Tamil Nadu 10th Std Model Question Paper Maths

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3562 (NS)
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!3562NSMathematics!
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Part - III
Pou® / MATHEMATICS
( uªÌ ©ØÖ® B[Q» ÁÈ / Tamil & English Version)

Põ» AÍÄ : 3.00 ©o ÷|µ® ] [ ö©õzu ©v¨ö£sPÒ : 100
Time Allowed : 3.00 Hours ] [Maximum Marks : 100

AÔÄøµPÒ : (1) AøÚzx ÂÚõUPЮ \›¯õP Aa_¨ £vÁõQ EÒÍuõ GߣuøÚ
\›£õºzxU öPõÒÍÄ®. Aa_¨£vÂÀ SøÓ°¸¨¤ß AøÓU
PsPõo¨£õÍ›h® EhÚi¯õP öu›ÂUPÄ®.
(2) }»® AÀ»x P¸¨¦ ø©°øÚ ©mk÷© GÊxÁuØS®
AiU÷PõikÁuØS® £¯ß£kzu ÷Ásk®. £h[PÒ ÁøµÁuØS
ö£ß]À £¯ß£kzuÄ®.
Instructions : (1) Check the question paper for fairness of printing. If there is any lack of
fairness, inform the Hall Supervisor immediately.
(2) Use Blue or Black ink to write and underline and pencil to draw diagrams.

SÔ¨¦ : CÆÂÚõzuõÒ |õßS £SvPøÍ öPõshx.
Note : This question paper contains four parts.

£Sv & I / PART - I

SÔ¨¦ : (i) AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 14x1=14

(ii) öPõkUP¨£mh |õßS ÂøhPÎÀ ªPÄ® Hئøh¯ Âøh°øÚ
÷uº¢öukzx SÔ±mkhß Âøh°øÚ²® ÷\ºzx GÊuÄ®.
Note : (i) Answer all the questions.
(ii) Choose the most appropriate answer from the given four alternatives and
write the option code and the corresponding answer.

[ v¸¨¦P / Turn over

3562 (NS) 2

1. A={1, 2}, B={1, 2, 3, 4}, C={5, 6} ©Ø-Ö® D={5, 6, 7, 8} GÛÀ R÷Ç öPõ-kU-P¨
£
- mh-øÁ-P-ÎÀ Gx \›-¯õÚ TØÖ ?

(A) (A×C) ⊂ (B×D) (B) (B×D) ⊂ (A×C)

(C) (A×B) ⊂ (A×D) (D) (D×A) ⊂ (B×A)

If A={1, 2}, B={1, 2, 3, 4}, C={5, 6} and D={5, 6, 7, 8}, then state which of the following
statement is true ?

(a) (A×C) ⊂ (B×D) (b) (B×D) ⊂ (A×C)

(c) (A×B) ⊂ (A×D) (d) (D×A) ⊂ (B×A)

2. f (x)=x2−x GÛÀ, f (x−1)−f (x+1)=

(A) 4x (B) 2−2x (C) 2−4x (D) 4x−2

Let f (x)=x2−x, then f (x−1)−f (x+1) is :

(a) 4x (b) 2−2x (c) 2−4x (d) 4x−2

3. ³U-Î-iß ÁSz-uÀ xønz ÷uØ-Óz-øu¨ £¯ß-£-kzv, G¢u ªøP •Ê-Âß
PÚz-øu-²® 9 &BÀ ÁSU-S® ÷£õx QøhU-S® «v-PÒ :

(A) 0, 1, 8 (B) 1, 4, 8 (C) 0, 1, 3 (D) 1, 3, 5

Using Euclid’s division lemma, if the cube of any positive integer is divided by 9, then
the possible remainders are :

(a) 0, 1, 8 (b) 1, 4, 8 (c) 0, 1, 3 (d) 1, 3, 5

3 3562 (NS)

4. A=2 65 ©Ø- Ö ® B=2 64 +2 63 +2 62 +⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅+2 0 GÚU öPõ- k U- P ¨- £ m- k ÒÍx.
¤ß-Á¸
- Á
- Ú
- Á
- Ø-ÔÀ Gx Esø© ?

(A) B &BÚx A &I Âh 264 Av-P®

(B) A ©Ø-Ö-® B \©®

(C) B &BÚ-x A &I Âh 1 Av-P®

(D) A &B-Úx B &I Âh 1 Av-P®

If A=265 and B=264+263+262+⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅+20, which of the following is true ?

(a) B is 264 more than A

(b) A and B are equal

(c) B is larger than A by 1

(d) A is larger than B by 1

a2 b2
5. + &ß ©v¨¦ :
a2 − b2 b2 − a2

(A) a−b (B) a+b

(C) a2−b2 (D) 1

a2 b2
+ =
a2 − b 2 b2 − a2

(a) a−b (b) a+b

(c) a2−b2 (d) 1

[ v¸¨¦P / Turn over

3562 (NS) 4

6. J¸ {µÀ Ao-°ß, {øµ {µÀ ©õØÖ Ao :
(A) A»S Ao (B) ‰ø»-Âmh Ao
(C) {µÀ Ao (D) {øµ Ao
Transpose of a column matrix is :
(a) unit matrix (b) diagonal matrix
(c) column matrix (d) row matrix

7. ∆LMN &À ∠L = 60 , ∠M = 50  . ÷©¾® ∆LMN ~ ∆PQR GÛÀ ∠R &ß ©v¨¦ :
(A) 408 (B) 708 (C) 308 (D) 1108
In ∆LMN, ∠L = 60 , ∠M = 50 . If ∆LMN ~ ∆PQR, then the value of ∠R is :
(a) 408 (b) 708 (c) 308 (d) 1108

8. £hz-vÀ EÒÍ-ÁõÖ O &øÁ ø©¯-©õ-PU öPõsh Ám-hz-vß P &À öuõ-k-÷Põk
PR GÛÀ, ∠POQ BÚx :

(A) 1208 (B) 1008 (C) 1108 (D) 908
In the figure, if PR is tangent to the circle at P and O is the centre of the circle, then
∠POQ is :

(a) 1208 (b) 1008 (c) 1108 (d) 908

5 3562 (NS)

9. x=11 GÚU öPõ-kU-P¨-£mh ÷|ºU-÷Põm-iß \©ß-£õ-hõ-Úx :

(A) x &Aa-_US Cøn

(B) y &Aa-_US Cøn

(C) Bv¨-¦Ò-Î ÁÈa-ö\À-¾®

(D) (0, 11) & GßÓ ¦ÒÎ ÁÈa-ö\À-¾®

The straight line given by the equation x=11 is :

(a) Parallel to x-axis

(b) Parallel to y-axis

(c) Passing through the origin

(d) Passing through the point (0, 11)

10. tan θ + cot θ = 2 GÛÀ tan2 θ+cot2 θ &ß ©v¨¦ :

(A) 0 (B) 1 (C) 2 (D) 4

If tan θ + cot θ = 2, then the value of tan2 θ+cot2 θ is :

(a) 0 (b) 1 (c) 2 (d) 4

11. 24 ö\.«. E¯-µ-•®, 6 ö\.«. Bµ-•® Eøh-¯ PÎ-©s-o-ÚõÀ ö\´-¯¨-£mh
J¸ T®-¤øÚ J¸ ]Öª ÷PõÍ-©õP ©õØ-Ô-ÚõÀ, ÷PõÍz-vß Bµ® :

(A) 24 ö\.«. (B) 12 ö\.«. (C) 6 ö\.«. (D) 48 ö\.«-.

A child reshapes a cone made up of clay of height 24 cm and radius 6 cm into a sphere,
then the radius of sphere is :

(a) 24 cm (b) 12 cm (c) 6 cm (d) 48 cm

[ v¸¨¦P / Turn over

3562 (NS) 6

12. r1 A»-S-PÒ Bµ-•ÒÍ J¸ ÷Põͨ-£¢x E¸U-P¨-£mk r2 A»-S-PÒ Bµ-•-øh¯
8 \© ÷PõÍ £¢-x-PÍõP BU-P¨-£-k-Q-Óx GÛÀ r1 : r2 :

(A) 2 : 1 (B) 1 : 2 (C) 4 : 1 (D) 1 : 4

A spherical ball of radius r1 units is melted to make 8 new identical balls each of radius
r2 units. Then r1 : r2 is :

(a) 2:1 (b) 1:2 (c) 4:1 (d) 1:4

13. 100 uµ-Ĩ ¦Ò-Î-P-Îß \µõ-\› 40 ©Ø-Ö® vm-h-Â-»U-P® 3 GÛÀ »U-P[-P-Îß
ÁºU-PU Tk-u-»õ-Úx :

(A) 40000 (B) 160900 (C) 160000 (D) 30000

The mean of 100 observations is 40 and their standard deviation is 3. The sum of
squares of all deviations is :

(a) 40000 (b) 160900 (c) 160000 (d) 30000

14. B[- Q » GÊz- x U- P Ò {a, b, c, ...., z} &°- ¼ - ¸ ¢x Kº GÊzx \©- Á õ´¨¦
•øÓ-°À ÷uºÄ ö\´-¯¨-£k - Q
- Ó
- x. A¢u GÊzx x &US •¢-øu¯ GÊz-xU-PÎ - À
Jß-ÓõP C¸¨-£-uØ-PõÚ {PÌ-u-PÄ :

12 1 23 3
(A) 13 (B) 13 (C) 26 (D) 26

If a letter is chosen at random from the English alphabets {a, b, c, ...., z}, then the
probability that the letters chosen precedes x, is :

12 1 23 3
(a) (b) (c) (d)
13 13 26 26

7 3562 (NS)

£Sv & II/PART - II

SÔ¨¦ : GøÁ÷¯Ý® 10 ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 28 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®. 10x2=20

Note : Answer any 10 questions. Question No. 28 is compulsory.

15. A×B={(3, 2) (3, 4) (5, 2) (5, 4)} GÛÀ, A ©Ø-Ö® B &I Põs-P.

If A×B={(3, 2) (3, 4) (5, 2) (5, 4)}, then find A and B.

16. f : N → N GßÓ \õº¦ f (m)=m2+m+3 GÚ Áøµ-¯-ÖU-P¨-£m-hõÀ Ax Jß-ÖUS
Jß-ÓõÚ \õº¦ GÚU Põm-k-P.

Show that the function f : N → N defined by f (m)=m2+m+3 is one-one function.

17. m ©Ø-Ö® n C¯À Gs-PÒ GÛÀ, G¢u m &ß ©v¨-¦-P-ÐUS 2n×5m GßÓ Gs
5 GßÓ C»U-Pz-øuU öPõsk •i-²® ?

If m, n are natural numbers, for what values of m, does 2n×5m end in 5 ?

 n 2 ; n J¸ JØøÓ Gs

18. J¸ öuõ- h º Á›- ø \- ° ß ö£õx EÖ¨¦ an =  2
n
 ; n J¸ Cµmøh Gs
 2

GÛÀ 3 &Áx ©Ø-Ö® 4 &Áx EÖ¨-¦-P-øÍU Põs-P.

 n 2 if n is odd

Find the 3rd and 4th terms of a sequence, if a n =  n 2 .
 if n is even
 2

[ v¸¨¦P / Turn over

3562 (NS) 8

19. 12+22+32+ ...... +102 &ß ©v¨¦ PõsP. Cv-¼-¸¢x 22+42+62+ ....+202 &ß
©v¨-¦ Põs-P.
Find the value of 12+22+32+ ...... +102 and hence deduce 22+42+62+ ....+202.

20. 9x2+3kx+4=0 GßÓ C¸-£-ia-\-©ß-£õm-iß ‰»[-PÒ ö©´ ©Ø-Ö® \©® GÛÀ
k &ß ©v¨¦ Põs-P.

Find the value of k for which the equation 9x2+3kx+4=0 has real and equal roots.

 7 −3 
 
21. A = − 5 2  GÛÀ −A &°ß {øµ-{-µÀ ©õØÖ Ao-ø¯U Põs-P.
 
 3 −5 
 

 7 −3 
 
If A = − 5 2  then find the transpose of −A.
 
 3 −5 
 

22. ¤ß-Á-¸-Á-Ú-ÁØ-ÔÀ ∆ABC &°À AD BÚx, ∠A &°ß C¸-\-©-öÁmi BS-©õ
GÚ ÷\õ-vU-P-Ä®.
AB=5 ö\.«., AC=10 ö\.«., BD=1.5 ö\.«. ©ØÖ® CD=3.5 ö\.«.

Check whether AD is bisector of ∠A of ∆ABC in the following.

AB=5 cm, AC=10 cm, BD=1.5 cm and CD=3.5 cm.

23. (14, 10) ©Ø- Ö ® (14, −6) BQ- ¯ ¦Ò- Î - P øÍ CønU- S ® ÷|ºU- ÷ Põm- i ß
\õ´-øÁU Põs-P.
Find the slope of a line joining the points (14, 10) and (14, −6).

9 3562 (NS)

1 + sinθ
24. {¹-¤U-P-Ä® : = sec θ + tan θ
1 − sinθ

1 + sinθ
Prove = sec θ + tan θ
1 − sinθ

25. J¸ ÷PõÍz-vß ¦Ó¨-£-µ¨¦ 154 \.«. GÛÀ Auß Âm-h® Põs-P.

Find the diameter of a sphere whose surface area is 154 m2.

26. J¸ vs© AøµU- ÷ PõÍz- v ß Ai¨- £ - µ ¨¦ 1386 \.«. GÛÀ Auß
¦Ó¨-£µ- ¨-¤ø
- ÚU Põs-P.

If the base area of a hemispherical solid is 1386 sq. metres, then find its total surface
area.

27. RÌU-Põ-q® uµ-Ä-P-ÐUS Ãa_ ©Ø-Ö® Ãa-_U öPÊ-øÁU Põs-P.

63, 89, 98, 125, 79, 108, 117, 68.

Find the range and coefficient of range of the data.

63, 89, 98, 125, 79, 108, 117, 68.

28. Kº EÒ- Ï hØÓ E¸- ø Í- ° ß E- ¯ - µ ®, Em- ¦ Ó ©Ø- Ö ® öÁΨ- ¦ Ó Bµ[- P Ò
•øÓ-÷¯ 9 ö\.«., 3 ö\.«. ©Ø-Ö® 5 ö\.«. BS®. E¸-øÍø¯ E¸-ÁõU-Pz
÷uøÁ¨-£-k® C¸®-¤ß PÚ AÍ-Â-øÚU Põs-P.

Find the volume of the iron used to make a hollow cylinder of height 9 cm and whose
internal and external radii are 3 cm and 5 cm respectively.

[ v¸¨¦P / Turn over

3562 (NS) 10

£Sv & III/PART - III
SÔ¨¦ : GøÁ÷¯Ý® 10 ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 42 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®. 10x5=50

Note : Answer any 10 questions. Question No. 42 is compulsory.

29. A Gß-£x 8 &I Âh SøÓ-ÁõÚ C¯À Gs-P-Îß Pn®,

B Gß-£-x 8 &I Âh SøÓ-ÁõÚ £Põ Gs-P-Îß Pn®,

©Ø- Ö ® Gß- £ - x Cµm- ø h¨- £ øh £- P õ Gs- P - Î ß Pn® GÛÀ,
C
(A∩B)×C=(A×C)∩(B×C) \›-£õºU-P-Ä®.

Let A=The set of all natural numbers less than 8

B=The set of all prime numbers less than 8

C=The set of even prime number. Verify that (A∩B)×C=(A×C)∩(B×C).

30. A = {1, 2, 3, 4} ©Ø-Ö® B = {2, 5, 8, 11, 14} Gß-£Ú C¸ Pn[-PÒ GßP. f : A → B
GÝ® \õº¦ f (x)=3x−1 GÚU öPõ-kU-P¨-£m-kÒÍx. Ca-\õº-¤øÚ,
(i) A®-¦U-S-Ô-£-h®
(ii) Am-h-Á-øn
(iii) Á›ø\ ÷\õ-i-P-Îß Pn®
(iv) Áøµ-£-h®
BQ-¯-ÁØ-ÓõÀ S-ÔU-P-Ä®.
Let A = {1, 2, 3, 4} and B = {2, 5, 8, 11, 14} be two sets. Let f : A → B be a function
given by f (x)=3x−1. Represent this function :

(i) by Arrow diagram

(ii) in a table form

(iii) as a set of ordered pairs

(iv) in a graphical form

11 3562 (NS)

31. 100 &U- S ® 1000
&U- S ® Cøh- ÷ ¯ 11 &BÀ ÁS- £ - k ® AøÚzx C¯À
Gs-P-Îß Tk-uÀ Põs-P.

Find the sum of all natural numbers between 100 and 1000 which are divisible by 11.

32. wºUP : 6x + 2y − 5z = 13

3x + 3y − 2z = 13

7x + 5y − 3z = 26

Solve : 6x + 2y − 5z = 13

3x + 3y − 2z = 13

7x + 5y − 3z = 26

33. ¤ß-Á-¸® £À-¾-Ö¨-¦U ÷Põ-øÁ-P-Îß «.ö£õ.Á. Põs-P.

x4+3x3−x−3, x3+x 2−5x+3

Find the GCD of the polynomials, x4+3x3−x−3 and x3+x2−5x+3.

x2 10 x 10 y y2
34. 2
− + 27 − + 2 GßÓ ÷Põ-øÁ-°ß ÁºU-P-‰-»® Põs-P.
y y x x

x2 10 x 10 y y2
Find the square root of the expression, 2
− + 27 − + 2
y y x x

[ v¸¨¦P / Turn over

3562 (NS) 12

 2 −1 
 1 2 1   T T T
35. A= 
2 1 1  ©ØÖ® B = −1 4  GÛÀ (AB) =B A Gß-£-øua \›-£õºU-P.
   0 2 
−


 2 −1 
 1 2 1  
If A =   and B = −1 4  show that (AB)T=BTAT.
 2 −1 1   0
 2 

36. ÷Põn C¸ \©-öÁmi ÷uØ-Óz-vøÚ GÊ-v {Ö-Ä-P.

State and prove Angle Bisector theorem.

37. (−4, −2), (−3, k), (3, −2) ©ØÖ® (2, 3) BQ-¯-ÁØøÓ •øÚ-PÍõ-PU öPõsh
|õØ-P-µz-vß £µ¨-£ÍÄ 28 \.A-»-S-PÒ GÛÀ k &°ß ©v¨¦ Põs-P.

Find the value of k, if the area of a quadrilateral is 28 sq. units, whose vertices are
(−4, −2), (−3, k), (3, −2) and (2, 3).

38. 60 «. E¯- µ - • ÒÍ ÷Põ- ¦ - µ z- v ß Ea- ] - ° - ¼ - ¸ ¢x ö\[- S z- u õP EÒÍ J¸
ÂÍU-SU P®-£z-vß Ea] ©Ø-Ö® Ai-°ß CÓU-PU ÷Põ-n[-PÒ •øÓ-÷¯
388 ©Ø- Ö ® 608 GÛÀ, ÂÍU- S U P®- £ z- v ß E¯- µ z- ø uU PõsP.
(tan 388 = 0.7813, 3 = 1.732 )

From the top of a tower 60 m high, the angles of depression of the top and bottom of a
vertical lamp post are observed to be 388 and 608 respectively. Find the height of the
lamp post. (tan 388 = 0.7813, 3 = 1.732 )

13 3562 (NS)

39. Âm-h® 20 ö\.«. EÒÍ Kº E¸øÍ Ái-ÁU Ps-nõi SÁ-øÍ-°À 9 ö\.«.
E¯-µz-vØS }º EÒÍx. Bµ® 5 ö\.«. ©Ø-Ö® E¯-µ® 4 ö\.«. Eøh-¯ Kº ]Ô¯
EÒ-Ï-hØÓ E÷»õP E¸-øÍ }›À •Ê-ø©-¯õP ‰Ì-S® ÷£õx HØ-£-k® }›ß
E¯º-øÁU PnU-Q-k-P.

A cylindrical glass with diameter 20 cm has water to a height of 9 cm. A small
non-hollow cylindrical metal of radius 5 cm and height 4 cm is immersed in it
completely. Calculate the rise of water in the glass.

40. 7 ÷£õm-i-P-ÎÀ J¸ Q›U-öPm õº Gkzu Km-h[-PÒ •øÓ-÷¯ 70, 80, 60, 50,
40, 90, 95. vmh »U-P® Põs-P.

The scores of a cricketer in 7 matches are 70, 80, 60, 50, 40, 90, 95. Find the standard
deviation.

41. Cµsk ^µõÚ £P-øh-PÒ •øÓ-¯õP J÷µ ÷|µz-vÀ E¸m-h¨-£-k-Q-Óx.

(i) Cµsk £P-øh-P-Î-¾® J÷µ •P©v¨¦ QøhU-P,

(ii) •P©v¨-¦-P-Îß ö£¸U-PØ-£-»ß £Põ Gs-nõ-PU QøhU-P,

(iii) •P©v¨-¦-P-Îß Tk-uÀ £Põ Gs-nõ-PU QøhU-P,

(iv) •P©v¨-¦-P-Îß Tk-uÀ 1 &BP C¸UP,

BQ¯ {PÌa]PÎß {PÌuPÄPøÍU PõsP.

Two unbiased dice are rolled once. Find the probability of getting :

(i) a doublet (equal numbers on both dice)

(ii) the product as a prime number

(iii) the sum as a prime number

(iv) the sum as 1

[ v¸¨¦P / Turn over

3562 (NS) 14

42. AB GßÓ ÷|ºU-÷Põk B¯ Aa-_-PøÍ A ©Ø-Ö® B ¦Ò-Î-P-ÎÀ öÁm-k-Q-Óx.
AB &ß |k¨-¦ÒÎ (2, 3) GÛÀ AB &ß \©ß-£õm-i-øÚU Põs-P.
A straight line AB cuts the co-ordinate axes at A and B. If the mid-point of AB is (2, 3),
find the equation of AB.

£Sv & IV/PART - IV
SÔ¨¦ : AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 2x8=16
Note : Answer the following questions.
6
43. (A) öPõ-kU-P¨-£mh •U-÷Põ-n® ABC &°ß Jzu £U-P[-P-Îß ÂQ-u® 5 GÚ

Aø©-²-©õÖ J¸ Ái-öÁõzu •U-÷Põ-n® Áøµ-P.  AÍÄ Põµo
6

 5

AÀ»x
(B) 5 ö\.«. Bµ-•ÒÍ Ám-hz-vß ø©¯z-v-¼-¸¢x 10 ö\.«. öuõ-ø»-Â-¾ÒÍ
¦Ò- Î - ° - ¼ - ¸ ¢x Ám- h z- v Ø- S z öuõ- k - ÷ Põ- k - P Ò Áøµ- ¯ - Ä ®. ÷©¾®
öuõ-k-÷PõkPÎß }Í[-PøÍU PnU-Q-k-P.
6
(a) Construct a triangle similar to a given triangle ABC with its sides equal to of
5

 6
the corresponding sides of the triangle ABC.  scale factor 
 5
OR
(b) Draw two tangents from a point which is 10 cm away from the centre of a circle
of radius 5 cm. Also measure the lengths of the tangents.

15 3562 (NS)

44. (A) x2−8x+16=0 GßÓ C¸-£-ia \©ß-£õm-iß Áøµ-£-h® Áøµ¢x wº-Âß
uß-ø©-ø¯U TÖ-P.
AÀ»x
(B) y=2x 2 −3x−5 &°ß Áøµ- £ - h ® Áøµ¢x Au- ø Ú¨ £¯ß- £ - k zv
2x2−4x−6=0 GßÓ \©ß-£õm-i-øÚz wºU-P-Ä®.

(a) Graph the quadratic equation x 2−8x+16=0 and state the nature of their
solution.

OR

(b) Draw the graph of y=2x2−3x−5 and hence solve 2x2−4x−6=0.

-o0o-

[ v¸¨¦P / Turn over