Karnataka 2nd PUC Model Question Paper 2022 for Basic Maths

II PUC BASIC MATHEMATICS MODEL QUESTION PAPER (ENGLISH VERSION) (FOR
THE YEAR 2021-22)

Time : 3:15 hrs Subject: Basic Mathematics (code:75) Marks: 100
Instructions:
(1) The question paper has 5 parts -A, B, C, D and E. Answer all the parts.
(2) Part A carries 10 marks; Part B carries 20 marks; Part C carries 30 marks;Part D carries 30 marks;Part
E carries 10 marks;
(3) Write the question number properly as indicated in the question paper.
PART - A
I. Answer any TEN questions 10 × 1 = 10


1
2 
1. If A    find 3A.

3 4 

400 404
2. Find the value of 408 412 .

3. In how many ways can 10 people be seated around a table ?
4. A bag contains 8 red and 4 green marbles. Find the probability that a marble selected at random
is red.

5. Write “Mathematics is easy and today is a holiday” in symbolic form.

6. Find the triplicate ratio of 3: 5.

7. How much stock can be bought for Rs. 3375 at Rs. 75?

8. Write the learning curve equation.

9. If tan A = 1, find tan 2A

10. If the length of Latus Rectum of parabola y2 = 8Kx is 4, find k.

4x
3
11. Evaluate lim .
x4 x
2
dy
12. Find if y = xe + ex ,
dx

dy cos x
13. Find if y  .
dx 1
sin x

14. Find

7 x  4 sec x
2 2

1
15. Evaluate
dx.
4x  3

BASIC MATHS / II PU / 2021-22 1

PART - B
II. Answer any TEN questions 10 × 2 = 10

 2 3
 2 x  2
4 1

16. Find x and y if

 .


7 5
y  1 5 

7 10 

17. Prove that “If in any determinant any two rows or columns are identical , then the value of
determinant is 0”.
18. In how many ways can 7 students and 4 teachers be seated in a row where no two teachers sit
together ?
3 1 A

19. If P(A) = , P(B) = , P(A ∪ B) = 1, find P
B  .
4 2

20. If the truth value of p → (q v r) is false, then find the truth value of p, q and r.
21. What must be added to each term of the ratio 4:5, so that it becomes 7:8 ?
22. If a : b = 2:3 and b : c = 6:13 , find a : b : c
23. True discount on a bill was Rs. 100 and Bankers Gain was Rs. 10. Find the face value of the bill.
24. How much income can be obtained from an investment of rupees 10,725 in 6.5% stock at 143?
Also find the amount of stock that can be purchased.
25. Reena paid Rs. 60 as sales tax on a watch worth Rs. 1200. Find the rate of sales tax.
1
26. Find the angle of elevation of the sun when the length of the shadow of a pole is times the
3
height of the pole.
3 −8 p p
27. If sin A = , cos B = , < A < p and < B < p then find the value of sin (A + B).
5 17 2 2
28. Find the equation of the parabola whose focus is (1, 0) and directrix is x = –1.
sin ax
bx
29. Evaluate lim
x  0 ax
sin bx
.
dy 1
30. If y  x
x
x
...... to
, prove that 
dx 2 y
1
31. If s = 5t 2 + 4t – 8 , find the initial velocity and acceleration.
32. Find the average cost and marginal cost for an output of 10 units if the total cost function of a
firm is given by C = 5x2 + 2x + 3 where x is output in units.
33. Evaluate ∫ log x dx
PART - C
III. Answer any TEN questions 10 × 3 = 30


2
1

3 2 
34. If A    ,B    then prove that(AB)ꞌ = BꞌAꞌ.

1 4 

1 4 
35. Solve by Cramer’s rule 2x + y = 1, x – 3y = 4.
36. A team of eight players has to be selected from 14 players . In how many ways can this be done if
(a) two particular players are always included
(ii) two particular players are always excluded ?

2 BASIC MATHS / II PU / 2021-22

37. One card is drawn at random from a pack of 52 cards. Find the probability that the card is
(i) a face card
(ii) either king or queen
(iii) not an ace card ?
38. Find the middle term in the expansion of

a  bx  .
12


x 
x 1
39. Resolve into partial fractions.

x
2

x
3

40. Write converse inverse and contrapositive of the proposition: “ If questions are easy then students
score better marks.”
41. Distribute 632 among A, B and C such that B gets 20% more than A, and C gets 20% less than B.
42. The Bankers Gain on a certain bill due after six months at 4% interest p.a. is Rs. 20. Find true
discount, Bill discount and face value of the bill.
43. Suraj helds Rs. 2100 of 3% stock. He sells at Rs. 121 and invests the proceeds in 5% stock.
Thereby his income increases by Rs. 14. Find the market price of 5% stock .
44. Prove that sin3A = 3sinA – 4sin3A.
45. Find the equation of a circle whose center is same as the circle x2 + y2 + 6x +2y + 1 = 0 and
passing through the point (–2,–1).

 e2 x
1
 , x0
46. Find k if the function f  x

 x is continuous at x = 0.
 K , x
0

dy
47. If xy = yx find, .
dx
dy
48. Find if x = e2t , y = log(2t + 1).
dx
49. Find the maximum and minimum value of f(x) = x3 – 3x.
4x  5
50. Evaluate  dx

x
1

x  2
.
1
51. Evaluate
x
2 log x  5

PART - D
IV. Answer any Six questions 6 × 5 = 30


1 2 3
4 2 3
4 1 2



52. If A =
1 4 1  , B =
2 4 1 , C =

0 3 1  verify (A + B)C = AC + BC.
 


0 5 3 

0 1 3 

1 3 4 
6
53. Find the term independent of x in the expansion of

3 x 2  1  .

2 3x 
2 x 2  16 x  29
54. Resolve into partial fractions.

x  3

x  4

2

BASIC MATHS / II PU / 2021-22 3

55. A box contains 5 red, 4 black and 3 white balls. How many selections of 8 balls can be made if
the selection contains
(a) Exactly 4 red, 2 black and 2 white balls
(b) Atleast 3 red, atleast 3 black and atleast 1 white ball.
56. Examine whether p ∨ (q ∧ r) and (p ∧ q) ∧ (q ∨ r) are logically equivalent.
57. A can do a piece of work in 20days, B in 30days and C in 60 days. All of them began to work
together. However, A left the job after 6 days and B quit work 6 days before the completion of
work. How many days did the work last.
58. xyz company supplies water tankers to the government the first water tanker takes 20000 labour
hours. The government auditors suggest that there should be a 90% learning effect rate. The
management expects an order of eight water tankers in the next year. What will be the labour cost
if the company will incur at the rate of Rs. 20 per hour?
59. Solve the following L.P.P graphically
Maximize Z = 6x + 8y;
Subject to the constraints 4x + 2y ≤ 20, 2x + 5y ≤ 24, x, y ≥ 0
60. A bill for ₹12900 was drawn on 3 Feb 2004 at 6 months and discounted on 13 March 2004 at 8%
p.a. For what sum was the bill discounted and how much did the banker gain in the transaction.
cos 7 x  cos 3 x
cos 5 x
cos x
61. Prove that
cot 2 x .
sin 7 x
sin 3 x
sin 5 x  sin x
d2y dy
62. If y = (x + a ) prove that
x  a

2 2 6
10 x
12 y  0 .
2 2

dx 2
dx
63. Find the area bounded between the curves y2 = x and x2 = y .
PART - E
V. Answer any ONE questions 10 × 1 = 10

xn
an
64. (a) Prove that lim
n a n
1 for all ‘n’. (6)
xa x
a
(b) Find the value of (0.99)4 using binomial theorem upto 4 decimal places. (4)
65. (a) Solve by matrix method :
x + y + z = 5, 2x + y – z = 2, 2x – y + z = 2 (6)
(b) From the top of a cliff, the angles of depression of two boats in the same vertical plane as
the observer are 30° and 45°. If the distance between the boats is 100 meters, find the height
of the cliff. (4)
66. (a) Show that (5, 5) (6, 4) (–1, –3) and (–2, –2) are concyclic. (6)
(b) A manufacturer produce 2 products P and Q. Each P requires 4 hrs on machine M1 and 2
hrs on machine M2. Each Q required 2 hrs on machine M1 and 5 hrs on machine M2. The
available total time on machine M1 is 20 hrs and on M2 is 24 hrs. Profit per unit of P is Rs6
and that of Q is Rs 8. Formulate the LPP maximizing his profit. (4)
* * *

4 BASIC MATHS / II PU / 2021-22

¢éwÃAiÀÄ ¦AiÀÄĹ ªÀiÁzÀj ¥Àæ±ÉߥÀwæPÉ
(FOR THE YEAR 2021-22)
¸ÀªÀÄAiÀÄ : 3:15 UÀAmÉ «µÀAiÀÄ: Basic Mathematics (PÉÆÃqï:75) CAPÀUÀ¼ÀÄ: 100

¸ÀÆZÀ£ÉUÀ¼ÀÄ:
(i) F ¥Àæ±ÉߥÀwæPÉAiÀÄ°è A, B, C, D ªÀÄvÀÄÛ E JA§ LzÀÄ ¨sÁUÀUÀ½ªÉ. J¯Áè ¨sÁUÀUÀ¼À£ÀÄß GvÀÛj¹.
(ii) ¨sÁUÀ-AUÉ 10 CAPÀUÀ¼ÀÄ, ¨sÁUÀ-BUÉ 20 CAPÀUÀ¼ÀÄ, ¨sÁUÀ-CUÉ 30 CAPÀUÀ¼ÀÄ, ¨sÁUÀ-DUÉ 30 CAPÀUÀ¼ÀÄ, ¨sÁUÀ-E UÉ 10 CAPÀUÀ½gÀÄvÀÛªÉ.
(iii) ¥Àæ±ÉßUÀ¼À ¸ÀASÉåUÀ¼À£ÀÄß ¥Àæ±ÉߥÀwæPÉAiÀÄ°è £ÀªÀÄÆ¢¹gÀĪÀAvÉ §gɬÄj.

¨sÁUÀ - A
I. F PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÁzÀgÀÆ ºÀvÀÄÛ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. 10 × 1 = 10


1
2 
1. A    DzÀgÉ ‘3A’ £À ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.

3 4 

400 404
2. £À ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.
408 412

3. ºÀvÀÄÛ ªÀÄA¢AiÀÄ£ÀÄß MAzÀÄ ªÀÈvÁÛPÁgÀzÀ ¸ÀÄvÀÛ®Æ PÀļÀîj¸À§ºÀÄzÁzÀ ¥À®èl£ÉUÀ¼ÉµÀÄÖ?
4. MAzÀÄ aîzÀ°è 8 PÉA¥ÀÄ ºÁUÀÆ 4 ºÀ¹gÀÄ UÉÆðUÀ¼ÀÄ EªÉ. EzÀgÀ°è£À MAzÀÄ UÉÆðAiÀÄ£ÀÄß AiÀiÁzÀÈaÒPÀªÁV
vÉUÉzÁUÀ CzÀÄ PÉA¥ÀÄ UÉÆðAiÀiÁUÀĪÀ ¸ÀA¨sÀªÀ¤ÃAiÀÄvÉ JµÀÄÖ?

5. “UÀtÂvÀ ¸ÀÄ®¨sÀ ªÀÄvÀÄÛ EAzÀÄ gÀeÉ ¢£À” JA§ GQÛAiÀÄ£ÀÄß aºÉßAiÀÄ°è §gɬÄj.

6. 3 : 5 AiÀÄ£ÀÄß ªÀÄĪÀÄär C£ÀÄ¥ÁvÀzÀ°è §gɬÄj.

7. 3375 gÀÆ.UÀ½AzÀ 75gÀÆ. UÀ½gÀĪÀ JµÀÄÖ ¸ÁÖPï (stock) C£ÀÄß Rjâ¸À§ºÀÄzÀÄ?

8. ®¤ðAUï PÀªïð (Learning cover) £À ¸À«ÄÃPÀgÀtªÀ£ÀÄß §gɬÄj.

9. tan A = 1 DzÀgÉ tan 2A £À ¨É¯É JµÀÄÖ?

10. y2 = 8Kx ¸À«ÄÃPÀgÀtªÀżÀî ¥ÀgÀªÀ®AiÀÄzÀ ®A§ £Á©üAiÀÄ GzÀݪÀÅ ‘4’ DzÀgÉ ‘K’ JµÀÄÖ?

4x
3
11. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj lim .
x4 x
2
dy
12. y = xe + ex DzÀgÉ, £À ¨É¯É K£ÀÄ?
dx

cos x dy
13. y  DzÀ°è C£ÀÄß PÀAqÀÄ»r¬Äj.
1
sin x dx

14. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj

7 x  4 sec x
2 2

1
15. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj
dx.
4x  3

BASIC MATHS / II PU / 2021-22 5

¨sÁUÀ - B
II. F PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÁzÀgÀÆ ºÀvÀÄÛ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. 10 × 2 = 20

 2 3
 2 x  2
4 1

16.
 
 DzÀgÉ ‘x’ ºÁUÀÆ ‘y’ £À ¨É¯É K£ÀÄ?

7 5
y  1 5 

7 10 
17. ‘MAzÀÄ ¤uÁðAiÀÄPÀ (determinant)£À°è AiÀiÁªÀÅzÁzÀgÀÄ JgÀqÀÄ ¸Á®ÄUÀ¼ÀÄ (rows) CxÀªÁ ¸ÀÛA¨sÀUÀ¼ÀÄ (columns)
MAzÉà vÀgÀºÀªÁVzÀÝgÉ D ¤uÁðAiÀÄPÀzÀ ¨É¯ÉAiÀÄÄ ¸ÉÆ£Éß’ JA§ÄzÀ£ÀÄß ¸Á¢ü¹.
18. 7 ²PÀëPÀgÀ£ÀÄß ºÁUÀÆ 4 «zÁåyðUÀ¼À£ÀÄß MAzÀÄ ¸Á°£À°è, AiÀiÁªÀ E§âgÀÄ ²PÀëPÀgÀÆ MnÖUÉ EgÀ¨ÁgÀzÀAvÉ JµÀÄÖ
«zsÀUÀ¼À°è PÀļÀîj¸À§ºÀÄzÀÄ?
3 1 A

19. P(A) = , P(B) = , P(A ∪ B) = 1DzÀgÉ P
 JµÀÄÖ?
4 2
B
20. p → (q v r) JA§ ¸ÀAAiÀÄÄPÉÆÛÃQÛAiÀÄ ¤d¨É¯É «ÄxÀåªÁzÀgÉ, p,q ªÀÄvÀÄÛ r £À ¤d¨É¯É K£ÀÄ?
21. 4 : 5 C£ÀÄ¥ÁvÀzÀ MAzÉÆAzÀÄ CAQUÀ½UÉ K£À£ÀÄß PÀÆr¹zÀgÉ ºÉƸÀ C£ÀÄ¥ÁvÀªÀÅ 7 : 8 DUÀÄvÀÛzÉ?
22. a : b = 2 : 3 ºÁUÀÆ b : c = 6 : 13 DzÀgÉ a : b : c JµÀÄÖ?
23. MAzÀÄ ºÀÄArAiÀÄ ¤d¸ÉÆÃr (TD) 100 gÀÆUÀ¼ÀÄ ºÁUÀÆ ¨ÁåAPÀj£À UÉÊ£ï (BG) 10 gÀÆUÀ¼ÁzÀgÉ CzÀgÀ ªÀÄÄR
¨É¯É JµÀÄÖ?
24. 10,725gÀÆ.UÀ¼À£ÀÄß 143 gÀAwgÀĪÀ ±Éà 6.5 ¸ÁÖQ£À°è ºÀÆrzÀgÉ CzÀjAzÀ ¹UÀĪÀ DzÁAiÀĪɵÀÄÖ ªÀÄvÀÄÛ CzÀgÀ
¨É¯É¬ÄAzÀ Rjâ¸À§ºÀÄzÁzÀ ¸ÁÖPï JµÀÄÖ JAzÀÄ PÀAqÀÄ»r¬Äj.
25. jãÁ¼ÀÄ 1200gÀÆ.UÀ½gÀĪÀ MAzÀÄ PÉÊUÀrAiÀiÁgÀPÉÌ 60gÀÆ.UÀ¼ÀÄ vÉjUÉ (sales tax) ¨sÀj¹zÀgÉ, CzÀgÀ ±ÉÃPÀqÀªÁgÀÄ vÉjUÉ
(rate of sales tax) PÀAqÀÄ»r¬Äj.
1
26. MAzÀÄ GzÀÞ£ÉAiÀÄ PÀA§zÀ £ÉgÀ½£À C¼ÀvÉAiÀÄÄ CzÀgÀ JvÀÛgÀzÀ gÀµÀÄÖ DVzÀÝ°è, ¸ÀÆAiÀÄð£À PÁgÀt¢AzÀ
3
¸ÀA¨sÀ«¸À®àqÀĪÀ PÉÆãÀ (angle of elevation) JµÀÄÖ?
3 −8 p p
27. sin A = , cos B = , < A < p ªÀÄvÀÄÛ < B < p EzÀÝgÉ, sin (A + B) £À ¨É¯É K£ÀÄ?
5 17 2 2

28. MAzÀÄ ¥ÀgÀªÀ®AiÀÄzÀ £Á©üAiÀÄÄ (1, 0) ªÀÄvÀÄÛ CzÀgÀ ¤¢ðµÀÖ gÉÃSÉAiÀÄÄ x = –1 DzÀgÉ D ¥ÀgÀªÀ®AiÀÄzÀ
¸À«ÄÃPÀgÀtªÉãÀÄ?
sin ax
bx
29. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj lim
x  0 ax
sin bx
:

dy 1
30. y  x
x
x
...... to
DzÀgÉ,  JAzÀÄ ¸Á¢ü¹.
dx 2 y
1

31. s = 5t 2 + 4t – 8 (s zÀÆgÀ, t ¸ÀªÀÄAiÀÄ) DzÀgÉ ¥ÁægÀA¨sÀzÀ°è£À ªÉÃUÀ (initial velocity) ºÁUÀÆ ªÉÃUÉÆÃvÀ̵ÀðªÀ£ÀÄß
PÀAqÀÄ»r¬Äj.
32. MAzÀÄ ¸ÀA¸ÉÜAiÀÄ MlÄÖ ªÉZÀѪÀÅ C = 5x2 + 2x + 3 gÀAvÉ EzÁÝUÀ (x GvÁàzÀ£ÉAiÀÄ WÀlPÀUÀ¼ÀÄ) CzÀgÀ ¸ÀgÁ¸Àj ªÉZÀѪÀÅ
ªÀÄvÀÄÛ ¹Ã«ÄvÀ ªÉZÀѪÀÅ x = 10 DzÀ°è K£ÁUÀÄvÀÛzÉ?

33. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj: ∫ log x dx

¨sÁUÀ - C
III. F PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÁzÀgÀÆ ºÀvÀÄÛ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. 10 × 3 = 30


2
1

3 2 
34. A    ,B    DzÀgÉ (AB)ꞌ = BꞌAꞌ JAzÀÄ ¸Á¢ü¹.

1 4 

1 4 
35. PÉæêÀÄgÀ£À ¤AiÀĪÀÄ¢AzÀ 2x + y = 1, x – 3y = 4 ¸À«ÄÃPÀgÀtUÀ¼À£ÀÄß ©r¹j.

6 BASIC MATHS / II PU / 2021-22

36. 14 DlUÁgÀjAzÀ 8 DlUÁgÀgÀ vÀAqÀªÀ£ÀÄß F PɼÀV£ÀAvÉ JµÀÄÖ jÃwAiÀÄ°è DAiÉÄÌ ªÀiÁqÀ§ºÀÄzÀÄ?
(i) JgÀqÀÄ ¤¢ðµÀÖ DlUÁgÀgÀÄ EgÀ¨ÉÃPÀÄ.
(ii) JgÀqÀÄ ¤¢ðµÀÖ DlUÁgÀgÀÄ EgÀ¨ÁgÀzÀÄ.
37. 52 PÁqïðUÀ¼À ¥ÁåPï¤AzÀ MAzÀÄ PÁqïð C£ÀÄß AiÀiÁzÀÈaÒPÀªÁV J¼ÉAiÀįÁUÀÄvÀÛzÉ.
(i) ¥sÉøïPÁqïð
(ii) gÁd CxÀªÁ gÁtÂ
(iii) J¸ï PÁqïð C®èzÀ
¸ÀA¨sÀªÀ¤ÃAiÀÄvÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.

38.

a  bx 
12
F «¸ÁÛgÀzÀ°è£À ªÀÄzsÀåªÀÄ ¥ÀzÀªÀ£ÀÄß PÀAqÀÄ»r¬Äj.

x 
x 1
39. EzÀ£ÀÄß «¨sÀfvÀ ©ü£ÀßgÁ²UÀ¼ÁV ¥ÀjªÀwð¹j.

x
2

x
3

40. “¥Àæ±ÉßUÀ¼ÀÄ ¸ÀÄ®¨sÀªÁVzÀÝgÉ «zÁåyðUÀ¼ÀÄ GvÀÛªÀÄ CAPÀUÀ¼À£ÀÄß ¥ÀqÉAiÀÄÄvÁÛgÉ”. F ¸ÀAAiÀÄÄPÉÆÛÃQÛAiÀÄ ¥Àæw¯ÉÆêÀÄ
(converse), «¯ÉÆêÀÄ (Inverse) ªÀÄvÀÄÛ ¥ÀæwzsÀ£À (contrapositive) §gɬÄj.
41. 632gÀÆUÀ¼À£ÀÄß A, B ªÀÄvÀÄÛ C UÉ, ºÁUÀÆ B UÉ A VAvÀ ±Éà 20 ºÉaÑgÀĪÀAvÉ ªÀÄvÀÄÛ C UÉ B VAvÀ ±ÉÃ.20 PÀrªÉÄ
¹UÀĪÀAvÉ «vÀj¹.
42. ¨ÁåAPÀgïUÀ¼ÀÄ DgÀÄ wAUÀ¼À £ÀAvÀgÀ MAzÀÄ ¤¢ðµÀÖ ©¯ï£À°è ªÁ¶ðPÀ 4 ¥Àæw±ÀvÀ §rØUÉ gÀÆ.20 UÀ½¸ÀÄvÁÛgÉ.
¤dªÁzÀ jAiÀiÁ¬Äw, ¨ÁåAPÀgïUÀ¼À jAiÀiÁ¬Äw ªÀÄvÀÄÛ ©¯ï£À ªÀÄÄR ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.
43. ¸ÀÆgÀeï ±ÉÃ.3 ¸ÁÖPï£À°è 2100gÀÆ.UÀ¼À£ÀÄß ºÉÆA¢zÁÝgÉ. CªÀgÀÄ gÀÆ 121PÉÌ ªÀiÁgÁl ªÀiÁqÀÄvÁÛgÉ ªÀÄvÀÄÛ DzsÁAiÀĪÀ£ÀÄß
5 ¥Àæw±ÀvÀ ¸ÁÖPï£À°è ºÀÆrPÉ ªÀiÁqÀÄvÁÛgÉ. vÀ£ÀÆä®PÀ CªÀ£À DzÁAiÀĪÀÅ 14gÀÆ.UÀ¼ÀµÀÄÖ ºÉZÁÑUÀÄvÀÛzÉ. 5 ¥Àæw±ÀvÀ
µÉÃgÀÄUÀ¼À ªÀiÁgÀÄPÀmÉÖ ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.
44. sin3A = 3sinA – 4sin3A JAzÀÄ ¸Á¢ü¹.
45. ªÀÈvÀÛzÀ ¸À«ÄÃPÀgÀtªÀ£ÀÄß PÀAqÀÄ»r¬Äj. CzÀgÀ PÉÃAzÀæªÀÅ x2 + y2 + 6x +2y + 1 = 0 ªÀÈvÀÛzÀ PÉÃAzÀæzÀAvÉAiÉÄà EgÀÄvÀÛzÉ
ªÀÄvÀÄÛ (–2,–1) ©AzÀÄ«£À ªÀÄÆ®PÀ ºÁzÀÄ ºÉÆÃUÀÄvÀÛzÉ.
 e2 x
1
 , x0
46. f  x

 x JA§ÄzÀÄ x = 0 gÀ°è D«aÒ£ÀߪÁVzÉ. `K’ ¨É¯É PÀAqÀÄ»r¬Äj.
 K , x
0

dy
47. xy = yx DzÀgÉ, AiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.
dx
dy
48. x = e2t , y = log(2t + 1) DzÀgÉ AiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj.
dx
49. f(x) = x3 – 3x EzÀgÀ UÀjµÀ× ªÀÄvÀÄÛ PÀ¤µÀ× ªÀiË®åªÀ£ÀÄß PÀAqÀÄ»r¬Äj.
4x  5
50. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj  dx

x
1

x  2
.
1
51. ¨É¯ÉAiÀÄ£ÀÄß PÀAqÀÄ»r¬Äj
x
2 log x  5

¨sÁUÀ - D
IV. F PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÁzÀgÀÆ DgÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹. 6 × 5 = 30

1 2 3
4 2 3
4 1 2



52. A =
1 4 1  , B =
2 4 1 , C =

0 3 1  DzÀgÉ, (A + B)C = AC + BC JAzÀÄ ¸Á¢ü¹.



0 5 3 

0 1 3 

1 3 4 
6
53.

3 x 2  1  F «¸ÀÛgÀuÉAiÀÄ°è, x ¤AzÀ ¸ÀévÀAvÀæ ¥ÀzÀªÀ£ÀÄß PÀAqÀÄ»r¬Äj.

2 3x 

BASIC MATHS / II PU / 2021-22 7

2 x 2  16 x  29
54. EzÀ£ÀÄß «¨sÀfvÀ ©ü£ÀßgÁ²UÀ¼ÁV ¥ÀjªÀwð¹.

x  3

x  4

2

55. MAzÀÄ ¥ÉnÖUÉAiÀÄÄ 5 PÉA¥ÀÄ, 4 PÀ¥ÀÄà ªÀÄvÀÄÛ 3 ©½ ZÉAqÀÄUÀ¼À£ÀÄß ºÉÆA¢gÀÄvÀÛzÉ. 8 ZÉAqÀÄUÀ¼À JµÀÄÖ DAiÉÄÌUÀ¼À£ÀÄß
ªÀiÁqÀ§ºÀÄzÀÄ. DAiÉÄÌAiÀÄÄ (i) ¤RgÀªÁV 4 PÉA¥ÀÄ, 2 PÀ¥ÀÄà ªÀÄvÀÄÛ 2 ©½ (ii) PÀ¤µÀ× 3 PÉA¥ÀÄ, PÀ¤µÀ× 3 PÀ¥ÀÄà
ªÀÄvÀÄÛ PÀ¤µÀ× 1 ©½ DzÀgÉ.
56. p ∨ (q ∧ r) ªÀÄvÀÄÛ (p ∧ q) ∧ (q ∨ r) vÁQðPÀªÁV ¸ÀªÀiÁ£ÀªÁVzÉAiÉÄà JAzÀÄ ¥ÀjÃQë¹.
57. A MAzÀÄ PÉ®¸ÀªÀ£ÀÄß 20 ¢£ÀUÀ¼À°è, B 30 ¢£ÀUÀ¼À°è ªÀÄvÀÄÛ C 60 ¢£ÀUÀ¼À°è ªÀiÁqÀ§ºÀÄzÀÄ. J®ègÀÆ MmÁÖV PÉ®¸À
ªÀiÁqÀ®Ä ¥ÁægÀA©ü¹zÀgÀÄ. DzÁUÀÆå A 6 ¢£ÀUÀ¼À £ÀAvÀgÀ PÉ®¸ÀªÀ£ÀÄß vÉÆgÉzÀgÀÄ ªÀÄvÀÄÛ B PÉ®¸ÀªÀÅ ¥ÀÆtðUÉƼÀÄîªÀ
6 ¢£ÀUÀ¼À ªÉÆzÀ®Ä PÉ®¸ÀªÀ£ÀÄß vÉÆgÉzÀgÀÄ PÁªÀÄUÁj JµÀÄÖ ¢£À £ÀqɬÄvÀÄ.
58. xyz PÀA¥À¤AiÀÄÄ ¤Ãj£À mÁåAPÀgïUÀ¼À£ÀÄß ¸ÀPÁðgÀPÉÌ MzÀV¸ÀÄvÀÛzÉ. ªÉÆzÀ® mÁåAPÀjUÉ 20,000 PÀÆ° UÀAmÉUÀ¼ÀÄ
¨ÉÃPÁUÀÄvÀÛzÉ. ¸ÀPÁðj ¯ÉPÀÌ ¥Àj±ÉÆÃzsÀPÀgÀÄ 90% PÀ°AiÀÄÄ«PÉ EgÀ¨ÉÃPÉAzÀÄ ¸ÀÆa¹zÁÝgÉ. PÀA¥À¤UÉ ªÀåªÀ¸ÁÜ¥ÀPÀgÀÄ
§gÀĪÀ ªÀµÀð 8 ¤Ãj£À mÁåAPÀgïUÀ¼À ¨ÉÃrPÉAiÀÄ ¤jÃPÉëAiÀÄ°èzÁÝgÉ. UÀAmÉUÉ 20gÀÆ. UÀ¼ÀAvÉ 8 mÁåAPÀgï ¤ÃgÀ£ÀÄß
MzÀV¸À®Ä ¨ÉÃPÁUÀĪÀ MlÄÖ PÀÆ° ªÉZÀѪÀ£ÀÄß PÀAqÀÄ»r¬Äj.
59. gÉÃSÁ£ÀPÉëAiÀÄ£ÀÄß G¥ÀAiÉÆÃV¹PÉÆAqÀÄ F PɼÀPÀAqÀ ¸ÀgÀ¼ÀgÉÃSÁvÀäPÀ PÁAiÀÄðPÀæªÀÄ ¸ÀªÀĸÉåAiÀÄ£ÀÄß ©r¹:
¥ÀgÀªÀiÁªÀ¢üÃPÀj¹: Z = 6x + 8y; 4x + 2y ≤ 20, 2x + 5y ≤ 24, x, y ≥ 0 ¤§AzsÀ£ÉUÉƼÀ¥ÀlÖAvÉ.
60. 3 ¥sɧæªÀj 2004 gÀAzÀÄ 6 wAUÀ¼ÀÄUÀ¼À°è 12900 gÀÆ.UÀ¼À ©¯ï C£ÀÄß qÁæªÀiÁqÀ¯ÁVzÉ ªÀÄvÀÄÛ 13 ªÀiÁZïð
2004gÀAzÀÄ ªÀµÀðPÉÌ 8 ¥Àæw±ÀvÀzÀµÀÄÖ jAiÀiÁ¬Äw ¤ÃqÀ¯Á¬ÄvÀÄ. AiÀiÁªÀ ªÉÆvÀÛPÉÌ ©¯ï£À°è jAiÀiÁ¬Äw ¤ÃqÀ¯ÁVzÉ
ªÀÄvÀÄÛ ªÀåªÀºÁgÀzÀ°è ¨ÁåAPÀgï£À ¯Á¨sÀ JµÀÄÖ.
cos 7 x  cos 3 x
cos 5 x
cos x
61.
cot 2 x JAzÀÄ ¸Á¢ü¹.
sin 7 x
sin 3 x
sin 5 x  sin x
d2y dy
62. y = (x + a ) DzÀgÉ
x  a

10 x
12 y  0 JAzÀÄ ¸Á¢ü¹.
2 2 6 2 2

dx 2
dx
63. ¥ÀgÀªÀ®AiÀÄUÀ¼ÀÄ y = x ªÀÄvÀÄÛ x = y £ÀqÀÄ«£À PÉëÃvÀæ ¥sÀ®ªÀ£ÀÄß PÀAqÀÄ»r¬Äj.
2 2

¨sÁUÀ - E
V. F PɼÀV£ÀªÀÅUÀ¼À°è AiÀiÁªÀÅzÁzÀgÀÆ MAzÀÄ ¥Àæ±ÉßUÉ GvÀÛj¹. 1 × 10= 10

xn
an
64. (J) J¯Áè ¥ÀÄuÁðAPÀ ‘n’ UÀ½UÉ lim
n a n
1 JAzÀÄ ¸Á¢ü¹. (6)
xa x
a
(©) (0.99)4 ¨É¯ÉAiÀÄ£ÀÄß ¢é¥ÀzÀ ¥ÀæªÉÄÃAiÀĪÀ£ÀÄß G¥ÀAiÉÆÃV¹ £Á®ÄÌ zÀ±ÁA±ÀPÉÌ PÀAqÀÄ»r¬Äj. (4)
65. (J) ªÀiÁvÀÈPÉ «zsÁ£À G¥ÀAiÉÆÃV¹ ©r¹:
x + y + z = 5, 2x + y – z = 2, 2x – y + z = 2 (6)

(©) §AqÉAiÀÄ ªÉÄïÁãUÀ¢AzÀ, «ÃPÀëPÀ£ÀAvÉAiÉÄà ®A§ ¸ÀªÀÄvÀ®zÀ°è JgÀqÀÄ zÉÆÃtÂUÀ¼À T£ÀßvÉAiÀÄ PÉÆãÀUÀ¼ÀÄ® 30°
ªÀÄvÀÄÛ 45°. zÉÆÃtÂUÀ¼À £ÀqÀÄ«£À CAvÀgÀªÀÅ 100m DVzÀÝgÉ, §AqÉAiÀÄ JvÀÛgÀªÀ£ÀÄß PÀAqÀÄ»r¬Äj. (4)
66. (J) ©AzÀÄUÀ¼ÀÄ (5, 5) (6, 4) (–1, –3) ªÀÄvÀÄÛ (–2, –2) MAzÉà ªÀÈvÀÛzÀ ¥ÀgÀ¢üAiÀÄ ªÉÄðzÉ JAzÀÄ zÀÈrüÃPÀj¹j. (6)
(©) vÀAiÀiÁgÀPÀgÀÄ P ªÀÄvÀÄÛ Q JgÀqÀÄ GvÀà£ÀßUÀ¼À£ÀÄß GvÁࢸÀÄvÁÛgÉ. ¥Àæw P, AiÀÄAvÀæ M1 £À°è 4 UÀAmÉUÀ¼ÀÄ ªÀÄvÀÄÛ
AiÀÄAvÀæ M2 £À°è 2 UÀAmÉUÀ¼À CUÀvÀå«zÉ. ¥Àæw Q, AiÀÄAvÀæ M1 £À°è 2 UÀAmÉUÀ¼ÀÄ ªÀÄvÀÄÛ AiÀÄAvÀæ M2 £À°è 5
UÀAmÉUÀ¼À CUÀvÀå«zÉ. M1 AiÀÄAvÀæzÀ°è ®¨sÀå«gÀĪÀ MlÄÖ ¸ÀªÀÄAiÀÄ 20 UÀAmÉUÀ¼ÀÄ ªÀÄvÀÄÛ M2 £À°è 25 UÀAmÉUÀ¼ÀÄ.
P AiÀÄ ¥Àæw AiÀÄƤmïUÉ ¯Á¨sÀªÀÅ 6 gÀÆ ªÀÄvÀÄÛ QAiÀÄ ¥Àæw AiÀÄƤmïUÉ ¯Á¨sÀªÀÅ 8 gÀÆ. CªÀ£À ¯Á¨sÀªÀ£ÀÄß
¥ÀgÀªÀiÁªÀ¢üÃPÀj¸À®Ä, ¸ÀgÀ¼ÀgÉÃSÁvÀäPÀ PÁAiÀÄðPÀæªÀÄ ¸ÀªÀĸÉåAiÀÄ£ÀÄß (LPP) gÀƦ¹. (4)
* * *

8 BASIC MATHS / II PU / 2021-22