Tamil Nadu 10th Std Model Question Paper 2026 Mathematics

Tamil Nadu Board Model Question Paper

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8512
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!8512Mathematics!
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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 £SvPøÍU öPõshx.
Note : This question paper contains four parts.

£Sv & I / PART - I
SÔ¨¦ : (i) AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 14x1=14

(ii) öPõkUP¨£mkÒÍ ©õØÖ ÂøhPÎÀ ªPÄ® Hئøh¯ Âøh°øÚz
÷uº¢öukzxU 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.

8512 2

1. R={(x, x 2 ) | x BÚx 13 I ÂhU SøÓÁõÚ £Põ GsPÒ } GßÓ EÓÂß
Ãa\P©õÚx :
(A) {2, 3, 5, 7} (B) {2, 3, 5, 7, 11}
(C) {4, 9, 25, 49, 121} (D) {1, 4, 9, 25, 49, 121}
The range of the relation R={(x, x2) | x is a prime number less than 13} is :
(a) {2, 3, 5, 7} (b) {2, 3, 5, 7, 11}
(c) {4, 9, 25, 49, 121} (d) {1, 4, 9, 25, 49, 121}

2. n(A)=m ©ØÖ® n(B)=n GÛÀ B &¼¸¢x A &US Áøµ¯ÖUP¨£mh \õº¦PÎß
ö©õzu GsoUøP :
(A) mn (B) n m (C) 2 mn − 1 (D) 2 mn
Let n(A)=m and n(B)=n, then the total number of functions that exist from B to A is :
(a) mn (b) nm (c) 2 mn − 1 (d) 2 mn

3. ³UÎiß ÁSzuÀ xønz ÷uØÓzvߣi, a ©ØÖ® b GßÓ ªøP •ÊUPÐUS,
uÛzu ªøP •ÊUPÒ q ©ØÖ® r, a=bq+r GßÓÁõÖ Aø©²©õÚõÀ, C[S
r BÚx :
(A) 1 < r < b (B) 0 < r < b (C) 0 ≤ r < b (D) 0 < r ≤ b
Euclid’s division lemma states that for positive integers a and b, there exist unique positive
integers q and r such that a=bq+r, where r must satisfy.
(a) 1<r<b (b) 0<r<b (c) 0≤r<b (d) 0<r≤b

4. J¸ Tmkz öuõhº Á›ø\°ß 6 Áx EÖ¨¤ß 6 ©h[S® 7 Áx EÖ¨¤ß
7 ©h[S® \©® GÛÀ, AUTmkz öuõhºÁ›ø\°ß 13 Áx EÖ¨¦ :
(A) 0 (B) 6 (C) 7 (D) 13
If 6 times of 6th term of an A.P. is equal to 7 times the 7th term, then the 13th term of the A.P.
is :
(a) 0 (b) 6 (c) 7 (d) 13

5. x2−2x−24 ©ØÖ® x2−kx−6&°ß «.ö£õ.Á (x−6) GÛÀ, k&°ß ©v¨¦ :
(A) 3 (B) 5 (C) 6 (D) 8
If (x−6) is the H.C.F of x2−2x−24 and x2−kx−6, then the value of k is :
(a) 3 (b) 5 (c) 6 (d) 8

3 8512

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. Ámhzvß öuõk÷Põk® Auß Bµ•® ö\[SzuõP Aø©²® Ch® :
(A) ø©¯® (B) öuõk ¦ÒÎ
(C) •i¼ (D) |õs
A tangent is perpendicular to the radius of a circle at the :
(a) Centre (b) Point of Contact
(c) Infinity (d) Chord

8. 7x−3y+4=0 GßÓ ÷|º÷PõmiØS Cøn¯õPÄ® Bv¨¦ÒÎ ÁÈa ö\À¾®
÷|º÷Põmiß \©ß£õk :
(A) 7x−3y+4=0 (B) 3x−7y+4=0
(C) 3x+7y=0 (D) 7x−3y=0
The equation of a line passing through the origin and parallel to the line 7x−3y+4=0 is :
(a) 7x−3y+4=0 (b) 3x−7y+4=0
(c) 3x+7y=0 (d) 7x−3y=0

9. (0, 0) ©ØÖ® (−8, 8) GßÓ ¦ÒÎPøÍ CønUS® ÷PõmiØSa ö\[SzuõÚ
÷Põmiß \õ´Ä :
1
(A) −1 (B) 1 (C) 3 (D) −8
The slope of the line which is perpendicular to a line joining the points (0, 0) and (−8, 8) is :
1
(a) −1 (b) 1 (c) (d) −8
3

10. J¸ ÷Põ¦µzvß E¯µzvØS® Auß {Ç¼ß }ÍzvØS® EÒÍ ÂQu® 3 : 1 GÛÀ,
`›¯øÚU Põq® HØÓU÷Põn AÍÁõÚx :
(A) 45E (B) 30E (C) 90E (D) 60E
If the ratio of the height of a tower and the length of its shadow is 3 : 1 , then the angle of
elevation of the sun has measure :
(a) 45E (b) 30E (c) 90E (d) 60E

8512 4

11. 15 ö\.«. E¯µ•® 16 ö\.«. Âmh•® öPõsh J¸ ÷|ºÁmhU T®¤ß ÁøÍ£µ¨¦ :
(A) 60 π \.ö\.«. (B) 68 π \.ö\.«.
(C) 120 π \.ö\.«. (D) 136 π \.ö\.«.
The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is :
(a) 60 π cm2 (b) 68 π cm2
(c) 120 π cm2 (d) 136 π cm2

12. 1 ö\.«. Bµ•®, 5 ö\.«. E¯µ•® öPõsh J¸ ©µ E¸øÍ°¼¸¢x AvP£m\U
PÚ AÍÄ öPõsh ÷PõÍ® öÁmi GkUP¨£kQÓx GÛÀ Auß PÚ AÍÄ.
(P.ö\.«. &À)
4 10 20
(A) 3 π (B) 3 π (C) 5 π (D) 3
π

The volume (in cm3) of the greatest sphere that can be cut off from a cylindrical log of wood
of base radius 1 cm and height 5 cm is :
4 10 20
(a) π (b) π (c) 5π (d) π
3 3 3

13. öPõkUP¨£mhøÁPÎÀ Gx uÁÓõÚx ?
(A) P(A) > 1 (B) 0 ≤ P(A) ≤ 1
(C) P(φ)=0 (D) P (A)+P ( A ) =1
Which of the following is incorrect ?
(a) P(A) > 1 (b) 0 ≤ P(A) ≤ 1

(c) P(φ)=0 (d) ( )
P(A)+P A =1

14. J¸ £n¨ø£°À ` 2000 ÷|õmkPÒ 10&®, ` 500 ÷|õmkPÒ 15 &®, ` 200 ÷|õmkPÒ
25 &® EÒÍÚ. J¸ ÷|õmk \©Áõ´¨¦ •øÓ°À GkUP¨£kQßÓx GÛÀ, A¢u
÷|õmk ` 500 ÷|õmhõP÷Áõ AÀ»x ` 200 ÷|õmhõP÷Áõ C¸¨£uØPõÚ {PÌuPÄ
GßÚ ?
1 3 2 4
(A) 5 (B) 10 (C) 3 (D) 5
A purse contains 10 notes of ` 2000, 15 notes of ` 500 and 25 notes of ` 200. One note is
drawn at random. What is the probability that the note is either a ` 500 note or ` 200 note ?
1 3 2 4
(a) (b) (c) (d)
5 10 3 5

5 8512

£Sv & II / PART - II

SÔ¨¦ : GøÁ- ÷ ¯Ý® 10 ÂÚõU- P - Ð US Âøh- ¯ - Î U- P - Ä ®. ÂÚõ Gs 28 &US
Pm-hõ-¯-©õP Âøh-¯-ÎU-PÄ®. 10x2=20
Note : Answer any ten questions. Question No. 28 is compulsory.

15. A={1, 2, 3} ©ØÖ® B={x|x Gߣx 10 &I Âha ]Ô¯ £Põ Gs} GÛÀ, A×B &IU
PõsP.
Let A={1, 2, 3} and B={ x|x is a prime number less than 10}. Find A×B

16. f (x)=2x+1 ©ØÖ® g (x)=x2−2 GÛÀ f o g ©ØÖ® g o f &IU PõsP.

Find f o g and g o f when f (x)=2x+1 and g (x)=x2−2.

17. J¸ |£›h® 532 §¢öuõmiPÒ EÒÍÚ. AÁº Á›ø\US 21 §¢öuõmiPÒ Ãu®
AkUP ¸®¤Úõº. GzuøÚ Á›ø\PÒ •Êø© ö£Ö® GÚÄ® ©ØÖ® GzuøÚ
§¢öuõmiPÒ «uª¸US® GÚÄ® PõsP.
A man has 532 flower pots. He wants to arrange them in rows such that each row contains
21 flower pots. Find the number of completed rows and how many flower pots are left over.

x3 y3
18. TmkP : +
x−y y−x

x3 y3
Add : +
x−y y−x

2 1 2 0
19. A=   , B=   GÛÀ AB ©ØÖ® BA PõsP. ÷©¾® AB=BA Gߣx \›¯õ
1 3  1 3
GÚ Bµõ´P.

2 1 2 0
If A =   , B=   find AB and BA. Verify if AB=BA.
1 3  1 3

8512 6

20. öPõkUP¨£mh £hzvÀ ∠A &°ß C¸\©öÁmi AD BS®. BD=4 ö\.«.,
DC=3 ö\.«. ©ØÖ® AB=6 ö\.«. GÛÀ AC -IU PõsP.

In the given diagram, AD is the bisector of ∠A. If BD=4 cm, DC=3 cm and AB=6 cm,
find AC.

1
21. (−2, a) ©ØÖ® (9, 3) GßÓ ¦ÒÎPÒ ÁÈaö\À¾® ÷|º÷Põmiß \õ´Ä − GÛÀ
2
a &°ß ©v¨¦ PõsP.
1
The line through the points (−2, a) and (9, 3) has slope − . Find the value of ‘a’.
2

22. x−2y+3=0, 6x+3y+8=0 BQ¯ ÷|º÷PõkPÒ JßÖUöPõßÖ ö\[SzuõÚøÁ
GÚU PõmkP.
Show that the straight lines x−2y+3=0 and 6x+3y+8=0 are perpendicular.

23. 50 3«. E¯µ•ÒÍ J¸ £õøÓ°ß Ea]°¼¸¢x 308 CÓUPU ÷PõnzvÀ
uøµ°¾ÒÍ ©QÊ¢x JßÖ £õºUP¨£kQÓx GÛÀ, ©QÊ¢vØS® £õøÓUS®
Cøh÷¯²ÒÍ öuõø»øÁU PõsP.
From the top of a rock 50 3 m high, the angle of depression of a car on the ground is
observed to be 308. Find the distance of the car from the rock.

7 8512

24. C¸ ÷PõÍ[PÎß Bµ[PÎß ÂQu® 4 : 7 GÛÀ AÁØÔß PÚ AÍÄPÎß ÂQu®
PõsP.
If the ratio of radii of two spheres is 4 : 7, find the ratio of their volumes.

25. J¸ T®¤ß CøhUPsha \õ²¯µ® 5 ö\.« BS®. Auß C¸ Bµ[PÒ 4 ö\.«
©ØÖ® 1 ö\.« GÛÀ, CøhUPshzvß ÁøÍ£µ¨ø£U PõsP.
The slant height of a frustum of a cone is 5 cm and the radii of its ends are 4 cm and 1 cm.
Find its curved surface area.

26. öPõkUP¨£mh £µÁ¼ß Ãa_U PõsP.

Á¯x
16 - 18 18 - 20 20 - 22 22 - 24 24 - 26 26 - 28
(Á¸h[PÎÀ)
©õnÁºPÎß
0 4 6 8 2 2
GsoUøP

Find the range of the following distribution :

Age (in years) 16 - 18 18 - 20 20 - 22 22 - 24 24 - 26 26 - 28

Number of Students 0 4 6 8 2 2

27. J¸ Tøh°¾ÒÍ 80 ©g\Ò, 70 ]Á¨¦ ©ØÖ® 50 öÁÒøͨ §UPμ¸¢x
\©Áõ´¨¦ •øÓ°À J¸ § ÷uº¢öukUS® ÷£õx Ax ©g\Ò {Ó¨ §ÁõP C¸UP
{PÌuPÄ PõsP.
A flower is selected at random from a basket containing 80 yellow, 70 red and 50 white
flowers. Find the probability of selecting a yellow coloured flower.

28. 2, 8, 18 , GßÓ öuõhºÁ›ø\ J¸ TmkzöuõhºÁ›ø\¯õ
32 , 50 , ......
CÀø»¯õ Gߣøu ÷\õvUP. Cx J¸ TmkzöuõhºÁ›ø\ GÛÀ Auß ö£õx
Âzv¯õ\® PõsP.
Check whether the sequence 2, 8, 18 , 32 , 50 , ...... is an A.P or not ? If it is an A.P,
then find the common difference.

8512 8

£Sv & III / PART - III

SÔ¨¦ : GøÁ÷¯Ý® 10 ÂÚõU- P - Ð US Âøh- ¯ - Î U- P - Ä ®. ÂÚõ Gs 42 &US
Pm-hõ-¯©õ-P Âøh-¯-ÎU-P-Ä®. 10x5=50
Note : Answer any ten questions. Question No. 42 is compulsory.

29. A Gߣx 8 -I ÂhU SøÓÁõÚ C¯À GsPÎß Pn®, B Gߣx 8 -I ÂhU
SøÓÁõÚ £Põ GsPÎß Pn® ©ØÖ® C Gߣx Cµmøh¨ £øh £Põ GsPÎß
Pn® GÛÀ (A ∩ B)×C=(A×C) ∩ (B×C) Gߣøu \›£õº.
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 numbers. Verify that (A ∩ B)×C=(A×C) ∩ (B×C).

x
30. f : A → B GßÓ \õº£õÚx f (x) = −1, GÚ Áøµ¯ÖUP¨£kQÓx. C[S,
2
A={2, 4, 6, 10, 12}, B={0, 1, 2, 4, 5, 9} BP C¸US®÷£õx \õº¦ f &I ¤ßÁ¸®
•øÓPÎÀ SÔUP.
(i) Á›ø\a ÷\õiPÎß Pn® (ii) AmhÁøn
(iii) A®¦USÔ £h® (iv) Áøµ£h®
x
Let f : A → B be a function defined by f ( x ) = −1, where A={2, 4, 6, 10, 12},
2
B={0, 1, 2, 4, 5, 9}. Represent f by
(i) set of ordered pairs (ii) a table
(iii) an arrow diagram (iv) a graph

x1 x x x
31. p p 2 × p 3 × p 4=113400
1 × 2 3 4
C[S p 1, p 2, p 3, p 4 Gß£Ú HÖ Á›ø\°À
Aø©¢u £Põ GsPÒ ©ØÖ® x1, x2, x3, x4 Gß£Ú •ÊUPÒ GÛÀ p1, p2, p3, p4
©ØÖ® x1, x2, x3, x4 BQ¯ÁØÔß ©v¨¦PøÍU PõsP.
x1 x x x
If p p 2 × p 3 × p 4=113400 where p , p , p , p are primes in ascending order
1 × 2 3 4 1 2 3 4
and x1, x2, x3, x4 are integers, find the value of p1, p2, p3, p4 and x1, x2, x3, x4.

32. TkuÀ PõsP. 62+72+82+⋅⋅⋅⋅+212.
Find the sum of the series 62+72+82+⋅⋅⋅⋅+212.

9 8512

 2 −1 
1 2 1
A= ©ØÖ® B = −1 4  GÛÀ (AB)T=BTAT Gߣøua \›£õºUP.
1 
33.
 2 −1  0
 2 

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

34. ÷Põn C¸\©öÁmiz ÷uØÓzøu GÊv {ÖÄP.
State and prove Angle Bisector Theorem.

35. (−4, −2), (−3, k), (3, −2) ©ØÖ® (2, 3) BQ¯ •øÚPøÍ Á›ø\¯õPU öPõsh
|õØPµzvß £µ¨¦ 28 \. A»SPÒ GÛÀ, k &°ß ©v¨ø£U PõsP.
Find the value of k, if the area of a quadrilateral is 28 sq. units, whose vertices are taken in the
order (−4, −2), (−3, k), (3, −2) and (2, 3).

36. (−3, 8) GßÓ ¦ÒÎ ÁÈa ö\ÀÁx®, B¯ Aa_PÎß ªøP öÁmkz xskPÎß
TkuÀ 7 Eøh¯x©õÚ ÷|º÷Põmiß \©ß£õmøhU PõsP.
A line makes positive intercepts on coordinate axes whose sum is 7 and it passes through
(−3, 8). Find its equation.

3 3 3 3
37. {¹¤UP : sin A + cos A + sin A − cos A = 2
sinA + cosA sinA − cosA

3 3 3 3
Prove that : sin A + cos A + sin A − cos A = 2
sinA + cosA sinA − cosA

38. |õuß GßÓ ö£õÔ°¯À ©õnÁº Kº E¸øÍ°ß C¸¦Ó•® T®¦PÒ EÒÍÁõÖ
©õv› JßøÓ E¸ÁõUQÚõº. ©õv›°ß }Í® 12 ö\.«. ©ØÖ® Âmh® 3 ö\.«.
BS®. JÆöÁõ¸ T®¤ß E¯µ•® 2 ö\.«. C¸US©õÚõÀ |õuß E¸ÁõUQ¯
©õv›°ß PÚ AÍøÁU PõsP.
Nathan, an engineering student was asked to make a model shaped like a cylinder with two
cones attached at its two ends. The diameter of the model is 3 cm and its length is 12 cm. If
each cone has a height of 2 cm, find the volume of the model that Nathan made.

8512 10

39. 16 ö\.«. Bµ•ÒÍ Kº E÷»õP¨ £¢x, E¸UP¨£mk 2 ö\.«. Bµ•ÒÍ ]Ö
£¢xPÍõUP¨£mhõÀ, GzuøÚ £¢xPÒ QøhUS® ?
A metallic sphere of radius 16 cm is melted and recast into small spheres each of radius 2 cm.
How many small spheres can be obtained ?

40. \z¯õ ©ØÖ® Âz¯õ C¸Á¸® 5 £õh[PÎÀ ö£ØÓ ö©õzu ©v¨ö£sPÒ
•øÓ÷¯ 460 ©ØÖ® 480 BS®. ÷©¾® Auß vmh »UP[PÒ •øÓ÷¯
4.6 ©ØÖ® 2.4 GÛÀ, ¯õ¸øh¯ ö\¯ÀvÓß, ªS¢u {ø»z ußø© öPõshx ?

The total marks scored by two students, Sathya and Vidhya in 5 subjects are 460 and 480
with standard deviation 4.6 and 2.4 respectively. Who is more consistent in performance ?

41. J¸ ø£°À 5 ]Á¨¦ {Ó¨ £¢xPЮ, 6 öÁÒøÍ {Ó¨ £¢xPЮ, 7 £aø\ {Ó¨
£¢xPЮ, 8 P¸¨¦ {Ó¨ £¢xPЮ EÒÍÚ. \©Áõ´¨¦ •øÓ°À ø£°¼¸¢x
J¸ £¢x GkUP¨£kQÓx. A¢u¨ £¢x,
(i) öÁÒøÍ
(ii) P¸¨¦ AÀ»x ]Á¨¦
(iii) öÁÒøͯõP CÀ»õ©À
(iv) öÁÒøͯõPÄ®, P¸¨£õPÄ® CÀ»õ©À
C¸¨£uØPõÚ {PÌuPÄPøÍU PõsP.
A bag contains 5 red balls, 6 white balls, 7 green balls, 8 black balls. One ball is drawn at
random from the bag. Find the probability that the ball drawn is

(i) white

(ii) black or red

(iii) not white

(iv) neither white nor black

42. (x−1)2+(x−2)2+(x−3)2 = 0 GßÓ \©ß£õmiß ‰»[PÎß ußø©ø¯ Bµõ´P.

Determine the nature of the roots of the equation :

(x−1)2+(x−2) 2+(x−3)2 = 0

11 8512

£Sv & IV / PART - IV
SÔ¨¦ : AøÚzx ÂÚõU-P-ÐU-S® Âøh-¯-ÎU-P-Ä®. 2x8=16
Note : Answer all the questions.

43. (A) - A i¨£UP® BC=8 ö\.«, ∠A=608 ©ØÖ® ∠A &°ß C¸\©öÁmi¯õÚx
BC &I D GßÓ ¦ÒΰÀ BD=6 ö\.« GßÓÁõÖ \¢vUQÓx GÛÀ,
•U÷Põn® ABC ÁøµP.
AÀ»x
(B) 4 ö\.« Bµ•ÒÍ Ámh® Áøµ¢x Auß ø©¯zv¼¸¢x 11 ö\.«
öuõø»Â¾ÒÍ J¸ ¦ÒÎø¯U SÔzx, A¨¦Òΰ¼¸¢x ÁmhzvØS
Cµsk öuõk÷PõkPÒ ÁøµP.
(a) Draw a triangle ABC of base BC=8 cm, ∠A=608 and the bisector of ∠A meets
BC at D such that BD=6 cm.
OR
(b) Take a point which is 11 cm away from the centre of a circle of radius 4 cm and draw
the two tangents to the circle from that point.

44. (A) -x2−9x+20=0 GßÓ C¸£ia \©ß£õmiß Áøµ£h® Áøµ¢x Auß ‰»®
wºÄPÎß ußø©ø¯U TÖP.
AÀ»x
(B) J¸ ÷£¸¢x 50 Q.«/©o GßÓ ^µõÚ ÷ÁPzvÀ £¯oUQÓx. Czöuõhº¦UPõÚ
yµ®&÷|µ® Áøµ£h® Áøµ¢x, ¤ßÁ¸ÁÚÁØøÓU PõsP.
(i) ÂQu\© ©õÔ¼ø¯U PõsP.
(ii) 90 {ªh[PÎÀ £¯oUS® yµ® GÆÁÍÄ ?
(iii) 300 Q.« yµzøu £¯oUP GÆÁÍÄ ÷|µ® BS® ?
(a) Graph the quadratic equation x2−9x+20=0 and state the nature of solution.
OR
(b) A bus is travelling at a uniform speed of 50 km/hr. Draw the distance - time graph and
hence find.
(i) the constant of variation.
(ii) how far will it travel in 90 minutes ?
(iii) the time required to cover a distance of 300 km.

-oOo-