Uttarakhand Board Class 12 Sample Paper 2023 Maths

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Roll No-
Ikzfrn”kZ iz”ui=
2023
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MATHEMATICS
le;% 3 ?k.Vs iw.kkZd% 80

Time: 3 hours Max Marks: 80
funsZ”k% ¼i½ lHkh ç”u vfuok;Z gSaA bl ç”ui= esa dqy 26 ç”u gSaA
¼ii½ ç”Uk la[;k 1 cgqfodYih; gS tks 8 ¼vkB½ [k.Mksa esa foHkä gSA çR;sd [k.M ,d vad dk gSaA
çR;sd [k.M ds mÙkj esa pkj fodYi fn;s x;s gSaA lgh fodYi viuh mÙkj iqfLrdk esa fyf[k;sA
(iii) ç”u la[;k 2ls 9 rd çR;ssd ç”u ,d vad dk gSA ç”u la[;k 10 ls 14 rd çR;sd ç”u nks
vadksa dk gSA ç”u la[;k 15 ls 20 rd çR;sd ç”u pkj vadksa dk gS rFkk ç”u la[;k 21 ls 26 rd
çR;sd
s ç”u ik¡p vadksa dk gSA
¼iv½ ç”u i= esa lexz ij dksbZ fodYi ugha gS] rFkkfi dqN ç”uksa esa vkarfjd fodYi çnku fd;s x;s
gSaSA ,sls ç”uksa ds dsoy ,d fodYi dk gh mÙkj nhft,A
¼V½ dSydqysVj ds ç;ksx dh vuqefr ugha gSA

Note: (i) All questions are compulsory. There are 26 questions in this question paper.
¼ii) Question No. 1 is multiple choice question divided into 8 (eight) parts. Each
part carry one mark. Four options are given in each part of question. Write the
correct option in your answer book.
¼iii½ Question No 2 to 9 carry one mark each. Question No 10 to 14 carry two
marks each Question No 15 to 20 carry four marks each and Question No 21 to
26 carry five marks each.
(iv) There is no overall choice in question paper, however, in some questions
internal choices is provided. You have to attempt only one of the given choices
in such questions.
¼v) Use of calculator is not permitted

1½ i) ;fn cos-1 x = y, rks& 1
If cos-1 x = y, then &

−𝜋 𝜋
(a) ≤𝑦≤ (b) 0< 𝑦 < 𝜋
2 2
−𝜋 𝜋
(c) 0≤ 𝑦 ≤ 𝜋 (d) <𝑦<
2 2

(ii) oxZ vkO;wg A= aij ftlesa aij =0 tc iǂ j dgykrk gS& 1
a) fod.kZ vkO;wg b) “kwU; vkO;wg
c) LrEHk vkO;wg d) iafä vkO;wg
Square matrix A= [aij] in which aij = o when iǂ j is called -
a) Diagonal b) Zero matrix
c) Column matrix d) Row matrix

(iii) Sin x2 dk vodyt gksxk- 1
Derivative of Sin x2 will be-
a) cos x2 b) 2x cos x2 c) 2x sin x2 d) sin x2

√3 1 cjkcj gS& 1
iv) ∫0 𝑑𝑥
1+𝑥 2

√3 1
∫0 𝑑𝑥 is equal to-
1+𝑥 2

𝜋 2𝜋 𝜋 𝜋
a) b) c) d)
3 3 6 12
v) vody lehdj.k 1
𝑑2 𝑦 𝑑𝑦 2 𝑑𝑦
+ + (𝑑𝑥 ) + 𝑠𝑖𝑛 (𝑑𝑥 ) + 1 = 0 dh ?kkr gS&
𝑑𝑥 2

a) 3 b) 2 c) 1 d) ifjHkkf’kr ugha gSA

𝑑2 𝑦 𝑑𝑦 2 𝑑𝑦
Degree of differential equation + + (𝑑𝑥 ) + 𝑠𝑖𝑛 (𝑑𝑥 ) + 1 = 0 is-
𝑑𝑥 2

a) 3 b) 2 c) 1 d) Not defined

(vi) nks lfn”kksa a. rFkk b ds chp dk dks.k] ftuds ifjek.k Øe”k% 3 rFkk 4 gSa ,oa a. b = 2 3
1
The angle between two vectors a and b with magnitudes 3 and 4 respectively
and a. b= 2 3 will be -
𝜋 𝜋 𝜋 5𝜋
a) b) c) d)
6 3 2 2

(vii) lery 2x-3y+4z-6 =0 dh ewy fcUnq ls nwjh gS& 1

Distance of plane 2x-3y+4z-6=0 from the origin is-
6
a) b) 6 c) √29 d) 3
√29

(viii) ;fn iklksa dk ,d tksMk+ mNkyk tkrk gS tks çR;sd ikls ij le vHkkT; la[;k çkIr djus dh
çkf;drk gS % 1
The probability of obtaining an even prime number on each die, when a pair of
dice rolled, is :
a) 0 b)1/3 c )1/12 d)1/36

2) D;k Qyu f(x)=x2 }kjk ijfHkkf’kr Qyu f:N N vkPNknd gSA 1
Whether the function f:N N, defined by f(x)=x2 is onto ?
3) lkjf.kd = -1 2 0 esa vo;o 3 dk milkjf.kd Kkr dhft,A 1
3 4 -5
0 6 1

In the determinant : = -1 2 0
3 4 -5 find the minor of element 3.
0 6 1
4) sin−1 𝑥+cos −1 𝑥 dk 𝑥 ds lkis{k vodyt Kkr dhft, A 1
Find the derivative of sin−1 𝑥+cos −1 𝑥 with respect to 𝑥.
5) x=2, ij oØ y= x3-x dh Li”kZ js[kk dh ço.krk Kkr dhft,A 1
Find the slope of tangent of curve y= x3-x, at x=2.
2
6) ∫0 𝑒 𝑥 𝑑𝑥 dk eku Kkr dhft, A 1
2
Evaluate ∫0 𝑒 𝑥 𝑑𝑥.
(log 𝑥)2
7) Qyu dk x ds lkis{k lekdyu Kkr dhft,A 1
𝑥
(log 𝑥)2
Integrate function with respect to x-
𝑥

8) ;fn a = 𝑖̂ + 3𝑗̂ − 2𝑘̂ gks rks |𝐴⃗| dk eku Kkr dhft,A 1

If 𝑎⃗ = 𝑖̂ + 3𝑗̂ − 2𝑘̂ then find the value of |𝐴⃗|
9) ,d js[kk ds fnd~&vuqikr -2,2,1 gSaA bldh fnd~ dksT;k,a Kkr dhft,A 1
Direction ratios of a line are -2,2,1, find its direction cosines.
1
10) ;fn sin (sin−1 + cos −1 𝑥) = 1, rks x dk eku Kkr dhft,A 2
5
If sin (sin-1 1 + cos-1 x) =1, then find the value of x.
5
11) ,d ?ku dk vk;ru 9 lseh3 @lsd.M dh nj ls c<+ jgk gSA ;fn blds dksj dh yEckbZ
5 lseh gS rks bldk dksj fdl nj ls ifjofrZr gks jgk gSA 2
The volume of a Cube is increasing at a rate of 9 cm3/sec. How fast is the side
of cube changing when the length of an edge is 5 cm.
fn[kkb, fd çnÙk Qyu f, f(x)= x3-3x2+4x, X ∈ R, R ij o/kZeku Qyu gSA

2
Show that the function f given by, f(x) = x3-3x2+4x, X ∈ R is increasing on R.
12) lery x+2y+3z = 6 }kjk v{kksa ij dkVs x;s vUr%[k.Mksa dh yEckbZ Kkr dhft,A
2
Find the intercepts cut off by the plane x+2y+3z = 6 on the axes.
x – v{k dk dkrhZ; lehdj.k f=foeh; :Ik esa Kkr dhft,A
Find the Cartesian equation of x-axis in 3-dimensional form.
1
13) Qyu dk x ds lkis{k lekdyu Kkr dhft,A 2
(x+2) (x+3)
1
Find the integration of the function with respect to x.
(x+2) (x+3)

1 1
14) ,d fo”ks’k ç”u dks A vkSj B }kjk Lora= :Ik ls gy djus dh çf;drk vkSj 3 gSA
2
çkf;drk Kkr dhft, fd ç”u gy gks tkrk gSA 2
1
Probability of solving a specific problem independently by A and B are and
2
1
respectively. If both try to solve the problem independently. Find the
3
probability that the problem is solved.

15) ,d gh ry esa fLFkr leLr lh/kh js[kkvksa ds leqPp; L esa laca/k R, R = { (L1, L2):
L1, L2 ds lekUrj gS } }kjk ifjHkkf’kr gSA fl) dhft, fd R ,d rqY;rk laca/k gSA
4
In a set L of all straight lines lying in a single plane, a relation R is defined by
R = {(L1, L2): L1 is parallel to L2} . Prove that R is an equivalence
relation.
𝑎𝑏
/kukRed ifjes; la[;kvksa ds leqPp; Q esa a * b = 2 }kjk ifjHkkf’kr f}vk/kkjh lafØ;k
* ds fy, rRled vo;o ,oa çfrykse Kkr dhft,A 4
Find identity element and inverse of a binary operation * on the set Q of positive
𝑎𝑏
rational numbers defined as a*b = .
2

16) çkjafHkd :ikUrj.k }kjk vkO;wg A= 2 3 dk O;qRØe Kkr dhft,A 4
5 7
Find the inverse of matrix A= 2 3 by the elementry oprations
5 7

17) f (x) = |𝑥 | , ;fn x # 0 dh laakrR;rk dh tkap dhft,A
𝑥
0, ;fn x=0 4

|𝑥|
Examine the continuity of the function f (x)=
𝑥
, if 𝑥 ≠ 0
0 , if 𝑥 = 0

𝑥+3
18) ∫ dx dk eku Kkr dhft, A 4
√5−4𝑥+𝑥 2
𝑥+3
Evaluate ∫ dx.
√5−4𝑥+𝑥 2

𝑎 √𝑥
∫0 𝑑𝑥 dk eku Kkr dhft, A 4
√𝑥+√𝑎−𝑥

𝑎 √𝑥
Evaluate ∫0 𝑑𝑥.
√𝑥+√𝑎−𝑥

19) ;fn 𝑎⃗, +𝑏⃗⃗, 𝑐⃗ ek=d lfn”k bl izdkj gSa fd 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = ⃗0⃗ rks 𝑎⃗.𝑏⃗⃗+𝑏⃗⃗.𝑐⃗+𝑐⃗.𝑎⃗ dk eku Kkr
dhft, A 4
If 𝑎⃗, +𝑏⃗⃗, 𝑐⃗ are unit vectors such that 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = ⃗0⃗ then find the value of
𝑎⃗.𝑏⃗⃗+𝑏⃗⃗.𝑐⃗+𝑐⃗.𝑎⃗

𝑥+3 𝑦−6 𝑧 𝑥+2 𝑦 𝑧−7
20) js[kkvksa = = ,oads e/; mHk;fu’B yEc dh yEckbZ ,oa lehdj.k Kkr dhft, A
= =
−4 3 2 −4 1 1
4
𝑥+3 𝑦−6 𝑧
Find the equation and length of common normal between the lines = =
−4 3 2
𝑥+2 𝑦 𝑧−7
and = = .
−4 1 1
-

ml lery dk lehdj.k Kkr dhft, tks fcUnq ¼&3]1]2½ ls xqtjrk gS rFkk leryksa x+2y+3z=5 vkSj
3x+3y+z=0 esa ls izR;sd ij yEc gS A

Find the equation of the plane passing through the point (-3,1,2) and is
perpandicular to each plane x+2y+3z=5 and 3x+3y+z=0.

2 −3 5
21) ;fn vkO;wg A=[3 2 −4] g]S rks 𝐴−1 Kkr dhft, A 𝐴−1 dk iz;ksx djds lehdj.k
1 1 −2
fudk; 2x-3y+5z=-11, 3x+2y-4z=-5, x+y-2z=-3 dks gy dhft, A 6
2 −3 5
If matrix A=[3 2 −4], Find 𝐴−1
1 1 −2
−1
Using 𝐴 solve the system of equations 2x-3y+5z=-11, 3x+2y-4z=-5, x+y-2z=-3.

22. fl) dhft, fd ,d “kadq ds vUrxZr egRre odzi`’B okys yEco`RRkh; csyu dh f=T;k “kadq dh
f=T;k dh vk/kh gksrh gS A 6
Prove that the radius of the right circular cylinder of greatest curved surface area
which can be inscribed in a given cone is half of that of the cone.
fl) dhft, fd odz x=y2 vkSj xy=k ,d nwljs dks ledks.k ij dkVrs gSa ;fn 8k2=1
Prove that the curves x=y2 and xy=k cut at right angles if 8k2=1.

𝑑𝑦
23. vody lehdj.k − 𝑦 = cos 𝑥 dk gy Kkr dhft, A 6
𝑑𝑥
𝑑𝑦
Solve the differential equation − 𝑦 = cos 𝑥.
𝑑𝑥
2 2
n”kkZb, fd vody lehdj.k (x -y )dx+(2xy)dy=0 le?kkrh; gS vkSj bls gy dhft, A
Show that the differential equation (x2-y2)dx+(2xy)dy=0 is homogeneous and solve it.

24. ijoy; y2=4ax ,oa js[kk y=mx ls f?kjs {ks= dk {ks=Qy Kkr dhft, A 6
Find the area bounded by the parabola y2=4ax and the line y=mx

25. vkys[kh; fof/k ls fuEu jSf[kd izksxkeu leL;k dks fn, x;s O;ojks/kksa ds vUrxZr gy dhft, ,oa
z=200x+500y dk U;wure eku Kkr dhft, A 6
x+2y≥10,
3x+4y≤ 24,
x≥ 0, y≥ 0
Solve the following linear programming problem graphically and find the minimum
value of z=200x+500y

Subject to the constraints-

x+2y≥10,
3x+4y≤ 24,
x≥ 0, y≥ 0.

26. iklksa ds ,d tksM+s dks rhu ckj mNkyus ij f}dksa dh la[;k dk izkf;drk caVu Kkr dhft, A 6
Find the probability distribution of number of doublets in three throws of a pair of
dice.
4
,d O;fDr }kjk lR; cksyus dh izkf;drk gS A ,d flDdk mNkyk tkrk gS vkSj ;g O;fDr crkrk gS fd
5
fpr izdV gqvk gSAa okLro esa fpr izdV gksus dh izkf;drk D;k gS \
4
The probability that a person speaks truth is . A coin is tossed and this person tells
5
that the head has appeared. What is the probability that the head has actually
appeared?