HBSE Class 12 Question Paper 2021 Maths

CLASS : 12th (Sr. Secondary) Code No. 231
Series : SS. April./2021
Roll No.

xf.kr GRAPH
MATHEMATICS
Hkkx – I
PART – I
¼vkRefu"B iz'u½
(Subjective Questions)
(Academic)
[ fgUnh ,oa vaxzsth ek/;e ]
[ Hindi and English Medium ]
(Only for Fresh/School Candidates)

le; : 2 21 ?k.Vs ] [ iw.kk±d : 80 ¼Hkkx–I : 40, Hkkx–II : 40½
Time allowed : 2 21 hours ] [ Maximum Marks : 80 (Part–I : 40, Part–II : 40)

iz'u&i= nks Hkkxkas eas foHkkftr gS % Hkkx–I ¼vkRefu"B½ ,oa Hkkx–II ¼oLrqfu"B½A
fu"B½A ijh{kkFkhZ dks nksuksa Hkkxkas ds iz'ukas
ds mÙkj dks viuh mÙkj iqfLrdk eas fy[kuk gSA iz'u&i= dk Hkkx–I ijh{kk vkjEHk gksus ij igys mÙkj&iqfLrdk
ds lkFk fn;k tk,xk rFkk Hkkx–II ds fy, vkf[kjh dk ,d ?kaVs dk le; fn;k tk,xk vFkkZr~ ijh{kk lekIr
gksus ls ,d ?kaVk iwoZ ijh{kkFkhZ dks Hkkx–II dk iz'u-i= fn;k tk,xkA
Hkkx–I ds iz'u&i= esa dqy 12 iz'u ,oa Hkkx–II ds iz'u&i= eas dqy 40 iz'u gSaA
Question paper is divided into two Parts : Part–I (Subjective type) and Part–II
(Objective type). Answer the questions of both parts in your answer-book. Part–I
of question paper with answer-book will be provided with starting of
Examination and last one hour of Examination will be given for Part–II i.e.
question paper of Part–II will be provided before one hour of the end of
Examination.
Total questions in question paper of Part–I are 12 and of Part–II are 40.

• Ñi;k tk¡p dj ysa fd Hkkx–I ds bl iz'u-i= esa eqfnzr i`"B 7 rFkk iz'u 12 gSaA
Please make sure that the printed pages in this question paper of Part–I are 7 in
number and it contains 12 questions.

231/ I P. T. O.

(2) 231
• iz'u-i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj dks Nk= mÙkj-iqfLrdk ds eq[;-i`"B ij fy[ksaA
The Code No. 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.

• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the question paper.

• d`i;k iz'uksa dk 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 question, 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 %
General Instructions :

(i) lHkh iz'u vfuok;Z gSaA
All questions are compulsory.
(ii) izR;sd iz'u ds vad mlds lkeus n'kkZ;s x, gSaA
Marks of each question are indicated against it.

(iii) vkids mÙkj vadkuqlkj gksus pkfg,A
Your answer should be according to marks.

231/ I

(3) 231
[k.M – v
SECTION – A

1. ;fn y = tan −1 2x 2  , rks dy Kkr dhft,A 2
1 − x  dx

 2x  dy
If y = tan−1 2
 , find .
1 − x  dx

2
d y dy
2. ;fn y = ae + be , rks n'kkZb;s fd + 6y = 0 A
3x 2x
2
−5 2
dx dx

3x 2x d 2y dy
If y = ae + be , then show that 2
−5 + 6y = 0 .
dx dx

π
3. vody lehdj.k dy = y cot x dks gy dhft,] ftlesa fn;k x;k gS x= , y = 1A 2
dx 2

dy π
Solve the equation = y cot x , given x = , y = 1 .
dx 2

4. ,d cYc ds ,d lky ds vUnj [kjkc gksus dh izkf;drk 0.05 gSA ;fn 5 cYc yx;s tk,¡ rks 3
cYcksa ds [kjkc gks tkus dh izkf;drk Kkr dhft,A 2

The probability that a bulb will fuse within an year is 0.05. Find the
probability that out of 5 bulbs 3 bulbs will fuse within the year.

231/ I P. T. O.

(4) 231

5. lfn'k a = iˆ + 3 ˆj + 7kˆ dk b = 7iˆ − ˆj + 8kˆ ij iz{ksi Kkr dhft,A 2

Find the projection of vector a = iˆ + 3 ˆj + 7kˆ on vector b = 7iˆ − ˆj + 8kˆ .

6. λ ds fdl eku ds fy, js[kk,¡ x − 1 = y − 2 = z − 3 vkSj x −1 y −1 z − 6
= = ,d nwljs ij
3 2k 2 − 3k 1 −5
yEc gSaA 2

x −1 y − 2 z − 3
For what value of λ the line = = is perpendicular to the line
3 2k 2
x −1 y −1 z − 6
= = .
− 3k 1 −5

[k.M – c

SECTION – B

7. Qyu f (x ) = x 3 − 6x 2 + 9x + 15 ftl varjky esa fujarj Ðkleku (Strictly Decreasing) gS
og Kkr dhft,A 4

Find the interval in which the function f (x ) = x 3 − 6x 2 + 9x + 15 is strictly
decreasing.

8. oØ y 2 = x vkSj x 2 = y ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 4

Find the area bounded by the curves y 2 = x and x 2 = y.

231/ I

(5) 231

9. js[kkvksa r = (iˆ + 2 ˆj + 3kˆ ) + λ(iˆ − 3 ˆj + 2kˆ ) vkSj r = 4iˆ + 5 ˆj + 6kˆ + µ(2iˆ + 3 ˆj + kˆ ) ds chp dh
U;wure nwjh (S.D) Kkr dhft,A 4

Find the shortest distance between the lines r = (iˆ + 2 ˆj + 3kˆ ) + λ(iˆ − 3 ˆj + 2kˆ ) and

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

10. ,d dkj[kkus esa nks e'khusa A vkSj B gSaA A e'khu dqy mRikn dk 60% vkSj B 40% mRiknu djrh
gSA A e'khu dk 2% vkSj B dk 1% mRikn =qfViw.kZ gSA ;fn dqy mRiknu ls ,d oLrq pquh tk;s vkSj
og =qfViw.kZ gks] rks mlds A }kjk mRikfnr gksus dh izkf;drk Kkr dhft,A 4

A factory has two machines A and B. A produces 60% and B 40% of the total
output. 2% product of machine A and 1% of machine B is defective. If one item
is chosen at random from the output and is found defective, find the
probability that it was produced by machine A.

[k.M – l

SECTION – C

11. n'kkbZ;s fd % 6

a −b −c 2a 2a
2b b −c −a 2b = (a + b + c )3
2c 2c c −a −b

231/ I P. T. O.

(6) 231

Show that :

a −b −c 2a 2a
2b b −c −a 2b = (a + b + c )3
2c 2c c −a −b

vFkok

OR

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

3x − 2y + 3z = 8

2x + y − z = 1

4x − 3y + 2z = 4

Solve the following equations by matrix method :

3x − 2y + 3z = 8

2x + y − z = 1

4x − 3y + 2z = 4

231/ I

(7) 231

12. fuEufyf[kr vojks/kksa ds vUrxZr z = 7 x + 6y dk vf/kdrehdj.k dhft,A vojks/k gS
x + 2y ≤ 50, 2x + y ≤ 40, x ≥ 0, y ≥ 0 A 6

Maximiz z = 7x + 6y subject to the constraints x + 2y ≤ 50, 2x + y ≤ 40,
x ≥ 0, y ≥ 0 .

vFkok

OR

z = 3x + 5y dk U;wurehdj.k] vojks/kksa x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0 ds vUrxZr
dhft,A

Minimize z = 3x + 5y subject to the constraints x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0 .

S
231/ I P. T. O.

CLASS : 12th (Sr. Secondary) Code No. 231
Series : SS – April/2021
Roll No.

xf.kr GRAPH
MATHEMATICS
Hkkx – II
PART – II
¼oLrqfu"B iz'u½
(Objective Questions)
(Academic)
[ fgUnh ,oa vaxzsth ek/;e ]
[ Hindi and English Medium ]
(Only for Fresh/School Candidates)

• Ñi;k tk¡p dj ysa fd Hkkx–II ds bl iz'u-i= esa eqfnzr i`"B 16 rFkk iz'u 40 gSaA
Please make sure that the printed pages in this question paper of Part-II are 16 in
number and it contains 40 questions.
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the question paper.
• d`i;k iz'uksa dk 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 question, 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 %
General Instructions :

(i) lHkh iz'u vfuok;Z gSaA
All questions are compulsory.

231/ II P. T. O.

(2) 231
(ii) iz'u Øekad 1 ls 40 rd oLrqfu"B iz'u gSaA izR;sd iz'u 1 vad dk gSA lgh mÙkj viuh
mÙkj&iqfLrdk eas fyf[k,A
Questions from 1 to 40 are objective type questions. Each question is of
1 mark. Write correct answer in your answer-book.

1. laca/k R tks R ij ifjHkkf"kr gS R = {(a, b ) : a ≤ b } gS %

(A) LorqY; vkSj lefer

(B) lefer vkSj laØfer

(C) LorqY; vkSj laØfer

(D) buesa ls dksbZ ugha

The relation on R defined R = {(a , b ) : a ≤ b } is :

(A) Reflexive and Symmetric

(B) Symmetric and Transitive

(C) Reflexive and Transitive

(D) None of these

2. ;fn f : R → R ij ifjHkkf"kr gS f (x ) = 3x }kjk] rks f gS %

(A) ,dSdh vkSj vkPNknd (B) cgq,dSdh vkSj vkPNknd

(C) ,dSdh ij vkPNknd ugha (D) u ,dSdh u vkPNknd

231/ II

(3) 231

If f : R → R defined by f (x ) = 3x , then f is :

(A) One-one onto

(B) Many-one, onto

(C) One-one not onto

(D) Neither one-one nor onto

3. ;fn ,d f}vk/kkjh lfØ;k * tks N ij bl izdkj ifjHkkf"kr gS fd a * b = a 2 + b 2 , rks fuEufyf[kr
esa ls lgh pqusa %

(A) lkgp;Z vkSj Øefofues;

(B) Øefofues; ij lkgp;Z ugha

(C) lkgp;Z ij Øefofues; ugha

(D) u lkgp;Z u Øefofues;

If a binary operation * on N defined as a * b = a 2 + b 2 , choose the correct
answer :

(A) Associative and Commutative

(B) Commutative but not Associative

(C) Associative but not Commutative

(D) Neither Associative nor Commutative

231/ II P. T. O.

(4) 231
 1
4. tan −1(1) + cos −1  −  cjkcj gS %
 2

π 7π
(A) − (B)
12 12

11π 5π
(C) (D)
12 12

 1
tan −1(1) + cos −1  −  is equal to :
 2

π 7π
(A) − (B)
12 12

11π 5π
(C) (D)
12 12

5. sin −1( 1 − x 2 ),| x | < 1 cjkcj gS %

(A) sin −1 x

(B) cos −1 x

(C) tan −1 x

(D) buesa ls dksbZ ugha

sin −1( 1 − x 2 ),| x | < 1 is equal to :

(A) sin −1 x

(B) cos −1 x

(C) tan −1 x

(D) None of these

231/ II

(5) 231
6. ;fn A ,d 2 × 3 dksfV dk vkO;wg gS vkSj B 3 × 2 dksfV dk] rks AB dh dksfV (Order) gS %

(A) 2×2 (B) 3×3

(C) ifjHkkf"kr ugha (D) buesa ls dksbZ ugha

If A is a matrix of order 2 × 3 and B is a matrix of order 3 × 2, then AB is of
order :

(A) 2×2 (B) 3×3

(C) Not defined (D) None of these

7. ;fn A vkSj B leku dksfV (Order) ds O;qRØe.kh; vkO;wg gSa] rks fuEu esa ls dkSu-lk lR; gS \

(A) (AB )−1 = B −1A −1 (B) (AB )−1 = A −1B −1

(C) ( A + B ) −1 = A −1 + B −1 (D) (A − B )−1 = A −1 − B −1

If A and B are two invertible matrices of some order, which of the following is
always true ?

(A) (AB )−1 = B −1A −1 (B) (AB )−1 = A −1B −1

(C) ( A + B ) −1 = A −1 + B −1 (D) (A − B )−1 = A −1 − B −1

x 6 6 −3
8. ;fn = , rks x dk eku gksxk %
8 2x 8 4

(A) 6 (B) 2

(C) 0 (D) lEHko ugha

231/ II P. T. O.

(6) 231
x 6 6 −3
If = , then the value of x is :
8 2x 8 4

(A) 6 (B) 2

(C) 0 (D) Not possible

9. ;fn Qyu f (x ) = ax + 3, x ≤ 5
= 18, x >5

x = 5 ij ,d lrr Qyu gS] rks a dk eku gS %

(A) 5 (B) 3

(C) 1 (D) buesa ls dksbZ ugha

The function f (x ) = ax + 3, x ≤ 5

= 18, x >5

is a continuous function at x = 5 , then the value of a is :

(A) 5 (B) 3

(C) 1 (D) None of these

10. ;fn y = log(cos e x ), rks dy cjkcj gS %
dx

(A) sec(e x ) (B) − sec(e x )e x

(C) − tan(e x ) e x (D) buesa ls dksbZ ugha

231/ II

(7) 231
dy
If y = log(cos e x ), then is equal to :
dx

(A) sec(e x ) (B) − sec(e x )e x

(C) − tan(e x ) e x (D) None of these

11. oØ y = 4x − 3 − 1 dh fcUnq (3, 2) ij izo.krk gS %

2 3
((A) (B)
3 2

1
(C) (D) buesa ls dksbZ ugha
6

The slope of the tangent to the curve y = 4x − 3 − 1 at (3, 2) is :

2 3
(A) (B)
3 2

1
(C) (D) None of these
6

12. tan1 x
∫ 1 + x 2 dx cjkcj gS %

(A) tan −1 x + c (B) (tan −1 x )2 + c

1
(C) (tan −1 x )2 + c (D) buesa ls dksbZ ugha
2

231/ II P. T. O.

(8) 231
tan1 x
∫ 1 + x 2 dx is equal to :

(A) tan −1 x + c (B) (tan −1 x )2 + c

1
(C) (tan −1 x )2 + c (D) None of these
2

1
dx
13. ∫ dk eku gS %
2
0 1− x

π π
(A) (B)
2 4

π
(C) − (D) 0
2

1
dx
∫ is :
0 1− x2

π π
(A) (B)
2 4

π
(C) − (D) 0
2

14. oØ y 2 = x , x − v{k ls Åij x = 0 , x = 1 }kjk f?kjs {ks= dk {ks=Qy gS %

1 2
(A) (B)
3 3

3
(C) 1 (D)
2

231/ II

(9) 231
Area bounded by the curve y 2 = x , above x-axis and x = 0 to x = 1 is :

1 2
(A) (B)
3 3

3
(C) 1 (D)
2

15. y = a sin(x + b ) tgk¡ a vkSj b LosPN vpj gS] ml dqy (family) dk vody lehdj.k gS %

d 2y
(A) −y = 0
dx 2

d 2y
(B) + ay = 0
dx

d 2y
(C) +y =0
dx 2

(D) buesa ls dksbZ ugha
The differential equation of the family of curves y = a sin(x + b ), where a and b
are arbitrary is :

d 2y
(A) −y = 0
dx 2

d 2y
(B) + ay = 0
dx

d 2y
(C) +y =0
dx 2

(D) None of these

231/ II P. T. O.

( 10 ) 231
16. ;fn E vkSj F nks Lora= ?kVuk,¡ gSa] rks fuEu esa ls dkSu-lk lR; ugha gS \

(A) P (E ∩ F ) = P (E ).P (F )

(B) P (E / F ) = P (E )

(C) P (E / F ) = P (F )

(D) P (E / F ) P (F ) = P (E ∩ F )

If E and F are independent events, then which of the following is not true ?

(A) P (E ∩ F ) = P (E ).P (F )

(B) P (E / F ) = P (E )

(C) P (E / F ) = P (F )

(D) P (E / F ) P (F ) = P (E ∩ F )

17. λ dk eku ftlds fy, lfn'k a = 2iˆ − ˆj + λkˆ lfn'k b = iˆ − 3 ˆj − 5kˆ ds yEcor gS] og gS %

(A) 0 (B) 1

(C) –5 (D) buesa ls dksbZ ugha

The value of λ for which the vectors a = 2iˆ − ˆj + λkˆ is perpendicular to the

vector b = iˆ − 3 ˆj − 5kˆ, is :

(A) 0 (B) 1

(C) –5 (D) None of these

231/ II

( 11 ) 231
18. ;fn ,d js[kk x-v{k ls 60 o vkSj y-v{k ls 45 o dk dks.k cukrh gS] rks mldk z-v{k ls mldk dks.k
cusxk og gS %

(A) 30 o (B) 45 o

(C) 60 o (D) 90 o

If a line makes angle 60 o with x-axis 45 o with y-axis, then this line will make
angle with z-axis is :

(A) 30 o (B) 45 o

(C) 60 o (D) 90 o

[kkyh LFkku Hkjsa %

Fill in the blanks :

19. ;fn f :R →R , g : R → [−1, 1] tgk¡ f (x ) = x 2 vkSj g(x) = sin x rks]
fog (x ) = ........ A (sin x 2 , sin 2 x , x 2 sin x ) A

If f (x ) = x 2 and g(x) = sin x where f : R → R and g : R → [ −1, 1], then

fog(x) = ………. . (sin x 2 , sin 2 x , x 2 sin x ).

  1   3 1 1 3
20. cos  2 sin −1  −   = ........ A  , ,− ,− 

  2   2 2 2 2 

  1   3 1 1 3
cos  2 sin −1  −   = ........ .  , ,− ,− 

  2   2 2 2 2 

231/ II P. T. O.

( 12 ) 231
 x 1 1 
21. cos(tan −1 x ) cjkcj gS …………….A  , , 
 2 2 2 
 1+ x 1+ x 1− x 

 x 1 1 
cos(tan −1 x ) is equal to ………………. .  , , 
 2 2 2 
 1+ x 1+ x 1− x 

22. ;fn A ,d 3 dksfV dh oxZ vkO;wg gS vkSj | A |= 5 gS] rks ( Adj A ½ = |Adj A| ds lkjf.kd dk
 1
eku gS ……………. A  5, 25, 125, 
 5

If A is a square matrix of order 3 with |A| = 5, then
 1
det (Adj A ) = | Adj A | = ……. .  5, 25, 125, 
 5

23. ;fn A vkSj B nks vkO;wg ,d-nwljs ds O;qRØe gSa rks fuEu esa ls dkSu-lk lR; gS \

(A) AB = BA (B) AB = BA = O

(C) AB = O, BA = I (D) AB = BA = I

If A and B are inverse of each other then which of the following is true ?
(A) AB = BA (B) AB = BA = O

(C) AB = O, BA = I (D) AB = BA = I

24. ;fn x = 2 at 2 , y = 4 at , rc dy =
dx

1 1
(A) (B) −
t t

1
(C) (D) buesa ls dksbZ ugha
t2

231/ II

( 13 ) 231
dy
If x = 2 at 2 , y = 4 at , than =
dx

1 1
(A) (B) −
t t

1
(C) (D) None of these
t2

25. ;fn f (x ) = tan 3x , x ≠ 0 x jsfM;u esa gSa
x

=k , x =0

vkSj f (x ) , x = 0 ij lrr gS rks k dk eku gS ………….. A

tan 3x
If f (x ) = , x ≠ 0 x is in radiang
x

=k , x =0

and f(x) is continuous at x = 0, then the value of k is ……………. .

∫ cot x dx dk eku gS :
2
26.

(A) cot x + x + c (B) − cot x − x + c

(C) tan x − x + c (D) buesa ls dksbZ ugha

∫ cot x dx
2
is :

(A) cot x + x + c (B) − cot x − x + c

(C) tan x − x + c (D) None of these

231/ II P. T. O.

( 14 ) 231
27.
∫ e (tan x + sec x ) dx dk eku gS = ………….. A
x 2

∫ e (tan x + sec x ) dx = ....... . .
x 2

28. ;fn A U;k¸; ikls dks Qsadk tkrk gS vkSj ?kVuk,¡ E = {1, 3, 5}, F = {2, 3}, rks P (F /E ) Kkr
dhft,A

A fair die is rolled. Consider the events E = {1, 3, 5}, F = {2, 3}, find the P (F /E ) .

x
29. ;fn f : [− 1, 1] → R , f (x ) = ls ifjHkkf"kr gks] rks f −1(x ) Kkr dhft,A
x +2

x
If f : [− 1, 1] → R , is given by f (x ) = , then find f −1(x ) .
x +2

1 3
30. tan −1 + tan −1 dk eku Kkr dhft, A
4 5

1 3
Find the value of tan −1 + tan −1 .
4 5

31. ;fn A ,d 3 dksfV dk oxZ vkO;wg gS ftldk | A | = 4, rks det | 2A | Kkr dhft,A

If A is a square matrix of order 3 and | A | = 4, then find det | 2A |.

32. ;fn x 3 + y 3 + 3axy = 0, rks dy Kkr dhft,A
dx

dy
If x 3 + y 3 + 3axy = 0 , then find .
dx

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( 15 ) 231
33. sin x – cos x dk vUrjky [0, π] esa mPpre eku gS ……………. A

The maximum value of sin x – cos x in the interval [0, π] is ……………… .

34. cos 2 x
∫ 1 + sin x dx cjkcj gS ………….. A

cos 2 x
∫ 1 + sin x dx equal to ………. .

1
35. ∫ x 2 + 4 dx dk eku Kkr dhft,A

1
Evaluate ∫ 2
dx .
x +4

36. nh?kZo`Ùk 4x 2 + y 2 = 4 ds prqFkk±'k dk {ks=Qy Kkr dhft,A

Find the area of a quadrant of an ellipse 4x 2 + y 2 = 4 .

4
 d 2y  3
37.   −  dy  = 0 vodyu lehdj.k dh dksfV gS %
 dx 2   dx 
 

(A) 4 (B) 3

(C) 2 (D) 1

4
 d 2y   dy 
3
The order of differential equation   −  = 0 is :
 dx 2   dx 
 

(A) 4 (B) 3
(C) 2 (D) 1

231/ II P. T. O.

( 16 ) 231
38. nks Lora= ?kVukvksa A vkSj B dh izkf;drk,¡ Øe'k% 1 rFkk 1 gS] rks P (A ∪ B ) Kkr dhft,A
2 3

1 1
The probabilities of two independent events A and B are and
2 3
respectively. Find the probability of P (A ∪ B ) .

39. ;fn a = iˆ − 7 ˆj + 7kˆ vkSj b = 3iˆ − 2 ˆj + 2kˆ rks a × b Kkr dhft,A

If vector a = iˆ − 7 ˆj + 7kˆ and b = 3iˆ − 2 ˆj + 2kˆ, then find a × b .

40. ;fn |a | = 2,|b | = 1 vkSj a × b ,d bdkbZ lfn'k gS] rks a vkSj b ds chp dk dks.k
2

crkb,A

1
If |a | = 2,|b | = and a × b is a unit vector, then find angle between
2

a and b .

S

231/ II