MP Board Class 12 Sample Paper 2022 Maths

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Model Question - Paper

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Higher Secondary School Certificate Exam 2021 -22

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Sub : Higher Mathematics

d{kk&12oha

Class : 12

le; % 3 ?kaVs iw.kkZad % 80
Time : 3 Hours Max.Marks: 80

“””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””””

funZs'k%&
1- lHkh ç'u gy djuk vfuok;Z gaSA
2- ç'uksa ds fy, vkoafVr vad muds lEeq[k vafdr gaSA
3- ç'u Ø- 1 ls ç'u Ø- 5 rd oLrqfu"V ç'u gaSA
4- ç'u Ø- 6 ls ç'u 23 rd çR;sd ç'u esa vkarfjd fodYi fn;k x;k gaSA
5- ç'u Ø- 6 ls 15 rd çR;sd ç'u 2 vad dk gaSA
6- ç'u Ø- 16 ls 19 rd çR;sd ç'u 3 vad dk gaSA
7- ç'u Ø- 20 ls 23 rd çR;sd ç'u 4 vad dk gaSA

Instructions :

1. All the questions are compulsory.

2. Marks allotted for the questions are mentioned against them.

3. Questions from 1 to 5 are objective type questions.

4. Internal choices have been provided for the questions from 6 to 23.

5. Questions from 6 to 15 each carries 2 marks.

6. Questions from 16 to 19 each carries 3 marks.

7.Questions from 20 to 23 each carries 4 marks.

1

iz-1- lgh fodYi pqudj fyf[k;sA 𝟏×𝟔=𝟔
(i). ekuk 𝑓: 𝑅 → 𝑅, bl izdkj ifjHkkf"kr gS fd 𝑓(𝑥 ) = 3𝑥 − 4 rc 𝑓 (𝑥 ) =
(𝑎 ) (𝑏 ) − 4 (𝑐) 3𝑥 + 4 (𝑑 ) buesa ls dksbZ ughaA
(ii). ekuk leqPp; 𝑁 esa 𝑅 = {(𝑎, 𝑏): 𝑎 = 𝑏 − 2, 𝑏 > 6} }kjk iznRr laca/k gS
rc fuEu esa ls lgh mRrj pqfu,%
(𝑎) (2, 4) ∈ 𝑅 (𝑏) (3, 8) ∈ 𝑅 (𝑐 ) (6, 8) ∈ 𝑅 (𝑑 )(8, 7) ∈ 𝑅
(iii). 𝑡𝑎𝑛 √3 − 𝑠𝑒𝑐 (−2) =
(𝑎 ) 𝜋 (𝑏) − (𝑐 ) (𝑑 )
(iv). 𝐴 = 𝑎 ,d oxZ vkO;wg gS ;fn
×
(𝑎 ) 𝑚 < 𝑛 (𝑏)𝑚 > 𝑛 (𝑐 ) 𝑚 = 𝑛 (𝑑 ) buesa ls dksbZ ugha
(v). ;fn 𝐴, 3 × 3 dksfV dk O;qRØe.kh; oxZ vkO;wg gS rks |𝑎𝑑𝑗𝐴| dk eku gSA
( 𝑎 ) | 𝐴| (𝑏) |𝐴| (𝑐 ) | 𝐴 | ( 𝑑 ) 3 | 𝐴|
(vi).5 dk 𝑥 ds lkis{k vodyu xq.kakd gksxkA
(𝑎) 5 𝑙𝑜𝑔 5 (𝑏) 5 𝑙𝑜𝑔 𝑒 (𝑐 ) 5 (𝑑 )

Q.1 Choose the correct option: 𝟏×𝟔= 𝟔
(i). let 𝑓: 𝑅 → 𝑅,be defined by𝑓(𝑥) = 3𝑥 − 4then𝑓 (𝑥) =
(𝑎) (𝑏) −4 (𝑐) 3𝑥 + 4 (𝑑) None of the above
(ii). let 𝑅 be the relation in the set N given by𝑅 = {(𝑎, 𝑏): 𝑎 = 𝑏 − 2, 𝑏 > 6}.
Choose the correct answer:
(𝑎) (2, 4) ∈ 𝑅 (𝑏) (3, 8) ∈ 𝑅 (𝑐) (6, 8) ∈ 𝑅 (𝑑)(8, 7) ∈ 𝑅
(iii). 𝑡𝑎𝑛 √3 − 𝑠𝑒𝑐 (−2) =
(𝑎)𝜋 (𝑏) − (𝑐) (𝑑)
(iv). 𝐴 = 𝑎 ×
is a square matrix, if
(𝑎)𝑚 < 𝑛 (𝑏)𝑚 > 𝑛 (𝑐) 𝑚 = 𝑛 (𝑑)None of the above
(v). let 𝐴be a non singular square matrix of order 3 × 3. Then |𝑎𝑑𝑗𝐴| is equal to :
(𝑎) |𝐴| (𝑏) |𝐴| (𝑐) |𝐴| (𝑑) 3|𝐴|
(vi). Derivative of 5 with respect to𝑥is :
(𝑎)5 𝑙𝑜𝑔 5 (𝑏)5 𝑙𝑜𝑔 𝑒 (𝑐)5 (𝑑)

2

. 2. fjDr LFkkuksa dh iwfrZ dhft, %& 𝟏×𝟕=𝟕

(i). ;fn f(x) = 1 − 𝑐𝑜𝑠𝑥 gS rks f ′ dk eku…………. gSA

(ii).∫ cjkcj ………………. gSA

(iii).vody lehdj.k 2𝑥 &3 + 𝑦 = 0 dh dksfV ………….gSA

(iv). ;fn nks lfn’kksa 𝑎⃗ vkSj 𝑏⃗ ds chp dk dks.k 𝜃 gS rks 𝑎⃗. 𝑏⃗ = 𝑎⃗ × 𝑏⃗
tc 𝜃 cjkcj………. gSA
(v). lqlaxr {ks= esa dksbZ fCkUnq tks mn~ns'; Qyu dk b"Vre eku nsrk gS---------dgykrk gSA

(vi). ;fn fdlh ,d ?kVuk ds ?kfVr gksus dh lwpuk nwljh ?kVuk ds ?kfVr gksus ij
dksbZ izHkko ugha Mkyrh gS rks ,slh ?kVuk,a -------------------dgykrh gSA
(vii). lqlaxr {ks= ds ckg~; Hkkx ds fdlh Hkh fcUnq dks…………… dgrs gSA

Q. 2. Fill in the blanks: 𝟏×𝟕= 𝟕

(i). If f(x) = 1 − cosxThen value off ′ is………….

(ii). ∫ is equal to……………….

(iii). Order of a differential equation 2𝑥 &3 + 𝑦 = 0 is………….

(iv). If angle between two vectors𝑎⃗ 𝑎𝑛𝑑 𝑏⃗ is 𝜃 then 𝑎⃗. 𝑏⃗ = 𝑎⃗ × 𝑏⃗
when 𝜃 is equal to……….
(v). Any point in feasible region that gives optimal value of the objective functions
is called……
(vi). If two events such that the probability of occurrence of one of them is not
affected by occurrence of the other; are called……………
(vii). Any point outside the feasible region is an……………

3

. 3. lgh tksM+h feykb, & 𝟏×𝟔= 𝟔
(i) ∫ (a) 𝑡𝑎𝑛 +𝑐
√

(ii) ∫ (b) 𝑙𝑜𝑔 +𝑐
√

(iii) ∫ (c) si𝑛 +𝑐

(iv) ∫ (d) 𝑙𝑜𝑔 𝑥 + √𝑥 − 𝑎 +𝑐

(v) ∫ √𝑥 + 𝑎 (e) 𝑥√𝑎 − 𝑥 + 𝑠𝑖𝑛 +𝑐

(vi) ∫ √𝑎 − 𝑥 (f) 𝑥√𝑥 + 𝑎 + 𝑙𝑜𝑔 𝑥 + √𝑥 + 𝑎 +𝑐

Q. 3. Match the columns :𝟏 × 𝟔 = 𝟔
(i) ∫ (a) 𝑡𝑎𝑛 +𝑐
√

(ii) ∫ (b) 𝑙𝑜𝑔 +𝑐
√

(iii) ∫ (c) si𝑛 +𝑐

(iv) ∫ (d) 𝑙𝑜𝑔 𝑥 + √𝑥 − 𝑎 +𝑐

(v) ∫ √𝑥 + 𝑎 (e) 𝑥√𝑎 − 𝑥 + 𝑠𝑖𝑛 +𝑐

(vi) ∫ √𝑎 − 𝑥 (f) 𝑥√𝑥 + 𝑎 + 𝑙𝑜𝑔 𝑥 + √𝑥 + 𝑎 +𝑐

4

. 4. ,d ’kCn@okD; esa mRrj nhft, %& 𝟏×𝟕=𝟕
6−𝑥 4
(i). 𝑥 ds fdl eku ds fy, vkO;wg ,d vO;qRØe.kh; vkO;wg gksxkA
3−𝑥 1
(ii). js[kk 𝑦 = 𝑚𝑥 + 1 odz 𝑦 = 4𝑥 dh Li’kZZ js[kk gS rks 𝑚 dk eku D;k gS \

(iii). ∫ x e dx dk eku fyf[k,&

(iv) 𝚤̂ × 𝚥̂ × 𝑘 + 𝚥̂. 𝚤̂ × 𝑘 + 𝑘. (𝚤̂ × 𝚥̂) dk eku D;k gS \

(v). 𝑦𝑧 −lery dk lehdj.k dk lehdj.k fyf[k,A

(vi). ,d ikals dk tksMk+ mNkyk tkrk gS rks izR;sd ikals ij le vHkkT; la[;k izkIr
djus dh izkf;drk D;k gS \
(vii) ;fn P(A) = , P(B) = 0 rks P(A/B) dk eku D;k gS \

Q. 4. Answer in one word / sentence: 𝟏×𝟕=𝟕

6−𝑥 4
(i). For what value of 𝑥, will the matrix be Singular.
3−𝑥 1
(ii). Find the value of 𝑚 , if the line 𝑦 = 𝑚𝑥 + 𝑐 is a tangent

to curve 𝑦 = 4𝑥.

(iii). Find the value of ∫ 𝑥 𝑒 𝑑𝑥

(iv). Find the value of 𝚤̂ × 𝚥̂ × 𝑘 + 𝚥̂. 𝚤̂ × 𝑘 + 𝑘 . (𝚤̂ × 𝚥̂)

(v). Write equation of a 𝑦𝑧-plane

(vi). When a pair of dice is rolled, what is the probability of obtaining

an even prime number on each die.

(vii) find P(A/B),if P(A) = and P(B) = 0

5

. 5. fuEufyf[kr esa ls lR;@vlR; fyf[k, : 𝟏×𝟔 = 𝟔
(i). ;fn 𝑓: 𝑅 → 𝑅, 𝑓(𝑥) = (3 − 𝑥 ) }kjk iznRr gS rks (𝑓𝑜𝑓) = 𝑥

(ii). 𝑐𝑜𝑠 𝑥 dk izkar 𝑅 − (−1, 1) gSA

(iii). ;fn cos ∝ , 𝑐𝑜𝑠𝛽, 𝑐𝑜𝑠𝛾 fdlh js[kk dh fnd dksT;k,a gks

rks 𝐶𝑜𝑠 ∝ +𝐶𝑜𝑠 𝛽 + 𝐶𝑜𝑠 𝛾 =

(iv). odz 𝑦 = 4𝑥 , 𝑦 −v{k ,oa js[kk 𝑦 = 3, ls f?kjs {ks= dk {ks=Qy oxZ bdkbZ gSA

(v). f=Hkqt dh rhuksa Hkqtkvksa dks dze esa ysus ij mudk lfn’k ;ksx 0 gksrk gSA

(vi). lery 2𝑥 − 𝑦 + 4𝑧 = 5 vkSj 5𝑥 − 2.5𝑦 + 10𝑧 = 6 ijLij lekarj gS

Q. 5. Write True / False for the following sentences: 𝟏×𝟔 = 𝟔

(i). If 𝑓: 𝑅 → 𝑅 𝑏𝑒 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓(𝑥) = (3 − 𝑥 ) then (𝑓𝑜𝑓) = 𝑥

(ii). 𝐷𝑜𝑚𝑎𝑖𝑛 of 𝑐𝑜𝑠 𝑥 is 𝑅 − (−1, 1)

(iii). 𝐼𝑓 𝑐𝑜𝑠 ∝ , 𝑐𝑜𝑠𝛽, 𝑐𝑜𝑠𝛾 𝑎𝑟𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑖𝑛𝑒𝑠 𝑜𝑓𝑎 𝑙𝑖𝑛𝑒

𝑡ℎ𝑒𝑛 𝐶𝑜𝑠 ∝ + 𝐶𝑜𝑠 𝛽 + 𝐶𝑜𝑠 𝛾 =

(iv). Area of region bounded by the curve 𝑦 = 4𝑥, 𝑦 − 𝑎𝑥𝑖𝑠

and the line 𝑦 = 3 is square units.

(v). The vector sum of the three sides of a triangle taken in order is 0.

(vi). The planes : 2𝑥 − 𝑦 + 4𝑧 = 5 and 5𝑥 − 2.5𝑦 + 10𝑧 = 0 are parallel.

.6. ;fn 𝐴 = {1, 2, 3, } , 𝐵 = {4, 5, 6, 7} rFkk 𝑓 = {(1, 4)(2, 5) (3, 6)}

𝐴 ls 𝐵 ij ,d Qyu gS rks fn[kkb, fd 𝑓 ,dSdh gSA 2

6

If A = {1, 2, 3}, B = {4, 5, 6, 7} and let 𝑓 = {(1, 4)(2, 5) (3, 6)} be a

function from A to B. Show that f is one-one.

vFkok (or)
fl) dhft, fd leqPp; {1, 2, 3} esa 𝑅 = {(1, 2)(2, 1)} }kjk iznRr laca/k 𝑅 lefer gS

Show that the relation in the set {1, 2, 3} given by𝑅 = {(1, 2)(2, 1)} is

symmetric.

𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 −𝑐𝑜𝑠𝜃
. 7.ljy dhft, 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 2
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 −𝑐𝑜𝑠𝜃
Simplify 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃

vFkok (or)
3 2 1 0
𝑋 dk eku Kkr dhft, ;fn 𝑌 = vkSj 2𝑋 + 𝑌 =
1 4 −3 2

3 2 1 0
Find the value of 𝑋 𝑖𝑓 𝑌 = and 2𝑋 + 𝑌 =
1 4 −3 2

𝑥 sin ; 𝑥≠0
.8.𝑘 dk eku Kkr dhft, ;fn Qyu 𝑓 (𝑥 ) = 𝑥 = 0 ij lrr~ gSA
𝑘 ; 𝑥=0
𝑥 sin ; 𝑥≠0
Find the value of 𝑘, if function 𝑓(𝑥) = 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥 = 0
𝑘 ; 𝑥=0
vFkok (or) 2
D;k Qyu 𝑓(𝑥 ) = 𝑥 − 𝑠𝑖𝑛𝑥 + 5 , 𝑥 = 𝜋 ij lrr~ gS \

Is function 𝑓(𝑥) = 𝑥 − 𝑠𝑖𝑛𝑥 + 5, 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥 = 𝜋

.9. o`Rr ds {ks=Qy ds ifjorZu dh nj bldh r`T;k 𝑟 ds Kkr dhft, tcfd 𝑟 = 5𝑐𝑚 2

7

Find the rate of change of area of a circle with respect to 𝑟

when 𝑟 = 5𝑐𝑚

vFkok (or)

odz + = 1 ij mu fcUnqvksa dks Kkr dhft, ftu ij Li’kZ js[kk,a 𝑥 −v{k ds lekUrj gS

Find points on the curve + = 1 at which the tangents are

parallel to 𝑥 − 𝑎𝑥𝑖𝑠

( )
.10. ∫ 𝑑𝑥 dk eku Kkr dhft,A 2

( )
Find the value of ∫ 𝑑𝑥

vFkok (or)
√
∫ 𝑑𝑥 dk eku Kkr dhft,A
√

√
Find the vaue of ∫ 𝑑𝑥
√

.11. ∫ 𝑆𝑖𝑛 𝑥 .𝐶𝑜𝑠 𝑥 𝑑𝑥dk eku Kkr dhft,A 2

Find the vaue of∫ 𝑆𝑖𝑛 𝑥 .𝐶𝑜𝑠 𝑥 𝑑𝑥

vFkok (or)

∫ 𝐶𝑜𝑠 𝑥 𝑑𝑥 dk eku Kkr dhft,A

Find the vaue of ∫ 𝐶𝑜𝑠 𝑥 𝑑𝑥

.12. lfn’k 𝑎⃗ = 𝚤̂ + 𝚥̂ + 2𝑘 ds vuqfn’k ,d ek=d lfn’k fyf[k,A 2

Find unit vector in the direction of the vector 𝑎⃗ = 𝚤̂ + 𝚥̂ + 2𝑘
8

vFkok (or)
lfn’kksa 𝚤̂ ± 2𝚥̂ + 3𝑘 ] vkSj 3𝚤̂ − 2𝚥̂ + 𝑘 ds chp dk dks.k Kkr dhft,A

Find angle between vectors 𝚤̂ + 2𝚥̂ + 3𝑘 and 3𝚤̂ − 2𝚥̂ + 𝑘

.13. lery 2𝑥 − 3𝑦 + 4𝑧 − 6 = 0 dh ewy fcUnq ls nwjh Kkr dhft,A 2

Find the distance of a plane 2𝑥 − 3𝑦 + 4𝑧 − 6 = 0 from the origin.

vFkok (or)

js[kk = = dk lfn’k lehdj.k Kkr dhft, A

Find the equation of line = = in vector form.

.14. ml lery dk lfn’k lehdj.k Kkr dhft, tks ewy fcUnq ls 7 ek=d nwjh ij 2

vkSj lfn’k 3𝚤̂ + 5𝚥̂ − 6𝑘 ij yEc gSA

Find the vector equation of the plane which is at a distance

of 7 units from the origin and its normal vector from

the origin is 3𝚤̂ + 5𝚥̂ − 6𝑘 .

vFkok (or)
ml lery dk lehdj.k Kkr dhft, tks 𝑥, 𝑦 vkSj 𝑧 −v{kksa ij dze'k% 2] 3 vkSj
4 var% [k.M dkVrk gSA

Find the equation of the plane with intercepts 2, 3 and 4 on the

𝑥, 𝑦 𝑎𝑛𝑑 𝑧 − 𝑎𝑥𝑖𝑠 respectively

.15. P(A/B)dk eku Kkr dhft, ;fn P(B) = 0.5 vkSj P(A ∩ B) = 0.32 2

Find value of P(A/B) if P(B)=0.5 andP(A ∩ B) = 0.32
9

vFkok(or)
;fn𝐴 vkSj 𝐵 Lora= ?kVuk,a nh xbZ gS tgka 𝑃(A ) = 0.3, 𝑃(B ) = 0.6 rks 𝑃(A ∩ 𝐵) Kkr
dhft,A

Find value ofP(A ∩ B) ,if𝑃(A ) = 0.3, 𝑃(B ) = 0.6and if A and B

are independent events.

.16. 𝑡𝑎𝑛 2𝑥 + 𝑡𝑎𝑛 3𝑥 = dks gy dhft, 3

Solve 𝑡𝑎𝑛 2𝑥 + 𝑡𝑎𝑛 3𝑥 =

vFkok (or)

-fln~/k dhft, fd 2𝑡𝑎𝑛 + 𝑡𝑎𝑛 = 𝑡𝑎𝑛

Prove that 2𝑡𝑎𝑛 + 𝑡𝑎𝑛 = 𝑡𝑎𝑛

2 3
1 −2 3
.17. ;fn𝐴 = vkSj 𝐵 = 4 5 gS n'kkZb, fd (𝐴𝐵) = 𝐵′𝐴′ 3
−4 2 5
2 1
2 3
1 −2 3
If𝐴 = 𝑎𝑛𝑑 𝐵 = 4 5 then show that (𝐴𝐵) = 𝐵′𝐴′
−4 2 5
2 1
vFkok(or)
25
oxZ vkO;wg dks lefer ,oa fo"ke lefer vkO;wgksa ds ;ksx ds :i esa iznf'kZr dhft,A
3
−6
5 2
Express the matrix as the sum of a symmetric and a skew
3 −6

symmetric matrix.

.18. Qyu sin 𝑥 + cos 𝑥 dk egÙke eku Kkr djksA 3

Find the maximum value of the function : sin 𝑥 + cos 𝑥

vFkok (or)
10

oØ 𝑥 = 1 − 𝑐𝑜𝑠𝜃 rFkk 𝑦 = 𝜃 − 𝑠𝑖𝑛𝜃 gS rks 𝜃 = ij vfHkyEc dk lehdj.k Kkr
dhft,

Find the equation of normal of the curve 𝑥 = 1 − 𝑐𝑜𝑠𝜃 and

𝑦 = 𝜃 − 𝑠𝑖𝑛𝜃 at 𝜃 =

.19. fuEufyf[kr O;ojks/kksa ds varxZr] 𝑍 = 3𝑥 + 2𝑦 dk U;wurehdj.k dhft,A 3

𝑥 + 𝑦 ≥ 8 , 3𝑥 + 5𝑦 ≤ 15 , 𝑥 ≥ 0, 𝑦 ≥ 0,

Minimise 𝑍 = 3𝑥 + 2𝑦, subject to the constraints :

𝑥 + 𝑦 ≥ 8 , 3𝑥 + 5𝑦 ≤ 15 , 𝑥 ≥ 0, 𝑦 ≥ 0,
vFkok (or)
fuEu O;ojks/kksa ds varxZr] 𝑍 = 3𝑥 + 4𝑦 dk vf/kdrehdj.k dhft,A
𝑥+𝑦 ≤4, 𝑥 ≥ 0, 𝑦 ≥ 0,

Maximise 𝑍 = 3𝑥 + 4𝑦 ,subject to the constraints :

𝑥 + 𝑦 ≤ 4 , 𝑥 ≥ 0, 𝑦 ≥ 0,
.20. xq.k/keksZ dk iz;ksx djds fln~/k dhft, fd 4
1+𝑎 1 1
1 1+𝑏 1 = 𝑎𝑏𝑐( 1 + + + )
1 1 1+𝑐

Applying the properties of determinants;

1+𝑎 1 1
and prove that : 1 1+𝑏 1 = 𝑎𝑏𝑐( 1 + + + )
1 1 1+𝑐
vFkok (or)
1 3 3
;fn 𝐴 = 1 4 3 gks rks lR;kfir dhft, fd A.(adjA) = |𝐴|. 𝐼
1 3 4

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1 3 3
𝐼𝑓𝐴 = 1 4 3 then prove that : A.(adjA) =|𝐴|. 𝐼
1 3 4
.21. 𝑥 ds lkis{k Qyu 𝑥 + (sin 𝑥) + dk vodyu dhft,A 4

Differentiate the function∶ 𝑥 + (sin 𝑥) + with respect to 𝑥
vFkok(or)
;fn𝑥 = 𝑎(𝜃 + 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃) gS rks Kkr dhft,A
Find if 𝑥 = 𝑎 (𝜃 + 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎 (1 − 𝑐𝑜𝑠𝜃)

.22. nh?kZo`Rr + = 1 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 4

Find the area enclosed by the ellipse + =1
vFkok (or)
js[kk 𝑥 = }kjk o`Rr 𝑥 + 𝑦 = 𝑎 ds NksVs Hkkx dk {ks=Qy Kkr dhft,A
√

Find the area of the smaller part of the circle 𝑥 + 𝑦 = 𝑎 cut off by line
𝑥=
√

.23. vody lehdj.k (𝑥 − 𝑦 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦 = 0dk O;kid gy Kkr dhft,A 4

Find the general solution of the differential equation ;
(𝑥 − 𝑦 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦 = 0
vFkok(or)
vody lehdj.k (𝑡𝑎𝑛 𝑦 − 𝑥)𝑑𝑦 + (1 + 𝑦 )𝑑𝑥 = 0 dks gy dhft,
Solve the differential equation : (𝑡𝑎𝑛 𝑦 − 𝑥 )𝑑𝑦 + (1 + 𝑦 )𝑑𝑥 = 0

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