Rajasthan Board 12th Model Paper 2024 Maths

MODEL
PAPER बोर्ड ऑफ सेकं डरी एज्युके शन राजस्थान
RAJASTHAN BOARD

2024

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mPp ek/;fed ijh{kk] 2024
Senior Secondary Examination, 2024
uewuk iz'u&i=
Model Paper
fo"k; & xf.kr
Sub : Mathematics
d{kk & 12oha
Class : 12th

le;% 3 ?k.Vs 15 feuV iw.kkZd% 80
ijh{kkÆFk;ksa ds fy, lkekU; funsZ'k%
GENERAL INSTRUCTION TO THE EXAMINEES :

1- ijh{kkFkÊ loZçFke vius ç'u i= ij ukekad vfuok;Zr% fy[ksaA
Candidate must write first his/her Roll No- on the question paper compulsorily.

2- lHkh ç'u djus vfuok;Z gSA
All the questions are compulsory.

3- çR;sd ç'u dk mÙkj nh xà mÙkj iqfLrdk es gh fy[ksaA
Write the answer to each question in the given answer book only.

4- ftu ç'uksa es vkUrfjd [k.M gS mu lHkh ds mÙkj ,d lkFk gh fy[ksaA
For questions having more than one part] the answers to those parts are to be written together in
continuity.

5- ç'u dk mÙkj fy[kus ls iwoZ ç'u dk Øekad vo'; fy[ksaA
Write down the serial number of the question before- attempting it.

6- ç'u i= ds fgUnh o vaxzsth :ikUrj.k esa fdlh çdkj dh =qfV@vUrj@fojksèkkHkkl gksus ij
fgUnh Hkk"kk ds ç'u dks gh lgÈ ekusaA

If there is any error/difference/Contradiction in Hindi & English versions of the question paper, the
question of Hindi version should be treated valid.

7- ç'u Øekad 16 ls 22 es vkUrfjd fodYi gSA
There are internal choices in Question No. 16 to 22 .

1- cgqfodYih; iz’u &

Multiple choice question

(i) eku yhft;s fd f:R R , f(x)= (x)3}kjk ifjHkkf"kr gS rks lgh fodYi dk p;u dhft;sA

v- 𝑓 ,dSdh vkPnknd gS c- 𝑓 cgq,dh vkPNknd gS

l- 𝑓 ,dSdh gS ij vkPNknd ugha gS n- 𝑓u rks ,dSdh gS vkSj uk gh vkPNknd gS(1)

Let f:R R , be defined as f(x)= (x)3 chose the correct answer

(a) f is one - one onto (b) f is many – one onto

(c) f is one - one but not onto (d) f is neither one one nor onto

(ii) { −
1
2
dk eku gS &

v- c- l- n-1 (1)

{ −
1
2
is equal to –

(a) (b) (c) (d) 1

(iii) eku yhft, fd X, Y, Z, W rFkk P Øe'k% 2 𝑋 𝑛, 3 𝑋 𝑘, 2 𝑋 𝑝, 𝑛 𝑋 3 rFkk 𝑝 𝑋 𝑘 dksfV;ks ds
vkO;wg gSA ;fn 𝑛 = 𝑝 rks vkO;wg 7𝑋 − 5𝑍 dh dksfV gS &
v- p x 2 c- 2 x n l- n x 3 n-p x n (1)
Assume X, Y, Z, W and P are matrix of order 2 𝑋 𝑛, 3 𝑋 𝑘, 2 𝑋 𝑝, 𝑛 𝑋 3 and 𝑝 𝑋 𝑘
respectively if 𝑛 = 𝑝 then the order of the matrix 7𝑋 − 5𝑍 is .
(a) 𝑝 x 2 (b) 2 x n (c) n x 3 (d) p x n

x 2 6 2
(iv) ;fn = rks 𝑋 cjkcj gS &
18 x 18 6
v- 6 c- ± 6 l- -6 n-0 (1)

x 2 6 2
if = then x is equal to –
18 x 18 6
(a) 6 (b) ± 6 (c) -6 (d) 0

(v) Qyucos(sinx) dk vodyt gS&

v- sin(sinx) c- sin(cosx) l- -sin(sinx) n-–cosx sin(sinx) (1)

the derivative of function cos(sinx) is

(a) sin(sinx) (b) sin(cosx) (c) -sin(sinx) (d) –cosx sin(sinx)

(vi) ,d o`r dh f=T;k 𝑟 = 6𝑐𝑚, ij 𝑟 ds lkis{k {ks=Qy esa ifjorZu dh nj gS&

v- 10π c- 12π l- 11π n-8π (1)

The rate of change of the area of a circle with respect to it’s radious r at
r=6cm is

(a) 10π (b) 12π (c) 11π (d) 8π

(vii) ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥 dk eku gS &

v- logx-x+c c- 1+logx+c l- x(logx-1)+c n-x(logx+1) +c (1)

The value of ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥 is -

(a) logx-x+c (b) 1+logx+c (c) x(logx-1)+c (d) x(logx+1) +c

(viii) vody lehdj.k 1+ = a dh ?kkr gS &

v- 1 c- 2 l- 3 n- 4 (1)

d2 y
The order of differential equation 1 + = a
dx 2

(a) 1 (b) 2 (c) 3 (d) 4
√
(ix) eku Ykhft;s ds nks lfn’k a⃗ rFkk b⃗ bl izdkj gSa fd |a⃗| = 3, b⃗ = rc a⃗ 𝑋 b⃗ ,d

ek=d lfn’k ;fn a⃗ rFkk b⃗ ds e/; dks.k gS &
v- c- l- n- (1)

√
Let the vector a⃗ and b⃗ be such that |a⃗| = 3, b⃗ = then a⃗ 𝑋 b⃗
Is a unit vector , if the angle between a⃗ 𝑎𝑛𝑑 b⃗ is
(a) (b) (c) (d)

(x) js[kk = = = λ ds fndvuqikr gS &

v- 2,4,3 c- 2 ,2 ,3 l- 2, 4 ,-3 n- 2,2,-3 (1)

Direction ratios of line = = = λ is

(a)2,4,3 (b) 2 ,2 ,3 (c) 2, 4 ,-3 (d) 2,2,-3

(xi) js[kkvksa = = 𝑎𝑛𝑑 = = ds e/; dks.k gSa &

v- 45 c- 30 l- 60 n- 90 (1)

The angle between the Straight lines

= = 𝑎𝑛𝑑 = = is

(a) 45 (b) 30 (c)60 (d) 90

(xii) o`rx + y = 4dk {ks=Qy gS &

v- 2π c- 16π l- 4π n- (1)

The area enclosed by the circle x + y = 4 is

(a) 2π (b) 16π(c)4π (d)

(xiii) odz y = 4x ,y v{k ,oa js[kk y=3 ls f?kjs {ks= dk {ks=Qy gS &

v- 2 c- l- n- (1)

Area of the region bounded by the curve y = 4x , y axis and the line y=3
is -

(a) 2 (b) (c) (d)

(xiv) ;fn P(A/B) > P(A) rks fuEu esa ls lR; gS &

v- P(B/A) < P(B) c- 𝑃(𝐴 ∩ 𝐵) < 𝑃(𝐴). 𝑃(𝐵)

l- P(B/A) > P(B) n- P(B/A) = P(B) (1)

if P(A/B) > P(A) then which of the following is correct

(a) P(B/A) < P(B) (b) 𝑃(𝐴 ∩ 𝐵) < 𝑃(𝐴). 𝑃(𝐵)

(c)P(B/A) > P(B)(d) P(B/A) = P(B)

(xv) ;fn P(A) = , P(B) = 0 rks P(A/B) gS &

v- 0 c- l- ifjHkkf"kr ugha n- 1 (1)

If P(A) = , P(B) = 0 then P(A/B) is

(a) 0 (b) (c)𝑛𝑜𝑡 𝑑𝑖𝑓𝑖𝑛𝑒 (d) 1

2 fjDr LFkkuksa dh iwfrZ dhft;s &

Fill in the blanks

(i) tan √3 − sec (−2) dk eku ----------------------------------------------gS A (1)

The value of tan √3 − sec (−2) is …………………………

(ii) cos cos dk eku ------------------------------------------------------gS A (1)

The value of cos cos is ……………………………..

(iii) tan + tan dk eku -------------------------------- gSA (1)

The value of tan + tan is ……………………………..

(iv) ;fn y= logax rks = ………………………………. (1)

If y= logax 𝑡ℎ𝑒𝑛 = ……………………………….

(v) ;fn f(x) = - I x+1I + 3 rks f(x) dk vf/kdre eku ----------------------------------gSA (1)

If f(x) = - I x+1I + 3 then the maximum value of f(x) is ………………….

(vi) vody lehdj.k + ysecx = tanx dk lekdy xq.kkad gS & (1)

The integrating factor of equation is + ysecx = tanx …………………

(vii) lfn’k ı̂ + Jdk lfn’k ı̂ − J ij iz{ksi --------------------------------gSA (1)

The projection of the vector ı̂ + J on the vector ı̂ − Jis …………..

3 vfry?kqRrjkRed iz’u &

Very short answer types questions

(i) lkfj.kd x − x + 1 x − 1 dk eku Kkr dhft, A (1)
x+1 x+1

Find the value of determinate x − x + 1 x − 1
x+1 x+1
2 4 2x 4
(ii) x dk eku Kkr dhft, ;fn = (1)
5 1 6 x
Find the value of x if
2 4 2x 4
=
5 1 6 x
(iii) varjky Kkr dhft;s ftlesa f(x) = cosx ls iznr Qyu f o/kZeku gS] tgk a0≤x≤2πA (1)

Find the interval in which the function f given by f(x) = cosx, where
0≤x≤2π is increasing function.

(iv) fdlh mRikn dh x bdkbZ;ksa ds fodz; ls izkIr dqy vk; R(x) :Ik;ksa esa R(x) =
13x2+26x+15 ls iznr gSA lhekUr vk; Kkr dhft;s tc x=7 gSA (1)

The total revenue in rupees received from the sale of x units of a product
is given by R(x) = 13x2+26x+15 .Find the marginal revenue when x=7

(v) ∫ dx dk eku Kkr dhft,A (1)

Find the value of ∫ dx

(vi) ∫ 𝑒 ( tan x+ ) dx dk eku Kkr dhft,A (1)

Find the value of ∫ 𝑒 ( tan x+ ) dx

(vii) pkj dksfV okys fdlh vody lehdj.k ds O;kid gy esa mifLFkr LosPN vpjksa dh la[;k
Kkr dhft;sA (1)

Find the number of arbitratry constants in the general solutions of a
differential equation of fourth order.

(viii) lfn’k a⃗ = 2ı̂ + 3J + kds vuqfn’k ek=d lfn’k Kkr dhft,A (1)

Find unit vector in the direction of vector

a⃗ = 2ı̂ + 3J + k

(ix) x rFkk y ds eku Kkr dhft, rkfd lfn’k 2ı̂ + 3J vkSj xı̂ + yJ leku gksaA (1)

Find the value of x and y so that the vectors 2ı̂ + 3J and xı̂ + yJ are equal .

(x) nks lfn’kksa a⃗rFkk b⃗ds ifj.kke dze’k% 1 vkSj 2 rFkk a⃗.b⃗=1A bu lfn’kksa ds e/; dks.k Kkr
dhft,A (1)

Find the angle between two vector a⃗ and b⃗with magnitudes 1 and 2
respectively and when a⃗.b⃗=1

[kaM & c
Section – B
y?kqmRrjh; iz’u &
Short answer type question –

4- tkWap dhft;s fd okLrfod la[;kvksa ds leqPp; 𝑅 esa 𝑅∗ = {(𝑎, 𝑏); 𝑎 ≤ 𝑏 ] }kjk
ifjHkkf"kr laca/k 𝑅∗ u rks LorqY;] u lefer vkSj u gh ladzked gSA (2)
Show that tha relation 𝑅∗ in the set R of real numbers defined as
𝑅∗ = {(𝑎, 𝑏); 𝑎 ≤ 𝑏 is neither reflexive, nor symmetric nor transitive.

𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼
5- ;fn 𝐴 = gks rks lR;kfir dhft;s fd 𝐴𝐴 = 1
−𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼
𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼
𝐼𝑓 𝐴 = then verify 𝐴𝐴 = 1
−𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼

3 −2 1 0
6- ;fn 𝐴 = rFkk 𝐼 = ,oa 𝐴 = 𝐾𝐴 − 2𝐼 gks rks 𝐾 dk eku Kkr
4 −2 0 1
dhft;sA (2)

3 −2 1 0
If 𝐴 = and 𝐼 = Then Find K, so that 𝐴 = 𝐾𝐴 − 2𝐼 .
4 −2 0 1

2 3 1 −2
7- ;fn 𝐴 = rFkk 𝐵 = rks lR;kfir dhft;s fd (𝐴𝐵) =
1 −4 −1 3
𝐵 𝐴 (2)
2 3 1 −2
If 𝐴 = and 𝐵 = , then prove that (𝐴𝐵) = 𝐵 𝐴
1 −4 −1 3

𝑘𝑥 + 1, 𝑥 ≤ 5
8- 𝐾 dk eku Kkr dhft;s ;fn 𝑓(𝑥) = ij lrr gksA (2)
3𝑥 − 5, 𝑥 > 5, 𝑥 = 5
𝑘𝑥 + 1, 𝑥 ≤ 5
Find the value of K so that , 𝑓(𝑥) = is continuous at
3𝑥 − 5, 𝑥 > 5
𝑥 = 5.

9- n’kkZb, fd 𝑓(𝑥) = |𝑐𝑜𝑠𝑥| }kjk ifjHkkf"kr Qyu ,d lrr Qyu gSA (2)
Show that tha function defined by 𝑓(𝑥) = |𝑐𝑜𝑠𝑥| is a continuous
function.

10- ;fn 𝑦 = sin rc dk eku Kkr dhft;sA (2)

If 𝑦 = sin then Find .

11- fn[kkb, fd iznr Qyu 𝑓, 𝑅 ij ,d o/kZeku Qyu gS & (2)
𝑓(𝑥) = 𝑥 − 3𝑥 + 4𝑥 , 𝑥𝜖𝑅
Show that the function given by 𝑓(𝑥) = 𝑥 − 3𝑥 + 4𝑥 , 𝑥𝜖𝑅 is increasing
on 𝑅.
12- ∫ 𝑑𝑥 dk eku Kkr dhft;sA (2)
Find the value of ∫ 𝑑𝑥

13- odz 𝑦 = 𝑥 ,oa js[kk 𝑦 = 4 ls f?kjs {ks= dk {ks=Qy Kkr dhft;sA (2)
Find the area of the region bounded by the curve 𝑦 = 𝑥 and line 𝑦 = 4.

14- fl} dhft;s fd nks lfn’kksa 𝑎⃗ o 𝑏⃗ ds fy;s lnSo 𝑎⃗. 𝑏⃗ ≤ |𝑎⃗| 𝑏⃗ (2)
Prove that for two vector 𝑎⃗ and 𝑏⃗, 𝑎⃗. 𝑏⃗ ≤ |𝑎⃗| 𝑏⃗

15- ;g fn;k x;k gS fd nks iklksa dks ,d lkFk Qsadus ij izkIr la[;k;sa fHkUu fHkUu gSA nksuksa

la[;kvksa dk ;ksx 4 gksus dh izkf;drk Kkr dhft;sA (2)
Given that the two numbers appearing on throwing two dice are
different. find tha probability of the event “ the sum of number on the
dice is 4”

[kaM & l
Section – C
nh?kZmRrjh; iz’u &
Long answer type question –

16- ∫ √𝑥 + 4𝑥 − 5 𝑑𝑥 dk eku Kkr dhft;sA (3)
Evaluate - ∫ √𝑥 + 4𝑥 − 5 𝑑𝑥
vFkok
Or

∫ sin 𝑑𝑥 dk eku Kkr dhft;sA
Evaluate ∫ sin 𝑑𝑥

17- fcanq (−2,3) ls xqtjus okys ,sls odz dk lehdj.k Kkr dhft, ftlds fdlh fcanq (𝑥, 𝑦)
ij Li’kZ js[kk dh izo.krk gSA (3)
Findthe equation of a curve passing through the point (-2,3) given that the
slope of the tangent to the curve at any point (𝑥, 𝑦)is .
vFkok
Or
fdlh cSad esa ewy/ku dh o`f} 𝑟% okf"kZd dh nj ls gksrh gSA ;fn 100 :Ik;s 10 o"kZ essa nksxqus
gks tkrs gSa rks 𝑟 dk eku Kkr dhft;sA(log 2 = 0.6931)
In a bank, principle increases continuously at the rate r% per year. Find
the value of r if Rs 100 double itself in 10 year. (log 2 = 0.6931)

18- js[kkvksa 𝑙 o 𝑙 ds chp esa U;wure nwjh Kkr dhft, ftuds lfn’k lehdj.k gS &
𝑟⃗ = 𝚤̂ + 𝚥̂ + ℷ (2 𝚤̂ − 𝚥̂ + 𝑘 )vkSj 𝑟⃗ = 2𝚤̂ + 𝚥̂ − 𝑘 + 𝜇(3 𝚤̂ − 5𝚥̂ + 𝑘) (3)

Find the shortest distance between the lines 𝑙 𝑎𝑛𝑑 𝑙 whose vector
equations are –

𝑟⃗ = 𝚤̂ + 𝚥̂ + ℷ (2 𝚤̂ − 𝚥̂ + 𝑘 )and𝑟⃗ = 2𝚤̂ + 𝚥̂ − 𝑘 + 𝜇(3 𝚤̂ − 5𝚥̂ + 𝑘 )

vFkok
Or

fn;s x;s js[kk ;qXe ds e/; dks.k Kkr dhft;s &

𝑟⃗ = 2𝚤̂ − 5𝚥̂ + 𝑘 + ℷ (3 𝚤̂ + 2𝚥̂ + 6𝑘 )vkSj
𝑟⃗ = 7𝚤̂ − 6𝑘 + 𝜇(𝚤̂ + 2𝚥̂ + 2𝑘 ) (3)
Find the angle between the pair of lines given by -
𝑟⃗ = 2𝚤̂ − 5𝚥̂ + 𝑘 + ℷ (3 𝚤̂ + 2𝚥̂ + 6𝑘 ) and
𝑟⃗ = 7𝚤̂ − 6𝑘 + 𝜇(𝚤̂ + 2𝚥̂ + 2𝑘 )

19- ‘,d FkSys esa 4 yky vkSj 4 dkyh xsansa gSaA ,d vU; FkSys esa 2 yky ,osa 6 dkyh xsansa gSaA nksukss
FkSyksa esa ls ,d ;kn`PN;k pquk tkrk gS ,oa ,d xsan fudkyh tkrh gS tks fd yky gSA bl
ckr dh D;k izkf;drk gS] fd xsan igys FkSys ls fudkyh x;h gSA (3)
In one bag, there are 4 red and 4 black balls. In another bag, there are 2
red and 6 black balls. One bag is randomly chosen, and a ball is drawn
from it, which happens to be red. What is the probability that the ball was
drawn from the first bag?

vFkok
Or
𝐴 }kjk lR; cksyus dh izkf;drk gSA ,d flDdk mNkyk tkrk gS rFkk 𝐴 crkrk gS fd
fpRr iznf’kZr gqvk gSA okLro esa fpRr iznf’kr gksus dh D;k izkf;drk gSA

The probability of telling the truth by A is . A coin is tossed, and it
indicates that the face is showing. What is the actual probability of the
face being shown?
[kaM & n
Section – D
fuca/kkRed iz’u &
Essay type question –
20. ∫ dk eku Kkr dhft;sA (4)

Evaluate ∫

vFkok
Or
∫ √𝑐𝑜𝑡𝑥 + √𝑡𝑎𝑛𝑥 𝑑𝑥dk eku Kkr dhft;sA

Find out ∫ √𝑐𝑜𝑡𝑥 + √𝑡𝑎𝑛𝑥 𝑑𝑥

21. js[kk;sa ftudk lfn’k lehdj.k fuEu gSa ds chp U;wure nwjh Kkr dhft;s & (4)

𝑟⃗ = (1 − 𝑡)𝚤̂ + (𝑡 − 2)𝚥̂ + (3 − 2𝑡)𝑘vkSj
𝑟⃗ = (𝑆 + 1)𝚤̂ + (2𝑆 − 1)𝚥̂ − (2𝑆 + 1)𝑘

Find the shortest distance between the lines whose vector equations are

𝑟⃗ = (1 − 𝑡)𝚤̂ + (𝑡 − 2)𝚥̂ + (3 − 2𝑡)𝑘and
𝑟⃗ = (𝑆 + 1)𝚤̂ + (2𝑆 − 1)𝚥̂ − 0(2𝑆 + 1)𝑘
vFkok
Or
𝑝 dk eku Kkr dhft;s rkfd js[kk;sa = = vkSj = =

ijLij yacor gksaA

Find the value of 𝑝 Show that lines = = and

= = are at right angle.

22. fuEufyf[kr O;ojks/kksa ds varxZr 𝑧 = −3𝑥 + 4𝑦 dk vkys[kh; fof/k ls U;wurehdj.k
dhft;s A (4)
𝑥 + 2𝑦 ≤ 8, 3𝑥 + 2𝑦 ≤ 12 , 𝑥 ≥ 0, 𝑦 ≥ 0

Minimize 𝑧 = −3𝑥 + 4𝑦 subject to constraints 𝑥 + 2𝑦 ≤ 8, 3𝑥 + 2𝑦 ≤
12, 𝑥 ≥ 0, 𝑦 ≥ 0 by using graphical method.

avFkok
Or
fuEufyf[kr O;ojks/kksa ds varxZr 𝑧 = 𝑥 + 𝑦 dk vkys[kh; fof/k ls vf/kdrehdj.k dhft;s A

𝑥 − 𝑦 ≤ −1, −𝑥 + 𝑦 ≤ 0 , 𝑥 ≥ 0, 𝑦 ≥ 0

Maximize 𝑧 = 𝑥 + 𝑦 subject to constraints 𝑥 − 𝑦 ≤ −1, −𝑥 + 𝑦 ≤ 0 , 𝑥 ≥ 0,

𝑦 ≥ 0 by using graphical method.