HBSE Class 12 Question Paper 2024 Mathematics

Haryana Board

QUESTIONPDF

PAPERS
2024

HBSE PYQP

CLASS : 12th (Sr. Secondary) Code No. 1232
Series : SS/Annual Exam.-2024
Roll No. SET : A

xf.kr GRAPH
MATHEMATICS
[ Hindi and English Medium ]
ACADEMIC/OPEN
(Only for Fresh/Re-appear/Improvement/Additional Candidates)

Time allowed : 3 hours ] [ Maximum Marks : 80

• Ñi;k tk¡p dj ysa fd bl iz'u&i= esa eqfnzr i`"B 24 rFkk iz'u 38 gSaA
Please make sure that the printed pages in this question paper are 24 in number
and it contains 38 questions.

• iz'u&i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj rFkk lsV dks Nk= mÙkj&iqfLrdk ds eq[;&i`"B ij
fy[ksaA
The Code No. and Set on the right side of the question paper should be written by
the candidate on the front page of the answer-book.
• Ñi;k iz'u dk mÙkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad vo'; fy[ksaA
Before beginning to answer a question, its Serial Number must be written.
• mÙkj&iqfLrdk ds chp esa [kkyh iUuk@iUus u NksMas+A
Don’t leave blank page/pages in your answer-book.
• mÙkj&iqfLrdk ds vfrfjDr dksbZ vU; 'khV ugha feysxhA vr% vko';drkuqlkj gh fy[ksa vkSj fy[kk mÙkj u
dkVsaA
Except answer-book, no extra sheet will be given. Write to the point and do not
strike the written answer.

1232/(Set : A) P. T. O.

(2) 1232/(Set : A)
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA jksy ua0 ds vfrfjDr iz'u&i= ij vU; dqN Hkh u
fy[ksa vkSj oSdfYid iz'uksa ds mÙkjksa ij fdlh izdkj dk fu'kku u yxk,¡A
Candidates must write their Roll No. on the question paper. Except Roll No. do not
write anything on question paper and don't make any mark on answers of objective
type questions.
• d`i;k iz'uksa ds mÙkj nsus lss iwoZ ;g lqfuf'pr dj ysa fd iz'u&i= iw.kZ o lgh gS] ijh{kk ds mijkUr bl
lEcU/k esa dksbZ Hkh nkok Lohdkj ugha fd;k tk;sxkA
Before answering the questions, ensure that you have been supplied the correct and
complete question paper, no claim in this regard, will be entertained after
examination.
lkekU; funsZ'k %
(i) lHkh iz'u vfuok;Z gSaA
(ii) bl ç'u-i= esa dqy 38 ç'u gSa] tksfd ik¡p [k.Mksa % ^v*]
^v* ^c*] ^l*] ^n* ,oa ^;* esa ck¡Vs x, gSa %
[k.M ^v* % bl [k.M esa ç'u la[;k 1 ls 20 rd dqy chl ç'u gSaA çR;sd ç'u 1 vad dk gSA

[k.M ^c
^c* % bl [k.M esa ç'u la[;k 21 ls 25 rd dqy ik¡p ç'u gSaA çR;sd ç'u 2 vadksa
dk gSA
[k.M ^l
^l* % bl [k.M esa ç'u la[;k 26 ls 31 rd dqy N% ç'u gSaA çR;sd ç'u 3 vadksa
dk gSA
[k.M ^n
^n* % bl [k.M esa ç'u la[;k 32 ls 35 rd dqy pkj ç'u gSaA çR;sd ç'u 5 vadksa dk
gSA
[k.M ^;
^;* % bl [k.M esa ç'u la[;k 36 ls 38 rd dqy rhu ç'u gSaA çR;sd ç'u 4 vadksa dk
gSA
(iii) bl ç'u&iz'u ds dqN ç'uksa esa vkarfjd fodYi fn;s x;s gSaA vkidks çR;sd esa ls ,d fodYi djuk
gSA
(iv) fn;s x;s xzkQ isij dks viuh mÙkj&iqfLrdk ds lkFk vo'; uRFkh dhft,A
(v) xzkQ isij ij viuh mÙkj&iqfLrdk dk Øekad vo'; fyf[k,A
General Instructions :
(i) All questions are compulsory.

1232/(Set : A)

(3) 1232/(Set : A)
(ii) This question paper consists of 38 questions, which are divided into five
Sections : 'A', 'B', 'C', 'D' and 'E':
Section 'A' : It contains twenty questions from 1 to 20. Each question
carries 1 mark.
Section 'B' : It contains five questions from 21 to 25. Each question carries
2 marks.
Section 'C' : It contains six questions from 26 to 31. Each question carries
3 marks.
Section 'D' : It contains four questions from 32 to 35. Each question carries
5 marks.
Section 'E' : It contains three questions from 36 to 38. Each question
carries 4 marks.
(iii) Internal choices are given in some questions of this question-paper. You have
to attempt one from each.
(iv) You must attach the given graph-paper along with your answer-book.
(v) You must write your answer-book Serial No. on the graph-paper.

[k.M – v
SECTION – A

1. eku yhft, fd N çkÑr la[;kvksa dk leqPp; gS rFkk f : N → N, f(n) = 2n + 3 ∀ n ∈ N }kjk
ifjHkkf"kr ,d Qyu gS] rks f gS % 1
(A) vkPNknh (B) ,dSd
(C) ,dSdh vkPNknh (D) buesa ls dksbZ ugha

Let N be the set of natural numbers and the function f : N → N be defined by
f(n) = 2n + 3 ∀ n ∈ N, then f is :
(A) Surjective (B) Injective
(C) Bijective (D) None of these

1232/(Set : A) P. T. O.

(4) 1232/(Set : A)
2. sin −1
2x dk çkar gS % 1

(A) [0, 1] (B) [−1, 1]
 1 1
(C) − 2 , 2  (D) [−2, 2]
 

The domain of sin −1 2x is :
(A) [0, 1] (B) [−1, 1]
 1 1
(C) − 2 , 2  (D) [−2, 2]
 

3. dksfV 3 × 3 ds lHkh lEHko vkO;wgksa dh la[;k] ftudh çR;sd çfof"V 2 ;k 0 gks] gksxh % 1

(A) 9 (B) 27

(C) 81 (D) 512

Total numbers of possible matrices of order 3 × 3 with each entry 2 or 0, are :

(A) 9 (B) 27

(C) 81 (D) 512

x 2 6 2
4. ;fn = gks] rks x cjkcj gS % 1
18 x 18 6
(A) 6 (B) ±6
(C) −6 (D) 0
x 2 6 2
If = , then x is equal to :
18 x 18 6

(A) 6 (B) ±6
(C) −6 (D) 0

1232/(Set : A)

(5) 1232/(Set : A)
5. ;fn A, 3 × 3 dksfV dk O;qRØe.kh; oxZ vkO;wg gS] rks |adj. A| dk eku gS --------------A 1

Let A be a non-singular square matrix of order 3 × 3. Then |adj. A| is equal
to …………… .

6. x ds lkis{k sin(log x) dk vodyt gS % 1

cos(log x )
(A)
x2

(B) cos (log x)

cos(log x )
(C)
x

(D) buesa ls dksbZ ugha

The derivative of sin(log x) w. r. t. x is :

cos(log x )
(A)
x2

(B) cos (log x)

cos(log x )
(C)
x

(D) None of these

7. cos x ds lkis{k sin x dk vodyt ------------ gSA 1

The derivative of sin x w. r. t. cos x is …………… .

1232/(Set : A) P. T. O.

(6) 1232/(Set : A)
∫ e sec x(1 + tan x )dx cjkcj gS %
x
8. 1

(A) e x cos x + c

(B) e x sec x + c

(C) e x sin x + c

(D) e x tan x + c

∫ e sec x(1 + tan x )dx is equal to :
x

(A) e x cos x + c

(B) e x sec x + c

(C) e x sin x + c

(D) e x tan x + c

π /2

∫ sin x dx dk eku cjkcj gS --------------A
5
9. 1
− π /2

π /2

∫ sin x dx is equal to ……………… .
5
The value of
− π /2

1232/(Set : A)

(7) 1232/(Set : A)
10. o`Ùk x + y
2 2
= 2 }kjk ifjc) {ks= dk {ks=Qy cjkcj gS % 1

(A) 4π oxZ bdkbZ

(B) 2 2π oxZ bdkbZ

(C) 4π2 oxZ bdkbZ

(D) 2π oxZ bdkbZ

The area enclosed by circle x 2 + y 2 = 2 is equal to :

(A) 4π sq. units

(B) 2 2π sq. units

(C) 4π2 sq. units

(D) 2π sq. units

2
  dy  2  d 2y
11. vody lehdj.k 1 +    = ds Øe'k% dksfV vkSj ?kkr gSa % 1
  dx   dx 2

(A) 1, 2 (B) 2, 2

(C) 2, 1 (D) 4, 2
2
  dy  2  d 2y
The order and degree of the differential equation 1 +    = respectively
  dx   dx 2
are :

(A) 1, 2 (B) 2, 2

(C) 2, 1 (D) 4, 2

1232/(Set : A) P. T. O.

(8) 1232/(Set : A)
12. vody lehdj.k x dy − y = sin x dk lekdyu xq.kd (I.F.) …………. gSA 1
dx
dy
Integrating factor of the differential equation x − y = sin x is ………….. .
dx

→ → 2 → →
13. eku yhft, lfn'k a→ vkSj b bl çdkj gSa fd | a→ | = 3 vkSj | b | = , rc a × b ,d ek=d
3
→
lfn'k gS] ;fn a→ vkSj b ds chp dk dks.k gS % 1

π π
(A) (B)
6 4

π π
(C) (D)
3 2

→ → → → 2 → →
Let the vectors a and b such that | a | = 3 and | b | = , then a × b is a
3
→ →
unit vector, if angle between a and b is :
π π
(A) (B)
6 4
π π
(C) (D)
3 2

14. ;fn lfn'k a→ = 2iˆ + λˆj + kˆ rFkk →
b = iˆ + 2 ˆj + 3kˆ ykafcd (orthogonal) gks] rks λ dk eku
gS ---------------A 1
→ →
If the vectors a = 2iˆ + λˆj + kˆ and b = iˆ + 2 ˆj + 3kˆ are orthogonal, then the value of
λ is ………….. .

1232/(Set : A)

(9) 1232/(Set : A)
15. vkdk'k ¼Lisl½ esa x-v{k dk lehdj.k gSa % 1

(A) x = 0, y = 0

(B) x = 0, z = 0

(C) x=0

(D) y = 0, z = 0

The equations of x-axis in space are :

(A) x = 0, y = 0

(B) x = 0, z = 0

(C) x=0

(D) y = 0, z = 0

16. ;fn iklksa dk ,d tksM+k mNkyk tkrk gS] rks çR;sd ikls ij le vHkkT; la[;k çkIr djus dh çkf;drk
-------------- gSA 1
The probability of obtaining an even prime number on each die, when a pair of
dice is rolled is …………. .

17. ,d FkSys esa 5 yky rFkk 3 uhyh xsansa gSaA ;fn 3 xsansa ;kn`PN;k fcuk çfrLFkkiu ds fudkyh tkrh gaS] rks
rF;r% ,d yky jax dh xsna ds fudkyus dh çkf;drk ----------- gSA 1

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without
replacement, the probability of getting exactly one red ball is ……….. .
1232/(Set : A) P. T. O.

( 10 ) 1232/(Set : A)
18. rk'k ds 52 iÙkksa dh Hkyh&Hkk¡fr QsaVh gqbZ fdlh xM~Mh ls nks iÙks çfrLFkkiu lfgr fudkys tkrs gSaA nksuksa
iÙkksa ds ^jkuh* gksus dh çkf;drk gS ------------A 1

Two cards are drawn from well shuffled deck of 52 playing cards with
replacement. The probability, that both cards are queens is …………… .

vfHkdFku ,oa dkj.k vk/kkfjr ç'u %
fuEufyf[kr ç'uksa (19 o 20) esa nks dFku gSa % vfHkdFku (A) vkSj dkj.k (R), ç'u ds uhps fn, x,
mi;qDr fodYi dk p;u djrs gq, mÙkj nsa %
Assertion-Reason Based Questions :

In the following questions (19 & 20) there are two statements : Assertion (A) and
Reason (R), answer the question by choosing the appropriate option given below :

19. vfHkdFku (A) : leqPp; A = {1, 2, 3, 4, 5} ij ifjHkkf"kr laca/k R = {(a, b); |a − b| < 2}
LorqY; (reflexive) gSA 1

dkj.k (R) : fdlh leqPp; A ij ifjHkkf"kr laca/k R LorqY; (reflexive) dgykrk gS] ;fn (a, b) ∈ R
rFkk (b, c) ∈ R ls (a, c) ∈ R çkIr gksA

(A) vfHkdFku (A) ,oa dkj.k (R) nksuksa lgh gSa rFkk dkj.k (R), vfHkdFku (A) dh lgh O;k[;k gSA

(B) vfHkdFku (A) ,oa dkj.k (R) nksuksa lgh gSa rFkk dkj.k (R), vfHkdFku (A) dh lgh O;k[;k ugha gSA

(C) vfHkdFku (A) lgh gS] fdUrq dkj.k (R) xyr gSA

(D) vfHkdFku (A) xyr gS] fdUrq dkj.k (R) lgh gSA

1232/(Set : A)

( 11 ) 1232/(Set : A)
Assertion (A) : A relation R = {(a, b); |a − b| < 2} defined on the set A = {1, 2, 3, 4, 5}
is reflexive.

Reason (R) : A relation R on the set A is said to be reflexive if for (a, b) ∈ R and
(b, c) ∈ R, we have (a, c) ∈ R.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct
explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true and Reason (R) is not the
correct explanation of Assertion (A).

(C) Assertion (A) is true, but Reason (R) is false.

(D) Assertion (A) is false, but Reason (R) is true.

20. vfHkdFku (A) : ljy js[kkvksa x + 1 = y − 2 = z + 3 rFkk x − 1 = y + 2 = z − 3 ds chp dk dks.k
2 5 4 1 2 −3
90° gSA 1

dkj.k (R) : fo"keryh; js[kk,¡ fofHkUu ryksa esa fLFkr js[kk,¡ gksrh gSa tks lekUrj vkSj çfrPNsnh gksrh gSaA
(A) vfHkdFku (A) ,oa dkj.k (R) nksuksa lgh gSa rFkk dkj.k (R), vfHkdFku (A) dh lgh O;k[;k gSA

(B) vfHkdFku (A) ,oa dkj.k (R) nksuksa lgh gSa rFkk dkj.k (R), vfHkdFku (A) dh lgh O;k[;k ugha gSA

(C) vfHkdFku (A) lgh gS] fdUrq dkj.k (R) xyr gSA

(D) vfHkdFku (A) xyr gS] fdUrq dkj.k (R) lgh gSA

1232/(Set : A) P. T. O.

( 12 ) 1232/(Set : A)
x +1 y − 2 z + 3
Assertion (A) : The angle between the straight lines = = and
2 5 4
x −1 y + 2 z − 3
= = is 90°.
1 2 −3

Reason (R) : Skew lines are lines in different planes which are parallel and
intersecting.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct
explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true and Reason (R) is not the
correct explanation of Assertion (A).

(C) Assertion (A) is true, but Reason (R) is false.

(D) Assertion (A) is false, but Reason (R) is true.

[k.M – c

SECTION – B

21. eku yhft, fd f : R → R, f(x) = sin x rFkk g : R → R, g(x) = x 2 }kjk ifjHkkf"kr gS] rks fog
rFkk gof Kkr dhft,A fn[kkb, fd % 2

fog ≠ gof

Let f : R → R be defined by f(x) = sin x and g : R → R be defined by g(x) = x 2 ,
then find fog and gof. Show that :

fog ≠ gof

1232/(Set : A)

( 13 ) 1232/(Set : A)
vFkok

OR

eku Kkr dhft, %

1 1
cos −1 + 2 sin −1
2 2

Find the value of :

1 1
cos −1 + 2 sin −1
2 2

;fn A = 
cos θ sin θ 
22. gks] rks lR;kfir dhft, fd % 2
− sin θ cos θ

A'A = I

 cos θ sin θ 
If A =  , then verify that :
− sin θ cos θ

A'A = I

23. x ds lkis{k vodyu dhft, % 2

 1 − cos x  π π
tan −1  ,
 <x<
 1 + cos x  4 4

1232/(Set : A) P. T. O.

( 14 ) 1232/(Set : A)
Differentiate w. r. t. x :

 1 − cos x  π π
tan −1  ,
 < x <
 1 + cos x  4 4

y2
24. lR;kfir dhft, fd Qyu xy = log y + c vody lehdj.k dy = , (xy ≠ 1) dk gy gSA 2
dx 1 − xy

Verify that the function xy = log y + c is a solution of the differential equation
dy y2
= , (xy ≠ 1) .
dx 1 − xy

vFkok
OR

vody lehdj.k dy = (1 + x 2 )(1 + y 2 ) dk O;kid gy Kkr dhft,A
dx
dy
Find the general solution of differential equation = (1 + x 2 )(1 + y 2 ) .
dx

5
25. P(A ∪ B) Kkr dhft,] ;fn 2P(A) = P(B) = vkSj P(A/B) = 2 A 2
13 5

5 2
Evaluate P(A ∪ B), if 2P(A) = P(B) = and P(A/B) = .
13 5

1232/(Set : A)

( 15 ) 1232/(Set : A)
[k.M – l

SECTION – C

26. fl) dhft, fd leLr f=Hkqtksa ds leqPp; A esa] R = {( T1 , T2 ) ; T1 le:i gS T2 ds} }kjk ifjHkkf"kr
laca/k R ,d rqY;rk laca/k gSA 3

Show that the relation R defined in the set A of all triangles as R = {( T1 , T2 ) ;
T1 is similar to T2 }, is equivalence relation.

vFkok

OR

fuEufyf[kr Qyu dks ljyre :i esa fyf[k, %

 3a
x − x 3 
2
−a a
tan −1  3 , a > 0; <x <
 2
3 3
 a − 3ax 

Write the following function in simplest form :

 3a 2 x − x 3 
tan −1  3 , a > 0; − a < x < a
 a − 3ax 2  3 3
 

;fn A = 
− 2
vkSj I = 
3 1 0
27.  ] tksfd A 2 = KA − 2I gS] rks K dk eku Kkr dhft,A 3
4 − 2 0 1

3 − 2 1 0
If A =   and I =   , find K, so that A 2 = KA − 2I .
4 − 2 0 1

1232/(Set : A) P. T. O.

( 16 ) 1232/(Set : A)
 K cos x π
 , ;fn x≠
28. K dk eku Kkr dhft,] rkfd çnÙk Qyu f (x ) =  π − 2x 2
π
fcUnq x = π ij
 3 , ;fn x= 2
 2
larr gSA 3
 K cos x π
 π − 2x , if x≠
Find the value of K, so that the function f (x ) =  2
π
 3 , if x=
 2
π
is continuous at x = .
2

29. ,d xqCckjk] tks lnSo xksykdkj jgrk gS] dk ifjorZu'khy O;kl 3 (2x + 1) gSA x ds lkis{k vk;ru ds
2
ifjorZu dh nj Kkr dhft,A 3
3
A balloon, which always remains spherical, has a variable diameter (2x + 1) .
2
Find the rate of change of its volume with respect to x.

30. lekdyu dhft, % 3

tan x
∫ sin x . cos x dx
Integrate :

tan x
∫ sin x . cos x dx

vFkok

OR

1232/(Set : A)

( 17 ) 1232/(Set : A)
lekdyu dhft, %

1
∫ x (x 4 − 1) dx

Integrate :

1
∫ x (x 4 − 1) dx

→
31. λ vkSj µ Kkr dhft,] ;fn (2iˆ + 6 ˆj + 27kˆ ) × (iˆ + λˆj + µkˆ ) = 0 A 3

→
Find λ and µ, if (2iˆ + 6 ˆj + 27kˆ ) × (iˆ + λˆj + µkˆ ) = 0 .

[k.M – n

SECTION – D

32. fuEufyf[kr jSf[kd lehdj.k fudk; dks vkO;wg fof/k ls gy dhft, % 5

2x + y + z = 1;

3
x − 2y − z = ;
2

3y − 5z = 9

1232/(Set : A) P. T. O.

( 18 ) 1232/(Set : A)
Solve the following system of linear equations, using matrix method :
2x + y + z = 1;
3
x − 2y − z = ;
2
3y − 5z = 9

33. fuEufyf[kr fuf'pr lekdyu dk eku Kkr dhft, % 5
π
4
∫ log(1 + tan x ) dx
0
Evaluate the following definite integral :
π
4
∫ log(1 + tan x ) dx
0

vFkok
OR

nh?kZo`Ùk 9x 2 + 4y 2 = 36 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A
Find the area of the region bounded by the ellipse 9x 2 + 4y 2 = 36 .

34. js[kk,¡] ftuds lfn'k lehdj.k fuEufyf[kr gSa] ds chp dh U;wure nwjh Kkr dhft, % 5

→
r = iˆ + 2 ˆj + 3kˆ + λ(iˆ − 3 ˆj + 2kˆ )

vkSj →
r = 4iˆ + 5 ˆj + 6kˆ + µ(2iˆ + 3 ˆj + kˆ )

Find the shortest distance between the lines whose vector equations are :

→
r = iˆ + 2 ˆj + 3kˆ + λ(iˆ − 3 ˆj + 2kˆ )

→
and r = 4iˆ + 5 ˆj + 6kˆ + µ(2iˆ + 3 ˆj + kˆ )

1232/(Set : A)

( 19 ) 1232/(Set : A)
vFkok
OR
js[kk,¡] ftuds lfn'k lehdj.k fuEufyf[kr gSa] ds chp dh U;wure nwjh Kkr dhft, %
→
r = (1 − t )iˆ + (t − 2) ˆj + (3 − 2t )kˆ

vkSj →
r = (s + 1)iˆ + (2s − 1) ˆj − (2s + 1)kˆ
Find the shortest distance between the lines whose vector equations are :
→
r = (1 − t )iˆ + (t − 2) ˆj + (3 − 2t )kˆ
→
and r = (s + 1)iˆ + (2s − 1) ˆj − (2s + 1)kˆ

35. fuEu vojks/kksa ds vUrxZr z = 5x + 10y dk U;wurehdj.k rFkk vf/kdrehdj.k dhft, % 5
x + 2y ≤ 120, x + y ≥ 60, x − 2y ≥ 0, x ≥ 0, y ≥ 0
Minimize and maximize z = 5x + 10y subject to constraints :
x + 2y ≤ 120, x + y ≥ 60, x − 2y ≥ 0, x ≥ 0, y ≥ 0

[k.M – ;
SECTION – E

36. ekuk f=T;k R okys ,d xksys ds vUrxZr ,d 'kadq gSA 'kadq dh Å¡pkbZ rFkk f=T;k Øe'k% h rFkk r gSaA
D

R
O
R
x

A B
C

1232/(Set : A) P. T. O.

( 20 ) 1232/(Set : A)
mijksDr tkudkjh ds vk/kkj ij fuEufyf[kr ç'uksa ds mÙkj nhft, %

(i) r vkSj R ds chp laca/k x ds inksa esa fy[ksaA 1

(ii) 'kadq dk vk;ru V dks R vkSj x ds inksa esa fy[ksaA 1

(iii) fn[kkb, fd x = R ij 'kadq dk vk;ru vf/kdre gSA 2
3

vFkok

;fn x = R ij 'kadq dk vk;ru V vf/kdre gS] rks V dk vf/kdre eku Kkr dhft,A tc 'kadq
3
dk vk;ru vf/kdre gS] rc 'kadq ds vk;ru rFkk xksys ds vk;ru dk vuqikr Kkr dhft,A 2

Let a cone is inscribed in a sphere of radius R. The height and radius of cone are
h and r respectively.

D

R

O
R
x

A B
C
On the basis of above information, answer the following questions :
(i) Write the relation between r and R in terms of x.
(ii) Write the volume V of the cone in terms of R and x.
R
(iii) Show that volume V of the cone is maximum, when x = .
3

1232/(Set : A)

( 21 ) 1232/(Set : A)
OR
R
If volume V of the cone is maximum at x = , then find the maximum
3
value of V and find the ratio of volume of cone and volume of sphere, when
volume of cone is maximum.

37. ,d ftys esa 50 K cPpksa dks iksfy;ks MªkWIl fiykbZ tkrh gSA iksfy;ks
y;ks MªkWIl nsus dh nj mu cPpksa dh la[;k
ds lekuqikrh gS ftUgsa MªkWIl ugha fiykbZ xbZ gSA nwljs lIrkg ds var rd vk/ks cPpksa dks iksfy;ks dh [kqjkd
nh tk pqdh gSA rhljs lIrkg ds var rd fdruksa dks MªkWIl nh xbZ gksxh] bldk vuqeku vody lehdj.k
dy
= λ(50 − y ) ds gy ls yxk;k tk ldrk gS] tgk¡ x lIrkgksa dh la[;k dks n'kkZrk gS vkSj y mu cPpksa
dx
dh la[;k dks n'kkZrk gS ftUgsa MªkWIl nh xbZ gSA

mijksDr tkudkjh ds vk/kkj ij fuEufyf[kr ç'uksa ds mÙkj nhft, %
(i) mijksDr vody lehdj.k dh dksfV Kkr dhft,A 1

1232/(Set : A) P. T. O.

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(ii) vody lehdj.k dy = λ(50 − y ) dks gy djus ds fy, fdl fof/k dk mi;ksx fd;k tk ldrk
dx
gS \ 1
dy
(iii) vody lehdj.k = λ(50 − y ) dks gy dhft,A 2
dx
vFkok
;fn λ = 0.049 rFkk y(0) = 0, rks vody lehdj.k dk fof'k"V gy Kkr dhft,A 2

Polio drops are delivered to 50 K children in a district. The rate at which polio
drops are given is directly proportional to the number of children who have not
been administered the drops. By the end of 2nd week, half the children have
been given the polio drops. How many will have been given the drops by the end
of 3rd week can be estimated using the solution to the differential equation
dy
= λ(50 − y ) , where x denotes the number of weeks and the y number of
dx
children who have been given the drops.

Based on the above information, answer the following questions :

(i) State the order of the above differential equation.

1232/(Set : A)

( 23 ) 1232/(Set : A)
(ii) Which method of solving a differential equation can be used to solve
dy
= λ(50 − y ) ?
dx
(iii) Solve the differential equation :

dy
= λ(50 − y )
dx

OR

If λ = 0.049 and y(0) = 0, then find the particular solution of differential
equation.

38. ,d Ldwy esa f'k{kd 3 Nk=ksa jfo] eksfgr vkSj lksfu;k ls ,d ç'u iwNrs gSAa jfo] eksfgr vkSj lksfu;k ds
ç'u gy djus dh çkf;drk Øe'k% 30%, 25% vkSj 45% gSA jfo] eksfgr vkSj lksfu;k }kjk =qfV djus
dh çkf;drk Øe'k% 1%, 1.2% vkSj 2% gSA

mijksDr tkudkjh ds vk/kkj ij fuEufyf[kr ç'uksa ds mÙkj nhft, %
(i) ç'u dks gy djus esa =qfV gksus dh dqy çkf;drk Kkr dhft,A 2
(ii) ;fn ç'u dk gy f'k{kd }kjk tk¡pk x;k gS vkSj mlesa dqN =qfV gS] rks çkf;drk Kkr dhft, fd
ç'u jfo }kjk gy ugha fd;k x;k gSA 2
1232/(Set : A) P. T. O.

( 24 ) 1232/(Set : A)
In a school, teacher asks a question to three students Ravi, Mohit and Sonia.
The probability of solving the question by Ravi, Mohit and Sonia are 30%, 25%
and 45% respectively. The probability of making error by Ravi, Mohit and Sonia
are 1%, 1.2% and 2% respectively.

Based on the above information, answer the following questions :

(i) Find the total probability of committing an error in solving the question.
(ii) If the solution of question is checked by teacher and has some error, then
find the probability that the question is not solved by Ravi.

S
1232/(Set : A)

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