MATHEMATICS

i1fu1a nme allowed : 3 hours J

r:;4mc1 'f{1F:{ : -3 E/TR 1

[ Maximum marks : 80

[�3icE:80

General Instructions :

(i) All questions are compulsory.

(ii) The question paper consists of 30 quest�ons divided into four sections­

A, B, C and D. Section A comprises of ten questions of 1 mark each,

CBSE QUESTION PAPER CLASS-X

Section B comprises of five questions of 2 marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6 marks each.

{iii) All questions in Section A are to be answered in one 11,;ord. one sentence or as per the exact requirement of the question.

(iv) There is no overall choice. However, an internal chofc£ l-!!E iJeen provided in one question of 2 marks each, three quesnor..: �--3 m!Ir.::: each and two questions of 6 marks each. You have to ar.2mp- :ir-: .,,.� of the alternative in all such questions.

(v) In question on construction, the drawings should be 1:E:II -=r.:i =--:r:;r;._as per the given measurements.

(vi) Use of calculators is not permitted.

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g � � ({� tfhr�g�-g� �* 6 �� f

{iii) 7sfrJg JJ # � � cnT 3fR � �, � cfTcP.[ 3--� p;:f cft

-j-f/ct:t44'd'3�-m w:TT \5fT "f{cRfT t I

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(v)

(vi)

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Section A

Question Number· 1 to 10 carry 1 mark each.

'Slv-1 � 1 "B 10 � � � cfil 1 � i I

1. The decimal expansion of the rational number ;3

3 , will terminate after

2 .5how many places of decimals ?

2. For what value of k, (- 4) is a zero of the polynomial �2 - x - (2k + 2) ?

3. For what value of p, are 2p- 1, 7 and 3 p three consecutive terms of an A.P. ?

4. In Fig. 1, CP and CQ are tangents to a circle with centre 0. ARB is another

tangent touching the circle at R. If CP = 11 cm, and BC = 7 cm, then find the

length of BR.

C

Fig. 1

• I

��Wt� fcfi "¥ cnl � R 1R � � t1 � CP = 11 w:fi am BC= 7 Bm %, m BR cfit � � cf.li)Q.1

C

5. In Fig. 2, L M = L N = 46° . Express x in terms of a, band cwhere a. b 2:".d:

ar� lengths of L�, MN and NK respectiYely.

M�b�N--<-- c--..,,..

Fig. 2

• 2 -ij, L M = L N = 46°, x cflltfR a, b crm c � fcif6Q. � a, b � c

6. If sine=½' then find the value of (2 cot2 e + 2).

1 � sin 0 =3 % d1, (2 cot2 0 + 2) cfiT l'.fR � �I

7. Find the value of a so that the point (3, a) lies on the line represented by

2x-3y = 5.

a cn1 � "tfR � ch1�Q. � � � (3, a), 2x - 3y = 5 IDU R(e;flla WT "9\

@@%1

8. A cylinder and a cone are of same base radius and of same height. Find the

ratio of the volume of cylinder to that of the cone.

��er��� 3TT� � 81�1�m i er�� Ji-c11��i mm i 1 �

� �cfiT�� �� ���I

9. Find the distance between the points ( �8 ,2) and (; ,2 )-

10. Write the median class of the following distribution :

Classes Frequency0-10 410-20 420:-30 830-40 1040-50 1250-60 860-70 4

0-10 4

10-20 4

20-30 8

30-40 10

40-50 12

50-60 8

60-70 4

Section B

��

Question Number 11 to 15 carry 2 marks each.

mmT 11 � 15ocfi��� 2 �t1

11. If the polynomial 6x4 + 8x3 + l 7x2 + 2 lx + 7 is divided by another polynomial3x2 + 4x + 1, the remainder comes out to be (ax+ b), find a and b.

� � 6x4 + 8x3 + l 7x2 + 2 lx + 7 cf,l � � � 3x2 + 4x + 1 B �

� 'IR �lt:flf>ci (ax+ b) � m cfT a o� b � lJR mo ch)f¾o_ I

12. Find the value(s) of kforwhich the pair of linear equations kx + 3y = k-2 and12x + ky = k has no solution.

k � 'cffl 1:JH mo cf>)�Q. � � fli:ficf>{o1 � 1cx + 3y = k - 2 3fu:

12x + ky = k i:f>T � m-f ";J m I

13. If Sn, the sum of first n terms of an A.P. is given by Sn= 3n2 4n, then find its

nth term.

� fcnut Bl4lrd{ � � � n tT<TT cflT <WT Sn= 3n2 - 4n �'ell� BqHR �

� ncff 1lG � ch1�Q. I

14. Two tangents PA and PB are drawn to a circle with centre O from an external

point P. Prove that LAPB = 2 LOAB.

p

Fig .. 3

Or

Prove that the parallelogram circumscribing a circle is a rhombus.

� O ci@vf ��� P �GTfW&rrtPA om PB ��ti m�

fcn LAPB = 2 LOAB.

p

3Wf@ 3

3t�

� � fcn fcnut vf � 1lWIT1 BqHR �$f �$f mID i I

15. . e 0

+. Simplify: . sin 3 0+cos3

0 cos 0

sm +cos

. 3 0 3 0 = i:h1NiO:

: sm + cos + . 0 0

""'' . 0 0 sm cos

sm +cos

Section C

� �

Question Number 16 to 25 carry 3 marks each.

m ti&n 16 "B 2s "ocn � � � 3 3Tcfi i1

16. Prove that Js is an irrational number.

17. · Solve the following pair of equations :

5 1 -+--=2x-1 y-2

6 3 ----=1x-1, y-2

RR l-141cfi{OI � cn7' � i:h1Ntll_: 5 1

x-l +y -2

=2

6 3 ---=1x-1 y-2

18. The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and I 0th terms

is 44. Find A.P.

fcnm l-ll-l HR ����om � i:rG1 cnr "lfllT 24 i ��om� i:rG1 cnr "lfllT44 ii l-ll-lHR � mcf �I

_ 9, Construct a ti ABC in which BC= 6.5 cm, AB= 4.5 cm and LABC = 60° .

Construct a triangle similar to this triangle

whose

sides are ! of the

corresponding sides of the triangle ABC.

� f;r� ABC c1;t � � � BC = 6 .5 wfr, AB = 4.5 wft -3lR LABC = 60° � I � � � �� ci;t � � f�rncli) � ti ABC

c1;t zjTra �3TT cl;t ! "ITT I

:o. In Fig. 4, ti ABC is right angled at C and DE l. AB. Prove that

ti ABC - ti ADE and hence find the lengths of AE and DE.

Fig. 4

Or

In Fig. 5;DEFG is a square. and LBAC = 90°. Show that DE2 = BD x EC.

Fig. 5

• 4 �, ABC � l-l4chlOI � % Nil-lc=hl ch7'CIT C l-l4ch101 % <fm DE .l AB

%1 ��-� ti ABC -ti ADE, 3'.ffi: AE <f2TT DE� ctkll��i mo�I

A

• 5 �, DEFG � q7f � <fm LBAC = 90° %1 � ch1NiQ. �DE2 = BD X EC.

B

C

F

A

21. Find the value of sin30° ge ometrically.

Or

Without using tri gonometrical table s, evaluate

cos58° sin22 ° cos38° cosec 52 °

sin32 ° + cos68° tanl8° .tan35 ° tan60°tan 72 °tan55 °

\J"lllfil@ mu sin30° cnT lfR" mo �I

,3{�

Gtcn"101filcfl� ffiUft cn1 � � w;n lfR" mo�=

cos58° sin22 ° cos38° cosec 52 °

--+ - -------------

sin32 0 cos68° tanl 8°.tan35 ° tan 60°tan 72 °tan55 °

22. Find the point on y-axis which is equidistant from the points (5, -2) and

(-3, 2).

Or

The line segment joining the points A (2,1) and B (5, -8) is trisected at the

points P and Q such that P is nearer to A. If P also lies on the line given by

2x- y + k= 0, find the value of k.

y-3l� "Cf\%�� ch)�Q_ -m rcn �m (5, -2) om (-3, 2) ��"Cf\ W-lo

i1

�m A (2,1) om B (5, -8) cf>T � cf@ {@l@Os cf>T � P am Q �

� TTrffl<f � t fcn � P �A�� WR t1 � � P oo

2x- y + k = 0 "Cl\ '4t W-lo i m k cn1 l=fR ��I

13. If P (.x, y) is any point on the line joining the points A ( a, 0) and B (0, b ), then

X y show that a+b= 1.

� P (x, y) �m A (a, 0), B (0, b) cf>T�cf@{@l@Os ��"ITTcTT�

�fcn ;+;= 1.

24. In Fig. 6, PQ = 24 cm, PR= 7 cm and O is the centre of the circle. Find the

area of shaded region ( take n = 3 .14)

p Fig. 6

• 6 �' PQ = 24 @ft, PR = 7 wit cf2TT O Vf cfiT �%I �lllifcha � cfiT

� � chlMQ,I (n = 3.14 �)

25. The king, queen and jack of clubs are removed from a deck of 52 playing

cards and the remaining cards are shuffled. A card is drawn from the remaining

cards. Find the probability of getting a card of (i) heart (ii) queen (iii) clubs.

s2 wcRtcfmcfTTTI:cf>�� l¾f%;q1 cfiT �,��,,�, fficf2TT��1mc121Tffl

�en)� m � TTml � ffl w � � TI:cf> -qm � TTml si1�cha1 �

� rcn � lfm -qm (i) "CfR cfiT -qm m (ii) � m (iii) l¾R;<-l, cfiT -qm m 1

Section D

�'G

Question Number 26 to_ 30 carry 6 marks each.

� zjwn 26 � 30 cf91 � � � 6 3rcfi t-1

26. The sum of the squares of two consecutive odd numbers is 394. Find t�e

numbers.

Or

Places A and B are 100 km apart on a highway. One car starts from A and

another from B at the same time. If the cars travel in the same direction at

different speeds, they meet in 5 hours. If they travel towards each other, they

meet in 1 hour. What are the speeds of the two cars ?

�

chlM Cit sfil-l!'ld 1q1:1� ��, eti ��1•1 394 QI B&-IIQ, � 7 Q,I

��

� U'l1l-li'T �GT� A� B, 100 PcnctiJ.ftc:( cfil��% 1 "Q,cf)m A Bom�

c1m B B"Q,cf)m���cRcft%1 ��cnRf�-��-u� m"Rffl

q �%<TT 5 m it�% I�� f¾qt)a "Rffl3TT it�% <TT� m it�

�I� cfllU cfit � mo chlMQ,I

2-. Prove that, if a line is drawn parallel to one sid.e of a triangle to intersect the

other two sides in 'distinct points, the other two sides are divided in the same

ratio.

Using the above result, do the following

In Fig. 7, DEIIBC and BD = CE. Prove that /J,. ABC is an isosceles triangle. A

B -'----------" C

Fig. 7

��' �fcfajtf;{�cfit"Q,cf) �� Bl-lHi( �GT �3TT���3TT

91: Sl@-ajRa cfi@��oo�m�f3f�cfit �GT �3lt�m �

it � cRcfi t I

�cf,T��, ��=-

• 7 it, DEIIBC elm BD = CE� I <TT m chlMQ, fen /J,. ABC � Bl-l�I§

&�ti A

28. A straight highway leads to the foot of a tower. A man standing at the top of

the tower observes a car at an angle of depression of 3 0°, which is approaching

the foot of the tower with a uniform speed. Six seconds later the angle .of

depression of the car is found to be 60° . Find the time taken by the car to

reach the foot of the tower from this point.

�mm {1'3!4Pf �l:ft;m:�'9TG<lcn�i1 l:ft;m:�·fu@:"Cf\�� ��

cnR cfiT 30° � � cfiTUf "Cf\� i "'311 fcn � � '9TG � 3TT{ Q,chfl4H .'q@

.B�i1�:��"ch"R��cfiTUT60° m\Jffil1%1w�*��

tfR <lcn � -ij "ch"R "[RT iwn Tfm ���I

29. From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity

of height 8 cm and of base radius 6 cm, is hollowed out. Find the volume of

the remaining solid correct to two places of decimals. Also find the total surface

area of the remaining solid. (taken = 3.1416)

Or

In Fig 8, ABC is a right triangle right angled at A. Find the area of shaded

region if AB = 6 cm, BC= 10 cm and O is the centre of the incircle of /1 ABC.

(taken = 3.14)

Fig. 8

� c3ffi��� 8 wftom� 6 wfti1 w-ij"B 8 wft �om 6 w:ft

��� �ic:fqlchlC� (cavity) cfi'R:lWTI�il ffi�c3ffi�Gl GW·ktcl

� <lcn 3WrcR ��om ffi c3ffi � � � �-m � �1

(n = 3.1416 �)

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