HBSE Class 12 Question Paper 2025 Mathematics

CLASS : 12th (Sr. Secondary) Code No. 2232
Series : SS/Annual Exam.-2025
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 NksMsa+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.

2232/(Set : A) P. T. O.

(2) 2232/(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 ^^cc* % 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 ^^ll* % 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 ^^nn* % 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

2232/(Set : A)

(3) 2232/(Set : A)
(iii) bl ç'u&i= 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
(vi) dSydqysVj ds iz;ksx dh vuqefr ugha gSA
General Instructions :

(i) All questions are compulsory.
(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.

(vi) Use of Calculator is not permitted.

2232/(Set : A) P. T. O.

(4) 2232/(Set : A)
[k.M – v
SECTION – A

1. eku yhft, f : R → R, f(x) = x4 }kjk ifjHkkf"kr gS] rc f(x) gS % 1
(A) ,dSdh vkPNknd
(B) cgq,d vkPNknd
(C) ,dSdh gS fdUrq vkPNknd ugha gS
(D) u rks ,dSdh gS u vkPNknd gS
Let f : R → R be defined by f(x) = x4, then f(x) is :
(A) one-one onto
(B) many one onto
(C) one-one but not onto
(D) neither one-one nor onto

 −1
2. sin−1  dk eq[; eku gS % 1
 2

(A) π (B) π
3π
4 4

(C) −π (D) − 3π
π
4 4

 −1
The principal value of sin−1  is :
 2

(A) π (B) π
3π
4 4

(C) −π (D) − 3π
π
4 4

2232/(Set : A)

(5) 2232/(Set : A)
3. 3 × 3 dksfV ds ,sls vkO;wgksa dh dqy fdruh la[;k gksxh] ftudh izR;sd izfof"V 0 ;k 2 gS \ 1
(A) 27 (B) 18
(C) 81 (D) 512
The total number of all possible matrices of order 3 × 3 with entry 0 or 2, is :
(A) 27 (B) 18
(C) 81 (D) 512

4. Qyu f(x) = [x]] tgk¡ [x] egÙke iw.kkZad Qyu dks O;Dr djrk gS] fuEufyf[kr ij larr gS % 1
(A) 4 (B) –2
(C) 1 (D) 1.5
The function f(x) = [x], where [x] denotes the greatest integer function, is
continuous at :
(A) 4 (B) –2
(C) 1 (D) 1.5

5. Qyu f(x) = |x| – |x + 1| % 1

(A) x = 0 rFkk x = –1 nksuksa ij larr gS
(B) x = –1 ij larr gS ijUrq x = 0 ij larr ugha gS

(C) x = 0 vkSj x = –1 nksuksa ij vlarr gS

(D) x = 0 ij larr gS ijUrq x = –1 ij larr ugha gS
The function f(x) = |x| – |x + 1| is :

(A) continuous at x = 0 as well as at x = –1

(B) continuous at x = –1 but not at x = 0
(C) discontinuous at x = 0 as well as at x = –1

(D) continuous at x = 0 but not at x = –1

2232/(Set : A) P. T. O.

(6) 2232/(Set : A)
 sin x − cos x 
2 2
6. ∫  sin2 x . cos2 x  dx cjkcj gS % 1
 
(A) tan x + cot x + c
(B) tan x – cot x + c
(C) –tan x + cot x + c
(D) –tan x – cot x + c
 sin2 x − cos2 x 
∫  sin2 x . cos2 x  dx is equal to :
 
(A) tan x + cot x + c
(B) tan x – cot x + c
(C) –tan x + cot x + c

(D) –tan x – cot x + c

x
7. ;fn f (x ) = ∫ 0 t sin t dt gS] rc f ′(x ) gksxk % 1

(A) cos x + x sin x
(B) x sin x
(C) x cos x
(D) sin x + x cos x
x
If f (x ) = ∫ t sin t dt , then f ′(x ) is :
0

(A) cos x + x sin x
(B) x sin x
(C) x cos x
(D) sin x + x cos x

2232/(Set : A)

(7) 2232/(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

9. nh xbZ f=T;k a ds lHkh o`Ùkksa ds vody lehdj.k dh dksfV gS % 1
(A) 1 (B) 2
(C) 3 (D) 4
The order of the differential equation of all circles of given radius a is :
(A) 1 (B) 2
(C) 3 (D) 4

10. λ dk og eku ftlds fy, lfn'k 2iˆ − ˆj + 2kˆ vkSj lfn'k 3iˆ + λˆj + kˆ yacor gS % 1
(A) 2 (B) 4
(C) 6 (D) 8
The value of λ for which the two vectors 2iˆ − ˆj + 2kˆ and 3iˆ + λ ˆj + kˆ are
perpendicular is :
(A) 2 (B) 4
(C) 6 (D) 8

2232/(Set : A) P. T. O.

(8) 2232/(Set : A)
11. lfn'k iˆ + ˆj − 2kˆ ds fnd~-dkslkbu gSa % 1
1 1 −2
(A) , ,
6 6 6
−1 −1 2
(B) , ,
6 6 6
(C) 1, 1, –2
(D) –1, –1, 2
The direction cosines of the vector iˆ + ˆj − 2kˆ are :
1 1 −2
(A) , ,
6 6 6
−1 −1 2
(B) , ,
6 6 6
(C) 1, 1, –2
(D) –1, –1, 2

12. ;fn A vkSj B nks ?kVuk,¡ bl izdkj gSa fd P(A/B) = P(B/A) ≠ 0] rc % 1

(A) A ⊂ B] ysfdu A ≠ B
(B) A=B
(C) A∩B=φ
(D) P(A) = P(B)
If A and B are events such that P(A/B) = P(B/A) ≠ 0, then :
(A) A ⊂ B, but A ≠ B
(B) A=B
(C) A∩B=φ
(D) P(A) = P(B)

2232/(Set : A)

(9) 2232/(Set : A)
13. ;fn 'kh"kZ (2, –6), (5, 4) vkSj (k, 4) okys f=Hkqt dk {ks=Qy 35 oxZ bdkbZ gks] rks k dk eku
gS ……….A 1
If area of triangle is 35 sq. units with vertices (2, –6), (5, 4) and (k, 4), then k
is ……….. .

14. ;fn ,d js[kk y vkSj z v{kksa esa ls izR;sd ls π dks.k cukrh gS] rks js[kk }kjk x-v{k ds lkFk cuk;k
4
x;k dks.k gksxk ………….A 1
π
If a line makes an angle ofwith each of y and z axis, then the angle which it
4
makes with x-axis is …………. .

15. ;fn A vkSj B ,slh Lora= ?kVuk,¡ gSa fd P(A) = p, P(B) = 2p rFkk P (A, B esa ls dsoy ,d) = 5 , rks
9
p = …………. gksxkA 1
If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one
5
of A, B) = , then p = …………. .
9

16. ;fn A ,d 3 × 3 dksfV dk vkO;wg gS] rc A ds lkjf.kd ds lHkh mi-lkjf.kdksa dh la[;k D;k gksxh \ 1

If A is a matrix of order 3 × 3, then what is the number of minors in determinant
of A ?

17. ewy fcUnq ls xqtjus okyh ljy js[kk ds dqy dk vody lehdj.k D;k gksxk \ 1

What is the differential equation of the family of lines passing through the origin ?

18. ;fn A vkSj B nks Lora= ?kVuk,¡ gS]a rks A vkSj B esa ls U;wure ,d ds gksus dh izkf;drk D;k gksxh \ 1

If A and B are two independent events, then what is the probability of occurrence
of at least one of A and B ?

2232/(Set : A) P. T. O.

( 10 ) 2232/(Set : A)
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) : ekuk f(x) = x2, g(x) = cos x, rc fog ≠ gof 1

dkj.k (R) : (fog) (x) = f(x) g(x)
(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] ijUrq dkj.k (R), vfHkdFku (A) dh lgh O;k[;k ugha gSA
(C) vfHkdFku (A) lgh gS] ijUrq dkj.k (R) xyr gSA
(D) vfHkdFku (A) xyr gS] ijUrq dkj.k (R) lgh gSA

Assertion (A) : Let f(x) = x2, g(x) = cos x, then fog ≠ gof

Reason (R) : (fog) (x) = f(x) g(x)

(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, but 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.

2232/(Set : A)

( 11 ) 2232/(Set : A)
x −1 y − 2 z +1
20. vfHkdFku (A) : js[kk = = lery 11x − 3z − 14 = 0 ij gSA 1
3 11 11

dkj.k (R) : ,d ljy js[kk ,d lery ij gksxh ;fn js[kk lery ds lekUrj gks vkSj js[kk dk ,d
fcUnq lery ij 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] ijUrq dkj.k (R), vfHkdFku (A) dh lgh O;k[;k ugha gSA
(C) vfHkdFku (A) lgh gS] ijUrq dkj.k (R) xyr gSA
(D) vfHkdFku (A) xyr gS] ijUrq dkj.k (R) lgh gSA
x −1 y − 2 z +1
Assertion (A) : Line = = lies in the plane 11x − 3z − 14 = 0 .
3 11 11

Reason (R) : A straight line lies in the plane if the line is parallel to the plane

and a point of the line lies in the plane.

(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, but 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.

2232/(Set : A) P. T. O.

( 12 ) 2232/(Set : A)
[k.M – c
SECTION – B

21. fl) dhft, fd leqPp; {1, 2, 3} esa R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} }kjk iznÙk lac/a k
LorqY; gS] ijUrq u rks lefer gS vkSj u laØked gSA 2
Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3),
(1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.
vFkok
OR

fl) dhft, fd IR esa R = {(a, b) : a ≤ b} }kjk ifjHkkf"kr laca/k LorqY; rFkk laØked gS fdUrq lefer
ugha gSA
Show that the relation R in IR (set of real numbers) defined by R = {(a, b) : a ≤ b},
is reflexive and transitive but not symmetric.

vFkok
OR

;ksX;rk vk/kkfjr iz'u %
Competency Based Question :

v/;kid us vius fo|kFkhZ ls tan–1 (–x) – tan–1  1  dk eku iwNkA fo|kFkhZ us dgk mijksDr expression
x
dk eku rHkh fudkyk tk ldrk gS ;fn x dk eku Kkr gksA D;k fo|kFkhZ lgh gS \ vius mÙkj dks
lR;kfir djsaA
1
Teacher asked his students to find the value of tan–1 (–x) – tan–1   . Students
x
said that the value of the above expression can be found only if x is known. Is
student correct ? Justify your answer.

2232/(Set : A)

( 13 ) 2232/(Set : A)
3 − 2 1 0
22. ;fn A =   vkSj I =   ,oa A = kA − 2I
2
gks] rks k dk eku Kkr dhft,A 2
 4 − 2  0 1 
3 − 2 1 0
If A =   and I =   , find k, so that A 2 = kA − 2I .
4 − 2 0 1 

 x + 1, ;fn x ≥1
23. Qyu f ds lHkh vlkarR; ds fcUnqvksa dks Kkr dhft,] tcfd f (x ) =  2 + 2
 x + 1, ;fn x <1

 x + 1, if x ≥1
Find the point of discontinuity of the function f, where f (x ) =  2
 x + 1, if x <1

24. vody lehdj.k (1 + x 2 )dy = (1 + y 2 )dx dk O;kid gy Kkr dhft,A 2

Find the general solution of the differential equation (1 + x 2 )dy = (1 + y 2 )dx .

vFkok
OR

dy y
vody lehdj.k + = x 2 dk gy Kkr dhft,A
dx x
dy y
Solve the differential equation + = x2 .
dx x

25. nks ikls ,d lkFk Qsads tkrs gSaA eku yhft, fd ?kVuk A ^igys ikls ij vad 6 izkIr gksuk* gS rFkk ?kVuk
B ^nwljs ikls ij vad 2 izkIr gksuk* gSA D;k A vkSj B Lora= ?kVuk,¡ gSa \ 2

Two dice are thrown together. Let A be the event 'getting 6 on the first die' and B
be the event 'getting 2 on the second die'. Are the events A and B independent ?

2232/(Set : A) P. T. O.

( 14 ) 2232/(Set : A)
[k.M – l

SECTION – C

26. fl) dhft, % 3

 2x  −1
 3x − x 3  1
tan−1 x + tan−1 2
= tan  2 
, |x | <
1 − x   1 − 3x  3

Prove that :

 2x  −1
 3x − x 3  1
tan−1 x + tan−1 2
= tan  2 
, |x | <
1 − x   1 − 3x  3

vFkok
OR

x ds os eku Kkr dhft, tks lehdj.k sin−1 x + sin−1(1 − x ) = cos −1 x dks larq"V djrs gSaA
Find the value of x which satisfy the equation sin−1 x + sin−1(1 − x ) = cos −1 x .

 2 3
27. iznf'kZr dhft, fd vkO;wg A =   lehdj.k A 2 − 4A + I = 0 ] tgk¡ I, 2 × 2 dksfV dk ,d
1 2
rRled vkO;wg gS vkSj 0, 2 × 2 dksfV dk ,d 'kwU; vkO;wg gSA bldh lgk;rk ls A–1 Kkr dhft,A 3
 2 3
Show that the matrix A =   satisfies the equation A 2 − 4A + I = 0 , where I
1 2
is 2 × 2 identity matrix and 0 is 2 × 2 zero matrix. Using equation, find A–1.

2232/(Set : A)

( 15 ) 2232/(Set : A)
dy
28. ;fn y = tan(x + y ) gS] rks Kkr dhft,A 3
dx
dy
If y = tan(x + y ) , find .
dx

29. varjky Kkr dhft, ftuesa iznÙk Qyu f(x) = sin 3x, x ∈ 0, π  esa o/kZeku ;k Ðkleku gSA 3
 2

 π
Find the intervals in which the function f(x) = sin 3x, x ∈ 0,  is increasing or
 2
decreasing.

30. Kkr djsa % 3

∫ e . sin x dx
x

Evaluate :

∫ e . sin x dx
x

vFkok

OR

Kkr djsa %

π dx
∫0 2 1 + cot x
Evaluate :

π dx
∫0 2 1 + cot x
2232/(Set : A) P. T. O.

( 16 ) 2232/(Set : A)
31. lfn'k a + b vkSj a − b esa ls izR;sd ds yacor ek=d lfn'k Kkr dhft,] tgk¡ a = iˆ + ˆj + kˆ vkSj

b = iˆ + 2 ˆj + 3kˆ gSA 3

Find a unit vector perpendicular to each of the vectors a + b and a − b , where

a = iˆ + ˆj + kˆ and b = iˆ + 2 ˆj + 3kˆ .

[k.M – n
SECTION – D

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

3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

Solve the following system of equations by matrix method :

3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

2232/(Set : A)

( 17 ) 2232/(Set : A)
vFkok
OR

 1 −1 2 
A =  0 2 − 3  vkO;wg dk A −1 Kkr dhft, vkSj bldk iz;ksx djds lehdj.k fudk; dks gy
 3 − 2 4 

dhft, %

x + 3z = 9

–x + 2y – 2z = 4

2x – 3y + 4z = –3

 1 −1 2 
Find A −1
from the matrix A =  0 2 − 3  and using this A − 1 , solve the
 
 3 − 2 4 

system of equations :

x + 3z = 9

–x + 2y – 2z = 4

2x – 3y + 4z = –3

2232/(Set : A) P. T. O.

( 18 ) 2232/(Set : A)
vFkok

OR

;ksX;rk vk/kkfjr iz'u %
Competency Based Question :

A(–2, 5), B(1, 2) vkSj C(k, 5) lef}ckgq ledks.k f=Hkqt ∆ABC ds fljs gSa] tgk¡ k /kukRed gSA

d.kZ AC dh yEckbZ 6 lseh gS] lkjf.kd dk iz;ksx djrs gq, k dk eku Kkr dhft,A

A(–2, 5), B(1, 2) and C(k, 5) are the vertices of isosceles right angled ∆ABC, where

k is positive. The hypotenuse AC is 6 cm long, use determinants to find the value

of k.

33. (x − 1)2 + y 2 = 1 and x 2 + y 2 = 1 }kjk f?kjs gq, {ks=Qy dks Kkr dhft,A 5

Find the area of the region bounded by (x − 1)2 + y 2 = 1 and x 2 + y 2 = 1

vFkok
OR

a
js[kk x = }kjk o`Ùk x 2 + y 2 = a 2 ds dkVs x, ,d y?kq o`Ùk[k.M dk {ks=Qy Kkr dhft,A
2

Find the area of a minor segment of the circle x 2 + y 2 = a 2 cut off by the line
a
x= .
2

2232/(Set : A)

( 19 ) 2232/(Set : A)
34. js[kkvksa l1 vkSj l2 ds chp dh U;wure nwjh Kkr djsa % 5

→
l1 : r = iˆ + ˆj + λ (2iˆ − ˆj + kˆ )

vkSj l 2 : →r = 2iˆ + ˆj − kˆ + µ(3iˆ − 5 ˆj + 2kˆ )

Find the shortest distance between the lines l1 and l2 :

→
l1 : r = iˆ + ˆj + λ (2iˆ − ˆj + kˆ )

→
and l 2 : r = 2iˆ + ˆj − kˆ + µ(3iˆ − 5 ˆj + 2kˆ )

vFkok

OR

x y −1 z − 2
js[kk = = esa fcUnq (1, 6, 3) dk izfrfcEc Kkr djsaA
1 2 3

x y −1 z − 2
Find the image of the point (1, 6, 3) in the line = = .
1 2 3

2232/(Set : A) P. T. O.

( 20 ) 2232/(Set : A)
35. vkys[kh; fof/k ls fuEu leL;k dks gy dhft, % 5

fuEu O;ojks/kksa ds varxZr Z = 3x + 9y dk U;wure vkSj vf/kdre eku Kkr dhft,

x + 3y ≤ 60, x + y ≥ 10, x ≤ y, x ≥ 0, y ≥ 0

Solve the following problem graphically :

Minimize and Maximize Z = 3x + 9y

Subject to the constraints

x + 3y ≤ 60, x + y ≥ 10, x ≤ y, x ≥ 0, y ≥ 0

vFkok
OR

,d izdkj ds dsd ds fy, 200 gm vkVk vkSj 25 gm olk dh vko';drk gksrh gS vkSj nwljs izdkj
ds dsd ds fy, 100 gm vkVk vkSj 50 gm olk dh vko';drk gksrh gSA dsd dh vf/kdre la[;k
Kkr dhft, tks 5 kg vkVs vkSj 1 kg olk ls cukbZ tk ldrh gSA ;g ekurs gq, fd dsd cukus esa
mi;ksx dh tkus okyh vU; lkexzh dh dksbZ deh ugha gSA
One kind of cake requires 200 gm of flour and 25 gm of fat and another kind of
cake requires 100 gm of flour and 50 gm of fat. Find the maximum number of
cakes which can be made from 5 kg of flour and 1 kg of fat, assuming that there
is no shortage of the other ingredients used in making the cakes.

2232/(Set : A)

( 21 ) 2232/(Set : A)
[k.M – ;
SECTION – E

36. ,d ekyh ,d vk;rkdkj Qwyksa dh D;kjh esa bl rjg ls Qwy yxkus dh ;kstuk cuk jgk gS fd
v/kZo`Ùkkdkj {ks= esa ,d vk;r gks ¼tSlk fd fp= esa fn[kk;k x;k gS½ v/kZo`Ùkkdkj {ks= dh f=T;k 30 m
gSA ekuk vk;r dh yackbZ PQ = x m gSA 4

O 30 m A
P x Q

mijksDr lwpuk ds vk/kkj ij fuEufyf[kr ds mÙkj nhft, %
(i) vk;rkdkj Qwyksa dh D;kjh dh pkSM+kbZ x esa D;k gksxh \

(ii) Qyu x esa vk;rkdkj {ks= dk {ks=Qy D;k gksxk \
(iii) ekyh vk;rkdkj Qwyksa dh D;kjh dk vf/kdre {ks=Qy pkgrk gSA blds fy, x dk eku D;k gksxk \
(iv) Qwyksa dh D;kjh dk vf/kdre {ks= gksus ds ckn cpk gqvk {ks= dk {ks=Qy D;k gksxk \
A gardener plans to plant flowers in a rectangular flower bed in such a way that
a rectangle is inscribed in the semi-circular field (as shown in the figure). Radius
of the semi-circular field is 30 m. Let the length of the rectangle be PQ = x m.

O 30 m A
P x Q
Based on above information answer the following :

2232/(Set : A) P. T. O.

( 22 ) 2232/(Set : A)
(i) What will be the breadth of the rectangular flower bed in terms of x ?
(ii) What will be the area of rectangular region as a function of x ?
(iii) Gardener wants maximum area far for the rectangular flower bed. For this
to happen, what will be the value of x ?
(iv) What will be the area of remaining field (in sq. m) after having the flower
bed of maximum area ?

37. fdlh thok.kq lewg esa thok.kqvksa dh la[;k 1,00,000 gSA thok.kqvksa dh o`f) nj muds mifLFkr la[;k ds
lekuqikrh gSA ekuk t le; ij thok.kqvksa dh la[;k x gSA 4

lekuqikrh vpj k ekurs gq,] mijksDr lwpukvksa ds vk/kkj ij fuEu iz'uksa ds mÙkj nhft, %
(i) bl leL;k dk vody lehdj.k D;k gS \
(ii) x vkSj t esa D;k laca/k gS \
(iii) ;fn 2 ?k.Vksa esa thok.kqvksa esa 10% dh o`f) gksrh gS] rks k dk eku Kkr dhft,A

(iv) thok.kqvksa dh la[;k 1,00,000 ls 2,00,000 gksus esa fdruk le; yxsxk \
In a culture, the bacteria count is 1,00,000. The growth of bacteria is
proportional to the number present. Let x be the number of bacteria at time t.
Based on the above information, answer the following questions (assuming k to
be the constant of proportionality) :

(i) What is the differential equation for this problem ?
(ii) What is the relation between x and t ?
(iii) If the bacteria increased 10% in 2 hours, then find k.
(iv) Find the time taken by the bacteria count increases from 1,00,000 to
2,00,000.

2232/(Set : A)

( 23 ) 2232/(Set : A)
38. rhu leku cDls I, II vkSj III fn, x, gSa ftuesa izR;sd esa nks flDds gSaA ckWDl-I esa nksuksa flDds lksus ds
gSaA ckWDl-II esa nksuksa flDds pk¡nh ds gSa vkSj ckWDl-III esa ,d lksus dk flDdk vkSj ,d pk¡nh dk flDdk
gSA ,d O;fDr ;kn`fPNd :i ls ,d ckWDl pqurk gS vkSj mlesa ls ,d flDdk fudkyrk gSA 4

G G S S G S

ckWDl-I ckWDl-II ckWDl-III

mijksDr tkudkjh ds vk/kkj ij fuEu iz'uksa ds mÙkj nhft, %
(i) ckWDl-I p;fur gksus ij lksus dk flDdk feyus dh izkf;drk D;k gksxh \
(ii) ;fn ckWDl-III pquk tkrk gS] rks lksus dk flDdk feyus dh izkf;drk D;k gksxh \
(iii) ;fn fudkyk x;k flDdk lksus dk flDdk gS] rks bldh izkf;drk D;k gksxh fd ;g ckWDl-II ls
fudkyk x;k gS \

(iv) ;fn fudkyk x;k flDdk lksus dk flDdk gS] rks ml ckWDl esa nwljk flDdk lksus dk gS] bldh
izkf;drk D;k gksxh \
Given three identical Boxes-I, II and III, each containing two coins. In Box-I both
coins are gold coins, in Box-II both coins are silver coins and in the Box-III,
there is one gold coin and one silver coin. A person chooses a Box at random
and takes out a coin.

G G S S G S

Box-I Box-II Box-III
Based on the above information answer the following :

(i) What is the probability of getting a gold coin if Box-I is selected ?

2232/(Set : A) P. T. O.

( 24 ) 2232/(Set : A)
(ii) What is the probability of getting a gold coin if Box-III is selected ?
(iii) If the coin drawn is gold, then what is the probability that it is drawn from
Box-II ?
(iv) If the coin drawn is gold coin, then what is the probability then the other
coin in the Box is also a gold coin ?

S

2232/(Set : A)