CBSE Class 12 Question Paper 2022 Maths (Solved)

Series ABCD1/2 Set No. 1
àíZ-nÌ H$moS>
Q.P. Code 65/2/1
AZwH«$_m§§H$
Roll No. narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$
_wI-n¥ð >na Adí` {bIo§ &
Candidates must write the Q.P. Code
on the title page of the answer-book.

7

14

15
10.15 10.15 10.30

Please check that this question paper contains 7 printed pages.
Q.P. Code given on the right hand side of the question paper should be
written on the title page of the answer-book by the candidate.
Please check that this question paper contains 14 questions.
Please write down the serial number of the question in the answer-book
before attempting it.
15 minute time has been allotted to read this question paper. The question
paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the
students will read the question paper only and will not write any answer on
the answer-book during this period.

J{UV
MATHEMATICS

:2 : 40
Time allowed : 2 hours Maximum Marks : 40

65/2/1 Page 1 of 7 P.T.O.

:
:
(i)
(ii)
(iii) 6 I 2
(iv) 4 II 3
(v) 4 4
(vi)
(vii) 14 2

IÊS> H$
1 6 2

1. AdH$b g_rH$aU
d dy
(xy 2 ) . y 0
dx dx

H$s H$mo{Q> Am¡a KmV H$m JwUZ\$b kmV H$s{OE & 2

2. (H$) kmV H$s{OE : 2

sin 3x
dx
sin x
AWdm
(I) _mZ kmV H$s{OE : 2
1
log 3
2
ex
dx
2x
e 1
0

3. Xmo _mÌH$ g{Xem| a Am¡a b Ho$ {bE |2a + 3 b | = | 3a 2 b | h¡ & a VWm b
Ho$ ~rM H$m H$moU kmV H$s{OE & 2
65/2/1 Page 2 of 7

General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper contains three sections Section A, B and C.
(ii) Each section is compulsory.
(iii) Section A has 6 short answer type I questions of 2 marks each.
(iv) Section B has 4 short answer type II questions of 3 marks each.
(v) Section C has 4 long answer type questions of 4 marks each.
(vi) There is an internal choice in some questions.
(vii) Question no. 14 is a case-study based question with 2 sub-parts of 2 marks
each.

SECTION A

Questions number 1 to 6 carry 2 marks each.

1. Find the product of the order and the degree of the differential equation
d dy
(xy 2 ) . y 0. 2
dx dx

2. (a) Find : 2
sin 3x
dx
sin x

OR

(b) Evaluate : 2
1
log 3
2
ex
dx
e 2x 1
0

3. a and b are two unit vectors such that |2a + 3 b | = | 3a 2 b |. Find
the angle between a and b . 2

65/2/1 Page 3 of 7 P.T.O.

4. nmgm| Ho$ EH$ `w½_ H$mo CN>mbm J`m & XmoZm| nmgm| na AmB© g§»`mAm| H$m `moJ\$b EH$ g_
g§»`m h¡ & H$_-go-H$_ EH$ nmgo na g§»`m 3 AmZo H$s àm{`H$Vm kmV H$s{OE & 2

5. EH$ {deof àíZ H$mo A Am¡a B Ûmam ñdV§Ì ê$n go hb H$aZo H$s àm{`H$VmE± H«$_e: 2 Am¡a
3 3
h¢ & `{X XmoZm| ñdV§Ì ê$n go àíZ H$mo hb H$aZo H$s H$mo{ee H$aVo h¢, Vmo àíZ Ho$ hb hmo
5
OmZo H$s àm{`H$Vm kmV H$s{OE & 2

6. aoIm PQ Omo {~ÝXþAm| P(2, 2, 1) VWm Q(5, 1, 2) go JwµOaVr h¡, H$m H$mVu` g_rH$aU
{b{IE & AV: aoIm PQ na Cg {~ÝXþ H$m y-{ZX©oem§H$ kmV H$s{OE {OgH$m z-{ZX©oem§H$ 2
h¡ & 2
IÊS> I
7 10 3
^ ^ ^ ^ ^
7. ABCD EH$ g_m§Va MVw^©wO h¡ {Og_| AC = i + j Am¡a BD = 2 i + j + k h¢ &
AB Am¡a AD kmV H$s{OE & g_m§Va MVw^©wO ABCD H$m joÌ\$b ^r kmV H$s{OE & 3

8. (H$) _mZ kmV H$s{OE : 3
1

tan 1 x dx
0
AWdm
(I) kmV H$s{OE : 3
2x
dx
2
x 3x 2

9. AdH$b g_rH$aU (y + 3x2) dx = x H$m {d{eîQ> hb kmV H$s{OE, {X`m J`m h¡ y = 1
dy
O~ x = 1. 3

10. (H$) {~ÝXþAm| (2, 1, 0), (3, 2, 2) Am¡a (1, 1, 7) go JwµOaZo dmbo g_Vb H$m
g_rH$aU kmV H$s{OE & Bg g_Vb H$s _yb-{~ÝXþ go Xÿar ^r kmV H$s{OE & 3
AWdm
(I) aoImAm| x = y 1 = z 2 VWm x + 1 = y 2 = z 1 Ho$ ~rM H$s Xÿar
2 3 2 3
kmV H$s{OE & 3
65/2/1 Page 4 of 7

4. A pair of dice is thrown. It is given that the sum of numbers appearing on
both dice is an even number. Find the probability that the number
appearing on at least one die is 3. 2
2 3
5. Probabilities of A and B solving a specific problem are and ,
3 5
respectively. If both of them try independently to solve the problem, then
find the probability that the problem is solved. 2
6. Write the cartesian equation of the line PQ passing through points
P(2, 2, 1) and Q(5, 1, 2). Hence, find the y-coordinate of the point on the
line PQ whose z-coordinate is 2. 2
SECTION B
Questions number 7 to 10 carry 3 marks each.
^ ^ ^ ^ ^
7. ABCD is a parallelogram such that AC = i + j and BD = 2 i + j + k .
Find AB and AD . Also, find the area of the parallelogram ABCD. 3

8. (a) Evaluate : 3
1

tan 1 x dx
0

OR
(b) Find : 3
2x
dx
x2 3x 2
dx
9. Find the particular solution of the differential equation (y + 3x2) = x,
dy
given that y = 1, when x = 1. 3
10. (a) Find the equation of the plane passing through points (2, 1, 0),
(3, 2, 2) and (1, 1, 7). Also, obtain its distance from the origin. 3
OR
y 1 z 2
(b) Find the distance between the lines x and
y 2 z 1 2 3
x 1 . 3
2 3

65/2/1 Page 5 of 7 P.T.O.

IÊS> J
11 14 4
x 2 y 1 z 2
11. aoIm Am¡a g_Vb x y+z=5 Ho$ à{VÀN>oXZ {~ÝXþ H$s
3 4 12
{~ÝXþ ( 1, 5, 10) go Xÿar kmV H$s{OE & 4
12. _mZ kmV H$s{OE : 4
1

x (1 x) n dx
0
13. (H$) g_mH$bZ Ho$ à`moJ go, dH«$ 4x2 + 4y2 = 9 Am¡a aoIm 2x + 2y = 3 Ho$ ~rM
{Kao bKw joÌ H$m joÌ\$b kmV H$s{OE & 4
AWdm
(I) `{X dH«$ y2 = 4ax Am¡a aoIm x = 4a go {Kao joÌ H$m joÌ\$b 256 dJ© BH$mB© h¡,
3
Vmo g_mH$bZ Ho$ à`moJ go a, Ohm± a > 0, H$m _mZ kmV H$s{OE & 4

àH$aU-AÜ``Z AmYm[aV àíZ
14. {H$gr ^r {H«$Ho$Q> Ho$ _¡M Ho$ Amaå^ _|,
OrVVr h¡ Cgo ~ëbo~mOr `m J|X~mOr MwZZo H$m _m¡H$m {_bVm h¡ & Eogm {
h¡ {Og_| {MV Am¡a nQ> Ho$ AmZo H$s àm{`H$VmE± g_mZ hmoVr h¢ &

Cnamoº$ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :
(H$) 2 ~ma CN>mbm OmE, Vmo nQ>m| H$s g§»`m Ho$ {bE àm{`H$Vm ~§Q>Z
kmV H$s{OE & 2
(I) Eogo 3 ~ma CN>mbZo na H$_-go-H$_ EH$ {MV AmZo H$s àm{`H$Vm kmV
H$s{OE & 2
65/2/1 Page 6 of 7

SECTION C

Questions number 11 to 14 carry 4 marks each.
11. Find the distance of the point ( 1, 5, 10) from the point of
x 2 y 1 z 2
intersection of the line and the plane x y + z = 5. 4
3 4 12
12. Evaluate : 4
1

x (1 x) n dx
0
13. (a) Using integration, find the area of the smaller region enclosed by
the curve 4x2 + 4y2 = 9 and the line 2x + 2y = 3. 4

OR
(b) If the area of the region bounded by the curve y2 = 4ax and the
256
line x = 4a is sq. units, then using integration, find the value
3
of a, where a > 0. 4

Case-Study Based Question
14. At the start of a cricket match, a coin is tossed and the team winning the
toss has the opportunity to choose to bat or bowl. Such a coin is unbiased
with equal probabilities of getting head and tail.

Based on the above information, answer the following questions :
(a) If such a coin is tossed 2 times, then find the probability
distribution of number of tails. 2
(b) Find the probability of getting at least one head in three tosses of
such a coin. 2
65/2/1 Page 7 of 7 P.T.O.