CBSE Class 12 Question Paper 2022 Applied Mathematics

SET-4
Series %BAB% àíZ-nÌ H$moS>
Q.P. Code 465
amob Z§. narjmWu àíZ-nÌ H$moS> H$mo CÎma-nwpñVH$m Ho$
Roll No. _wI-n¥ð >na Adí` {bIo§ &
Candidates must write the Q.P. Code on
the title page of the answer-book.

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ &
àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS H$mo narjmWu CÎma-nwpñVH$m Ho$ _wI-n¥>ð> na
{bI| &
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >14 àíZ h¢ &
H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ Adí`
{bI| &
Bg àíZ-nÌ 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-
10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-
Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &
Please check that this question paper contains 11 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.

ì`dhm[aH$ J{UV
APPLIED MATHEMATICS

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

465 Page 1 P.T.O.

:

:

(i)

(ii)

(iii) 6 I 2

(iv) 4 II 3

(v) 4 4

(vi)

IÊS> H$

1 6 2

1. (H$) _mZ kmV H$s{OE : 2
1
xex
dx
(x 1)2
0

AWdm
(I) {ZåZ AdH$b g_rH$aU H$mo hb H$s{OE : 2
dy
= ex+y + x2ey
dx

2. < 18,000 Ho$ AZ§V-H$mb (perpetuity), Omo 6 _mh ~mX Xo` h¡, H$m dV©_mZ _mZ kmV
H$s{OE O~{H$ YZam{e, 8% dm{f©H$ MH«$d¥{Õ ã`mO A{O©V H$aVr h¡ O~{H$ ã`mO
AY©dm{f©H$ g§{MV hmoVm h¡ & 2

465 Page 2 P.T.O.

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.

SECTION A

Questions number 1 to 6 carry 2 marks each.

1. (a) Evaluate : 2
1
xex
dx
(x 1)2
0

OR

(b) Solve the following differential equation : 2
dy
= ex+y + x2ey
dx

2. Find the present value of a perpetuity of < 18,000 payable at the end of
6 months, if the money is worth 8% p.a. compounded semi-annually. 2

465 Page 3 P.T.O.

3. (H$) 10% dm{f©H$ MH«$d¥{Õ Zm__mÌ Xa H$s g_mZ à^mdr Xa kmV H$s{OE, O~{H$ Xa
_m{gH$ MH«$d{Õ ã`mO go g§{MV hmoVr h¡ & 2
[{X`m J`m h¡ {H$ : (1·00833)12 = 1·1047]

AWdm
(I) A^` EH$ _mo~mBb \$moZ < 30,000 H$m IarXVm h¡ & 3 df© níMmV² Bg _mo~mBb
\$moZ H$m ñH«¡$n _yë` < 3,000 hmoZo H$m AZw_mZ h¡ & a¡{IH$ _yë`õmg {d{Y Ho$
à`moJ go, 2 df© níMmV² Bg _mo~mBb \$moZ H$m nwñVH$ _yë` kmV H$s{OE & 2

4. {ZåZ n[aH$ënZm na {dMma H$s{OE :
H0 : = 35
H1 : 35

81 _Xm| Ho$ EH$ Z_yZo H$m _mÜ` 37·5 Am¡a _mZH$ {dMbZ 5 h¡ & 5% gmW©H$Vm Ho$ ñVa na
n[aH$ënZm H$m narjÊm H$s{OE & 2
[{X`m J`m h¡ {H$ : Xmo-ny±N> narjU Ho$ {bE 5% gmW©H$Vm ñVa na Z H$m H«$m§{VH$ _mZ 1·96 h¡]

5. Moamny±Or, _oKmb` Ho$ {bE dm{f©H$ dfm© ({__r _|) {ZåZ gmaUr _| A{^{b{IV h¡ :
dfm©
df©
({__r _|)
2001 1·2
2002 1·9
2003 2
2004 1·4
2005 2·1
2006 1·3
2007 1·8
2008 1·1
2009 1·3

3-dfu` J{V_mZ _mÜ` Ûmam dfm© H$s àd¥{Îm kmV H$s{OE & 2

465 Page 4 P.T.O.

3. (a) Find the effective rate which is equivalent to nominal rate of
10% p.a. compounded monthly. 2
[Given that : (1·00833)12 = 1·1047]
OR
(b) Abhay bought a mobile phone for < 30,000. The mobile phone is
estimated to have a scrap value of < 3,000 after a span of 3 years.
Using the linear depreciation method, find the book value of the
mobile phone at the end of 2 years. 2

4. Consider the following hypothesis :
H0 : = 35
H1 : 35
A sample of 81 items is taken whose mean is 37·5 and the standard
deviation is 5. Test the hypothesis at 5% level of significance. 2
[Given : Critical value of Z for a two-tailed test at 5% level of significance
is 1·96]

5. The following table shows the annual rainfall (in mm) recorded for
Cherrapunji, Meghalaya :
Rainfall
Year
(in mm)
2001 1·2
2002 1·9
2003 2
2004 1·4
2005 2·1
2006 1·3
2007 1·8
2008 1·1
2009 1·3
Determine the trend of rainfall by 3-year moving average. 2
465 Page 5 P.T.O.

6. {ZåZ ì`damoYm| Ho$ A§VJ©V
x y 1
x+y 0
x, y 0
z = 3x + 4y H$m A{YH$V_rH$aU H$s{OE, `{X g§^d hmo & 2

IÊS> I

7 10 3

7. (H$) EH$ dñVw H$m ny{V© \$bZ 100p = (x + 20)2 h¡ & BgH$m CËnmXH$ A{Yeof (PS)
kmV H$s{OE, O~ ~mµOma ^md < 25 h¡ & 3

AWdm
(I) kmV H$s{OE : 3

2x 2 1
dx
x2 3x 2

8. gab aoIr` àd¥{Îm {\$Q> H$s{OE Am¡a
df© 2008 Ho$ {bE àd¥{Îm _mZ kmV H$s{OE : 3

CËnmXZ
df©
(bmI Q>Zm| _|)
2001 30
2002 35
2003 36
2004 32
2005 37
2006 40
2007 36

465 Page 6 P.T.O.

6. Maximize z = 3x + 4y, if possible,
subject to the constraints :
x y 1
x+y 0
x, y 0 2

SECTION B

Questions number 7 to 10 carry 3 marks each.

7. (a) The supply function of a commodity is 100p = (x + 20)2. Find the
P Surplus (PS), when the market price is < 25. 3

OR
(b) Find : 3

2x 2 1
dx
x2 3x 2

8. Fit a straight line trend by the method of least squares and find the trend
value for the year 2008 for the following data : 3

Production
Year
(in lakh tonnes)
2001 30

2002 35

2003 36

2004 32

2005 37

2006 40

2007 36

465 Page 7 P.T.O.

9. EH$ ñdMm{bV n¡qH$J _erZ go Xg {S>ã~o `mÑÀN>`m {bE JE & BZH$m Am¡gV ewÕ dµOZ
11·8 {H$bmoJ«m_ Am¡a _mZH$ {dMbZ 0·15 12 {H$bmoJ«m_ Ho$ BamXVZ
_mÜ` go `h Z_yZm _mÜ`, gmW©H$Vm Ho$ ñVa na, {^Þ h¡ ? 3
[{X`m J`m h¡ {H$ d.f. = 9 Ho$ {bE t0·05 = 2·26]

10. _Yw AnZr nwamZr H$ma, {OgH$s H$s_V < 1,50,000 bJmB© JB© h¡, XoH$a EH$ ZB© H$ma
{OgH$m _yë` < 6,50,000 h¡, IarXVr h¡ & dh < x H$m A{J«_ ^wJVmZ H$aVr h¡ Am¡a
~Mr am{e H$m ^wJVmZ < 21,000 H$s 20 _m{gH$ g_mZ {H$ñVm| _| H$aVr h¡ & Cgo 9%
dm{f©H$ Xa Ho$ ã`mO H$m àñVmd {X`m OmVm h¡ & x H$m _mZ kmV H$s{OE & 3
[{X`m J`m h¡ : (1·0075) 20 = 0·86118985]

IÊS> J

11 14 4

11. {H$gr OrdmUw g_yh _| OrdmUwAm| H$s d¥{Õ H$s Xa CZH$s CnpñWV g§»`m Ho$ g_mZwnmVr h¡ &
`h nm`m OmVm h¡ {H$ 3 K§Q>m| Ho$ níMmV² OrdmUwAm| H$s g§»`m 10,000 VWm 5 K§Q>m| Ho$
níMmV² `h g§»`m 40,000 h¡ & ewéAmV _| CnpñWV OrdmUwAm| H$s g§»`m kmV H$s{OE & 4

12. (H$) < 5,00,000 Ho$ CYma, {OgH$s ã`mO H$s Xa 10% dm{f©H$ h¡, VWm g_` 5 df© h¡,
Ho$ {bE âb¡Q> Xa àUmbr go EMI kmV H$s{OE & 4
AWdm
(I) < 2,00,000 _yë` H$s EH$ _erZ H$s à^mdr Am`w 7 df© h¡ Am¡a Bg _erZ H$m
ñH«¡$n _yë` < 30,000 h¡ & H$ånZr {H$VZr YZam{e G$U emoYZ {Z{Y _| aIo, {Oggo
5% dm{f©H$ H$s H$_mB© hmoVr h¡, {Oggo {H$ H$ånZr Bg _erZ H$mo CgH$s Cn`moJr
Am`w Ho$ Cnam§V ~Xb gHo$ ? _mZ br{OE {H$ 7 dfmªo ~mX ZB© _erZ H$m _yë`
< 3,00,000 hmoJm & 4
[{X`m J`m h¡ : (1·05)7 = 1·407]

13. EH$ ñQ>mQ>©-An H$ånZr Zo 5 dfmªo Ho$ {bE eo`am| _| < 3,00,000 H$m {Zdoe {H$`m & Xÿgao
df© Ho$ AÝV _|, Bg {Zdoe H$m _yë` < 3,50,000 Wm, Vrgao df© Ho$ AÝV _|, Bg${Zdoe H$m
_yë` < 3,80,000 < 4,50,000 hmo J`m & {Zdoe
na MH«$d¥{Õ dm{f©H$ d¥{Õ Xa (CAGR) H$s JUZm H$s{OE & 4

[{X`m J`m h¡ : (1·5)1/5 = 1·084]

465 Page 8 P.T.O.

9. Ten cartons are taken at random from an automatic packing machine.
The mean net weight of the ten cartons is 11·8 kg and standard deviation
is 0·15 kg. Does the sample mean differ significantly from the intended
mean of 12 kg ? 3
[Given that for d.f. = 9, t0·05 = 2·26]

10. Madhu exchanged her old car valued at < 1,50,000 with a new one priced
at < 6,50,000. She paid < x as down payment and the balance in
20 monthly equal instalments of < 21,000 each. The rate of interest
offered to her is 9% p.a. Find the value of x. 3
[Given that : (1·0075) 20 = 0·86118985]

SECTION C

Questions number 11 to 14 carry 4 marks each.

11. In a certain culture of bacteria, the rate of increase of bacteria is
proportional to the number present. It is found that there are
10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of
5 hours. Determine the number of bacteria present in the beginning. 4

12. (a)
< 5,00,000 with 10% annual interest rate for 5 years. 4
OR
(b) A machine costing < 2,00,000 has effective life of 7 years and its
scrap value is < 30,000. What amount should the company put
into a sinking fund earning 5% p.a., so that it can replace the
machine after its usual life ? Assume that a new machine will cost
< 3,00,000 after 7 years. 4
[Given that : (1·05)7 = 1·407]

13. A start-up company invested < 3,00,000 in shares for 5 years. The value
of this investment was < 3,50,000 at the end of second year, < 3,80,000
at the end of third year and on maturity, the final value stood at
< 4,50,000. Calculate the Compound Annual Growth Rate (CAGR) on the
investment. 4
[Given that : (1·5)1/5 = 1·084]
465 Page 9 P.T.O.

14. EH$ Amhma{dX² Xmo àH$ma Ho$ ^moÁ`m§o F1 Am¡a F2 H$mo Bg àH$ma {_bmZm MmhVm h¡ {H$ {_lU
_| {dQ>m{_Z A H$s _mÌm H$_-go-H$_ 8 _mÌH$ Am¡a {dQ>m{_Z C H$s H$_-go-H$_ 10 _mÌH$
hm| & ^moÁ` F1 _| à{V {H$bmoJ«m_ 2 _mÌH$ {dQ>m{_Z A VWm 1 _mÌH$ {dQ>m{_Z C H$s _mÌm
gpå_{bV h¡, O~{H$ ^moÁ` F2 _| à{V {H$bmoJ«m_ 1 _mÌH$ {dQ>m{_Z A VWm 2 _mÌH$
{dQ>m{_Z C H$s _mÌm gpå_{bV h¡ & ^moÁ` F1 H$m IarX _yë` < 5 à{V {H$bmoJ«m_ Am¡a
^moÁ` F2 H$m IarX _yë` < 7 à{V {H$bmoJ«m_ h¡ &
Cnamoº$ OmZH$mar na AmYm[aV hmoVo hþE, {ZåZ àíZm| H$m CÎma Xr{OE : 4
(H$) {_lU H$m Ý`yZV_ IarX _yë` kmV H$aZo hoVw, Cnamoº$ g_ñ`m Ho$ {bE EH$ a¡{IH$
àmoJ«m_Z g_ñ`m ~ZmBE &
(I) {_lU H$m Ý`yZV_ IarX _yë` kmV H$s{OE &

465 Page 10 P.T.O.

14. A dietician wishes to mix two types of foods F1 and F2 in such a way that
the vitamin content of the mixture contains at least 8 units of vitamin A
and 10 units of vitamin C. Food F1 contains 2 units/kg of vitamin A and
1 unit/kg of vitamin C, while Food F2 contains 1 unit/kg of vitamin A and
2 units/kg of vitamin C. It costs < 5 per kg to purchase Food F1 and
< 7 per kg to purchase Food F2.

Based on the above information, answer the following questions : 4

(a) To find out the minimum cost of such a mixture, formulate the
above problem as a LPP.
(b) Determine the minimum cost of the mixture.

465 Page 11 P.T.O.