MP Board Class 12 Sample Paper 2026 Maths

dsoy vH;kl gsr qu ewuk ç'u i=
Sample Question Paper for Practice only
gk;j lsd s.Mjh ijh{kk−2026
Higher Secondary Examination −2026
fo"k; −mPp xf.kr
Subject Name −Higher Mathematics
(Hindi & English Versions)
Total Questions Total Printed Pages Time Maximum Marks
23 13 3 Hours 80

funs'Z k:

(i). lHkh ç'u vfuok;Z gSaA

(ii). ç'u la[;k 1 ls 5 rd ds miç'u çR;sd 1 vad ds gSaA

(iii). ç'u la[;k 6 ls 15 rd çR;sd 2 vad ds gSaA

(iv). ç'u la[;k 16 ls 19 rd çR;sd 3 vad ds gSaA

(v). ç'u la[;k 20 ls 23 rd çR;sd 4 vad ds gSaA

Instructions:

(i) All questions are compulsory.
(ii) Sub-questions of Question numbers 1 to 5 carry 1 mark each.
(iii) Question numbers 6 to 15 carry 2 marks each.
(iv) Question numbers 16 to 19 carry 3 marks each.
(v) Question numbers 20 to 23 carry 4 marks each.

1

(1) lgh fodYi pqu dj fyf[k, : 1x6=6
i) ;fn 𝐴 = {1,2,3} gks rks ,sls laca/k ftuesa vo;o (1,2) rFkk (1,3) gks vkSj
tks LorqY; rFkk lefer gSa fdarq laØked ugha gS fd la[;k gS& -
𝑎) 0 𝑏) 1 𝑐) 2 𝑑) 3
ii) 𝑐𝑜𝑠 −1 𝑥 dh eq[; 'kk[kk dk ifjlj gS&
𝜋 𝜋
𝑎) (0, 𝜋) 𝑏) [− , ]
2 2

𝑐) 𝑅 𝑑) [0, 𝜋]
iii) ;fn ,d vkO;qg 𝐴 ds fy, 𝐴 = − 𝐴′ rks
𝑎) ,d lefer vO;wg gSA 𝑏) ,d fo"ke lefer vkO;wg gSA
𝑐) ,d 'kwU; vkO;wg gSA 𝑑) ,d rR~led vkO;wg gSA
1 1
iv) ;fn 𝑃(𝐴) = , 𝑃(𝐵) = gks vkSj 𝐴 oa 𝐵 Lora= ?kVuk,s gS rks 𝑃(𝐴). 𝑃(𝐵) =
2 4

1 1 3
𝑎) 𝑏) 𝑐) 𝑑)0
8 2 4
v) ;fn 𝐴 ,d LraHk vkO;wg gS rks 𝐴 dk ifjorZ gksxk
𝑎) ,d LraHk vkO;wg 𝑏) ,d iafä vkO;wg
𝑐) ,d oxZ vkO;wg 𝑑) 'kwU; vkO;wg
𝜋
vi) varjky [0, ] esa 𝑠𝑖𝑛𝑥 dk mPpre eku gS
4
√3 1 1
𝑎) 𝑏) 1 𝑐) 𝑑)
2 2 √2

Choose and write correct option -
i) If 𝐴 = {1,2,3} then number of relations containing (1,2) and (1,3)
which are reflexive and symmetric but not transitive are:
𝑎) 0 𝑏)1 𝑐)2 𝑑) 3
−1
ii) Range of principal value of 𝑐𝑜𝑠 𝑥 is
𝜋 𝜋
𝑎) (0, 𝜋) 𝑏)[− , ]
2 2

𝑐)𝑅 𝑑)[0, 𝜋]

2

iii) For any matrix 𝐴, If 𝐴 = − 𝐴′ then:

𝑎)𝐴 is a symmetric matrix

𝑏) 𝐴 is a skew symmetric matrix

𝑐)𝐴 is a zero matrix

𝑑)𝐴 is an identity matrix
1 1
iv) If 𝑃(𝐴) = ,𝑃(𝐵) = and 𝐴 and 𝐵 are independent events
2 4
then 𝑃(𝐴). 𝑃(𝐵) =
1 1 3
𝑎) 𝑏) 𝑐) 𝑑)0
8 2 4

v) If 𝐴 is a column matrix then transpose of 𝐴 will be-

a) A Column matrix. b) A row matrix.

c) A Square matrix d) zero matrix
𝜋
vi) Maximum value of 𝑠𝑖𝑛𝑥 in interval [0, ] is
4

√3 1 1
a) b) 1 c) d)
2 2 √2

(2) fjDr LFkkuksa dh iwf rZ dhft,& 1x6=6
i) 𝑓: 𝑋 → 𝑌 ,d vkPNknd Qyu gS ;fn vkSj ;fn 𝑓 dk ifjlj =……
ii) nks O;qRdze.kh; vkO;wg 𝐴 vkSj 𝐵 ds fy, (𝐴𝐵)−1 =……
𝑑𝑦
iii) vody lehdj.k = 2 dk O;kid gy ….. gksxkA
𝑑𝑥

iv) lfn’k 𝑎⃗ = 2𝑖̂ + 3𝑗̂ dh fn’kk esa bdkbZ lfn’k …… gksxkA
𝑑2 𝑦
v) vody lehdj.k 2𝑥 + 5 = 0 dh ?kkr …… gS A
𝑑𝑥2

vi) 𝑠𝑖𝑛2𝑥 dk vodyu xq.kkd ....gksrk gSA

3

Fill in the blanks -
i) 𝑓: 𝑋 → 𝑌 is an onto function if and only if, Range of 𝑓 =……
ii) For two invertible matrices 𝐴 and 𝐵 (𝐴𝐵)−1 =……
𝑑𝑦
iii) General solution of differential equation = 2 will be …..
𝑑𝑥
iv) Unit vector, in the direction of vector 𝑎⃗ = 2𝑖̂ + 3𝑗̂ will be……
𝑑2 𝑦
v) Degree of differential equation 2𝑥 + 5 = 0 is …..
𝑑𝑥2
vi) Differential Coefficient of 𝑠𝑖𝑛2𝑥 is....

(3) lR;@vlR; fyf[k,& 1x6=6

i) ,d Qyu 𝑓: 𝑋 → 𝑌 ,dSdh Qyu gS ;fn 𝑓(𝑥1 ) = 𝑓(𝑥2)⇒ 𝑥1 = 𝑥2、

𝑥1 , 𝑥2 ∈ 𝑋
𝑑𝑦
ii) vody lehdj.k 𝑥 − 𝑦 = 2𝑥² dk lekdyu xq.kd 𝑒𝑥 gSA
𝑑𝑥
1 𝜋
iii) ∫0 √1 − 𝑥 2 𝑑𝑥 dk eku ds cjkcj gSA
4

iv) lfn’k 𝑎⃗ = 0î + ĵ + 0k̂ rFkk 𝑏⃗⃗ = î + 0ĵ + 0k̂ ds e/; dks.k dk eku 𝜋 ds
cjkcj gSA
v) 𝑋 −v{k ds fnd~ dkslkbu 1,0,0 gSA
vi) Qyu 𝑓(𝑥 ) = 5𝑥 3 − 7𝑥 + 13 varjky [−5,5] esa ,d larr Qyu gSA
𝐖𝐫𝐢𝐭𝐞 𝐭𝐫𝐮𝐞 𝐚𝐧𝐝 𝐟𝐚𝐥𝐬𝐞-
i) A function 𝑓: 𝑋 → 𝑌 is one-one function if 𝑓(𝑥1 ) = 𝑓(𝑥2)⇒ 𝑥1 = 𝑥2、
𝑥1 , 𝑥2 ∈ 𝑋
𝑑𝑦
ii) Integrating factor of differential equation 𝑥 − 𝑦 = 2𝑥² is 𝑒𝑥
𝑑𝑥
1 𝜋
iii) Value of ∫0 √1 − 𝑥 2 𝑑𝑥 is equal to the
4
iv) Angle between the vectors 𝑎⃗ = 0î + ĵ + 0k̂ and 𝑏⃗⃗ = î + 0ĵ + 0k̂ is
equal to 𝜋
v) Direction cosine of 𝑋 − 𝑎𝑥𝑖𝑠 are 1,0,0
vi) Function 𝑓(𝑥 ) = 5𝑥 3 − 7𝑥 + 13 is a continuous function in interval
[−5,5]

4

(4) lgh tksM+h cukb;s& 1x7=7
LrEHk v LrEHk c
𝜋
2
i) ∫ 𝑐𝑜𝑡𝑥𝑑𝑥 a) √2 − 1
𝜋
4
𝜋
𝑎2 𝜋 𝜋3
ii) ∫0 𝑠𝑖𝑛𝑥𝑑𝑥
2 b) +
2 24
2 1 𝜋
iii) ∫1 √𝑥 2 − 1 𝑑𝑥 c) 𝑡𝑎𝑛−1
𝑎 2𝑎
𝜋 1
1 d) 𝑙𝑜𝑔2
iv) ∫12 𝑑𝑥 2
𝑥
𝜋
v) ∫02 2
1
𝑑𝑥 e) 1
𝑎 +𝑥2
𝜋 1
vi) ∫02(𝑎2 + 𝑥 2 )𝑑𝑥 f) √3 − log(2 + √3)
2
𝜋 𝜋
vii) ∫0 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥𝑑𝑥
4 g) 𝑙𝑜𝑔
2

Match the correct column -
𝐶𝑜𝑙𝑢𝑚𝑛 𝐴 𝐶𝑜𝑙𝑢𝑚𝑛 𝐵
𝜋
2
i) ∫ 𝑐𝑜𝑡𝑥𝑑𝑥 a) √2 − 1
𝜋
4
𝜋
𝑎2 𝜋 𝜋3
ii) ∫0 𝑠𝑖𝑛𝑥𝑑𝑥
2 b) +
2 24
2 1 𝜋
iii) ∫1 √𝑥 2 − 1 𝑑𝑥 c) 𝑡𝑎𝑛−1
𝑎 2𝑎
𝜋 1
1 d) 𝑙𝑜𝑔2
iv) ∫1 2 𝑑𝑥 2
𝑥
𝜋
v) ∫0 2
1
𝑑𝑥 e) 1
𝑎2 +𝑥2
𝜋 1
vi) ∫0 (𝑎2 + 𝑥 2 )𝑑𝑥
2 f) √3 − log(2 + √3)
2
𝜋 𝜋
vii) ∫04 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥𝑑𝑥 g) 𝑙𝑜𝑔
2

5

(5) ,d okD;@'kCn esa mRrj fyf[k,& 1x7=7
1
i) 2sin−1 dk eku Kkr dhft, A
2
2 0
ii) ;fn 𝐴 = [ ] rks 𝑎𝑑𝑗 𝐴 dk eku fyf[k,A
0 3
1 0 3 0
iii) ;fn 𝐴 = [ ] rFkk 𝐵 = [ ] rks 𝐴′𝐵 dk eku fyf[k, A
0 0 0 0
iv) o`r dh f=T;k ds lkis{k {ks=Qy esa ifjorZu dh nj fyf[k,A
𝑑𝑦
v) vody lehdj.k = 𝑒𝑥 +𝑦 dk O;kid gy fyf[k,A
𝑑𝑥

vi) ;fn lfn’k 𝑎⃗ = î − 2ĵ + 3k̂ vkSj 𝑏⃗⃗ = 3î − 2ĵ + k̂ rks 𝑎⃗.𝑏⃗⃗ Kkr dhft,A
vii) 𝑃(𝐴 ∣ 𝐵) dk eku Kkr dhft, ;fn 𝑃(𝐵) = 0.7 rFkk 𝑃(𝐴 ∩ 𝐵) = 0.1
𝐖𝐫𝐢𝐭𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐧 𝐨𝐧𝐞 𝐰𝐨𝐫𝐝/𝐬𝐞𝐧𝐭𝐞𝐧𝐜𝐞-
1
i) Find the value of 2sin−1
2

2 0
ii) If 𝐴 = [ ] then write the value of 𝑎𝑑𝑗 𝐴
0 3
1 0 3 0
iii) If 𝐴 = [ ] and 𝐵 = [ ] then write the value of 𝐴′𝐵
0 0 0 0
iv) Write the change in area of a circle with respect to its radius
𝑑𝑦
v) Write general solution of differential equation = 𝑒𝑥 +𝑦
𝑑𝑥
vi) If vector 𝑎⃗ = î − 2ĵ + 3k̂ and 𝑏⃗⃗ = 3î − 2ĵ + k̂ then find 𝑎⃗.𝑏⃗⃗
vii) Find the value of 𝑃(𝐴 ∣ 𝐵) if 𝑃(𝐵) = 0.7 and 𝑃(𝐴 ∩ 𝐵) = 0.1

(6) fl) dhft, fd çnÙk Qyu 𝑓: 𝑅 → 𝑅 𝑓(𝑥) = 2𝑥 ,dSdh vkPNknd gSA 2

Prove that the function 𝑓: 𝑅 → 𝑅 given by 𝑓(𝑥) = 2𝑥 is one-one and onto.

fn[kkb;s fd ,d vkPNknd Qyu 𝑓: {1 ,2,3} → {1 , 2, 3} ges'kk ,dSdh gSA

Show that an onto function 𝑓: {1 ,2,3} → {1 , 2, 3} is always one-one.

6

1 1
(7) fn[kkb;s fd 3 sin−1 𝑥 = sin−1(3𝑥 − 4𝑥 3 ), 𝑥 ∈ [− , ] 2
2 2

1 1
Show that 3 sin−1 𝑥 = sin−1(3𝑥 − 4𝑥 3 ) , 𝑥 ∈ [− , ]
2 2

1
cot −1 ,|𝑥 | > 1 dk ljyre :i fyf[k, A
√𝑥2 −1

1
write the simplest form of cot −1 ,|𝑥 | > 1
√𝑥2 −1

3𝜋
(8) sin−1 (𝑠𝑖𝑛 ) dk eku Kkr dhft,A 2
5

3𝜋
Find the values of sin−1 (𝑠𝑖𝑛 )
5

13𝜋
cos −1(𝑐𝑜𝑠 ) dk eku Kkr dhft,A
6

13𝜋
Find the value of cos −1 (𝑐𝑜𝑠 )
6

(9) Qyu 𝑐𝑜𝑠[𝑠𝑖𝑛(𝑥3 )] dk 𝑥 ds lkis{k vodyu xq.kkad Kkr dhft,A 2

find differential coefficient of function 𝑐𝑜𝑠[𝑠𝑖𝑛(𝑥 3 )] with respect to 𝑥.

Qyu 𝑠𝑖𝑛[𝑐𝑜𝑠(𝑥4 )] dk 𝑥 3 ds lkis{k vodyu xq.kkad Kkr dhft,A

find differential coefficient of function 𝑠𝑖𝑛[𝑐𝑜𝑠(𝑥 4 )] with respect to 𝑥 3.

(10) fn[kkb;s dh çnÙk Qyu 𝑓(𝑥 ) = 𝑥 2 − 3𝑥 + 17 , 𝑥 ∈ 𝑅, 𝑅 esa o/kZeku Qyu gSA 2

show that the function given by 𝑓(𝑥 ) = 𝑥 2 − 3𝑥 + 17 , 𝑥 ∈ 𝑅, is

increasing function on R.

7

fn[kkb;s dh çnÙk Qyu 𝑓(𝑥 ) = 17𝑥 − 5 , 𝑅 esa o/kZeku Qyu gSA

Show that the function given by 𝑓(𝑥) = 17𝑥 − 5 is increasing function on 𝑅.

(11) ,d o`Ùk dh f=T;k esa ifjorZu dh nj 0.7𝑐𝑚/𝑠 gSAmldh ifjf/k esa ifjorZu dh nj Kkr dhft, A 2

The radius of a circle change at the rale of 0.7 𝑐𝑚/𝑠𝑒𝑐 what is the rate of change of
its circumference

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

𝑑𝑦
(12) vody lehdj.k 𝑥 + 2𝑦 = 𝑥 2 𝑙𝑜𝑔𝑥 dk lekdyu xq.kd Kkr dhft,A 2
𝑑𝑥

𝑑𝑦
Find Integrating factor of differential equation 𝑥 + 2𝑦 = 𝑥 2 𝑙𝑜𝑔𝑥.
𝑑𝑥

𝑑𝑦 1+𝑦2
vody lehdj.k = dk O;kid gy Kkr dhft,A
𝑑𝑥 1+𝑥2

𝑑𝑦 1+𝑦2
find the general solution of differential equation =
𝑑𝑥 1+𝑥2

(13) lfn'k ̂i + 2ĵ + 3k̂ dh fnd dksT;k, Kkr dhft,A 2

find the direction cosines of the vector ̂i + 2ĵ + 3k̂

8

lfn'k ̂i + 3ĵ + 7k̂ dk ç{ksi lfn'k 7î − ̂j + 8k̂ ij Kkr dhft,A

Find the projection of the vector ̂i + 3ĵ + 7k̂ on the Vector 7î − ̂j + 8k̂

(14) foUnq (5,2, −4) ls tkus okyh vkSj lfn'k 3î + 2ĵ − 8k̂ ds lekUrj ljy js[kk dk lfn'k lehdj.k
Kkr dhft,A 2

Find vector Equation of straight line passing through the point (5,2, −4) and parallel
to the vector 3î + 2ĵ − 8k̂

𝑥+3 𝑦+1 𝑧+3 𝑥+1 𝑦−4 𝑧−5
js[kkvks = = vkSj = = ds chp dk dks.k Kkr dhft,A
3 5 4 1 1 2

𝑥+3 𝑦+1 𝑧+3 𝑥+1 𝑦−4 𝑧−5
Find the angle between the pair of lines = = and = =
3 5 4 1 1 2
(15) ;fn 𝑎⃗ = ̂i + 2ĵ − 3k̂ और 𝑏⃗⃗ = 2î − ̂j है तो |𝑎⃗ × 𝑏⃗⃗| dk eku Kkr dhft,

If 𝑎⃗ = ̂i + 2ĵ − 3k̂ and 𝑏⃗⃗ = 2î − ̂j then find the value of |𝑎⃗ × 𝑏⃗⃗| 2

;fn (𝑎⃗ + 𝑏⃗⃗). (𝑎⃗ − 𝑏⃗⃗) = 8 vkSj |𝑎⃗| = 8|𝑏⃗⃗| rks |𝑎⃗| vkSj |𝑏⃗⃗| dk eku Kkr dhft,A

If (𝑎⃗ + 𝑏⃗⃗). (𝑎⃗ − 𝑏⃗⃗) = 8 and |𝑎⃗| = 8|𝑏⃗⃗| then find the value of |𝑎⃗| and |𝑏⃗⃗|.
1 2 3 9 12 15
(16) vkO;wg 𝑋 Kkr dhft,] ;fn 𝑋 [ ]= [ ] 3
4 5 6 19 26 33
1 2 3 9 12 15
Find the matrix 𝑋 If 𝑋 [ ]=[ ]
4 5 6 19 26 33
1
;fn vkO;wg 𝐴 = [−4] vkSj 𝐵 = [−1 2 1] gSa]rks ,lR;kfir dhft, fd (𝐴𝐵)’ = 𝐵’𝐴′
3

9

1
If matrix 𝐴 = [−4] and 𝐵 = [−1 2 1] then verify that (𝐴𝐵)’ = 𝐵’𝐴′
3
𝑥2 𝑦2
(17) lekdyu dk iz;ksx djrs gq, nh?kZo`Rr + = 1 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 3
4 9
𝑥2 𝑦2
By using integration Find the area bounded by the ellipse + =1
4 9

𝑥 = 0 ,oa 𝑥 = 2 𝜋 ds e/; oØ 𝑦 = 𝑠𝑖𝑛𝑥 ls f?kjs {ks= dk {ks=Qy Kkr dhft, A

Find the the area bounded by curve 𝑦 = 𝑠𝑖𝑛𝑥 between 𝑥 = 0 and 𝑥 = 2 𝜋

(18) vkys[kh; fof/k }kjk fuEu jSf[kd çksxkz eu leL;k dks gy dhft,∶
fuEu O;ojks/kksa ds varxZr 5𝑥 + 𝑦 ≤ 100 ,𝑥 + 𝑦 ≤ 60, 𝑥 ≥ 0, 𝑦 ≥ 0
𝑍 = 75𝑥 + 𝑦 dk vf/kdrehdj.k dhft,A 3

Solve the following linear programming problem by graphical method :
Maximize 𝑍 = 75𝑥 + 𝑦 Subject to constraints:
5𝑥 + 𝑦 ≤ 100 ,𝑥 + 𝑦 ≤ 60, 𝑥 ≥ 0, 𝑦 ≥ 0

vkys[kh; fof/k }kjk fuEu jSf[kd çksxkz eu leL;k dks gy dhft,A
fuEu O;ojks/kksa ds varxZr 𝑥 + 𝑦 ≤ 50,3𝑥 + 𝑦 ≤ 90, 𝑥 ≥ 0, 𝑦 ≥ 0
𝑧 = 4𝑥 + 𝑦 dk vf/kdrehdj.k dhft,:
Solve the following linear programming problem by graphical method :
Maximize 𝑍 = 4𝑥 + 𝑦 subject to the constraints :
𝑥 + 𝑦 ≤ 50, 3𝑥 + 𝑦 ≤ 90, 𝑥 ≥ 0, 𝑦 ≥ 0

10

(19) ,d O;fä ds ckjs esa Kkr gS fd og 4 esa ls 3 ckj lR; cksyrk gSA og ,d ikls dks mNkyrk gS
vkSj crykrk gS fd ml ij vkus okyh la[;k 5 gSA bldh çkf;drk Kkr dhft, fd ikls ij
vkus okyh la[;k okLro esa 5 gSA 3
A man is known to speak truth 3 out of 4 times. He throws a die and reports
that it is a 5 .Find the probability that it is actually a 5

,d ikls dks nks ckj mNkyk x;k vkSj çdV gqbZ la[;kvksa dk ;ksx 6 ik;k x;k A la[;k 4 ds
U;wure ,d ckj çdV gksus dh lçfrca/k çkf;drk Kkr dhft, A
A die is thrown twice and the sum of the appearing is observed to be 6.what
is the conditional probability that the number 4 has appeared at least once.

(20) 𝑎 rFkk 𝑏 ds e/; laca/k Kkr dhft, ftlls iznRr Qyu 4
5 ;𝑥 ≤ 2
𝑓 (𝑥 ) = { 𝑎𝑥 + 𝑏 ; 2 < 𝑥 < 10
21 ; 𝑥 ≥ 10
larr gksA
Find the relation between 𝑎 and 𝑏 for which the given function
5 ;𝑥 ≤ 2
𝑓(𝑥 ) = { 𝑎𝑥 + 𝑏 ; 2 < 𝑥 < 10
21 ; 𝑥 ≥ 10
is continuous
𝑑𝑦
;fn 𝑥 3 + 𝑥 2 𝑦 + 𝑥𝑦 2 + 𝑦 3 = 81 rks dk eku Kkr dhft,A
𝑑𝑥
𝑑𝑦
If 𝑥 3 + 𝑥 2 𝑦 + 𝑥𝑦 2 + 𝑦 3 = 81 then find the value of
𝑑𝑥

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(21) fn, x;s lehdj.k fudk; dks vkO;wg fof/k ls gy dhft, & 4
2 3 10
+ + =4
𝑥 𝑦 𝑧
4 6 5
− + =1
𝑥 𝑦 𝑧
6 9 20
+ − =2
𝑥 𝑦 𝑧
Solve the following system of equation by matrix method
2 3 10
+ + =4
𝑥 𝑦 𝑧
4 6 5
− + =1
𝑥 𝑦 𝑧
6 9 20
+ − =2
𝑥 𝑦 𝑧

4 kg I;kt] 3 kg xsgw¡ vkSj 2 kg&pkoy dk ewY; # 60 gSA 2 kg I;kt] 4 kg xsgwa vkSj
6 kg pkoy dk ewY; #- 90 gSA 6 kg I;kt] 2 kg xsgw¡ vkSj 3 kg pkoy dk ewY; # 70 gS
vkO;wg fof/k }kjk çR;sd dk ewY; çfr kg Kkr dhft, A

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(22) ∫2 |𝑥 − 5|𝑑𝑥 dk eku Kkr dhft,A 4
8
Find the value of ∫2 |𝑥 − 5|𝑑𝑥

∫ √1 + 3𝑥 − 𝑥 2 𝑑𝑥 dk eku Kkr dhft,A
Find the value of ∫ √1 + 3𝑥 − 𝑥 2 𝑑𝑥

(23) ;fn dksbZ js[kk 𝑙1 foUnq (1,2, −4)a ls tkrh gS rFkk 2𝑖̂ + 3𝑗̂ + 6𝑘̂ ds lekUrj gS rFkk
𝑙2dk lfn'k lehdj.k 𝑟⃗ = (2𝑖̂ + 4𝑗̂ + 5𝑘̂) +++μ(3𝑖̂ + 4𝑗̂ + 5𝑘̂) gSs rks] 𝑙1 rFkk
𝑙2 ds chp dh U;wure bjh Kkr dhft,A 4
If a line 𝑙1, passes through the point (1,2, −4) and parallel to 2𝑖̂ + 3𝑗̂ + 6𝑘̂
and vector equation of the line 𝑙2 is given by 𝑟⃗ = (2𝑖̂ + 4𝑗̂ + 5𝑘̂) +μ(3𝑖̂ + 4𝑗̂ + 5𝑘̂)
then find the minimum distance between 𝑙1 and 𝑙2

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𝑥−8 𝑦+19 𝑧−10 𝑥−15 𝑦−29 𝑧−5
fcUnq (1,2, −4)Z ls tkus okyh vkSj js[kkvksa = = rFkk = = ij
3 −16 7 3 8 −5

yac js[kk dk lfn'k lehdj.k Kkr dhft, A
Find the vector equation of the line which passes through the point (1,2, −4) and
𝑥−8 𝑦+19 𝑧−10 𝑥 −15 𝑦−29 𝑧−5
perpendicular to the lines = = and = =
3 −16 7 3 8 −5

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