Karnataka 2nd PUC Question Paper 2025 Maths

Government of Karnataka
Karnataka Secondary Education Examination Board

Question Papers

£ÉÆÃAzÀt ¸ÀASÉå :
Registration No. :

X1 – 2025
«µÀAiÀÄ ¸ÀAPÉÃvÀ /
35 (NS)
Subject Code

UÀtÂvÀ±Á¸ÀÛç / MATHEMATICS
(Kannada and English Versions)

[¸ÀªÀÄAiÀÄ: 3 UÀAmÉUÀ¼ÀÄ] [MlÄÖ ¥Àæ±ÉßUÀ¼À ¸ÀASÉå : 47] [UÀjµÀ× CAPÀUÀ¼ÀÄ : 80]
[Time : 3 Hours] [Total No. of questions : 47] [Max. Marks : 80]

(Kannada Version)
¸ÀÆZÀ£ÉUÀ¼ÀÄ : 1. F ¥Àæ±Éß ¥ÀwæPÉAiÀÄ°è A, B, C, D ªÀÄvÀÄÛ E JA§
LzÀÄ «¨sÁUÀUÀ½ªÉ. J¯èÁ «¨sÁUÀUÀ¼À£ÀÄß GvÀÛj¹.
2. «¨sÁUÀ-A zÀ°è 15 §ºÀÄ DAiÉÄÌ ¥Àæ±ÉßUÀ¼ÀÄ, 5 ©lÖ ¸ÀܼÀ
vÀÄA§ÄªÀ ¥Àæ±ÉßUÀ½zÀÄÝ, ¥ÀæwAiÉÆAzÀÄ ¥Àæ±ÉßAiÀÄÄ
1 CAPÀzÁÝVzÉ.

3. «¨sÁUÀ-A zÀ°è §gÀĪÀ ¥Àæ±ÉßUÀ½UÉ ¥ÀæxÀªÀÄ GvÀÛgÀªÀ£ÀÄß
ªÀiÁvÀæ ¥ÀjUÀt¹ CAPÀUÀ¼À£ÀÄß ¤ÃqÀ¯ÁUÀĪÀÅzÀÄ.
4. «¨sÁUÀ-E zÀ°è §gÀĪÀ `gÉÃTÃAiÀÄ ¥ÉÆæÃUÁæ å«ÄAUï' ¥Àæ±ÉßUÉ
¤ªÀÄUÉ MzÀV¹gÀĪÀ £ÀPÉëAiÀÄ£ÀÄß G¥ÀAiÉÆÃV¹ GvÀÛj¹.
5. ¥Àæ±Éß ¥ÀwæPÉAiÀÄ PÉÆ£ÉAiÀÄ°è, zÀ馅 «PÀ®ZÉÃvÀ£À
«zÁåyðUÀ½UÁV avÀæ / £ÀPÉë EgÀĪÀ ¥Àæ±ÉßUÀ½UÉ, «¨sÁUÀ-F £À°è
¥ÀAiÀiÁðAiÀÄ ¥Àæ±ÉßUÀ¼À£ÀÄß ¤ÃqÀ¯ÁVzÉ.

P.T.O.

35 (NS) -2-

«¨sÁUÀ- A
I. J¯Áè §ºÀÄ DAiÉÄÌAiÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. (15 × 1 = 15)

1) MAzÀÄ UÀt A AiÀÄ°è ¸ÀA§AzsÀ R ¥Àæw¥sÀ®£ÀªÁUÀ¨ÉÃPÁzÀgÉ
a) ¥ÀæwAiÉÆAzÀÄ a ∈ A EgÀĪÁUÀ (a, a ) ∈ R
b) AiÀiÁªÀÅzÁzÀgÉÆAzÀÄ a ∈ A EgÀĪÁUÀ (a, a ) ∈ R
c) (a, b ) ∈ R EgÀĪÁUÀ (b, a ) ∈ R

d) (a, b ) ∈ R ªÀÄvÀÄÛ (b, c ) ∈ R EgÀĪÁUÀ (a, c ) ∈ R

 1 
2) sin −1  gÀ ¥ÀæzsÁ£À ¨É¯ÉAiÀÄÄ
 2
π π
a) b)
2 3
π π
c) d)
4 6

3) ¥ÀnÖ - I £ÀÄß ¥ÀnÖ - II gÀ eÉÆvÉUÉ ºÉÆA¢¹.
¥ÀnÖ - I ¥ÀnÖ - II
A) sin −1 x £À PÉëÃvÀæ −π π 
i)  , 
 2 2
B) tan −1 x £À ªÁå¦Û ii) [0, π ]

C) cos −1 x £À ªÁå¦Û iii) [ −1, 1]

F PɼÀUÉ PÉÆnÖgÀĪÀ DAiÉÄÌUÀ¼À°è ¸ÀjAiÀiÁzÀ GvÀÛgÀªÀ£ÀÄß Dj¹ :

a) A-i, B-ii, C-iii b) A-iii, B-ii, C-i
c) A-ii, B-i, C-iii d) A-iii, B-i, C-ii

-3- 35 (NS)

4) aij = 2 i − jJA§ CA±ÀUÀ½gÀĪÀ A = [aij ] JA§ ªÀiÁvÀÈPÉAiÀÄÄ 2 × 2
zÀeÉðAiÀÄ ªÀiÁvÀÈPÉAiÀiÁzÁUÀ A =
2 3 1 0
a) 1 b)
 2 3 2
 
1 1  1 2
c) 2 d)
 2 2 1
 
5) A MAzÀÄ 3 × 3 zÀeÉðAiÀÄ ¥Àæw¯ÉÆêÀÄ PÉÆñÀªÁVzÁÝUÀ adj A =
a) A b) 3A
3 2
c) A d) A

π 
6) f ( x ) = cos 2 x DVzÀÝ°è f ′   =
4
a) 2 b) –2
c) 2 d) − 2
7) PÉÆnÖgÀĪÀ avÀæPÉÌ F PɼÀV£À ºÉýPÉ 1 ªÀÄvÀÄÛ ºÉýPÉ 2 UÀ¼À£ÀÄß ¥ÀjUÀt¹.

ºÉýPÉ 1 : x = 1 gÀ°è y = f ( x ) £À JqÀ ¤µÀá£ÀߪÀÅ –1 DVgÀÄvÀÛzÉ.
ºÉýPÉ 2 : x = 1 gÀ°è y = f ( x ) ¤µÀá£ÀßvÉ ºÉÆA¢gÀÄvÀÛzÉ.
F PɼÀV£ÀªÀÅUÀ¼À°è ¸ÀjAiÀiÁzÀÄzÀÝ£ÀÄß DAiÉÄÌ ªÀiÁr.
a) ºÉýPÉ 1 ¸ÀjAiÀiÁVzÉ, ºÉýPÉ 2 vÀ¥ÁàVzÉ
b) ºÉýPÉ 1 vÀ¥ÁàVzÉ, ºÉýPÉ 2 ¸ÀjAiÀiÁVzÉ
c) ºÉýPÉ 1 ªÀÄvÀÄÛ ºÉýPÉ 2, JgÀqÀÆ ¸ÀjAiÀiÁVªÉ
d) ºÉýPÉ 1 ªÀÄvÀÄÛ ºÉýPÉ 2, JgÀqÀÆ vÀ¥ÁàVªÉ

35 (NS) -4-

8) f ( x ) = x 3 , x ∈ [ −2, 2] f GvÀà£ÀßzÀ ¤gÀ¥ÉÃPÉë UÀjµÀ× ¨É¯ÉAiÀÄÄ ____________

a) 2 b) 0
c) –2 d) 8

 e (sin x − cos x ) dx =
x
9)

a) − e x cos x b) e x cos x
c) e x sin x d) e x sin 2 x

dy
d 3y d 2y
10) 3
+ 2 + e dx = 0 CªÀPÀ®£À ¸À«ÄÃPÀgÀtzÀ ¥ÀæªÀiÁt (ªÀÄlÖ) =
dx dx
—————————
a) 1 b) 3
c) 2 d) ªÁåSÁ夸ÀĪÀÅ¢®è

→
11) a = iˆ − ˆj + 2 kˆ ¸À¢±ÀzÀ ¢±Á PÉƸÉÊ£ïUÀ¼ÀÄ

1 −1 2 1 −1 2
a) , , b) , ,
5 5 5 6 6 6
1 −1 2 −1 1 2
c) , , d) , ,
6 6 6 6 6 6

→ → → → → →
12) a = 3 , b = 2 ªÀÄvÀÄÛ a⋅ b = 6 DzÁUÀ a ªÀÄvÀÄÛ b ¸À¢±ÀUÀ¼À £ÀqÀÄ«£À

PÉÆãÀªÀÅ _______________
π π
a) b)
6 3
π π
c) d)
4 2

13) ªÀÄÆgÀÄ DAiÀiÁªÀÄzÀ°è y-CPëÀzÀ ¸À«ÄÃPÀgÀtªÀÅ _______________
a) x = 0, y = 0 b) x = 0, z = 0
c) y = 0, z = 0 d) y=0

-5- 35 (NS)

1 2
14) P ( A) = , P (B | A) = DzÀgÉ P ( A ∩ B ) = _______________
2 3
1 1
a) b)
3 2
3
c) 1 d)
5

15) ¸ÀªÀÄxÀð£É [A] : E ªÀÄvÀÄÛ F WÀl£ÉUÀ½UÉ P (E ) = 1 , P (F ) = 1 ªÀÄvÀÄÛ
5 2
1
P (E | F ) = DVzÀÝ°è E ªÀÄvÀÄÛ F UÀ¼ÀÄ ¸ÀévÀAvÀæ
5
WÀl£ÉU¼
À ÁVgÀÄvÀÛª.É

PÁgÀt [R] : WÀl£É E ªÀÄvÀÄÛ F UÀ¼ÀÄ ¸ÀévÀAvÀæªÁVzÀÝ°è P (F | E ) = P (F )

DVgÀĪÁUÀ PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÀÄ ¸ÀjAiÀiÁVzÉ?
a) [A] ¸ÀjAiÀiÁVzÉ ªÀÄvÀÄÛ [R] vÀ¥ÁàVzÉ

b) [A] ªÀÄvÀÄÛ [R] JgÀqÀÆ vÀ¥ÁàVªÉ

c) [A] ªÀÄvÀÄÛ [R] JgÀqÀÆ ¸ÀjAiÀiÁVªÉ

d) [A] vÀ¥ÁàVzÀÄÝ [R] ¸ÀjAiÀiÁVzÉ

II. DªÀgÀtzÀ°è PÉÆnÖgÀĪÀ DAiÉÄÌUÀ½AzÀ ¸ÀÆPÀÛªÁzÀ GvÀÛgÀªÀ£ÀÄß Dj¹ PɼÀV£À
©lÖ ¸ÀܼÀUÀ¼À£ÀÄß vÀÄA©j. (5 × 1 = 5)

5
[0, 2, 1, , –1, 6]
9

  3 
16) cos  sec −1(2) − sin −1   EzÀgÀ ¨É¯ÉAiÀÄÄ _______________

  2 

dy
17) y = sin −1(cos x ) DzÀgÉ = _______________
dx

35 (NS) -6-

13
18)  1 dx EzÀgÀ ¨É¯ÉAiÀÄÄ _______________
7

19) iˆ + ˆj ¸À¢±ÀzÀ iˆ − ˆj ªÉÄÃ¯É ¸À¢±ÀzÀ ¨ÁUÀÄ«PÉAiÀÄÄ _______________

4
20) P ( A ∩ B ) = ªÀÄvÀÄÛ P (B ) = 9 DzÀgÉ P ( A′ | B ) = _______________
13 13

«¨sÁUÀ – B

III. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ DgÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. (6 × 2 = 12)

21) (1, 2) ªÀÄvÀÄÛ (3, 6) ©AzÀÄUÀ½AzÀ GAmÁzÀ ¸ÀgÀ¼ÀgÉÃSÉAiÀÄ
¸À«ÄÃPÀgÀtªÀ£ÀÄß ¤zsÁðgÀPÀªÀ£ÀÄß G¥ÀAiÉÆÃV¹ PÀAqÀÄ»r¬Äj.

dy y
22) x + y = 10 DzÀgÉ + = 0 JAzÀÄ vÉÆÃj¹.
dx x

23) §zÀ¯ÁUÀÄwÛgÀĪÀ wædåªÀ£ÀÄß ºÉÆA¢gÀĪÀ, AiÀiÁªÁUÀ®Æ
UÉÆüÁPÁgÀzÀ°ègÀĪÀ §®Æ¤£À wædåªÀÅ 10 ¸ÉA.«ÄÃ. DVzÁÝUÀ wædåPÉÌ
¸ÀA§A¢ü¹zÀAvÉ CzÀgÀ WÀ£À¥sÀ® (volume) zÀ §zÀ¯ÁªÀuÉAiÀÄ ºÉZÀѼÀzÀ
zÀgÀªÀ£ÀÄß PÀAqÀÄ»r¬Äj.

24) f ( x ) = 4 x 3 − 6 x 2 − 72 x + 30 GvÀà£ÀߪÀÅ QëÃt¸ÀĪÀ CAvÀgÁ¼ÀªÀ£ÀÄß
PÀAqÀÄ»r¬Äj.

25)  cot x ⋅ log (sin x ) dx PÀAqÀÄ»r¬Äj.
d 2y
26) + y = 0 CªÀPÀ®£À ¸À«ÄÃPÀgÀtPÉÌ, GvÀà£Àß y = a sin x + b cos x ¸ÁªÀiÁ£Àå
dx 2
¥ÀjºÁgÀªÉà JAzÀÄ vÁ¼É£ÉÆÃr.

-7- 35 (NS)
→ → →
27) a = iˆ + ˆj + kˆ, b = 2 iˆ − ˆj + 3 kˆ ªÀÄvÀÄÛ c = iˆ − 2 ˆj + kˆ DzÀgÉ ¸À¢±À
→ → →
2 a − b + 3 c UÉ ¸ÀªÀiÁ£ÁAvÀgÀ KPÀ ¸À¢±ÀªÀ£ÀÄß PÀAqÀÄ»r¬Äj.

x −1 y − 2 z − 3
28) = = ªÀÄvÀÄÛ x − 1 = y − 1 = z − 6 gÉÃSÉUÀ¼ÀÄ ¥ÀgÀ¸ÀàgÀ
−3 2k 2 3k 1 −5
®A§ªÁVzÀÝ°è k AiÀÄ ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.

29) MAzÀÄ ºÀÆfAiÀÄ°è 10 PÀ¥ÀÄà ªÀÄvÀÄÛ 5 ©½ ZÉAqÀÄUÀ½ªÉ, ºÀÆf¬ÄAzÀ
JgÀqÀÄ ZÉAqÀÄUÀ¼À£ÀÄß, MAzÀgÀ £ÀAvÀgÀ ªÀÄvÉÆÛAzÀgÀAvÉ, ZÉAqÀ£ÀÄß ¥ÀÅ£ÀB
ºÀÆfAiÀÄ°è ºÁPÀzÉ vÉUÉAiÀįÁVzÉ. JgÀqÀÆ ZÉAqÀÄUÀ¼ÀÄ PÀ¥ÀÄà
§tÚzÁÝVgÀĪÀ ¸ÀA¨sÀªÀ¤ÃAiÀÄvÉ JµÀÄÖ?

«¨sÁUÀ – C

IV. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ DgÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹j. (6 × 3 = 18)

30) ªÁ¸ÀÛ«PÀ ¸ÀASÁåUÀt £À°è ¸ÀA§AzsÀ R £ÀÄß R = {(a, b ) : a ≤ b 3 } JAzÀÄ
ªÁåSÁ夹zÀgÉ, CzÀÄ ¥Àæw¥sÀ®£À, ¸ÀªÀiÁAUÀvÀ ªÀÄvÀÄÛ ªÁºÀPÀ
¸ÀA§AzsÀUÀ¼ÁVªÉAiÉÄà JAzÀÄ ¥ÀjÃQë¹.

63 5 3
31) tan −1   = sin −1   + cos −1   JAzÀÄ ¸Á¢ü¹.
 16   13  5

 1 5
32)   ªÀiÁvÀÈPÉAiÀÄ£ÀÄß ¸ÀªÀiÁAUÀ ªÀÄvÀÄÛ C¸ÀªÀiÁAUÀ ªÀiÁvÀÈPÉUÀ¼À
 − 1 2
ªÉÆvÀÛªÉAzÀÄ ¸ÀàµÀÖ¥Àr¹.

  t  dy
33) x = a  cos t + log  tan   ªÀÄvÀÄÛ y = a sin t DzÀgÉ £ÀÄß PÀAqÀÄ»r¬Äj.
  2  dx

34) x ªÀÄvÀÄÛ y JgÀqÀÄ zsÀ£ÁvÀäPÀ ¸ÀASÉåUÀ¼ÁVzÀÄÝ x + y = 60 ªÀÄvÀÄÛ xy 3 ¨É¯É
UÀjµÀתÁVgÀ¨ÉÃPÁzÀgÉ, D ¸ÀASÉåUÀ¼À£ÀÄß PÀAqÀÄ»r¬Äj.

35 (NS) -8-

2x
35)  x 2 + 3 x + 2 dx PÀAqÀÄ»r¬Äj.

36) MAzÀÄ wæPÉÆãÀ ABC AiÀÄ°è A, B, C UÀ¼À ¸ÁÜ£À ¸À¢±ÀUÀ¼ÀÄ iˆ − ˆj + 2 kˆ ,
2 ˆj + kˆ, ˆj + 3 kˆ DVzÀÝ°è, D wæPÉÆãÀzÀ «¹ÛÃtðªÀ£ÀÄß PÀAqÀÄ»r¬Äj.

→
37) zÀvÀÛ ¸À¢±À b UÉ ¸ÀªÀiÁ£ÁAvÀgÀªÁV ºÁUÀÆ zÀvÀÛ ©AzÀÄ«£À ªÀÄÆ®PÀ
ºÁzÀĺÉÆÃUÀĪÀ gÉÃSÉAiÉÆAzÀgÀ ¸À«ÄÃPÀgÀtªÀ£ÀÄß ¸À¢±À gÀÆ¥ÀzÀ°è
¥Àæw¥Á¢¹ (derive).

38) PÉÆnÖgÀĪÀ JgÀqÀÄ MAzÉà jÃwAiÀÄ qÀ§âUÀ¼À°è, qÀ§â I gÀ°è JgÀqÀÄ a£ÀßzÀ
£ÁtåUÀ¼ÀÄ, qÀ§â II gÀ°è MAzÀÄ a£ÀßzÀ £Átå ªÀÄvÀÄÛ MAzÀÄ ¨É½îAiÀÄ £Átå
EªÉ. M§â ªÀåQÛAiÀÄÄ MAzÀÄ qÀ§âªÀ£ÀÄß AiÀiÁzÀÈaÒPÀªÁV DAiÉÄÌ ªÀiÁqÀĪÀ£ÀÄ
ªÀÄvÀÄÛ MAzÀÄ £ÁtåªÀ£ÀÄß ºÉÆgÀUÉ vÉUÉAiÀÄĪÀ£ÀÄ. MAzÀÄ ªÉÃ¼É vÉUÉzÀ
£ÁtåªÀÅ a£ÀßzÁÝVzÀÝgÉ, qÀ§âzÀ°è E£ÉÆßAzÀÄ £ÁtåªÀÅ PÀÆqÁ
a£ÀßzÁÝVgÀĪÀ ¸ÀA¨sÀªÀ¤ÃAiÀÄvÉ JµÀÄÖ?

«¨sÁUÀ – D
V. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ £Á®ÄÌ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. (4 × 5 = 20)

39) A = − {3} ªÀÄvÀÄÛ B= − {1} DVgÀ°, GvÀà£Àß f :A→B AiÀÄ£ÀÄß
 x − 2
f (x) =   JAzÀÄ ªÁåSÁ夹zÉ. f MAzÀÄ KPÀ-KPÀ ªÀÄvÀÄÛ ªÉÄît
 x −3
GvÀà£ÀߪÉÃ? ¤ªÀÄä GvÀÛgÀªÀ£ÀÄß ¸ÀªÀÄyð¹.

 1
40) A = − 4 ªÀÄvÀÄÛ B = [− 1 2 1] DzÀgÉ, ( AB )′ = B′A′ C£ÀÄß ¥Àj²Ã°¹.
 
 3 

-9- 35 (NS)

41) 4 x + 3 y + 2z = 60 , 2 x + 4 y + 6z = 90 , 6 x + 2y + 3z = 70 ¸À«ÄÃPÀgÀtUÀ¼À£ÀÄß
PÉÆñÀzÀ «zsÁ£À¢AzÀ ¸À«ÄÃPÀj¹.

42) y = (tan −1 x )2 DzÀgÉ ( x 2 + 1)2 y 2 + 2 x ( x 2 + 1) y 1 = 2 JAzÀÄ ¸Á¢ü¹.

1
43) GvÀà£Àß £À C£ÀÄPÀ°vÀªÀ£ÀÄß ‘x’ UÉ C£ÀÄUÀÄtªÁV PÀAqÀÄ»r¬Äj.
x + a2
2

1
EzÀ£ÀÄß G¥ÀAiÉÆÃV¹  2 dx £ÀÄß PÀAqÀÄ»r¬Äj.
x − 6 x + 13

44) C£ÀÄPÀ°vÀÀ «zsÁ£À¢AzÀ x 2 + y 2 = a2 , ªÀÈvÀÛzÀ «¹ÛÃtðªÀ£ÀÄß
PÀAqÀÄ»r¬Äj.

dy  π
45) cos 2 x + y = tan x  0 ≤ x <  CªÀPÀ°vÀ ¸À«ÄÃPÀgÀtzÀ ¸ÁªÀiÁ£Àå
dx  2
¥ÀjºÁgÀªÀ£ÀÄß PÀAqÀÄ»r¬Äj.

«¨sÁUÀ – E
VI. F PɼÀV£À ¥Àæ±ÉßUÀ¼À£ÀÄß GvÀÛj¹j.
a a
46)  f ( x ) dx =  f (a − x ) dx JAzÀÄ ¸Á¢ü¹, CzÀ£ÀÄß G¥ÀAiÉÆÃV¹
0 0
π
4
 log (1 + tan x ) dx ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj. (6)
0

CxÀªÁ
F PɼÀV£À gÉÃTÃAiÀÄ ¥ÉÆæÃUÁæ å«ÄAUï ¸ÀªÀĸÉåAiÀÄ£ÀÄß £ÀPÁëvÀäPÀªÁV ©r¹.
x + 2y ≤ 120,
x + y ≥ 60,
x − 2y ≥ 0,
x ≥ 0, y ≥ 0
¤§AzsÀ£ÉUÉƼÀ¥ÀlÄÖ G¢ÝµÀÖ GvÀà£Àß Z = 5 x + 10 y C£ÀÄß UÀjµÀ× ªÀÄvÀÄÛ
PÀ¤µÀ×UÉƽ¹.

35 (NS) -10-

 3 1
47) A =   DzÀgÉ A2 − 5 A + 7 I = O JAzÀÄ ¸Á¢ü¹ ªÀÄvÀÄÛ EzÀ£ÀÄß
 − 1 2
G¥ÀAiÉÆÃV¹ A −1 £ÀÄß PÀAqÀÄ»r¬Äj. (4)

CxÀªÁ
 k cos x , x ≠ π
 π
f ( x ) =  π − 2x 2 GvÀà£ÀߪÀÅ x= £À°è C«aÒ£ÀߪÁzÀgÉ k AiÀÄ
 3,
π 2
x=
 2
¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.

«¨sÁUÀ – F

VII. zÀ馅 «PÀ®ZÉÃvÀ£À «zÁåyðUÀ½UÁV

7) ºÉýPÉ 1 : x = 0 gÀ°è f ( x ) = x GvÀà£ÀßzÀ JqÀ ¤µÀá£ÀߪÀÅ –1 DVgÀÄvÀÛzÉ.
ºÉýPÉ 2 : x = 0 gÀ°è f ( x ) = x GvÀà£ÀߪÀÅ ¤µÀá£ÀßvÉ ºÉÆA¢gÀÄvÀÛzÉ.
ªÉÄð£À ºÉýPÉUÀ½UÉ F PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÀÄ ¸Àj?
a) ºÉýPÉ 1 ¸ÀjAiÀiÁVzÉ ªÀÄvÀÄÛ ºÉýPÉ 2 vÀ¥ÁàVzÉ
b) ºÉýPÉ 1 vÀ¥ÁàVzÉ ªÀÄvÀÄÛ ºÉýPÉ 2 ¸ÀjAiÀiÁVzÉ
c) ºÉýPÉ 1 ªÀÄvÀÄÛ 2, JgÀqÀÆ ºÉýPÉUÀ¼ÀÄ ¸ÀjAiÀiÁVªÉ
d) ºÉýPÉ 1 ªÀÄvÀÄÛ 2, JgÀqÀÆ ºÉýPÉUÀ¼ÀÄ vÀ¥ÁàVªÉ

––––––––––––––––

-11- 35 (NS)

(English Version)

Instructions : 1. The question paper has five Parts namely A, B, C, D
and E. Answer all parts.

2. PART-A has 15 M.C.Q.’s, 5 Fill in the blanks of 1 mark
each.

3. For PART-A questions, only the first written answers will
be considered for awarding marks.

4. Use graph sheet for question on Linear Programming in
PART-E.

5. For questions having figure / graph, alternate questions are
given at the end of question paper in separate PART-F for
visually challenged students.

PART – A

I. Answer all the multiple choice questions : (15 × 1 = 15)

1) A relation R in a set A is called Reflexive relation if
a) (a, a ) ∈ R for all a ∈ A
b) (a, a ) ∈ R for atleast one a ∈ A
c) (a, b ) ∈ R implies (b, a ) ∈ R
d) (a, b ) ∈ R and (b, c ) ∈ R implies (a, c ) ∈ R

 1 
2) The principal value of sin −1  is
 2
π π
a) b)
2 3
π π
c) d)
4 6

35 (NS) -12-

3) Match List - I with List - II.

List - I List - II
A) Domain of sin −1 x −π π 
i)  , 
 2 2
B) Range of tan −1 x ii) [0, π ]
C) Range of cos −1 x iii) [−1, 1]

Choose the correct answer from the options given below :

a) A-i, B-ii, C-iii b) A-iii, B-ii, C-i
c) A-ii, B-i, C-iii d) A-iii, B-i, C-ii

4) For a 2 × 2 matrix A = [aij ] whose elements are given by aij = 2 i − j then
A is equal to
2 3  1 0
a)  1 2 b) 3 2
   
 1 1  1 2
c) 2 2  d) 2 1
   

5) Let A be a nonsingular matrix of order 3 × 3, then adj A is equal to

a) A b) 3A
3 2
c) A d) A

π 
6) If f ( x ) = cos 2 x , then f ′   is
4
a) 2 b) –2
c) 2 d) − 2

-13- 35 (NS)

7) For the given figure consider the following statements 1 and 2 :

Statement 1 : Left hand derivative of y = f (x ) at x = 1 is –1.
Statement 2 : The function y = f (x ) is differentiable at x = 1.
Then which of the following are true?
a) Statement 1 is true, Statement 2 is false
b) Statement 1 is false, Statement 2 is true
c) Both Statements 1 and 2 are true
d) Both Statements 1 and 2 are false

8) The absolute maximum value of the function f given by f ( x ) = x 3 ,
x ∈ [−2, 2] is
a) 2 b) 0
c) –2 d) 8

 e (sin x − cos x ) dx is
x
9)

a) − e x cos x b) e x cos x
c) e x sin x d) e x sin 2 x

dy
d 3y d 2y
10) The degree of differential equation 3
+ 2 + e dx = 0 is
dx dx
a) 1 b) 3
c) 2 d) not defined

35 (NS) -14-

→
11) The direction cosines of the vector a = iˆ − ˆj + 2 kˆ are
1 −1 2 1 −1 2
a) , , b) , ,
5 5 5 6 6 6
1 −1 2 −1 1 2
c) , , d) , ,
6 6 6 6 6 6

→ → → →
12) The angle between two vectors a and b with a = 3, b = 2 and
→ →
a ⋅ b = 6 is
π π
a) b)
6 3
π π
c) d)
4 2
13) The equation of y-axis in space is
a) x = 0, y = 0 b) x = 0, z = 0
c) y = 0, z = 0 d) y=0

1 2
14) If P ( A) = , P (B | A) = then P ( A ∩ B ) is
2 3
1 1
a) b)
3 2
3
c) 1 d)
5

1 1
15) Assertion [A] : For two events E and F if P (E ) = , P (F ) = and
5 2
1
P (E | F ) = then E and F are independent events.
5

Reason [R] : If E and F are two independent events then
P (F | E ) = P (F )
Then which of the following are true?
a) [A] is true but [R] is false b) Both [A] and [R] are false
c) Both [A] and [R] are true d) [A] is false but [R] is true

-15- 35 (NS)

II. Fill in the blanks by choosing the appropriate answer from those given in the
bracket : (5 × 1 = 5)
5
[0, 2, 1, , –1, 6]
9

  3 
16) The value of cos  sec −1(2) − sin −1   is _________.

  2 

dy
17) If y = sin −1(cos x ) then = __________.
dx
13
18) The value of  1 dx = _________.
7

19) The projection of vector iˆ + ˆj along the vector iˆ − ˆj is __________.

4 9
20) If P ( A ∩ B ) = and P (B ) = then P ( A′ | B ) = _________.
13 13

PART – B

III. Answer any six of the following questions : (6 × 2 = 12)
21) Find the equation of the line through the points (1, 2) and (3, 6) using
determinants.

dy y
22) If x + y = 10 then show that + =0.
dx x
23) A balloon which is always remains spherical has a variable radius. Find
the rate at which its volume is increasing with radius when the radius is
10 cms.

24) Find the interval in which the function given by f ( x ) = 4 x 3 − 6 x 2 − 72 x + 30
is decreasing.

25) Find  cot x ⋅ log (sin x ) dx .

26) Verify that the function y = a sin x + b cos x is a solution of differential
d 2y
equation + y = 0.
dx 2

35 (NS) -16-

→  →
27) If a = iˆ + jˆ + kˆ, b = 2 iˆ − jˆ + 3 kˆ and c = iˆ − 2 ˆj + kˆ then find unit vector
→ → →
parallel to the vector 2 a − b + 3 c .

x −1 y − 2 z − 3 x −1 y −1 z − 6
28) If the lines = = and = = are
−3 2k 2 3k 1 −5
perpendicular to each other, then find the value of k.
29) An urn contains 10 black and 5 white balls. Two balls are drawn from the
urn one after the other without replacement. What is the probability that
both drawn balls black?

PART – C

IV. Answer any six of the following questions : (6 × 3 = 18)

30) Check whether the relation R in defined by R = {(a, b ) : a ≤ b 3 } is
reflexive, symmetric and transitive.

63 5 3
31) Prove that tan −1   = sin −1   + cos −1   .
 16   13  5

 1 5
32) Express   as the sum of a symmetric and a skew-symmetric
 − 1 2
matrix.

dy   t 
33) Find if x = a  cos t + log  tan   and y = a sin t .
dx   2 

34) Find the two positive numbers x and y such that x + y = 60 and xy 3 is
maximum.

2x
35) Evaluate  2
dx .
x + 3x + 2

36) Find the area of triangle ABC where position vectors of A, B, C are
iˆ − ˆj + 2 kˆ , 2 ˆj + kˆ, ˆj + 3 kˆ respectively.

-17- 35 (NS)

37) Derive the equation of a line in space through a given point and parallel to
→
a given vector b in the vector form.

38) In two identical boxes, box I contains 2 gold coins, while box II contains
one gold and one silver coin. A person chooses a box at random and
takes out a coin. If the coin is of gold, what is the probability that the other
coin in the box is also a gold?

PART – D

V. Answer any four of the following questions : (4 × 5 = 20)

39) If A = − {3} and B = − {1} and f : A → B is a function defined by
 x − 2
f (x) =   . Is f one-one and onto? Justify your answer.
 x −3

 1
40) If A = − 4 and B = [− 1 2 1], verify that ( AB )′ = B′A′ .
 
 3 

41) Solve the following system of linear equations by matrix method
4 x + 3 y + 2z = 60 , 2 x + 4 y + 6z = 90 , 6 x + 2y + 3z = 70 .

42) If y = (tan −1 x )2 then show that ( x 2 + 1)2 y 2 + 2 x ( x 2 + 1) y 1 = 2 .

1
43) Find the integral of with respect to ‘x’ and hence find
x 2 + a2
1
 x 2 − 6 x + 13 dx .

44) Find the area of circle x 2 + y 2 = a 2 by method of integration.

dy  π
45) Solve the differential equation cos 2 x + y = tan x  0 ≤ x <  .
dx  2

35 (NS) -18-

PART – E
VI. Answer the following questions :
π
a a 4
46) Prove that  f ( x ) dx =  f (a − x ) dx and hence evaluate  log (1 + tan x ) dx .
0 0 0
(6)
OR
Solve the following Linear Programming Problem graphically :
Minimise and Maximise Z = 5 x + 10 y
Subject to
x + 2y ≤ 120,
x + y ≥ 60,
x − 2y ≥ 0,
x ≥ 0, y ≥ 0.

 3 1
47) If A =   , show that A 2 − 5 A + 7 I = O and hence find A −1 . (4)
 − 1 2
OR
Determine the value of k if
 k cos x , x ≠ π

f ( x ) =  π − 2x 2
 3,
π
x=
 2
π
is continuous at x = .
2
PART – F
VII. For visually challenged students only
7) If Statement 1 : Left hand derivative of f ( x ) = x at x = 0 is –1.
Statement 2 : The derivative of f ( x ) = x exists at x = 0.
Then which of the following is true?
a) Statement 1 is true, Statement 2 is false
b) Statement 1 is false, Statement 2 is true
c) Statement 1 and 2 both are true
d) Statement 1 and 2 both are false
————————

-19- 35 (NS)

35 (NS) -20-

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