MP Board Class 12 Sample Paper 2023 Maths

vH;kl gsrq lsEiy iz’ui=
ek-f’k eaMy e-iz-Hkksiky
gk;j lsdsUMjh ijh{kk&2022&23
Higher Secondary Examination (main) 2022-23
mPp xf.kr
HIGHER MATHEMATICS
(Hindi & English Versions)
Time: 3 Hours Maximum Marks: 80

funsZ'k&%
(i) lHkh iz'u vfuok;Z gSaA
(ii) iz'u Øekad1 ls 5 rd ds izR;sd miiz'u ij 1&1 vad fu/kkZfjr gSaA
(iii) iz'u Øekad 6 ls 15 rd izR;sd 2 vad dk gSA
a
(iv) iz'u Øekad 16 ls 19 rd izR;sd 3 vad dk gSA a
(v) iz'u Øekad 20 ls 23 rd izR;sd 4 vad dk gSA a
Instructions:
(i) All questions are compulsory.
(ii) Sub question of question Nos 1 to 5 carry 1 marks each.
(iii) Question Nos 6 to 15 carry 2 marks each.
(iv) Question Nos 16 to 19 carry 3 marks each.
(v) Question Nos 20 to 23 carry 4 marks each.

1) lgh fodYi pqudj fyf[k;s % 1x6=6
𝟕𝝅
(i) 𝐬𝐢𝐧−𝟏 (𝐬𝐢𝐧 ) dk eku gS %
𝟔
𝝅 𝝅
(a) 𝟐 (b) 𝟔
−𝝅 −𝝅
(c) (d)
𝟐 𝟔

(ii) 𝐭𝐚𝐧−𝟏 (𝟏) − 𝐜𝐨𝐭 −𝟏 (−𝟏) dk eku gS %
𝝅 −𝝅
(a) (b)
𝟐 𝟐
−𝟑𝝅 𝟑𝝅
(c) (d)
𝟐 𝟐

(iii) ;fn 𝑨 vkSj 𝑩 nks mi;qDr dksfV ds vkO;qg gS rc (𝑨𝑩)′ cjkcj gS %
(a) 𝑨′ 𝑩′ (b) 𝑨′ −𝑩′
(c) 𝑨′ +𝑩′ (d) 𝑩′ 𝑨′
(iv) ;fn 𝑨 vkSj 𝑩 Lora= ?kVuk,a gSa rFkk 𝑷(𝑨) = 𝟎. 𝟑 , 𝑷(𝑩) = 𝟎. 𝟒 rc 𝑷(𝑨 ∩ 𝑩) dk eku gS %
𝟕 𝟑
(a) (b)
𝟏𝟎 𝟐𝟓
𝟔 𝟑
(c) 𝟐𝟓 (d) 𝟏𝟎
𝒅𝒚
(v) vody lehdj.k 𝒅𝒙 + 𝑷𝒚 = 𝑸 esa lekdyu xq.kkad (𝑰𝑭) dk eku gS %
(a) 𝒆∫ 𝑷𝒅𝒙 (b) 𝒆∫ 𝑷𝒅𝒚
(c) 𝒆∫ 𝑸𝒅𝒙 (d) 𝒆∫ 𝑸𝒅𝒚
(vi) ;fn js[kk dh fnd~dksT;k,a 𝒍, 𝒎, 𝒏 gSa] rks a 𝒍, 𝒎, 𝒏 ds chp lgh laca/k gksxk %
(a) 𝒍𝟐 + 𝒎𝟐 + 𝒏𝟐 = 1 (b) 𝒍 + 𝒎 + 𝒏 = 𝟏
𝟐 𝟐 𝟐
(c) 𝒍 + 𝒎 + 𝒏 = 0 (d) 𝒍 + 𝒎 + 𝒏 = 𝟎

Choose and write the correct options %
𝟕𝝅
(i) The value of 𝐬𝐢𝐧−𝟏 (𝐬𝐢𝐧 𝟔 ) is %
𝝅 𝝅
(a) (b)
𝟐 𝟔
−𝝅 −𝝅
(c) (d)
𝟐 𝟔

(ii) The value of 𝐭𝐚𝐧−𝟏 (𝟏) − 𝐜𝐨𝐭 −𝟏 (−𝟏) is %
𝝅 −𝝅
(a) 𝟐 (b) 𝟐
−𝟑𝝅 𝟑𝝅
(c) (d)
𝟐 𝟐

(iii) If A and B are two matrices of suitable order then(𝑨𝑩)′ is equal
(a) 𝑨′ 𝑩′ (b) 𝑨′ −𝑩′
(c) 𝑨′ +𝑩′ (d) 𝑩′ 𝑨′
(iv) If 𝑨 and 𝑩 are independent event and 𝑷(𝑨) = 𝟎. 𝟑 , 𝑷(𝑩) = 𝟎. 𝟒 than value of
𝑷(𝑨 ∩ 𝑩) is %
𝟕 𝟑
(a) 𝟏𝟎 (b) 𝟐𝟓
𝟔 𝟑
(c) 𝟐𝟓 (d) 𝟏𝟎
𝒅𝒚
(v) The value of integrating factor (𝑰𝑭) in Differential equation 𝒅𝒙 + 𝑷𝒚 = 𝑸 is %
(a) 𝒆∫ 𝑷𝒅𝒙 (b) 𝒆∫ 𝑷𝒅𝒚
(c) 𝒆∫ 𝑸𝒅𝒙 (d) 𝒆∫ 𝑸𝒅𝒚
(vi) If 𝒍, 𝒎, 𝒏 aare direction Cosine of a line then correct relation between 𝒍, 𝒎, 𝒏 will be
(a) 𝒍𝟐 + 𝒎𝟐 + 𝒏𝟐 =1 (b) 𝒍 + 𝒎 + 𝒏 = 𝟏
𝟐 𝟐 𝟐
(c) 𝒍 + 𝒎 + 𝒏 =0 (d) 𝒍 + 𝒎 + 𝒏 = 𝟎

2) fjDr LFkkuksa dh iwfrZ dhft,A 1x7=7
(i) dksbZ Qyu 𝒇: 𝒙 → 𝒚 ,dSdh Qyu gksxk ;fn 𝒇(𝒙𝟏 ) = 𝒇(𝒙𝟐 ) rc ---------------------------------------A
(ii) 𝐜𝐨𝐬 −𝟏 (𝒙) dk eq[; eku -----------------------------esa fLFkr gksxkA
(iii) vkO;wg 𝑨 = [𝒂𝒊𝒋 ] ds fy, 𝑨′ =...................A
𝒎×𝒏
(iv) 𝒄𝒐𝒔√𝒙 dk 𝒙 ds lkis{k vodyt --------------------------------gksxkA
(v) ;fn Qyu 𝒇: 𝒙 → 𝒚 eas 𝒇 vkPNknd Qyu gks rks mlds fy, lg izkar dk eku ifjlj ds ---------------gksxkA
(vi) lfn'k esa 𝒂⃗ = 𝟐𝒊̂ +̇ 𝟑𝒋̂ + 𝒌
̂ ds vuqfn'k ek=d lfn'k --------------------------gksxkA
(vii) vkO;wg 𝑨 vkSj 𝑩 ,d nwljs ds O;qRdze gksx a s ;fn 𝑨𝑩 = 𝑩𝑨 =. . . . . . . ..

Fill in the blanks:
(i) A function 𝒇: 𝒙 → 𝒚 will be one one function when 𝒇(𝒙𝟏 ) = 𝒇(𝒙𝟐 ) implies ------------------------------------
---
(ii) The Principal value of 𝐜𝐨𝐬 −𝟏 (𝒙) lie in -----------------------------
(iii) The value of 𝑨′ when matrix 𝑨 = ⌈𝒂𝒊𝒋 ⌉ -----------------------
𝒎×𝒏
(iv) Differentiation of 𝒄𝒐𝒔√𝒙 with respect to 𝒙 will be--------------------------------
(v) If function 𝐟: 𝐱 → 𝐲 is onto function then relation between codomaln and
range .................
(vi) The unit vector along vector 𝒂 ⃗ = 𝟐𝒊̂ +̇ 𝟑𝒋̂ + 𝒌
̂ is.......
(vii) Matrices A and 𝑩 will be inverse to each other if 𝑨𝑩 = 𝑩𝑨 = ........

3) lgh tksM+h cukb, % 1x6=6
LraHk ′𝑨′ LraHk ′𝑩′
𝒅𝒙
(i) ∫ 𝟐 𝟐 (a) 𝒍𝒐𝒈 |𝐬𝐞𝐜 𝒙| + 𝑪
𝒙 +𝒂
𝟏 𝒙
(ii) ∫ 𝒕𝒂𝒏𝒙 𝒅𝒙 (b) 𝒂 𝐭𝐚𝐧−𝟏 𝒂 + 𝑪
𝒅𝒙
(iii) ∫ 𝒂𝟐 −𝒙𝟐 (c) 𝒆𝒙 (𝒙 − 𝟏) + 𝑪
𝟏 𝒂+𝒙
(iv) ∫ 𝒙𝒆𝒙 𝒅𝒙 (d) 𝒍𝒐𝒈 |𝒂−𝒙| + 𝑪
𝟐𝒂
𝒅𝒙 𝟏 𝟏
(v) ∫ 𝒙𝟐 −𝒂𝟐 (e) (𝒙 − 𝟐 𝐬𝐢𝐧 𝟐𝒙) + 𝑪
𝟐
𝟏 𝒙−𝒂
(vi) ∫ 𝒔𝒊𝒏𝟐 𝒙 𝒅𝒙 (f) 𝟐𝒂 𝒍𝒐𝒈 |𝒙+𝒂| + 𝑪
(g) 𝐥𝐨𝐠 |𝒄𝒐𝒔𝒆𝒄 𝒙| + 𝑪
Match the correct column :
𝑪𝒐𝒍𝒖𝒎𝒏 ′𝑨′ 𝑪𝒐𝒍𝒖𝒎𝒏 ′𝑩′
𝒅𝒙
(i) ∫ 𝒙𝟐 +𝒂𝟐 (a) 𝒍𝒐𝒈 |𝐬𝐞𝐜 𝒙| + 𝑪
𝟏 𝒙
(ii) ∫ 𝒕𝒂𝒏𝒙 𝒅𝒙 (b) 𝒂 𝐭𝐚𝐧−𝟏 𝒂 + 𝑪
𝒅𝒙
(iii) ∫ 𝒂𝟐 −𝒙𝟐 (c) 𝒆𝒙 (𝒙 − 𝟏) + 𝑪
𝟏 𝒂+𝒙
(iv) ∫ 𝒙. 𝒆𝒙 𝒅𝒙 (d) 𝒍𝒐𝒈 |𝒂−𝒙| + 𝑪
𝟐𝒂
𝒅𝒙 𝟏 𝟏
(v) ∫ 𝒙𝟐 −𝒂𝟐 (e) (𝒙 − 𝟐 𝐬𝐢𝐧 𝟐𝒙) + 𝑪
𝟐
𝟏 𝒙−𝒂
(vi) ∫ 𝒔𝒊𝒏𝟐 𝒙 𝒅𝒙 (f) 𝟐𝒂 𝒍𝒐𝒈 |𝒙+𝒂| + 𝑪
(g) 𝐥𝐨𝐠 |𝒄𝒐𝒔𝒆𝒄 𝒙| + 𝑪

4) izR;sd dk ,d 'kCn@okD; esa mRRkj fyf[k;s % 1x7=7
(i) 𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 dk eku fyf[k, tcfd 𝒙𝒚 > 𝟏, 𝒙 > 𝟎 𝒚 > 𝟎
(ii) 𝑨 = [𝒂𝒊𝒋 ]
𝒎×𝒏
,d oxZ vkO;wg gS rks 𝒎 o 𝒏 ds e/; laca/k fyf[k, A
𝐜𝐨𝐬 𝜽 − 𝐬𝐢𝐧 𝜽
(iii) | | dk eku fyf[k, A
𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽
𝒅𝒚
(iv) ;fn 𝒙 − 𝒚 = 𝝅 rks dk eku fyf[k, A
𝒅𝒙
𝒅𝒙
(v) + 𝒙 = 𝐜𝐨𝐬 𝒚 eas Lora= pj fyf[k,A
𝒅𝒚
𝒅𝟒 𝒚
(vi) vody lehdj.k + 𝐬𝐢𝐧 𝒙 = 𝟎 dh dksfV fyf[k, A
𝒅𝒙𝟒
(vii) 𝒇(𝒙) = 𝐜𝐨𝐬 𝒙 varjky [𝟎 𝝅] ds e/; fdl izdkj dk Qyu gS o/kZeku vFkok âklekuA
Write Answer in one word / sentance of each
(i) Write the value of 𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 when 𝒙𝒚 > 𝟏, 𝒙 > 𝟎 𝒚 > 𝟎
(ii) Write the relation between 𝒎 and 𝒏. If 𝑨 = [𝒂𝒊𝒋 ] is a square matrix
𝒎×𝒏
𝐜𝐨𝐬 𝜽 −𝐬𝐢𝐧 𝜽
(iii) Write the value of | |
𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽
𝒅𝒚
(iv) If 𝒙 − 𝒚 = 𝝅 then write the value of 𝒅𝒙
𝒅𝒙
(v) Write independent variable in differential equation 𝒅𝒚 + 𝒙 = 𝐜𝐨𝐬 𝒚
𝒅𝟒 𝒚
(vi) Write the order of differential equation 𝒅𝒙𝟒 + 𝐬𝐢𝐧 𝒙 = 𝟎
(vii) Which types of given function 𝒇(𝒙) = 𝐜𝐨𝐬 𝒙 in [𝟎 𝝅] increasing or decreasing.
5) fuEufyf[kr esa lR;@vlR; fyf[k, % 1x6=6
(i) ;fn fdlh lkjf.kd esa dksbZ nks LrEHk ds vo;o leku gks rks lkjf.kd dk eku 'kwU; gksxkA
(ii) Qyu 𝒇(𝒙) varjky 𝑰 esa o/kZeku Qyu gksxkA
tcfd 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) > 𝒇(𝒙𝟐 ) 𝒙𝟏, , 𝒙𝟐 ∈ 𝑰

𝑨
(iii) ;fn ?kVukvksa 𝑨 o 𝑩 ds fy, 𝑨 ⊂ 𝑩 gks rks 𝑷 ( ) = 𝑷(𝑨)
𝑩
(iv) ;fn 𝒂 ⃗ rks 𝒂
⃗ = 𝒌𝒃 ⃗ , ⃗𝒃 lajs[k lfn'k gksxA
sa
(v) leku ifjek.k okys nks lfn'k lnSo leku lfn'k gksrs gSaA
(vi) 𝒇(𝒙) = 𝒙𝟐 , 𝒇: 𝑵 → 𝑵 ds fy, Qyu ,dSd rFkk vkPNknd gSA
Write True/False in the following %
(i) If two column are identical then the value of determinant is zero.
(ii) A function 𝒇(𝒙) will be an increasing function in given interval 𝑰
when 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) > 𝒇(𝒙𝟐 ) 𝒙𝟏, , 𝒙𝟐 ∈ 𝑰
𝑨
(iii) If for two events 𝑨 and 𝑩 , 𝑨 ⊂ 𝑩 then 𝑷 (𝑩) = 𝑷(𝑨)
(iv) If 𝒂⃗ = 𝒌𝒃⃗ then 𝒂 ⃗ are collinear.
⃗, 𝒃
(v) Two vector having same magnitude is always equal to each other.
(vi) Function 𝒇(𝒙) = 𝒙𝟐 𝒇𝒐𝒓 𝒇: 𝑵 → 𝑵 is one –one onto. `

6) fn[kkb, fd leqPPk; {𝟏, 𝟐, 𝟑} esa iznŸk laca/k 𝑹 = {(𝟏, 𝟐), (𝟐, 𝟏)} ,d LorqY; laca/k ugha gSA 2
Show that the relation R given by 𝑹 = {(𝟏, 𝟐), (𝟐, 𝟏)} in the set {𝟏, 𝟐, 𝟑} is not a reflexive
relation.
fn[kkb, fd okLrfod la[;k ds leqPp; esa 𝑹 = {(𝒂, 𝒃) ∶ 𝒂 ≤ 𝒃} ,d lefer laca/k ugha gSA
Show that the relation esa 𝑹 = {(𝒂, 𝒃) ∶ 𝒂 ≤ 𝒃} in the set of real numbers is not a
symmetric relation
𝟏 𝟏 𝝅
7) fn[kkb, fd 𝐭𝐚𝐧−𝟏 (𝟐) + 𝐭𝐚𝐧−𝟏 (𝟑) = 𝟒 2
𝟏 𝟏 𝝅
Show that 𝐭𝐚𝐧−𝟏 (𝟐) + 𝐭𝐚𝐧−𝟏 (𝟑) = 𝟒
𝟏
𝐜𝐨𝐭 −𝟏 ( ) , 𝒙 > 𝟏 dks blds ljyre :i eas fyf[k,A
√𝒙𝟐 −𝟏
𝟏
Write 𝐜𝐨𝐭 −𝟏 ( ) , 𝒙 > 𝟏 in its simplest from.
√𝒙𝟐 −𝟏

8) 𝑿 rFkk 𝒀 Kkr dhft, ;fn 2
𝟗 𝟎 𝟐 𝟎
𝑿+𝒀= [ ] rFkk 𝑿 − 𝒀 = [ ]
𝟐 𝟕 𝟎 𝟐
Find 𝑿 and 𝒀 if
𝟗 𝟎 𝟐 𝟎
𝑿+𝒀 = [ ] and 𝑿 − 𝒀 = [ ]
𝟐 𝟕 𝟎 𝟐
;fn 𝑨 = [𝟐 𝟒] rFkk 𝑩 = [−𝟐 𝟓] rks 𝟑𝑨 − 𝑩 Kkr dhft,A
𝟑 𝟐 𝟑 𝟒

𝟐 𝟒 −𝟐 𝟓
If 𝑨 = [ ] and 𝑩= [ ] then find 𝟑𝑨 − 𝑩
𝟑 𝟐 𝟑 𝟒

9) 𝒆√𝒙 dk 𝒙 ds lkis{k vodyu dhft,A 2
Differentiate 𝒆√𝒙 with respect to 𝒙
𝐥𝐨𝐠(𝐜𝐨𝐬 𝒙) dk 𝒙 ds lkis{k vodyu dhft,A
Differentiate 𝐥𝐨𝐠(𝐜𝐨𝐬 𝒙) with respect to 𝒙
10) fn[kkb, fd iznŸk Qyu 𝒇 , 𝒇(𝒙) = 𝟑𝒙 + 𝟐, 𝒙𝝐𝑹, 𝑹 ij o/kZeku Qyu gSA 2
Show that the function 𝒇 given by 𝒇(𝒙) = 𝟑𝒙 + 𝟐, 𝒙𝝐𝑹, is increasing function on 𝑹.

o`Ÿk ds {ks=Qy ds ifjorZu dh nj bldh f=T;k 𝒓 ds lkis{k Kkr dhft, tc 𝒓 = 𝟕 lseh- gS
Find the rate of change of the area of a circle with respect to its radius 𝒓 when 𝒓 = 𝟕 𝒄𝒎.
𝟏
11) gok ds ,d cqycys dh f=T;k [email protected] dh nj ls c<+ jgh gSA cqycqys dk vk;ru fdl nj ls c<+
𝟓
jgk gS tcfd f=T;k 𝟐 lseh- gSA 2
𝟏
The radius of an air bubble is increasing at the rate of 𝟓 𝒄𝒎/𝒔𝒆𝒄. At what rate ,the
volume of the bubble increasing when the radius is 𝟐 𝒄𝒎.
fdlh o`Ÿk dh f=T;k 𝟎. 𝟐 [email protected] dh nj ls c<+ jgh gS blds ifjf/k dh o`f) nj Kkr dhft,A
The radius of a circle is increasing at rate of 𝟎. 𝟐 𝒄𝒎/𝒔𝒆𝒄. What is the rate of increase of
its circumference.

12) eku Kkr dhft,& ∫(𝐬𝐢𝐧−𝟏 𝒙 + 𝐜𝐨𝐬 −𝟏 𝒙)𝒅𝒙 2
Evalute ∫ (𝐬𝐢𝐧−𝟏 𝒙 + 𝐜𝐨𝐬 −𝟏
𝒙)𝒅𝒙
eku Kkr dhft,& ∫(𝟏 − 𝒙)√𝒙 𝒅𝒙
Evalute ∫(𝟏 − 𝒙)√𝒙 𝒅𝒙
13) ;fn ,d js[kk ds fnd~vuqikr −𝟑, 𝟐, −𝟏 gSa] rks bldh fnd~dkslkbu Kkr dhft,A 2
If a line has direction ratios −𝟑, 𝟐, −𝟏 then determine its direction Cosines.
rhu ledksf.kd v{kksa dh fnd~dkslkbu Kkr dhft,A
Determine direction cosines of three rectangular axis
14) lfn'k 𝒂 ̂ dk lfn'k 𝒃
⃗ = 𝒊̂ + 𝟐 𝒋̂ + 𝟑𝒌 ⃗ = 𝒊̂ + 𝟑𝒋̂ + 𝒌
̂ ij iz{ksi Kkr dhft,A 2
̂ on the vector ⃗𝒃 = 𝒊̂ + 𝟑𝒋̂ + 𝒌
⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌
Find the projection of the vector 𝒂 ̂
;fn 𝒂 ̂ vkSj ⃗𝒃 = 𝟐𝒊̂ − 𝒋̂ − 𝒌
⃗ = 𝒋̂ + 𝟑𝒌 ̂ rks |𝒂 ⃗ | dk eku Kkr dhft,A
⃗ −𝒃
̂ and ⃗𝒃 = 𝟐𝒊̂ − 𝒋̂ − 𝒌
⃗ = 𝒋̂ + 𝟑𝒌
If 𝒂 ̂ then find the value of |𝒂
⃗ − ⃗𝒃|

15) lfn'k 𝒂 ̂ ds vuqfn'k ,d ,slk lfn'k Kkr dhft, ftldk ifjek.k 𝟓 bdkbZ gks A 2
⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝟓𝒌
̂ that has magnitude 5 units.
⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝟓𝒌
Find a vector in the direction of vector 𝒂
fn[kkb, fd fdlh lfn'k dk oxZ mlds ifjek.k ds oxZ ds cjkcj gksrk gSA
Show that square of any vector is equal to square of its magnitudes.

𝒙𝟐 𝒚𝟐
16) nh?kZo`Ÿk 𝟒 + 𝟗 = 𝟏 ls f?kjs {ks= dk {ks=QYk Kkr dhft,A 3
𝒙𝟐 𝒚𝟐
Find the area of the region bounded by the ellipse 𝟒 + 𝟗 = 𝟏
oØ 𝒚 = 𝟒𝒙 ,oa js[kk 𝒙 = 𝟑 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A
𝟐

Find the area of the region bounded by the curve 𝒚𝟐 = 𝟒𝒙 and the line 𝒙 = 𝟑
17) vody lehdj.k (𝒆𝒙 + 𝒆−𝒙 )𝒅𝒚 − (𝒆𝒙 − 𝒆−𝒙 )𝒅𝒙 = 𝟎 dks gy dhft,A 3

Solve the differential equation (𝒆𝒙 + 𝒆−𝒙 )𝒅𝒚 − (𝒆𝒙 − 𝒆−𝒙 )𝒅𝒙 = 𝟎
fdlh cSad esa ewy/ku dh o`f) 𝟓% okf"kZd dh nj ls gksrh gSA bl cSad esa ₹1000 tek djk, tkrs gSaA Kkr
dhft, fd 10 c"kZ ckn ;g jkf'k fdruh gks tk;sxhA(𝒆𝟎.𝟓 = 𝟏. 𝟔𝟒𝟖)

In a bank Principal Increases continously at the rate of 𝟓% per year amount of ₹1000 is
deposited with this bank, how much will it worth after 10 year.(𝒆𝟎.𝟓 = 𝟏. 𝟔𝟒𝟖)

18) fuEUk jSf[kd izkx
s keu leL;k dks gy dhft,] fuEu O;ojks/kksa ds varZxr 3
𝒙 + 𝟐𝒚 ≥ 𝟏𝟎
𝟑𝒙 + 𝟒𝒚 ≤ 𝟐𝟒
𝒙≥𝟎, 𝒚≥𝟎
𝒛 = 𝟐𝟎𝟎𝒙 + 𝟓𝟎𝟎𝒚 dk U;wure eku Kkr dhft,A
Solve the following linear programming.problem
minimise 𝒛 = 𝟐𝟎𝟎𝒙 + 𝟓𝟎𝟎𝒚
Subject to the constraints
𝒙 + 𝟐𝒚 ≥ 𝟏𝟎
𝟑𝒙 + 𝟒𝒚 ≤ 𝟐𝟒
𝒙≥𝟎, 𝒚≥𝟎

fuEu vojks/kks ds varxZr 𝒛 = 𝟑𝒙 + 𝟓𝒚 dk vf/kdrehdj.k dhft, %
𝟑𝒙 + 𝟓𝒚 ≤ 𝟏𝟓 , 𝟓𝒙 + 𝟐𝒚 ≤ 𝟏𝟎 , 𝒙 ≥ 𝟎 , 𝒚 ≥ 𝟎
maximise 𝒛 = 𝟑𝒙 + 𝟓𝒚 subject to
𝟑𝒙 + 𝟓𝒚 ≤ 𝟏𝟓 , 𝟓𝒙 + 𝟐𝒚 ≤ 𝟏𝟎 , 𝒙≥𝟎, 𝒚≥𝟎

19) ,d ikals dks rhu ckj mNkyk tkrk gS rks de ls de ,d ckj fo"ke la[;k izkIr gksus dh izkf;drk Kkr
dhft,A 3
A dia is tossed thrice.Find the probability of getting an odd number at least once.
𝟔 𝟓 𝟕 𝑨
;fn 𝑷(𝑨) = 𝟏𝟏 , 𝑷(𝑩) = 𝟏𝟏 vkSj 𝑷(𝑨𝑼𝑩) = 𝟏𝟏 rks Kkr dhft, 𝑷 (𝑩)
𝟔 𝟓 𝟕 𝑨
If 𝑷(𝑨) = 𝟏𝟏 , 𝑷(𝑩) = 𝟏𝟏 and 𝑷(𝑨𝑼𝑩) = 𝟏𝟏 then find 𝑷 (𝑩)
20) lkjf.kdks ds xq.k/keksZ dk iz;ksx djds fl) dhft, fd& 4
𝒃+𝒄 𝒂 𝒂
| 𝒃 𝒄+𝒂 𝒃 | = 𝟒𝒂𝒃𝒄
𝒄 𝒄 𝒂+𝒃

Using the property of determinants prove that -
𝒃+𝒄 𝒂 𝒂
| 𝒃 𝒄+𝒂 𝒃 | = 𝟒𝒂𝒃𝒄
𝒄 𝒄 𝒂+𝒃
lkjf.kdks ds xq.k/keksZ dk iz;ksx djds fl) dhft, fd
𝒂−𝒃−𝒄 𝟐𝒂 𝟐𝒂
| 𝟐𝒃 𝒃−𝒄−𝒂 𝟐𝒃 | = (𝒂 + 𝒃 + 𝒄)𝟑
𝟐𝒄 𝟐𝒄 𝒄−𝒂−𝒃
Using the property of determinants prove that -
𝒂−𝒃−𝒄 𝟐𝒂 𝟐𝒂
| 𝟐𝒃 𝒃−𝒄−𝒂 𝟐𝒃 | = (𝒂 + 𝒃 + 𝒄)𝟑
𝟐𝒄 𝟐𝒄 𝒄−𝒂−𝒃

यदि 𝒙 ≥ 𝟏
21) Qyu 𝒇(𝒙) = { 𝒙𝟐 + 𝟏 gks rks 𝒇(𝒙) dh 𝒙 = 𝟏 ij lkrR;rk dh tkWp dhft,A 4
𝒙 +𝟏 यदि 𝒙 < 𝟏

Examine the continuity of function 𝒇(𝒙) at 𝒙 = 𝟏 where
𝒙 + 𝟏 𝒊𝒇 𝒙 ≥ 𝟏
𝒇(𝒙) = { 𝟐
𝒙 + 𝟏 𝒊𝒇 𝒙 < 𝟏
𝒅𝒚 𝒄𝒐𝒔𝒙
;fn 𝒚 = (𝐥𝐨𝐠 𝒙)𝐜𝐨𝐬 𝒙 gks rks fl) dhft, fd&𝒅𝒙 = (𝐥𝐨𝐠 𝒙)𝐜𝐨𝐬 𝒙 [𝒙 𝐥𝐨𝐠 𝒙 − 𝐬𝐢𝐧 𝒙 𝒍𝒐𝒈(𝐥𝐨𝐠 𝒙) ]

𝒅𝒚 𝒄𝒐𝒔𝒙
If 𝒚 = (𝐥𝐨𝐠 𝒙)𝐜𝐨𝐬 𝒙 then show that 𝒅𝒙 = (𝐥𝐨𝐠 𝒙)𝐜𝐨𝐬 𝒙 [𝒙 𝐥𝐨𝐠 𝒙 − 𝐬𝐢𝐧 𝒙 𝒍𝒐𝒈(𝐥𝐨𝐠 𝒙) ]

22) Kkr dhft,& ∫ 𝒙𝟐 𝐥𝐨𝐠 𝒙 𝒅𝒙 4
Find ∫ 𝒙𝟐 𝐥𝐨𝐠 𝒙 𝒅𝒙
Kkr dhft,& ∫ 𝒔𝒊𝒏𝟑𝒙 𝒄𝒐𝒔𝟒𝒙 𝒅𝒙
Find ∫ 𝒔𝒊𝒏𝟑𝒙 𝒄𝒐𝒔𝟒𝒙 𝒅𝒙
𝒙+𝟏 𝒚+𝟏 𝒛+𝟏 𝒙−𝟑 𝒚−𝟓 𝒛−𝟕
23) js[kkvksa 𝟕
= −𝟔 = 𝟏
rFkk 𝟏
= −𝟐 = 𝟏
ds chp dh U;wure nwjh Kkr dhft,A 4
𝒙+𝟏 𝒚+𝟏 𝒛+𝟏 𝒙−𝟑 𝒚−𝟓 𝒛−𝟕
Find the shortest distance between the lines = −𝟔 = and = −𝟐 =
𝟕 𝟏 𝟏 𝟏
n'kkZb, fd fcUnqvksa (𝟏, −𝟏, 𝟐)और(𝟑, 𝟒, −𝟐) ls gksdj tkus okyh js[kk fcUnqvksa (𝟎, 𝟑, 𝟐) vkSj (𝟑, 𝟓, 𝟔) ls
tkus okyh js[kk ij yEc gSA
Show that the line through the points (𝟏, −𝟏, 𝟐), (𝟑, 𝟒, −𝟐) is perpendicular to the line
through the points (𝟎, 𝟑, 𝟐) and (𝟑, 𝟓, 𝟔)