 40 km upstream as in going 40 km downstream. Find the speed ofthe stream.

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O R

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1In a competitive examination, one mark is awarded for each correct answer while " 2 markis deducted for each wrong answer. Jayanti answered 120 questions and got 90 marks.How many questions did she answer correctly?If a , f 3 are zeroes of the polynomial x'-2x-15, then form a quadratic polynomial whosezeroes are ( 2 a ) and ( 2 f 3 ) .

23. Prove that (coseeS-sinS)( sece-cose) = . 1 .tan8+cot824. If cosS+sinS = .j2 cosS, show that cosS-sinS=.j2 sinS

In Fig. 7, AB j_ BC, FG j_ BC andDE .1AC. Prove that

c

25.

MDE-6GCF26. MBC and 6DBC are on the same base

BC and on opposite sides of BC and 0isthe point of intersections of AD and BC.

Fig.7

area(MBC) AOProve that area(6DBC) = DO

27. Find mean of the following frequency distribution, using step-deviation method:Class 0-10 10-20 20-30 30-40 40-50Frequency 7 12 13 10 8

O RThe mean of the following frequency distribution is 25. Find the value of p.Class 0-10 10-20 20-30 30-40 40-50Frequency 2 3 5 3 P

28. Find the median of the following dataClass 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100Frequency 5 3 4 3 3 4 7 9 7 8

SECTION DQuestion numbers 29 to 34 carry 4 marks each29. Find other zeroes of the polynomial p(x) = 2X4+7x3_19x2_14x+30 if two of its zeroes are

J2 and - J2 .5


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

ORProve that in a triangle, if the square of one side is equal to the sum of the squares of theother two sides, then the angle opposite to the fi rst side is a right angle.

31. Prove that sec S+tanS-1tan8-sec8+1

coss---1-sin8

OR

E I t sec8cosec(900-8)-tan8cot(900-S)+sin'55+sin'35vauae-----~-~---~-~-------tan 10 tan 20 tan 60 tan 70 tan 80p'-1If sec8+tan8 =, prove that sin8 =--p'+1

Draw the graphs of following equations:2x-y = 1, x+2y = 13 and(i) find the solution of the equations from the graph.(ii) shade the triangular region formed by the lines and the y-axis

34. The following table gives the production yield per hectare of wheat of1 00 farms of avillage:

32.33.

Production yield 50-55 55-60 60-65 65-70 70-75 75-80in kg/hectareNumber of farms 2 8 12 24 38 16Change the above distribution to more than type distribution and draw its ogive.


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Marking SchemeMathematicsClass X (SA~I)

Section A1. (C) 2 . (8) 3. (C) 4. (8) 5. (C)6. (D) 7. (C) 8. (A) 9. (8) 10. (D)

1x10=10

SECTION B11. 7x5x3x2+3 =3(7x5x2+1)

= 3x71 ..... (i)By Fundamental Theorem of Arithmetic, every composite numbercan be expressed as product of primes in a unique way, apart fromthe order of factors.

12.}

In case of division of a polynomial by another polynomial }the degree of remainder (polynomial) is always less than that of divisor.', (x-2) can not be the remainder when p(x) is divided by (2x+3) as degree is same

.'. (i) is a composite number

13. opposite sides of a rectangle are equalx+y=12 ...(i) andx-y=8 ...(ii)

Adding (i) and (ii), we get 2x=20 or x=1 0and y=2x=10, y=2

14. 7sin28+3cos'8=4 or 3(sin'H+cos28)+4sin'H= 4.. 28 1::::} sin =-4

1. tanH = lan30o=-. . J : 3O R

15 .cot8=~ (given)8

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2 (1+ sin9)(1-sin9)Given expression = = cot 292(1 + cos9)(1- coss)22564

15. OE I i AC => B E =B OE C O A ....(i )B F B OOF I I AE => - = - (.ii)E F O A ....

BE BF CE FEFrom (i) and (ii) -=-- or -=-_ -EC EF BE BF16. Let SD=x =? CD=3x, In right triangleADC

CN=CD'+AD' (i)and AS' = AD2+BD2=? AD' =AS' - SO' (ii)Substituting (ii) in (i),

CN = CD'+AB'-BD'O R 2CN = 2AS2+2(9x2-x')= 2AB2+BCZ (. : BC=4x)=? 2CN=2AB'+BCZ

17. Less thanDaily income 120 140 160 180 200Number of wo rks 12 26 34 40 50

18. Modal Class = 30-4032-12.. Mode = 30+ 64 _ 32 x10 = 30+6 ..25 = 36.25

SECTION C19. Let a be a positive odd integer

By Euclid's Division algorithm a=4q+rWhere q, r are positive integes and 0 < r


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_ i Q _ = 3x40 =? x = 2..55-x 5+x:. Speed of the stream = 2.5 km/hourO R

1+%

But 4q and 4q+2 are both even=? a is of the form 4q+1 or 4q+3

20. 2.J3 ..Let5= x where x IS a rational number=? 2./3 "" 5x or . J 3 = 5x (i)2 ...

5xAs x is a rational number, so is2:. J3 is also rational which is a contradiction }

as . J 3 is an irrational

.. 2.J3 is irrational5OR Let 5-J2 = y , where y is a rational number }

:. 5-y = J2 .....(i )As y is a rational number, so is 5-y.. from (i), .J 2 is also rational which }

is a contradiction as .J 2 is irrational: . 5 - J 2 is irrational

21. Let the speed of stream be x km/hour:. Speed of the boat rowing

upstream = (5-x) km/hourdownstream = (5+x) km/hour }

:. According to the question,

Let the number of correct answers be x:. wrong answers are (120-x) in number

1:. x--(120-x) =9029


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3x::::}-",150 ::::}x=1002:. The number of correctly answered questions = 100

22. p(x) =x2-2x-15 ...(i)As a, {3are zeroes of (i), ::::} 0.+{3= 2 and a{3 = -15zeroes of the required polynom ial are 20. and 2{3.. sum of zeroes = 2(0.+{3) = 4

Product of zeroes = 4( -15) =~60 }:. The required polynomial is x'-4x-60.

23. LHS can be written as (_._1_ - sin S)(_1_ - COSS)sinS cosH

(1-sinz8)(1-cosZ8) . 8 S= = sin cossin8cosSsine cos8

sin2S+cos2S sin'S cos'S----+----sinScosS sinS cosS

tana+cots24. sin8+cos8=.J2cos8::::} sin8= (.J2-1)COS8

(.fi -1) (.J2 +1)or sin e = cos8( J 2 + 1 )or sinS = cosS ::::} cosS - sin S= . J 2 sin e.fi +1

25.LA+LC = 90Also LA+L2=90 ::::}LC=L2 }Similarly, LA=L1.. 8'S ADE and GCF are equiangular. . L ' I . ADE - 8 GCF

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26. Draw AL .lBC and. OM.lBC[\.'s AOL and DOM are similar

AO AL00 OM

1.ALArea(MBC) _ l i _ AOArea(L'.BCO) - 1DM - 0027. Class 0-10 10-20 20-30 30-40 40-50

Class marks (Xi) 5 15 25 35 45Frequency (f) 7 12 13 10 8 >x-25d=-'- -2 -1 0 1 2, 10fd -14 -12 0 10 16

x = A.M + I,f,d, x10 = 25+0 = 25.0. I,fi

. %+1OR

Class 0-10 10-20 20-30 30-40 40-50Frequency (fil 2 3 5 3 PClass mark (x.) 5 15 25 35 45fixi 10 45 125 105 45p

Lfi = 13+p, I, ~x, = 285+45pMean = 25 (given).. 25x(13+p) = 285+45p

=> 20p = 40 => p=228. Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Frequency 5 3 4 3 3 4 7 9 7 8Cum. 5 8 12 15 18 22 29 38 45 53Frequency

Median Class is 60-70

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( ~ - C f )Median = 1 + 2 xhfSECTION D

29. p(x) = 2x4+7x'-19x'-14x+30If two zeroes of p(x) are J 2 and -J2: . ( x + J 2 )( x - J 2 ) or x'-2 is a factor of p(x)p(x)Cc(x2-2)= [2x4+7x"-19X'-14X+30].,. (x'-2) =2X2+7x-15Now 2x'+ 7x-15 = 2x'+1 Ox-3x-15

= (2x-3)(x+5)3.. other two zeroes of p(x) are "2and -5

30. 1Correctly stated given, to prove, construction and correct figure 4x"2Correct proofO R

1Correctly stated given, to prove, construction and correct figure 4x"2correct proof

31. LHS= sec8+tan8-1tan8 - sec8 + 1sec8+tan8 - (sec'8 -tan28)

lan8 - sec8 + 1(secS+tanS)[ 1- secS+ tanS] 1 + sin 8

= = sec8+tanS = - - - 1 1(1-sec8+tanS) cosS +

(1+ sin8)(1- sin S) cosS(1- sin8)cosS 1- sin 8

O Rcosec (900 - 8) = sees. '. col (900 -. S) =. taonS, sin 55 = cos 35ltan 80 = col 10, tan 70 = cot 20, tan 60 = .j3 f

12

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1%

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... (sec '8-tan'8)+ (sin235 + cos '35)Given Expression becomes tan100cot100tan200cot200.J3

32. 1+ sinSseca+tane ep}---=peosS(1 + Sin8)2 2 (1+ Sin8t - cos'8=} --- =p =}cosS (1+ sin8/ + cos' 8

p'-1p'+1

(1-ms28) +sin'S+ 2sin8 p"-1cr"-------'-----2+2sinS p"+12sin8(1 + sin8) p2_1or =-"-2 (1+ sin8) p'+1

P '-1orsin8 =--p'+133.

y1412

10 2raph

II~ -8-10-12-14

y

(i) x = 3, y=5(ii) Shaded region is shown in figure

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34. Classes Frequency Cumulative Frequency (More than type)50-55 2 50 or more than 50 10055-60 8 55 or more than 55 9860-65 12 60 or more than 60 9065-70 24 65 or more than 65 7870-75 38 70 or more than 70 5475-80 16 75 or more than 75 16

10090

' " 80E. . . .~ 70 60~ 50QE 40:JZ 3020 (75, 16)10

o 102.0 30 40 50 60 70 80 70100V ie Id in kg I hectare

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