NIOS Class 10 Question Paper 2023 Maths

This Question Paper consists of 36 questions and 12 printed pages.
Bg àíZ-nÌ ‘| 36 àíZ VWm 12 ‘w{ÐV n¥ð> h¢& Code No.
65/AS/3
u
H$moS> Z§0
Roll No.
Set / goQ>
AZwH«$‘m§H$

MATHEMATICS
J{UV
(211)
Day and Date of Examination .....................................................................................
(narjm H$m {XZ d {XZm§H$)
Signature of Invigilators 1. .....................................................................................
({ZarjH$m| Ho$ hñVmja)
2. .....................................................................................

General Instructions :

1. Candidate must write his/her Roll Number on the first page of the Question
Paper.

2. Please check the Question Paper to verify that the total pages and total
number of questions contained in the Question Paper are the same as those
printed on the top of the first page. Also check to see that the questions are
in sequential order.

3. For the objective type of questions, you have to choose any one of the four
alternatives given in the question, i.e., (A), (B), (C) or (D) and indicate your
correct answer in the Answer-Book given to you.

4. All the questions including objective-type questions are to be answered within
the allotted time and no separate time limit is fixed for answering objective-
type questions.

5. Making any identification mark in the Answer-Book or writing Roll Number
anywhere other than the specified places will lead to disqualification of the
candidate.

6. Write your Question Paper Code No. 65/AS/3, Set u on the Answer-Book.

211/AS/3/403A [ P.T.O.

7. (a) The Question Paper is in English/Hindi medium only. However, If you
wish, you can answer in any one of the languages listed below :
English, Hindi, Urdu, Punjabi, Bengali, Tamil, Malayalam, Kannada,
Telugu, Marathi, Odia, Gujarati, Konkani, Manipuri, Assamese, Nepali,
Kashmiri, Sanskrit and Sindhi.
You are required to indicate the language you have chosen to answer in
the box provided in the Answer-Book.
(b) If you choose to write the answer in the language other than Hindi and
English, the responsibility for any errors/mistakes in understanding the
questions will be yours only.

gm‘mݶ AZwXoe …
1. narjmWu àíZ-nÌ Ho$ nhbo n¥ð> na AnZm AZwH«$‘m§H$ Adí¶ {bI|&
2. H¥$n¶m àíZ-nÌ H$mo Om±M b| {H$ àíZ-nÌ Ho$ Hw$b n¥ð>m| VWm àíZm| H$s CVZr hr g§»¶m h¡ {OVZr àW‘ n¥ð> Ho$ g~go
D$na N>nr h¡& Bg ~mV H$s Om±M ^r H$a b| {H$ àíZ H«${‘H$ ê$n ‘| h¢&
3. dñVw{Zð> àíZm| _o§ AmnH$mo Mma {dH$ënm| (A), (B), (C) VWm (D) _| go H$moB© EH$ CÎma MwZZm h¡ VWm Xr JB©
CÎma-nwpñVH$m _| Amn ghr CÎma {bI|Ÿ&
4. dñVw{Zð> àíZm| Ho$ gmW-gmW g^r àíZm| Ho$ CÎma {ZYm©[aV Ad{Y Ho$ ^rVa hr XoZo h¢Ÿ& dñVw{Zð> àíZm| Ho$ {bE AbJ go
g_` Zht {X`m OmEJmŸ&
5. CÎma-nwpñVH$m ‘| nhMmZ-{M• ~ZmZo AWdm {Z{X©ï> ñWmZm| Ho$ A{V[aº$ H$ht ^r AZwH«$‘m§H$ {bIZo na narjmWu H$mo
A¶mo½¶ R>ham¶m OmEJm&
6. AnZr CÎma-nwpñVH$m na àíZ-nÌ H$m H$moS> Z§0 65/AS/3, goQ u, {bI|&
7. (H$) n«íZ-nÌ Ho$db {hÝXr/A§J«oOr ‘| h¡& {’$a ^r, ¶{X Amn Mmh| Vmo ZrMo Xr JB© {H$gr EH$ ^mfm ‘| CÎma Xo
gH$Vo h¢ …
A§J«oOr, {hÝXr, CXÿ©, n§Om~r, ~§Jbm, V{‘b, ‘b¶mb‘, H$Þ‹S>, VobwJy, ‘amR>r, C{‹S>¶m, JwOamVr, H$m|H$Ur,
‘{Unwar, Ag{‘¶m, Zonmbr, H$í‘rar, g§ñH¥$V Am¡a {gÝYr&
H¥$n¶m CÎma-nwpñVH$m ‘| {XE JE ~m°³g ‘| {bI| {H$ Amn {H$g ^mfm ‘| CÎma {bI aho h¢&
(I) ¶{X Amn {hÝXr Ed§ A§J«oOr Ho$ A{V[aº$ {H$gr Aݶ ^mfm ‘| CÎma {bIVo h¢, Vmo àíZm| H$mo g‘PZo ‘| hmoZo dmbr
Ìw{Q>¶m|/Jb{V¶m| H$s {Oå‘oXmar Ho$db AmnH$s hmoJr&

211/AS/3/403A 2

MATHEMATICS
J{UV
(211)
Time : 2½ Hours ] [ Maximum Marks : 85
g‘¶ … 2½ KÊQ>o ] [ nyUmªH$ … 85

Note : (i) Question Numbers (1 to 10) are Multiple Choice Questions. Each
question carries one mark. For each question, four alternative choices
(A), (B), (C) and (D) are provided of which only one is correct. You have
to select the correct alternative and indicate it in the Answer-Book
provided to you by writing (A), (B), (C) or (D) as the case may be.
Question Numbers (11 to 15) also carry one mark each.

(ii) Question Numbers (16–25) carry 2 marks each.

(iii) Question Numbers (26–33) carry 4 marks each.

(iv) Question Numbers (34–36) carry 6 marks each.

(v) All questions are compulsory.

{ZX}e … (i) àíZ g§»¶m (1 go 10) VH$ ~hþ{dH$ënr àíZ (Multiple Choice Questions) h¢& à˶oH$ àíZ
EH$ A§H$ H$m h¡& à˶oH$ àíZ ‘| Mma {dH$ën (A), (B), (C) Am¡a (D) {XE JE h¢, {OZ‘| go Ho$db
EH$ ghr h¡& AmnH$mo ghr {dH$ën MwZZm h¡ VWm à˶oH$ àíZ Ho$ CÎma AnZr CÎma-nwpñVH$m ‘|
(A), (B), (C) AWdm (D), Omo ^r hmo, {bIH$a Xem©Zm h¡& àíZ g§»¶m (11 go 15) à˶oH$ àíZ
^r EH$ A§H$ H$m h¡&

(ii) àíZ g§»¶m (16–25) VH$ à˶oH$ Ho$ 2 A§H$ h¢&

(iii) àíZ g§»¶m (26–33) VH$ à˶oH$ Ho$ 4 A§H$ h¢&

(iv) àíZ g§»¶m (34–36) VH$ à˶oH$ Ho$ 6 A§H$ h¢&

(v) g^r àíZ A{Zdm¶© h¢&

211/AS/3/403A 3 [ P.T.O.

1. Representation of 3  6 in rational number form is
11 3
(A) (B)
3 11

36 33
(C) (D) 1
10 10

3  6 H$m n[a‘o¶ g§»¶m Ho$ ê$n ‘| {Zê$nU h¡
11 3
(A) (B)
3 11

36 33
(C) (D)
10 10

2
 2  3 
2. The value of    is
 3  

243 64
(A) (B)
32 729

729 32
(C) (D) 1
64 243
2
 2  3 
   H$m ‘mZ h¡
 3  

243 64
(A) (B)
32 729

729 32
(C) (D)
64 243

3. The pair of equations x  2y  3 and 5x  ky  7  0 has no solution, if

(A) k  10 (B) k  10

7 7
(C) k   (D) k  1
3 6

211/AS/3/403A 4

g‘rH$aU ¶w½‘ x  2y  3 Am¡a 5x  ky  7  0 H$m H$moB© hb Zht h¡, ¶{X
(A) k  10 (B) k  10

7 7
(C) k   (D) k 
3 6

4. The price of an article is increased from < 550 to < 605. The percent
increase in price is

44
(A) 55% (B) %
5
(C) 10% (D) 15% 1
EH$ dñVw Ho$ ‘yë¶ H$mo < 550 go ~‹T>mH$a < 605 H$a {X¶m OmE, Vmo ‘yë¶ ‘| à{VeV ~‹T>moVar h¡
44
(A) 55% (B) %
5
(C) 10% (D) 15%

5. A triangle and a parallelogram are on the same base and between same
parallels, then the ratio of the area of triangle to the area of parallelogram
is
(A) 1 : 3 (B) 1 : 2
(C) 3 : 1 (D) 1 : 4 1
EH$ {Ì^wO VWm EH$ g‘m§Va MVw^w©O, g‘mZ AmYma na h¢ VWm EH$ hr g‘m§Va aoImAm| Ho$ ‘ܶ ‘| h¢,
Vmo {Ì^wO VWm g‘m§Va MVw^w©O Ho$ joÌ’$b ‘| AZwnmV h¡
(A) 1 : 3 (B) 1 : 2
(C) 3 : 1 (D) 1 : 4

6. ABCD is a rhombus. If ACB  40 , then the measure of ADB is
(A) 40° (B) 45°
(C) 50° (D) 60° 1
ABCD EH$ g‘MVw^w©O h¡& ¶{X ACB  40 h¡, Vmo ADB H$m ‘mZ h¡
(A) 40° (B) 45°
(C) 50° (D) 60°

211/AS/3/403A 5 [ P.T.O.

7. The distance (in cm) of a chord of length 18 cm from the centre of a circle
of radius 15 cm is
(A) 306 (B) 17

(C) 12 (D) 99 1

15 go.‘r. {ÌÁ¶m dmbo EH$ d¥Îm H$s 18 go.‘r. b§~r EH$ Ordm H$s d¥Îm Ho$ H|$Ð go Xyar (go.‘r. ‘|)
h¡
(A) 306 (B) 17

(C) 12 (D) 99

8. If (a, 4) is the mid-point of the line segment joining the points A(– 6, 5)
and B(– 2, 3), then the value of a is
(A) – 1 (B) – 4
(C) – 6 (D) – 12 1
¶{X q~XþAm| A(– 6, 5) VWm B(– 2, 3) H$mo {‘bmZo dmbo aoIm-IÊS> H$m ‘ܶq~Xþ (a, 4) h¡, Vmo
a H$m ‘mZ h¡
(A) – 1 (B) – 4
(C) – 6 (D) – 12

9. If sec 4 A  cosec ( A  10) , 4A  90 , then the measure of A is
(A) 20° (B) 30°
(C) 10° (D) 50° 1

¶{X sec 4A  cosec (A  10) h¡, O~{H$ 4A  90 h¡, Vmo A H$m ‘mZ h¡
(A) 20° (B) 30°
(C) 10° (D) 50°

10. sin2 A  cosec 2A  cos 2 A  cot 2 A is equal to
(A) 1 (B) – 1
(C) 0 (D) 2 1

sin2 A  cosec 2A  cos 2 A  cot 2 A ~am~a h¡
(A) 1 (B) – 1
(C) 0 (D) 2

211/AS/3/403A 6

11. Find the value of k for which – 3 is a zero of the polynomial x 2  11x  k . 1

k H$m dh ‘mZ kmV H$s{OE, {OgHo$ {bE – 3, ~hþnX x 2  11x  k H$m EH$ eyݶH$ hmo&

12. Simplify : 1

 3 2  7 11
   
 5 7  8 15

gab H$s{OE …
 3 2  7 11
   
 5 7  8 15

13. If one root of the equation 2x 2  5x  (  6)  0 is the reciprocal of the
other, then find the value of  . 1

¶{X g‘rH$aU 2x 2  5x  (  6)  0 H$m EH$ ‘yb, BgHo$ Xygao ‘yb H$m ì¶wËH«$‘ h¡, Vmo  H$m
‘mZ kmV H$s{OE&

14. A pair of socks with listing price < 80 is available for < 64. Find the
discount percent offered. 1
< 80 A§{H$V ‘yë¶ dmbo Owamdm| H$m EH$ Omo‹S>m < 64 ‘| CnbãY h¡& {X¶m OmZo dmbm à{VeV ~Å>m
kmV H$s{OE&

15. If one angle of a parallelogram is 54 th of its adjacent angle, then find the
angles of the parallelogram. 1

¶{X EH$ g‘m§Va MVw^w©O H$m EH$ H$moU BgHo$ AmgÞ H$moU H$m 54 h¡, Vmo g‘m§Va MVw^w©O H$m H$moU
kmV H$s{OE&

16. A shopkeeper marks his goods 50% more than their cost price and allows
a discount of 40% on their sale. Find his gain or loss percent. 2
EH$ XþH$mZXma AnZo gm‘mZm| H$m A§{H$V ‘yë¶, CZHo$ H«$¶ ‘yë¶ go 50% A{YH$ aIVm h¡ Am¡a BZHo$
~oMZo na 40% ~Å>m XoVm h¡& CgH$m bm^ ¶m hm{Z à{VeV kmV H$s{OE&

211/AS/3/403A 7 [ P.T.O.

17. In Fig. 1 D, E and F are the mid-points of the sides of ABC . Show that
3
BE  CF  BC . 2
2

3
Fig. 1 ‘|, D, E VWm F, ABC H$s ^wOmAm| Ho$ ‘ܶq~Xþ h¢& Xem©BE {H$ BE  CF  2 BC .

Fig. 1

18. In Fig. 2, PAB is a secant and PT is a tangent to the circle from an external
point P. If PT  x cm, PA  4 cm and AB  5 cm, then find the value
of x. 2
Fig. 2 ‘|, PAB d¥Îm H$s N>oXH$ aoIm VWm PT d¥Îm H$s ñne©aoIm h¡, Omo {H$ EH$ ~mø q~Xþ P go
ItMr JB© h¡& ¶{X PT  x go.‘r., PA  4 go.‘r. VWm AB  5 go.‘r. h¡, Vmo x H$m ‘mZ kmV
H$s{OE&

Fig. 2

19. Find the area of the sector of a circle of radius 7 cm with central
angle 45°. 2
7 go.‘r. {ÌÁ¶m dmbo EH$ d¥Îm Ho$ Cg {ÌÁ¶IÊS> H$m joÌ’$b kmV H$s{OE, {OgH$m H|$Ðr¶ H$moU 45°
h¡&

211/AS/3/403A 8

20. The volume of a hemispherical bowl is 2425·5 cm3. Find its radius. 2
EH$ AY©JmobmH$ma H$Q>moao H$m Am¶VZ 2425·5 go‘r3 h¡& BgH$s {ÌÁ¶m kmV H$s{OE&

4 sin A  3 cos A 1
21. If 3 cot A  2 , then show that  . 2
2sin A  6 cos A 3

4 sin A  3 cos A 1
¶{X 3 cot A  2 h¡, Vmo Xem©BE {H$ 2sin A  6 cos A  3 h¡&

22. Prove : 2

cos  1  sin 
  2sec 
1  sin  cos 

{gÕ H$s{OE …
cos  1  sin 
  2sec 
1  sin  cos 

23. Diagonals of a trapezium ABCD with AB  DC intersect each other at O.
If AB  2CD , find the ratio of the areas of triangles AOB and COD. 2

EH$ g‘b§~ ABCD, {Og‘| AB  DC h¡, Ho$ {dH$U© nañna O na H$mQ>Vo h¢& ¶{X AB  2CD
h¡, Vmo {Ì^wOm| AOB VWm COD Ho$ joÌ’$bm| H$m AZwnmV kmV H$s{OE&

24. The percentage of marks obtained by 100 students in an examination are
given below :

% marks obtained : 30 35 40 45 50 55 60
Number of students : 14 16 18 23 18 8 3

Determine the median percentage of marks. 2
100 {dÚm{W©¶m| Ûmam EH$ narjm ‘| àmá à{VeV A§H$ {ZåZ h¢ …

àmám§H (à{VeV)$ : 30 35 40 45 50 55 60
{dÚm{W¶©mo§ H$s g§»¶m$ : 14 16 18 23 18 8 3
‘mܶH$ à{VeV A§H$ kmV H$s{OE&

211/AS/3/403A 9 [ P.T.O.

25. Two different dice are thrown simultaneously. What is the probability that
the sum of numbers appearing on the dice is
(a) less than 5;
(b) at least 11? 2
Xmo {d{^Þ nmgm| H$mo EH$ gmW CN>mbm J¶m& XmoZmo§ nmgm| na AmB© g§»¶mAm| H$m ¶moJ {ZåZ hmoZo H$s àm{¶H$Vm
³¶m h¡?
(H$) 5 go H$‘
(I) H$‘-go-H$‘ 11

26. A motor boat whose speed in still water is 9 km/h, goes 15 km down-
stream and comes back at the same point in a total time of 3 hours
45 minutes. Find the speed of the motor boat. 4
EH$ ‘moQ>a~moQ>, {OgH$s pñWa Ob ‘| Mmb 9 {H$.‘r./K§Q>m h¡, 15 {H$.‘r. H$s Xÿar Ymam H$s {Xem ‘|
MbZo Ho$ ~mX dmng Cgr ñWmZ na AmZo ‘| 3 K§Q>o 45 {‘ZQ> H$m Hw$b g‘¶ boVr h¡& ‘moQ>a~moQ> H$s
Mmb kmV H$s{OE&

27. If the sum of first 6 terms of an AP is 36 and that of the first 16 terms
is 256, find the sum of its first 10 terms. 4
¶{X {H$gr g‘m§Va lo‹T>r Ho$ àW‘ 6 nXm| H$m ¶moJ 36 h¡ VWm àW‘ 16 nXm| H$m ¶moJ 256 h¡, Vmo
CgHo$ àW‘ 10 nXm| H$m ¶moJ kmV H$s{OE&

28. Find the sum of money which will amount to < 26,460 in six months at
20% per annum, when the interest is compounded quarterly. 4
dh am{e kmV H$s{OE, Omo N>… ‘hrZo ‘| 20% dm{f©H$ Xa go < 26,460 hmo OmEJr, O~{H$ ã¶mO
{V‘mhr g§¶mo{OV hmoVm h¡&

29. Prove that the tangents drawn at the ends of a diameter of a circle are
parallel. 4
{gÕ H$s{OE {H$ {H$gr d¥Îm Ho$ EH$ ì¶mg Ho$ {gam| na ItMr JB© ñne©aoImE± g‘m§Va hmoVr h¢&

211/AS/3/403A 10

30. Draw a circle of diameter 6 cm. From a point P outside the circle at a
distance of 7 cm from the centre, draw two tangents to the circle. 4
6 go.‘r. ì¶mg H$m EH$ d¥Îm It{ME& d¥Îm Ho$ H|$Ð go 7 go.‘r. H$s Xÿar na Ho$ EH$ ~mø q~Xþ P go
d¥Îm na Xmo ñne©aoImE± It{ME&

Or / AWdm

( For Visually Impaired Learners only )
( Ho$db Ñ{ï> {dH$bm§J {dÚm{W©¶m| Ho$ {bE )

Write only the steps of construction for the following :
{ZåZ Ho$ {bE Ho$db aMZm Ho$ MaU {b{IE …
Draw a circle of diameter 6 cm. From a point P outside the circle at a
distance of 7 cm from the centre, draw two tangents to the circle.
6 go.‘r. ì¶mg H$m EH$ d¥Îm It{ME& d¥Îm Ho$ H|$Ð go 7 go.‘r. H$s Xÿar na Ho$ EH$ ~mø q~Xþ P go
d¥Îm na Xmo ñne©aoImE± It{ME&

31. A person standing on the bank of a river, observes that the angle of
elevation of the top of a tree standing on the opposite bank is 60°. When
he moves 40 metres away from the bank, he finds the angle to be 30°.
Find the height of the tree and the width of the river. 4
EH$ ì¶{º$ ZXr Ho$ EH$ {H$Zmao na I‹S>m hmoH$a Xygao {H$Zmao na bJo EH$ no‹S> Ho$ {eIa H$m CÞ¶Z H$moU
60° nmVm h¡& O~ dh 40 ‘rQ>a nrN>o hQ> OmVm h¡, Vmo H$moU 30° H$m hmo OmVm h¡& no‹S> H$s D±$MmB©
Am¡a ZXr H$s Mm¡‹S>mB© kmV H$s{OE&

32. Find the mean of the following data : 4

Class : 156–158 158–160 160–162 162–164 164–166 166–168
Frequency : 2 4 8 16 14 6

{ZåZ Am±H$S>m| H$m ‘mܶ kmV H$s{OE :

dJ©$ : 156–158 158–160 160–162 162–164 164–166 166–168
~ma§ ~maVm $: 2 4 8 16 14 6

211/AS/3/403A 11 [ P.T.O.

33. In a bag, there are 44 identical cards with figure of a circle or a square
on them. There are 24 circles of which 9 are blue and rest are green, and
20 squares of which 11 are blue and rest are green. One card is drawn
at random from the bag. Find the pobability that it has a figure of
(a) a square (b) green colour (c) a blue circle (d) a green square. 4
EH$ W¡bo ‘| 44 EH$ O¡go H$mS>© h¢, {OZ‘| à˶oH$ na d¥Îm AWdm dJ© H$m {MÌ ~Zm h¡& 24 d¥Îm h¢
{OZ‘| 9 Zrbo a§J Ho$ h¢ VWm Aݶ hao a§J Ho$ h¢ Am¡a 20 dJ© h¢, {OZ‘| 11 Zrbo a§J Ho$ h¢ VWm
Aݶ hao h¢& W¡bo ‘| go ¶mÑÀN>¶m EH$ H$mS>© {ZH$mbm J¶m& àm{¶H$Vm kmV H$s{OE {H$ {ZH$mbo JE H$mS>©
H$m {MÌ (H$) dJ© hmo, (I) hao a§J H$m hmo, (J) Zrbm d¥Îm hmo (K) ham dJ© hmo&

34. The sum of a 2-digit number and the number formed by interchanging the
digits is 132. If 12 is added to the number, the new number becomes
5 times the sum of the digits. Find the number. 6
EH$ 2-A§H$s¶ g§»¶m VWm CgHo$ A§H$m| H$m ñWmZ nbQ>Zo na àmá g§»¶m H$m ¶moJ 132 h¡& ¶{X g§»¶m
‘| 12 Omo‹S> {XE OmE±, Vmo àmá ZB© g§»¶m, ‘yb g§»¶m Ho$ A§H$m| Ho$ ¶moJ Ho$ 5 JwZo Ho$ ~am~a hmo OmEJm&
‘yb g§»¶m kmV H$s{OE&

35. Points A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD.
If E is the mid-point of CD, find the area of ADE . 6
q~Xþ A(6, 1), B(8, 2) VWm C(9, 4) EH$ g‘m§Va MVw^w©O ABCD Ho$ erf© h¢& ¶{X E, ^wOm CD
H$m ‘ܶq~Xþ h¡, Vmo ADE H$m joÌ’$b kmV H$s{OE&

36. Water is flowing through a cylindrical pipe of internal diameter 2 cm into
a cylindrical tank of base radius 40 cm, at the rate of 0·4 m per second.
Determine the rise in the level of water in the tank in half an hour. 6
2 go.‘r. Am§V[aH$ {ÌÁ¶m dmbr EH$ ~obZmH$ma nmBn ‘| nmZr 0·4 ‘r./go. H$s J{V go ~hVm hþAm,
40 go.‘r. {ÌÁ¶m dmbo EH$ ~obZmH$ma Q>¢H$ ‘| {Ja ahm h¡& AmYo K§Q>o ‘| Q>¢H$ ‘| nmZr Ho$ Vb ‘| ~‹T>moVar
kmV H$s{OE&

  

211/AS/3/403A [V23] 12

This Question Paper consists of 36 questions and 12 printed pages.
Bg àíZ-nÌ ‘| 36 àíZ VWm 12 ‘w{ÐV n¥ð> h¢& Code No.
65/AS/3
v
H$moS> Z§0
Roll No.
Set / goQ>
AZwH«$‘m§H$

MATHEMATICS
J{UV
(211)
Day and Date of Examination .....................................................................................
(narjm H$m {XZ d {XZm§H$)
Signature of Invigilators 1. .....................................................................................
({ZarjH$m| Ho$ hñVmja)
2. .....................................................................................

General Instructions :

1. Candidate must write his/her Roll Number on the first page of the Question
Paper.

2. Please check the Question Paper to verify that the total pages and total
number of questions contained in the Question Paper are the same as those
printed on the top of the first page. Also check to see that the questions are
in sequential order.

3. For the objective type of questions, you have to choose any one of the four
alternatives given in the question, i.e., (A), (B), (C) or (D) and indicate your
correct answer in the Answer-Book given to you.

4. All the questions including objective-type questions are to be answered within
the allotted time and no separate time limit is fixed for answering objective-
type questions.

5. Making any identification mark in the Answer-Book or writing Roll Number
anywhere other than the specified places will lead to disqualification of the
candidate.

6. Write your Question Paper Code No. 65/AS/3, Set v on the Answer-Book.

211/AS/3/403B [ P.T.O.

7. (a) The Question Paper is in English/Hindi medium only. However, If you
wish, you can answer in any one of the languages listed below :
English, Hindi, Urdu, Punjabi, Bengali, Tamil, Malayalam, Kannada,
Telugu, Marathi, Odia, Gujarati, Konkani, Manipuri, Assamese, Nepali,
Kashmiri, Sanskrit and Sindhi.
You are required to indicate the language you have chosen to answer in
the box provided in the Answer-Book.
(b) If you choose to write the answer in the language other than Hindi and
English, the responsibility for any errors/mistakes in understanding the
questions will be yours only.

gm‘mݶ AZwXoe …
1. narjmWu àíZ-nÌ Ho$ nhbo n¥ð> na AnZm AZwH«$‘m§H$ Adí¶ {bI|&
2. H¥$n¶m àíZ-nÌ H$mo Om±M b| {H$ àíZ-nÌ Ho$ Hw$b n¥ð>m| VWm àíZm| H$s CVZr hr g§»¶m h¡ {OVZr àW‘ n¥ð> Ho$ g~go
D$na N>nr h¡& Bg ~mV H$s Om±M ^r H$a b| {H$ àíZ H«${‘H$ ê$n ‘| h¢&
3. dñVw{Zð> àíZm| _o§ AmnH$mo Mma {dH$ënm| (A), (B), (C) VWm (D) _| go H$moB© EH$ CÎma MwZZm h¡ VWm Xr JB©
CÎma-nwpñVH$m _| Amn ghr CÎma {bI|Ÿ&
4. dñVw{Zð> àíZm| Ho$ gmW-gmW g^r àíZm| Ho$ CÎma {ZYm©[aV Ad{Y Ho$ ^rVa hr XoZo h¢Ÿ& dñVw{Zð> àíZm| Ho$ {bE AbJ go
g_` Zht {X`m OmEJmŸ&
5. CÎma-nwpñVH$m ‘| nhMmZ-{M• ~ZmZo AWdm {Z{X©ï> ñWmZm| Ho$ A{V[aº$ H$ht ^r AZwH«$‘m§H$ {bIZo na narjmWu H$mo
A¶mo½¶ R>ham¶m OmEJm&
6. AnZr CÎma-nwpñVH$m na àíZ-nÌ H$m H$moS> Z§0 65/AS/3, goQ v, {bI|&
7. (H$) n«íZ-nÌ Ho$db {hÝXr/A§J«oOr ‘| h¡& {’$a ^r, ¶{X Amn Mmh| Vmo ZrMo Xr JB© {H$gr EH$ ^mfm ‘| CÎma Xo
gH$Vo h¢ …
A§J«oOr, {hÝXr, CXÿ©, n§Om~r, ~§Jbm, V{‘b, ‘b¶mb‘, H$Þ‹S>, VobwJy, ‘amR>r, C{‹S>¶m, JwOamVr, H$m|H$Ur,
‘{Unwar, Ag{‘¶m, Zonmbr, H$í‘rar, g§ñH¥$V Am¡a {gÝYr&
H¥$n¶m CÎma-nwpñVH$m ‘| {XE JE ~m°³g ‘| {bI| {H$ Amn {H$g ^mfm ‘| CÎma {bI aho h¢&
(I) ¶{X Amn {hÝXr Ed§ A§J«oOr Ho$ A{V[aº$ {H$gr Aݶ ^mfm ‘| CÎma {bIVo h¢, Vmo àíZm| H$mo g‘PZo ‘| hmoZo dmbr
Ìw{Q>¶m|/Jb{V¶m| H$s {Oå‘oXmar Ho$db AmnH$s hmoJr&

211/AS/3/403B 2

MATHEMATICS
J{UV
(211)
Time : 2½ Hours ] [ Maximum Marks : 85
g‘¶ … 2½ KÊQ>o ] [ nyUmªH$ … 85

Note : (i) Question Numbers (1 to 10) are Multiple Choice Questions. Each
question carries one mark. For each question, four alternative choices
(A), (B), (C) and (D) are provided of which only one is correct. You have
to select the correct alternative and indicate it in the Answer-Book
provided to you by writing (A), (B), (C) or (D) as the case may be.
Question Numbers (11 to 15) also carry one mark each.

(ii) Question Numbers (16–25) carry 2 marks each.

(iii) Question Numbers (26–33) carry 4 marks each.

(iv) Question Numbers (34–36) carry 6 marks each.

(v) All questions are compulsory.

{ZX}e … (i) àíZ g§»¶m (1 go 10) VH$ ~hþ{dH$ënr àíZ (Multiple Choice Questions) h¢& à˶oH$ àíZ
EH$ A§H$ H$m h¡& à˶oH$ àíZ ‘| Mma {dH$ën (A), (B), (C) Am¡a (D) {XE JE h¢, {OZ‘| go Ho$db
EH$ ghr h¡& AmnH$mo ghr {dH$ën MwZZm h¡ VWm à˶oH$ àíZ Ho$ CÎma AnZr CÎma-nwpñVH$m ‘|
(A), (B), (C) AWdm (D), Omo ^r hmo, {bIH$a Xem©Zm h¡& àíZ g§»¶m (11 go 15) à˶oH$ àíZ
^r EH$ A§H$ H$m h¡&

(ii) àíZ g§»¶m (16–25) VH$ à˶oH$ Ho$ 2 A§H$ h¢&

(iii) àíZ g§»¶m (26–33) VH$ à˶oH$ Ho$ 4 A§H$ h¢&

(iv) àíZ g§»¶m (34–36) VH$ à˶oH$ Ho$ 6 A§H$ h¢&

(v) g^r àíZ A{Zdm¶© h¢&

211/AS/3/403B 3 [ P.T.O.

1. A triangle and a parallelogram are on the same base and between same
parallels, then the ratio of the area of triangle to the area of parallelogram
is
(A) 1 : 3 (B) 1 : 2
(C) 3 : 1 (D) 1 : 4 1
EH$ {Ì^wO VWm EH$ g‘m§Va MVw^w©O, g‘mZ AmYma na h¢ VWm EH$ hr g‘m§Va aoImAm| Ho$ ‘ܶ ‘| h¢,
Vmo {Ì^wO VWm g‘m§Va MVw^w©O Ho$ joÌ’$b ‘| AZwnmV h¡
(A) 1 : 3 (B) 1 : 2
(C) 3 : 1 (D) 1 : 4

2. ABCD is a rhombus. If ACB  40 , then the measure of ADB is
(A) 40° (B) 45°
(C) 50° (D) 60° 1
ABCD EH$ g‘MVw^w©O h¡& ¶{X ACB  40 h¡, Vmo ADB H$m ‘mZ h¡
(A) 40° (B) 45°
(C) 50° (D) 60°

3. If sec 4 A  cosec ( A  10) , 4A  90 , then the measure of A is
(A) 20° (B) 30°
(C) 10° (D) 50° 1

¶{X sec 4 A  cosec ( A  10) h¡, O~{H$ 4A  90 h¡, Vmo A H$m ‘mZ h¡
(A) 20° (B) 30°
(C) 10° (D) 50°

4. sin2 A  cosec 2A  cos 2 A  cot 2 A is equal to
(A) 1 (B) – 1
(C) 0 (D) 2 1

sin2 A  cosec 2A  cos 2 A  cot 2 A ~am~a h¡
(A) 1 (B) – 1
(C) 0 (D) 2

211/AS/3/403B 4

5. Representation of 3  6 in rational number form is
11 3
(A) (B)
3 11

36 33
(C) (D) 1
10 10

3  6 H$m n[a‘o¶ g§»¶m Ho$ ê$n ‘| {Zê$nU h¡
11 3
(A) (B)
3 11

36 33
(C) (D)
10 10

2
 2  3 
6. The value of    is
 3  

243 64
(A) (B)
32 729

729 32
(C) (D) 1
64 243
2
 2 3 
   H$m ‘mZ h¡
 3  

243 64
(A) (B)
32 729

729 32
(C) (D)
64 243

7. The pair of equations x  2y  3 and 5x  ky  7  0 has no solution, if

(A) k  10 (B) k  10

7 7
(C) k   (D) k  1
3 6

211/AS/3/403B 5 [ P.T.O.

g‘rH$aU ¶w½‘ x  2y  3 Am¡a 5x  ky  7  0 H$m H$moB© hb Zht h¡, ¶{X
(A) k  10 (B) k  10

7 7
(C) k   (D) k 
3 6

8. The price of an article is increased from < 550 to < 605. The percent
increase in price is

44
(A) 55% (B) %
5
(C) 10% (D) 15% 1
EH$ dñVw Ho$ ‘yë¶ H$mo < 550 go ~‹T>mH$a < 605 H$a {X¶m OmE, Vmo ‘yë¶ ‘| à{VeV ~‹T>moVar h¡
44
(A) 55% (B) %
5
(C) 10% (D) 15%

9. The distance of a chord (in cm) of length 18 cm from the centre of a circle
of radius 15 cm is

(A) 306 (B) 17

(C) 12 (D) 99 1

15 go.‘r. {ÌÁ¶m dmbo EH$ d¥Îm H$s 18 go.‘r. b§~r EH$ Ordm H$s d¥Îm Ho$ H|$Ð go Xyar (go.‘r. ‘|)
h¡
(A) 306 (B) 17

(C) 12 (D) 99

10. If (a, 4) is the mid-point of the line segment joining the points A(– 6, 5)
and B(– 2, 3), then the value of a is
(A) – 1 (B) – 4
(C) – 6 (D) – 12 1

211/AS/3/403B 6

¶{X q~XþAm| A(– 6, 5) VWm B(– 2, 3) H$mo {‘bmZo dmbo aoIm-IÊS> H$m ‘ܶq~Xþ (a, 4) h¡, Vmo
a H$m ‘mZ h¡
(A) – 1 (B) – 4
(C) – 6 (D) – 12

11. If one root of the equation 2x 2  5x  (  6)  0 is the reciprocal of the
other, then find the value of  . 1

¶{X g‘rH$aU 2x 2  5x  (  6)  0 H$m EH$ ‘yb, BgHo$ Xygao ‘yb H$m ì¶wËH«$‘ h¡, Vmo  H$m
‘mZ kmV H$s{OE&

12. A pair of socks with listing price < 80 is available for < 64. Find the
discount percent offered. 1
< 80 A§{H$V ‘yë¶ dmbo Owamdm| H$m EH$ Omo‹S>m < 64 ‘| CnbãY h¡& {X¶m OmZo dmbm à{VeV ~Å>m
kmV H$s{OE&

13. If one angle of a parallelogram is 54 th of its adjacent angle, then find the
angles of the parallelogram. 1

¶{X EH$ g‘m§Va MVw^w©O H$m EH$ H$moU BgHo$ AmgÞ H$moU H$m 54 h¡, Vmo g‘m§Va MVw^w©O H$m H$moU
kmV H$s{OE&

14. Simplify : 1

 3 2  7 11
   
 5 7  8 15

gab H$s{OE …
 3 2  7 11
   
 5 7  8 15

211/AS/3/403B 7 [ P.T.O.

15. Find the value of k for which – 3 is a zero of the polynomial x 2  11x  k . 1

k H$m dh ‘mZ kmV H$s{OE, {OgHo$ {bE – 3, ~hþnX x 2  11x  k H$m EH$ eyݶH$ hmo&

16. Diagonals of a trapezium ABCD with AB  DC intersect each other at O.
If AB  2CD , find the ratio of the areas of triangles AOB and COD. 2

EH$ g‘b§~ ABCD, {Og‘| AB  DC h¡, Ho$ {dH$U© nañna O na H$mQ>Vo h¢& ¶{X AB  2CD
h¡, Vmo {Ì^wOm| AOB VWm COD Ho$ joÌ’$bm| H$m AZwnmV kmV H$s{OE&

17. The percentage of marks obtained by 100 students in an examination are
given below :

% marks obtained : 30 35 40 45 50 55 60
Number of students : 14 16 18 23 18 8 3

Determine the median percentage of marks. 2
100 {dÚm{W©¶m| Ûmam EH$ narjm ‘| àmá à{VeV A§H$ {ZåZ h¢ …

àmám§H (à{VeV)$ : 30 35 40 45 50 55 60
{dÚm{W¶©mo§ H$s g§»¶m$ : 14 16 18 23 18 8 3
‘mܶH$ à{VeV A§H$ kmV H$s{OE&

18. Two different dice are thrown simultaneously. What is the probability that
the sum of numbers appearing on the dice is
(a) less than 5;
(b) at least 11? 2
Xmo {d{^Þ nmgm| H$mo EH$ gmW CN>mbm J¶m& XmoZmo§ nmgm| na AmB© g§»¶mAm| H$m ¶moJ {ZåZ hmoZo H$s àm{¶H$Vm
³¶m h¡?
(H$) 5 go H$‘
(I) H$‘-go-H$‘ 11

211/AS/3/403B 8

19. In Fig. 1 D, E and F are the mid-points of the sides of ABC . Show that
3
BE  CF  BC . 2
2

3
Fig. 1 ‘|, D, E VWm F, ABC H$s ^wOmAm| Ho$ ‘ܶq~Xþ h¢& Xem©BE {H$ BE  CF  2 BC .

Fig. 1

4 sin A  3 cos A 1
20. If 3 cot A  2 , then show that  . 2
2sin A  6 cos A 3

4 sin A  3 cos A 1
¶{X 3 cot A  2 h¡, Vmo Xem©BE {H$  h¡&
2sin A  6 cos A 3

21. In Fig. 2, PAB is a secant and PT is a tangent to the circle from an external
point P. If PT  x cm, PA  4 cm and AB  5 cm, then find the value
of x. 2
Fig. 2 ‘|, PAB d¥Îm H$s N>oXH$ aoIm VWm PT d¥Îm H$s ñne©aoIm h¡, Omo {H$ EH$ ~mø q~Xþ P go
ItMr JB© h¡& ¶{X PT  x go.‘r., PA  4 go.‘r. VWm AB  5 go.‘r. h¡, Vmo x H$m ‘mZ kmV
H$s{OE&

Fig. 2

211/AS/3/403B 9 [ P.T.O.

22. A retailer buys books from a wholesaler at the rate of < 200 per book and
marked them at < 300 each. He allows some discount and gets a profit
of 20% on the cost price. What percent discount does he allow to his
customers? 2
EH$ ’w$Q>H$a {dH«o$Vm EH$ WmoH$ {dH«o$Vm go < 200 à{V nwñVH$ Ho$ ^md go nwñVH|$ IarXVm h¡ Am¡a à˶oH$
nwñVH$ na < 300 H$m ‘yë¶ A§{H$V H$aVm h¡& dh CÝh| ~Å>o na ~oMH$a 20% H$m bm^ A{O©V H$aVm
h¡& kmV H$s{OE {H$ dh J«mhH$m| H$mo {H$VZo à{VeV ~Å>m XoVm h¡&

23. Find the area of the sector of a circle of radius 6·3 cm with central
angle 60°. 2
6·3 go.‘r. {ÌÁ¶m dmbo EH$ d¥Îm Ho$ Cg {ÌÁ¶IÊS> H$m joÌ’$b kmV H$s{OE, {OgH$m H|$Ðr¶ H$moU 60°
h¡&

24. Three cubes each of side 6 cm are joined end to end in a row. Find the
surface area and the volume of the resulting cuboid. 2
^wOm 6 go.‘r. dmbo VrZ KZm| H$mo {gao go {gam {‘bmH$a EH$ n§{º$ ‘§o aIm OmVm h¡& Bg àH$ma ~Zo
KZm^ H$m n¥ð>r¶ joÌ’$b Am¡a Am¶VZ kmV H$s{OE&

25. Prove : 2
(cosec   sin )(sec   cos )(tan   cot )  1

{gÕ H$s{OE …
(cosec   sin )(sec   cos )(tan   cot )  1

26. Find the sum of money which will amount to < 26,460 in six months at
20% per annum, when the interest is compounded quarterly. 4
dh am{e kmV H$s{OE, Omo N>… ‘hrZo ‘| 20% dm{f©H$ Xa go < 26,460 hmo OmEJr, O~{H$ ã¶mO
{V‘mhr g§¶mo{OV hmoVm h¡&

27. Prove that the tangents drawn at the ends of a diameter of a circle are
parallel. 4
{gÕ H$s{OE {H$ {H$gr d¥Îm Ho$ EH$ ì¶mg Ho$ {gam| na ItMr JB© ñne©aoImE± g‘m§Va hmoVr h¢&

28. Find the mean of the following data : 4

Class : 156–158 158–160 160–162 162–164 164–166 166–168
Frequency : 2 4 8 16 14 6

211/AS/3/403B 10

{ZåZ Am±H$S>m| H$m ‘mܶ kmV H$s{OE :
dJ©$ : 156–158 158–160 160–162 162–164 164–166 166–168
~ma§ ~maVm $: 2 4 8 16 14 6

29. In a bag, there are 44 identical cards with figure of a circle or a square
on them. There are 24 circles of which 9 are blue and rest are green, and
20 squares of which 11 are blue and rest are green. One card is drawn
at random from the bag. Find the pobability that it has a figure of
(a) a square (b) green colour (c) a blue circle (d) a green square. 4
EH$ W¡bo ‘| 44 EH$ O¡go H$mS>© h¢, {OZ‘| à˶oH$ na d¥Îm AWdm dJ© H$m {MÌ ~Zm h¡& 24 d¥Îm h¢
{OZ‘| 9 Zrbo a§J Ho$ h¢ VWm Aݶ hao a§J Ho$ h¢ Am¡a 20 dJ© h¢, {OZ‘| 11 Zrbo a§J Ho$ h¢ VWm
Aݶ hao h¢& W¡bo ‘| go ¶mÑÀN>¶m EH$ H$mS>© {ZH$mbm J¶m& àm{¶H$Vm kmV H$s{OE {H$ {ZH$mbo JE H$mS>©
H$m {MÌ (H$) dJ© hmo, (I) hao a§J H$m hmo, (J) Zrbm d¥Îm hmo (K) ham dJ© hmo&
30. A person standing on the bank of a river, observes that the angle of
elevation of the top of a tree standing on the opposite bank is 60°. When
he moves 40 metres away from the bank, he finds the angle to be 30°.
Find the height of the tree and the width of the river. 4
EH$ ì¶{º$ ZXr Ho$ EH$ {H$Zmao na I‹S>m hmoH$a Xygao {H$Zmao na bJo EH$ no‹S> Ho$ {eIa H$m CÞ¶Z H$moU
60° nmVm h¡& O~ dh 40 ‘rQ>a nrN>o hQ> OmVm h¡, Vmo H$moU 30° H$m hmo OmVm h¡& no‹S> H$s D±$MmB©
Am¡a ZXr H$s Mm¡‹S>mB© kmV H$s{OE&
31. Draw a circle of diameter 6 cm. From a point P outside the circle at a
distance of 7 cm from the centre, draw two tangents to the circle. 4
6 go.‘r. ì¶mg H$m EH$ d¥Îm It{ME& d¥Îm Ho$ H|$Ð go 7 go.‘r. H$s Xÿar na Ho$ EH$ ~mø q~Xþ P go
d¥Îm na Xmo ñne©aoImE± It{ME&
Or / AWdm

( For Visually Impaired Learners only )
( Ho$db Ñ{ï> {dH$bm§J {dÚm{W©¶m| Ho$ {bE )
Write only the steps of construction for the following :
{ZåZ Ho$ {bE Ho$db aMZm Ho$ MaU {b{IE …
Draw a circle of diameter 6 cm. From a point P outside the circle at a
distance of 7 cm from the centre, draw two tangents to the circle.
6 go.‘r. ì¶mg H$m EH$ d¥Îm It{ME& d¥Îm Ho$ H|$Ð go 7 go.‘r. H$s Xÿar na Ho$ EH$ ~mø q~Xþ P go
d¥Îm na Xmo ñne©aoImE± It{ME&

211/AS/3/403B 11 [ P.T.O.

32. How many terms of the AP 2, 4, 6, 8, ... are needed to get a sum of 210? 4
g‘m§Va lo‹T>r 2, 4, 6, 8, ... Ho$ {H$VZo nXm| H$m ¶moJ 210 h¡?

33. The hypotenuse of a right-angled triangle is 17 cm. If the difference of
remaining two sides is 7 cm, find the lengths of the remaining two sides. 4
EH$ g‘H$moU {Ì^wO Ho$ H$U© H$s b§~mB© 17 go.‘r. h¡& ¶{X CgH$s Aݶ Xmo ^wOmAm| H$s b§~mB¶m| H$m
AÝVa 7 go.‘r. h¡, Vmo eof Xmo ^wOmAm| H$s bå~mB¶m± kmV H$s{OE&

34. Points A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD.
If E is the mid-point of CD, find the area of ADE . 6
q~Xþ A(6, 1), B(8, 2) VWm C(9, 4) EH$ g‘m§Va MVw^w©O ABCD Ho$ erf© h¢& ¶{X E, ^wOm CD
H$m ‘ܶq~Xþ h¡, Vmo ADE H$m joÌ’$b kmV H$s{OE&

35. The sum of a 2-digit number and the number formed by interchanging the
digits is 132. If 12 is added to the number, the new number becomes
5 times the sum of the digits. Find the number. 6
EH$ 2-A§H$s¶ g§»¶m VWm CgHo$ A§H$m| H$m ñWmZ nbQ>Zo na àmá g§»¶m H$m ¶moJ 132 h¡& ¶{X g§»¶m
‘| 12 Omo‹S> {XE OmE±, Vmo àmá ZB© g§»¶m, ‘yb g§»¶m Ho$ A§H$m| Ho$ ¶moJ Ho$ 5 JwZo Ho$ ~am~a hmo OmEJm&
‘yb g§»¶m kmV H$s{OE&

36. Water is flowing through a cylindrical pipe of internal diameter 2·8 cm into
a rectangular tank of length 44 cm and breadth 14 cm, at the rate of
0·5 m/sec. Determine the rise in level of water in the tank in
10 minutes. 6
2·8 go.‘r. Am§V[aH$ {ÌÁ¶m dmbr EH$ ~obZmH$ma nmBn go nmZr 0·5 ‘r./go. H$s J{V go ~hVm hþAm
EH$ 44 go.‘r. bå~o VWm 14 go.‘r. Mm¡‹S>o Am¶VmH$ma Q>¢H$ ‘| {Ja ahm h¡& 10 {‘ZQ> ‘| Q>¢H$ ‘| nmZr
Ho$ Vb ‘o§ d¥{Õ kmV H$s{OE&

  

211/AS/3/403B [V23] 12

This Question Paper consists of 36 questions and 12 printed pages.
Bg àíZ-nÌ ‘| 36 àíZ VWm 12 ‘w{ÐV n¥ð> h¢& Code No.
65/AS/3
w
H$moS> Z§0
Roll No.
Set / goQ>
AZwH«$‘m§H$

MATHEMATICS
J{UV
(211)
Day and Date of Examination .....................................................................................
(narjm H$m {XZ d {XZm§H$)
Signature of Invigilators 1. .....................................................................................
({ZarjH$m| Ho$ hñVmja)
2. .....................................................................................

General Instructions :

1. Candidate must write his/her Roll Number on the first page of the Question
Paper.

2. Please check the Question Paper to verify that the total pages and total
number of questions contained in the Question Paper are the same as those
printed on the top of the first page. Also check to see that the questions are
in sequential order.

3. For the objective type of questions, you have to choose any one of the four
alternatives given in the question, i.e., (A), (B), (C) or (D) and indicate your
correct answer in the Answer-Book given to you.

4. All the questions including objective-type questions are to be answered within
the allotted time and no separate time limit is fixed for answering objective-
type questions.

5. Making any identification mark in the Answer-Book or writing Roll Number
anywhere other than the specified places will lead to disqualification of the
candidate.

6. Write your Question Paper Code No. 65/AS/3, Set w on the Answer-Book.

211/AS/3/403C [ P.T.O.

7. (a) The Question Paper is in English/Hindi medium only. However, If you
wish, you can answer in any one of the languages listed below :
English, Hindi, Urdu, Punjabi, Bengali, Tamil, Malayalam, Kannada,
Telugu, Marathi, Odia, Gujarati, Konkani, Manipuri, Assamese, Nepali,
Kashmiri, Sanskrit and Sindhi.
You are required to indicate the language you have chosen to answer in
the box provided in the Answer-Book.
(b) If you choose to write the answer in the language other than Hindi and
English, the responsibility for any errors/mistakes in understanding the
questions will be yours only.

gm‘mݶ AZwXoe …
1. narjmWu àíZ-nÌ Ho$ nhbo n¥ð> na AnZm AZwH«$‘m§H$ Adí¶ {bI|&
2. H¥$n¶m àíZ-nÌ H$mo Om±M b| {H$ àíZ-nÌ Ho$ Hw$b n¥ð>m| VWm àíZm| H$s CVZr hr g§»¶m h¡ {OVZr àW‘ n¥ð> Ho$ g~go
D$na N>nr h¡& Bg ~mV H$s Om±M ^r H$a b| {H$ àíZ H«${‘H$ ê$n ‘| h¢&
3. dñVw{Zð> àíZm| _o§ AmnH$mo Mma {dH$ënm| (A), (B), (C) VWm (D) _| go H$moB© EH$ CÎma MwZZm h¡ VWm Xr JB©
CÎma-nwpñVH$m _| Amn ghr CÎma {bI|Ÿ&
4. dñVw{Zð> àíZm| Ho$ gmW-gmW g^r àíZm| Ho$ CÎma {ZYm©[aV Ad{Y Ho$ ^rVa hr XoZo h¢Ÿ& dñVw{Zð> àíZm| Ho$ {bE AbJ go
g_` Zht {X`m OmEJmŸ&
5. CÎma-nwpñVH$m ‘| nhMmZ-{M• ~ZmZo AWdm {Z{X©ï> ñWmZm| Ho$ A{V[aº$ H$ht ^r AZwH«$‘m§H$ {bIZo na narjmWu H$mo
A¶mo½¶ R>ham¶m OmEJm&
6. AnZr CÎma-nwpñVH$m na àíZ-nÌ H$m H$moS> Z§0 65/AS/3, goQ w, {bI|&
7. (H$) n«íZ-nÌ Ho$db {hÝXr/A§J«oOr ‘| h¡& {’$a ^r, ¶{X Amn Mmh| Vmo ZrMo Xr JB© {H$gr EH$ ^mfm ‘| CÎma Xo
gH$Vo h¢ …
A§J«oOr, {hÝXr, CXÿ©, n§Om~r, ~§Jbm, V{‘b, ‘b¶mb‘, H$Þ‹S>, VobwJy, ‘amR>r, C{‹S>¶m, JwOamVr, H$m|H$Ur,
‘{Unwar, Ag{‘¶m, Zonmbr, H$í‘rar, g§ñH¥$V Am¡a {gÝYr&
H¥$n¶m CÎma-nwpñVH$m ‘| {XE JE ~m°³g ‘| {bI| {H$ Amn {H$g ^mfm ‘| CÎma {bI aho h¢&
(I) ¶{X Amn {hÝXr Ed§ A§J«oOr Ho$ A{V[aº$ {H$gr Aݶ ^mfm ‘| CÎma {bIVo h¢, Vmo àíZm| H$mo g‘PZo ‘| hmoZo dmbr
Ìw{Q>¶m|/Jb{V¶m| H$s {Oå‘oXmar Ho$db AmnH$s hmoJr&

211/AS/3/403C 2

MATHEMATICS
J{UV
(211)
Time : 2½ Hours ] [ Maximum Marks : 85
g‘¶ … 2½ KÊQ>o ] [ nyUmªH$ … 85

Note : (i) Question Numbers (1 to 10) are Multiple Choice Questions. Each
question carries one mark. For each question, four alternative choices
(A), (B), (C) and (D) are provided of which only one is correct. You have
to select the correct alternative and indicate it in the Answer-Book
provided to you by writing (A), (B), (C) or (D) as the case may be.
Question Numbers (11 to 15) also carry one mark each.

(ii) Question Numbers (16–25) carry 2 marks each.

(iii) Question Numbers (26–33) carry 4 marks each.

(iv) Question Numbers (34–36) carry 6 marks each.

(v) All questions are compulsory.

{ZX}e … (i) àíZ g§»¶m (1 go 10) VH$ ~hþ{dH$ënr àíZ (Multiple Choice Questions) h¢& à˶oH$ àíZ
EH$ A§H$ H$m h¡& à˶oH$ àíZ ‘| Mma {dH$ën (A), (B), (C) Am¡a (D) {XE JE h¢, {OZ‘| go Ho$db
EH$ ghr h¡& AmnH$mo ghr {dH$ën MwZZm h¡ VWm à˶oH$ àíZ Ho$ CÎma AnZr CÎma-nwpñVH$m ‘|
(A), (B), (C) AWdm (D), Omo ^r hmo, {bIH$a Xem©Zm h¡& àíZ g§»¶m (11 go 15) à˶oH$ àíZ
^r EH$ A§H$ H$m h¡&

(ii) àíZ g§»¶m (16–25) VH$ à˶oH$ Ho$ 2 A§H$ h¢&

(iii) àíZ g§»¶m (26–33) VH$ à˶oH$ Ho$ 4 A§H$ h¢&

(iv) àíZ g§»¶m (34–36) VH$ à˶oH$ Ho$ 6 A§H$ h¢&

(v) g^r àíZ A{Zdm¶© h¢&

211/AS/3/403C 3 [ P.T.O.

1. If (a, 4) is the mid-point of the line segment joining the points A(– 6, 5)
and B(– 2, 3), then the value of a is
(A) – 1 (B) – 4
(C) – 6 (D) – 12 1
¶{X q~XþAm| A(– 6, 5) VWm B(– 2, 3) H$mo {‘bmZo dmbo aoIm-IÊS> H$m ‘ܶq~Xþ (a, 4) h¡, Vmo
a H$m ‘mZ h¡
(A) – 1 (B) – 4
(C) – 6 (D) – 12

2. If sec 4 A  cosec ( A  10) , 4A  90 , then the measure of A is
(A) 20° (B) 30°
(C) 10° (D) 50° 1

¶{X sec 4A  cosec (A  10) h¡, O~{H$ 4A  90 h¡, Vmo A H$m ‘mZ h¡
(A) 20° (B) 30°
(C) 10° (D) 50°

3. sin2 A  cosec 2A  cos 2 A  cot 2 A is equal to
(A) 1 (B) – 1
(C) 0 (D) 2 1

sin2 A  cosec 2A  cos 2 A  cot 2 A ~am~a h¡
(A) 1 (B) – 1
(C) 0 (D) 2

4. A triangle and a parallelogram are on the same base and between same
parallels, then the ratio of the area of triangle to the area of parallelogram
is
(A) 1 : 3 (B) 1 : 2
(C) 3 : 1 (D) 1 : 4 1
EH$ {Ì^wO VWm EH$ g‘m§Va MVw^w©O, g‘mZ AmYma na h¢ VWm EH$ hr g‘m§Va aoImAm| Ho$ ‘ܶ ‘| h¢,
Vmo {Ì^wO VWm g‘m§Va MVw^w©O Ho$ joÌ’$b ‘| AZwnmV h¡
(A) 1 : 3 (B) 1 : 2
(C) 3 : 1 (D) 1 : 4

211/AS/3/403C 4

5. ABCD is a rhombus. If ACB  40 , then the measure of ADB is
(A) 40° (B) 45°
(C) 50° (D) 60° 1
ABCD EH$ g‘MVw^w©O h¡& ¶{X ACB  40 h¡, Vmo ADB H$m ‘mZ h¡
(A) 40° (B) 45°
(C) 50° (D) 60°

6. The distance (in cm) of a chord of length 18 cm from the centre of a circle
of radius 15 cm is

(A) 306 (B) 17

(C) 12 (D) 99 1

15 go.‘r. {ÌÁ¶m dmbo EH$ d¥Îm H$s 18 go.‘r. b§~r EH$ Ordm H$s d¥Îm Ho$ H|$Ð go Xyar (go.‘r. ‘|)
h¡
(A) 306 (B) 17

(C) 12 (D) 99

7 The price of an article is increased from < 550 to < 605. The percent
increase in price is

44
(A) 55% (B) %
5
(C) 10% (D) 15% 1
EH$ dñVw Ho$ ‘yë¶ H$mo < 550 go ~‹T>mH$a < 605 H$a {X¶m OmE, Vmo ‘yë¶ ‘| à{VeV ~‹T>moVar h¡
44
(A) 55% (B) %
5
(C) 10% (D) 15%

8. The pair of equations x  2y  3 and 5x  ky  7  0 has no solution, if

(A) k  10 (B) k  10

7 7
(C) k   (D) k  1
3 6

211/AS/3/403C 5 [ P.T.O.

g‘rH$aU ¶w½‘ x  2y  3 Am¡a 5x  ky  7  0 H$m H$moB© hb Zht h¡, ¶{X
(A) k  10 (B) k  10

7 7
(C) k   (D) k 
3 6

2
 2  3 
9. The value of    is
 3  

243 64
(A) (B)
32 729

729 32
(C) (D) 1
64 243
2
 2  3 
   H$m ‘mZ h¡
 3  

243 64
(A) (B)
32 729

729 32
(C) (D)
64 243

10. Representation of 3  6 in rational number form is

11 3
(A) (B)
3 11

36 33
(C) (D) 1
10 10

3  6 H$m n[a‘o¶ g§»¶m Ho$ ê$n ‘| {Zê$nU h¡
11 3
(A) (B)
3 11

36 33
(C) (D)
10 10

211/AS/3/403C 6

11. If one angle of a parallelogram is 54 th of its adjacent angle, then find the
angles of the parallelogram. 1

¶{X EH$ g‘m§Va MVw^w©O H$m EH$ H$moU BgHo$ AmgÞ H$moU H$m 54 h¡, Vmo g‘m§Va MVw^w©O H$m H$moU
kmV H$s{OE&

12. Find the value of k for which – 3 is a zero of the polynomial x 2  11x  k . 1

k H$m dh ‘mZ kmV H$s{OE, {OgHo$ {bE – 3, ~hþnX x 2  11x  k H$m EH$ eyݶH$ hmo&

13. A pair of socks with listing price < 80 is available for < 64. Find the
discount percent offered. 1
< 80 A§{H$V ‘yë¶ dmbo Owamdm| H$m EH$ Omo‹S>m < 64 ‘| CnbãY h¡& {X¶m OmZo dmbm à{VeV ~Å>m
kmV H$s{OE&

14. Simplify : 1

 3 2  7 11
   
 5 7  8 15

gab H$s{OE …
 3 2  7 11
   
 5 7  8 15

15. If one root of the equation 2x 2  5x  (  6)  0 is the reciprocal of the
other, then find the value of  . 1

¶{X g‘rH$aU 2x 2  5x  (  6)  0 H$m EH$ ‘yb, BgHo$ Xygao ‘yb H$m ì¶wËH«$‘ h¡, Vmo  H$m
‘mZ kmV H$s{OE&

211/AS/3/403C 7 [ P.T.O.

16. In Fig. 1 D, E and F are the mid-points of the sides of ABC . Show that
3
BE  CF  BC . 2
2

3
Fig. 1 ‘|, D, E VWm F, ABC H$s ^wOmAm| Ho$ ‘ܶq~Xþ h¢& Xem©BE {H$ BE  CF  2 BC .

Fig. 1

17. In Fig. 2, PAB is a secant and PT is a tangent to the circle from an external
point P. If PT  x cm, PA  4 cm and AB  5 cm, then find the value
of x. 2
Fig. 2 ‘|, PAB d¥Îm H$s N>oXH$ aoIm VWm PT d¥Îm H$s ñne©aoIm h¡, Omo {H$ EH$ ~mø q~Xþ P go
ItMr JB© h¡& ¶{X PT  x go.‘r., PA  4 go.‘r. VWm AB  5 go.‘r. h¡, Vmo x H$m ‘mZ kmV
H$s{OE&

Fig. 2

18. Diagonals of a trapezium ABCD with AB  DC intersect each other at O.
If AB  2CD , find the ratio of the areas of triangles AOB and COD. 2

EH$ g‘b§~ ABCD, {Og‘| AB  DC h¡, Ho$ {dH$U© nañna O na H$mQ>Vo h¢& ¶{X AB  2CD
h¡, Vmo {Ì^wOm| AOB VWm COD Ho$ joÌ’$bm| H$m AZwnmV kmV H$s{OE&

211/AS/3/403C 8

19. The percentage of marks obtained by 100 students in an examination are
given below :

% marks obtained : 30 35 40 45 50 55 60
Number of students : 14 16 18 23 18 8 3

Determine the median percentage of marks. 2
100 {dÚm{W©¶m| Ûmam EH$ narjm ‘| àmá à{VeV A§H$ {ZåZ h¢ …

àmám§H (à{VeV)$ : 30 35 40 45 50 55 60
{dÚm{W¶©mo§ H$s g§»¶m$ : 14 16 18 23 18 8 3
‘mܶH$ à{VeV A§H$ kmV H$s{OE&

20. Two different dice are thrown simultaneously. What is the probability that
the sum of numbers appearing on the dice is
(a) less than 5;
(b) at least 11? 2
Xmo {d{^Þ nmgm| H$mo EH$ gmW CN>mbm J¶m& XmoZmo§ nmgm| na AmB© g§»¶mAm| H$m ¶moJ {ZåZ hmoZo H$s àm{¶H$Vm
³¶m h¡?
(H$) 5 go H$‘
(I) H$‘-go-H$‘ 11

4 sin A  3 cos A 1
21. If 3 cot A  2 , then show that  . 2
2sin A  6 cos A 3

4 sin A  3 cos A 1
¶{X 3 cot A  2 h¡, Vmo Xem©BE {H$  h¡&
2sin A  6 cos A 3

22. A dealer buys a table listed at < 3,000 and gets a discount of 25%. He
spends < 100 on transportation and sells it at a profit of 16%. Find its
selling price. 2
EH$ ì¶mnmar < 3,000 A§{H$V ‘yë¶ dmbm EH$ ‘oO 25% ~Å>o na IarXVm h¡& dh < 100 Tw>bmB©
na IM© H$aVm h¡ VWm Bgo 16% bm^ na ~oM XoVm h¡& ‘oO H$m {dH«$¶ ‘yë¶ kmV H$s{OE&

211/AS/3/403C 9 [ P.T.O.

23. Find the area of the sector of a circle of radius 6 cm when the length of
the arc of the sector is 22 cm. 2
EH$ d¥Îm Ho$ Cg {ÌÁ¶IÊS> H$m joÌ’$b kmV H$s{OE, {OgH$s {ÌÁ¶m 6 go.‘r. h¡ O~H$s {ÌÁ¶IÊS>
H$s Mmn H$s b§~mB© 22 go.‘r. h¡&

24. A cubic centimetre gold is drawn into a wire of diameter 0·2 mm. Find the
length of wire. [Use   3  14 ] 2
EH$ KZ go.‘r. gmoZo H$mo 0·2 {‘.‘r. ì¶mg H$s EH$ Vma Ho$ ê$n ‘| ItMm J¶m& Vma H$s b§~mB© kmV
H$s{OE& [   3  14 br{OE]

25. If sin2 A  sin A  1 , then show that cos 2 A  cos 4 A  1 . 2

¶{X sin2 A  sin A  1 h¡, Vmo Xem©BE {H$ cos2 A  cos 4 A  1 h¡&

26. In a bag, there are 44 identical cards with figure of a circle or a square
on them. There are 24 circles of which 9 are blue and rest are green, and
20 squares of which 11 are blue and rest are green. One card is drawn
at random from the bag. Find the pobability that it has a figure of
(a) a square (b) green colour (c) a blue circle (d) a green square. 4
EH$ W¡bo ‘| 44 EH$ O¡go H$mS>© h¢, {OZ‘| à˶oH$ na d¥Îm AWdm dJ© H$m {MÌ ~Zm h¡& 24 d¥Îm h¢
{OZ‘| 9 Zrbo a§J Ho$ h¢ VWm Aݶ hao a§J Ho$ h¢ Am¡a 20 dJ© h¢, {OZ‘| 11 Zrbo a§J Ho$ h¢ VWm
Aݶ hao h¢& W¡bo ‘| go ¶mÑÀN>¶m EH$ H$mS>© {ZH$mbm J¶m& àm{¶H$Vm kmV H$s{OE {H$ {ZH$mbo JE H$mS>©
H$m {MÌ (H$) dJ© hmo, (I) hao a§J H$m hmo, (J) Zrbm d¥Îm hmo (K) ham dJ© hmo&

27. Find the mean of the following data : 4

Class : 156–158 158–160 160–162 162–164 164–166 166–168
Frequency : 2 4 8 16 14 6

{ZåZ Am±H$S>m| H$m ‘mܶ kmV H$s{OE :
dJ©$ : 156–158 158–160 160–162 162–164 164–166 166–168
~ma§ ~maVm $: 2 4 8 16 14 6

211/AS/3/403C 10

28. A person standing on the bank of a river, observes that the angle of
elevation of the top of a tree standing on the opposite bank is 60°. When
he moves 40 metres away from the bank, he finds the angle to be 30°.
Find the height of the tree and the width of the river. 4
EH$ ì¶{º$ ZXr Ho$ EH$ {H$Zmao na I‹S>m hmoH$a Xygao {H$Zmao na bJo EH$ no‹S> Ho$ {eIa H$m CÞ¶Z H$moU
60° nmVm h¡& O~ dh 40 ‘rQ>a nrN>o hQ> OmVm h¡, Vmo H$moU 30° H$m hmo OmVm h¡& no‹S> H$s D±$MmB©
Am¡a ZXr H$s Mm¡‹S>mB© kmV H$s{OE&

29. Draw a circle of diameter 6 cm. From a point P outside the circle at a
distance of 7 cm from the centre, draw two tangents to the circle. 4
6 go.‘r. ì¶mg H$m EH$ d¥Îm It{ME& d¥Îm Ho$ H|$Ð go 7 go.‘r. H$s Xÿar na Ho$ EH$ ~mø q~Xþ P go
d¥Îm na Xmo ñne©aoImE± It{ME&
Or / AWdm

( For Visually Impaired Learners only )
( Ho$db Ñ{ï> {dH$bm§J {dÚm{W©¶m| Ho$ {bE )

Write only the steps of construction for the following :
{ZåZ Ho$ {bE Ho$db aMZm Ho$ MaU {b{IE …
Draw a circle of diameter 6 cm. From a point P outside the circle at a
distance of 7 cm from the centre, draw two tangents to the circle.
6 go.‘r. ì¶mg H$m EH$ d¥Îm It{ME& d¥Îm Ho$ H|$Ð go 7 go.‘r. H$s Xÿar na Ho$ EH$ ~mø q~Xþ P go
d¥Îm na Xmo ñne©aoImE± It{ME&

30. Prove that the tangents drawn at the ends of a diameter of a circle are
parallel. 4
{gÕ H$s{OE {H$ {H$gr d¥Îm Ho$ EH$ ì¶mg Ho$ {gam| na ItMr JB© ñne©aoImE± g‘m§Va hmoVr h¢&

31. Find the sum of money which will amount to < 26,460 in six months at
20% per annum, when the interest is compounded quarterly. 4
dh am{e kmV H$s{OE, Omo N>… ‘hrZo ‘| 20% dm{f©H$ Xa go < 26,460 hmo OmEJr, O~{H$ ã¶mO
{V‘mhr g§¶mo{OV hmoVm h¡&

211/AS/3/403C 11 [ P.T.O.

32. Two different dice are tossed together. Find the probability of getting
(a) a doublet (b) odd number on both (c) a sum of 9. 4
Xmo {d{^Þ nmgm| H$mo EH$ gmW CN>mbm J¶m& (H$) EH$ {×H$, (I) XmoZm| nmgm| na {df‘ g§»¶m,
(J) XmoZm| g§»¶mAm| H$m ¶moJ 9 AmZo {H$ à{¶H$Vm kmV H$s{OE&

33. The 12th term of an AP is – 28 and the 18th term is – 46. Find the sum
of its first 10 terms. 4
EH$ g‘m§Va lo‹T>r H$m 12dm± nX – 28 VWm 18dm± nX – 46 h¡& BgHo$ àW‘ 10 nXm| H$m ¶moJ kmV
H$s{OE&

34. Points A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD.
If E is the mid-point of CD, find the area of ADE . 6
q~Xþ A(6, 1), B(8, 2) VWm C(9, 4) EH$ g‘m§Va MVw^w©O ABCD Ho$ erf© h¢& ¶{X E, ^wOm CD
H$m ‘ܶq~Xþ h¡, Vmo ADE H$m joÌ’$b kmV H$s{OE&

35. Water is flowing through a cylindrical pipe of internal diameter 2 cm into
a cylindrical tank of base radius 40 cm, at the rate of 0·4 m per second.
Determine the rise in the level of water in the tank in half an hour. 6
2 go.‘r. Am§V[aH$ {ÌÁ¶m dmbr EH$ ~obZmH$ma nmBn ‘| nmZr 0·4 ‘r./go. H$s J{V go ~hVm hþAm,
40 go.‘r. {ÌÁ¶m dmbo EH$ ~obZmH$ma Q>¢H$ ‘| {Ja ahm h¡& AmYo K§Q>o ‘| Q>¢H$ ‘| nmZr Ho$ Vb ‘| ~‹T>moVar
kmV H$s{OE&

36. The length of a rectangular garden is 7 m more than its breadth. If the
area of the garden is 144 m2, find the perimeter of the garden. 6
EH$ Am¶VmH$ma ~mJ H$s b§~mB©, BgH$s Mm¡‹S>mB© go 7 ‘r. A{YH$ h¡& ¶{X Bg ~mJ H$m joÌ’$b
144 dJ© ‘r. h¡, Vmo BgH$m n[a‘mn kmV H$s{OE&

  

211/AS/3/403C [V23] 12