CBSE Class 12 Applied Mathematics Question Paper 2020

H$moS> Z§.
Code No. 364
amob Z§. narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð
Roll No. >na Adí` {bIo§ &
Candidates must write the Code on the
title page of the answer-book.

ZmoQ> NOTE
(I) H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV (I) Please check that this question
n¥ð> 15 h¢ & paper contains 15 printed pages.

(II) àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS (II) Code number given on the right
>Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na hand side of the question paper
{bI| & should be written on the title page of
the answer-book by the candidate.
(III) H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| (III) Please check that this question
>34 àíZ h¢ & paper contains 34 questions.

(IV) H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go (IV) Please write down the Serial
nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ Number of the question in the
Adí` {bI| & answer-book before attempting it.
(V) Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m (V) 15 minute time has been allotted to
g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU read this question paper. The
nydm©• _| 10.15 ~Oo {H$`m OmEJm & question paper will be distributed
10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db at 10.15 a.m. From 10.15 a.m. to
10.30 a.m., the students will read the
àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ question paper only and will not
do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & write any answer on the
answer-book during this period.

AZwà`wŠV$J{UV
APPLIED MATHEMATICS

{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 70
Time allowed : 3 hours Maximum Marks : 70

.364 1 P.T.O.

gm_mÝ` {ZX}e :
{ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE :
(i) àíZ-nÌ Mma IÊS>m| _| {d^m{OV {H$`m J`m h¡  H$, I, J Ed§ K & Bg àíZ-nÌ _|
34 àíZ h¢ & g^r àíZ A{Zdm`© h¢ &
(ii) IÊS> H$ _| 20 àíZ h¢, àË`oH$ H$m 1 A§H$ h¡ &
(iii) IÊS> I _| 5 àíZ h¢, àË`oH$ Ho$ 2 A§H$ h¡ &
(iv) IÊS> J _| 5 àíZ h¢, àË`oH$ Ho$ 4 A§H$ h¡ &
(v) IÊS> K _| 4 àíZ h¢, àË`oH$ Ho$ 5 A§H$ h¡ &
(vi) H$moB© ^r ì`mnH$ {dH$ën Zht h¡ & VWm{n Am§V[aH$ {dH$ën àXmZ {H$E JE h¡, 1 A§H$ Ho$ VrZ
àíZm| _§o, 2 A§H$m| Ho$ Xmo àíZm| _§o, 4 A§H$m| Ho$ Xmo àíZm| _§o Am¡a 5 A§H$m| Ho$ Xmo àíZm| _§o &
AmnH$mo Bg Vah Ho$ àíZm| _| {g\©$ EH$ {dH$ën H$m hr CÎma XoZm h¡ &
(vii) BgHo$ A{V[aº$, Amdí`H$VmZwgma, àË`oH$ IÊS> Am¡a àíZ Ho$ gmW `Wmo{MV {ZX}e {X`m JE h¢ &
(viii) H¡$ëHw$boQ>a Ho$ à`moJ H$s AZw_{V Zht h¡ &
IÊS> H$
ZmoQ> : àíZ g§»`m 1 go 10 ~hþ{dH$ënr` àíZ h¢ {OZH$m CÎma ghr {dH$ën MwZH$a XoZm h¡ &
1. VrZ AZ{^ZV nmgm| H$mo CN>mbm J`m & A{YH$-go-A{YH$ Xmo ~ma ‘nQ>’ AmZo H$s àm{`H$Vm
Š`m h¡ ? 1
1
(A)
8
3
(B)
8
5
(C)
8
7
(D)
8
2. nmgm| Ho$ EH$ Omo‹S>o H$mo EH$ ~ma CN>mbZo na, `moJ\$b 8 AmZo H$s KQ>Zm _§o à{VXe© {~ÝXþAm|
H$s g§»`m h¡ 1
(A) 6
(B) 5
(C) 4
(D) 3
3. {~ÝXþAm| (– 4, 1) Am¡a (2, 3) go g_mZ Xÿar na y-Aj na pñWV {~ÝXþ H$s H$mo{Q>
(ordinate) h¡ 1
(A) 2
(B) 1
(C) –1
(D) –2
.364 2

General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper comprises four sections  A, B, C and D. There are
34 questions in the question paper. All questions are compulsory.
(ii) Section A comprises of 20 questions of 1 mark each.
(iii) Section B comprises of 5 questions of 2 marks each.
(iv) Section C comprises of 5 questions of 4 marks each.
(v) Section D comprises of 4 questions of 5 marks each.
(vi) There is no overall choice in the question paper. However, an internal choice
has been provided in 3 questions of one mark, 2 questions of two marks,
2 questions of four marks and 2 questions of five marks. Only one of the choices
in such questions have to be attempted.
(vii) In addition to this, separate instructions are given with each section and
question, wherever necessary.
(viii) Use of calculator is not permitted.
SECTION A
Note : Question numbers 1 to 10 are multiple choice type questions. Select the
correct option.
1. Three unbiased coins are tossed. What is the probability of getting at
most two tails ? 1
1
(A)
8
3
(B)
8
5
(C)
8
7
(D)
8
2. In a single throw of a pair of dice, the number of elements in the event of
getting the sum as 8 on them is 1
(A) 6
(B) 5
(C) 4
(D) 3
3. The ordinate of the point on y-axis, which is equidistant from (– 4, 1) and
(2, 3) is 1
(A) 2
(B) 1
(C) –1
(D) –2
.364 3 P.T.O.

4. `{X EH$ aoIm, {~ÝXþAm| (x, 5) VWm (3, 4) go JwµOaVo hþE, x-Aj H$s YZmË_H$ {Xem go
135 H$m H$moU ~ZmVr h¡, Vmo x ~am~a h¡ 1
(A) 1
(B) 2
(C) –4
(D) –3
5. p  q H$s gË`_mZ gmaUr _| F (False) H$s g§»`m h¡ 1
(A) 4
(B) 3
(C) 2
(D) 1
6. loUr 7, 24, 67, 148, 279, ? H$m AJbm nX h¡ 1
(A) 472
(B) 478
(C) 556
(D) 474
 1
 4 x – 3, x
7. `{X f (x )  
2
Ûmam n[a^m{fV \$bZ f, x = 1 na gVV²² h¡, Vmo k
kx 3  1, 1 2
 x
2
~am~a h¡ 1
(A) 16
(B) 8
(C) –8
(D) – 16

 x  4 dx ~am~a h¡
x –1
8. 1

2
(A) (x + 4)3/2 – 10 x  4  c
3
2
(B) (x + 4)3/2  10 x  4  c
3
2
(C) x  4  (x + 4)3/2  c
3
2
(D) x  4 – (x + 4)3/2  c
3
.364 4

4. If the line passing through (x, 5) and (3, 4) makes 135 angle with the
positive direction of x-axis, then x is equal to 1
(A) 1
(B) 2
(C) –4
(D) –3
5. The number of F’s (False) in the Truth table of p  q is 1
(A) 4
(B) 3
(C) 2
(D) 1
6. The next term in the series 7, 24, 67, 148, 279, ? is 1
(A) 472
(B) 478
(C) 556
(D) 474
 1
 4 x – 3, x
2 1
7. If f (x)   is continuous at x = , then k is equal to 1
kx 3  1, 1 2
 x
2
(A) 16
(B) 8
(C) –8
(D) – 16

 x  4 dx is equal to
x –1
8. 1

2
(A) (x + 4)3/2 – 10 x  4  c
3
2
(B) (x + 4)3/2  10 x  4  c
3
2
(C) x  4  (x + 4)3/2  c
3
2
(D) x  4 – (x + 4)3/2  c
3

.364 5 P.T.O.

– i 0 2 1 0 
9. `{X A , (i  – 1) , B    VWm AB = aI2 h¡, Vmo ‘a’ ~am~a
0 i 0 – 1
h¡ 1
(A) 1
(B) –1
(C) –i
(D) i

1 2 4
10. gma{UH$ –1 3 0 H$m _mZ h¡ 1
4 1 0
(A) 13
(B) – 13
(C) 52
(D) – 52

ZmoQ> : àíZ g§»`m 11 go 15 VH$ Ho$ àíZm| _| Imbr ñWmZm| H$mo ^[aE :
11. y-Aj Ho$ g_mÝVa d (– 3, – 2) go JwµOaZo dmbr aoIm H$m g_rH$aU h¡ ___________ & 1

12. 3 nwéfm|, 2 _{hbmAm| Am¡a 4 ~ƒm| Ho$ EH$ g_yh go `mÑÀN>`m 4 ì`{º$`m| H$m M`Z {H$`m
OmVm h¡ & R>rH$ 2 ~ƒm| Ho$ M`Z hmoZo H$s àm{`H$Vm h¡ ___________ & 1
2 – 1
13. `{X A  h¡, Vmo A2 = ___________ . 1
0 1 
AWdm
 5 6 – 3
 
`{X A  [a ij ]33   – 4 3 2  h¡, Vmo a12 H$m gh-I§S> h¡ _________ & 1
– 4 –7 3 

14. gyMH$m§H$, BH$mB© go ___________ hmoVm h¡ & 1
15. `{X DANGER H$mo H$moS> ^mfm _| 8 – 5 – 18 – 11 – 9 – 22 {bIm OmVm h¡, Vmo
SIGNAL H$mo {bIm OmEJm ___________ & 1
AWdm
_mZm p ‘‘ê$~r YZr h¡’’ Am¡a q ‘‘ê$~r Iwe h¡’’ Ho$ àVrH$ h¢ & Vmo ‘‘ê$~r Z Vmo YZr h¡ Am¡a Z
hr Iwe h¡’’ H$m àVrH$ h¡ ___________ & 1

.364 6

– i 0 2 1 0 
9. If A   , (i  – 1) , B    and AB = aI2, then ‘a’ is equal to 1
0 i 0 – 1
(A) 1
(B) –1
(C) –i
(D) i

1 2 4
10. The value of the determinant –1 3 0 is 1
4 1 0
(A) 13
(B) – 13
(C) 52
(D) – 52

Note : Fill in the blanks in question numbers 11 to 15.
11. Equation of the line through (– 3, – 2) and parallel to y-axis is _________ . 1
12. Four persons are to be chosen at random from a group of 3 men, 2 women
and 4 children. The probability of selecting exactly 2 children is ________ . 1

2 – 1 2
13. If A    , then A = ___________ . 1
0 1 

OR
 5 6 – 3
 
If A  [a ij ]33   – 4 3 2  , then co-factor of a12 is __________ . 1
– 4 –7 3 

14. Index numbers are ____________ from units. 1
15. If DANGER is coded as 8 – 5 – 18 – 11 – 9 – 22, then code of SIGNAL is
____________ . 1
OR
Let p be ‘Ruby is rich’ and q be ‘Ruby is happy’. Then the symbolic form of
‘Ruby is neither rich nor happy’ is __________ . 1

.364 7 P.T.O.

ZmoQ> : àíZ g§»`m 16 go 20 VH$ Ho$ àíZm| Ho$ CÎma Xr{OE :
16. {gÕ H$s{OE {H$ : 1
pqpq

x –1
17. lim H$m _mZ kmV H$s{OE, `{X ApñVËd _| h¡ & 1
x1 | x – 1|

AWdm
 1
lim x –  H$m _mZ kmV H$s{OE, `{X ApñVËd _§o h¡, Ohm± [x] Cg _hÎm_ nyUmªH$ H$mo
x 
1 2
2
àH$Q> H$aVm h¡, Omo x go H$_ `m CgHo$ ~am~a h¡ & 1

18. {~ÝXþAm| (10, – 5) VWm (2, 3) H$mo {_bmZo dmbo aoImI§S> Ho$ b§~-{Û^mOH$ H$m g_rH$aU
kmV H$s{OE & 1

19. Xmo nañna AndOu KQ>ZmAm| H$m EH$ CXmhaU Xr{OE & 1

20. Ag_rH$aU 2x – y  1 go {Zê${nV joÌ H$mo A§{H$V H$s{OE & 1

IÊS> I
21. Amì`yh X kmV H$s{OE, {OgHo$ {bE {ZåZ Amì`yh g_rH$aU ghr hmo : 2
 2 – 1 – 1 –8 – 10
   
 1 0  X  1 –2 –5
   
– 3 4   9 22 15 

22. EH$ eãX _| 8 Aja h¢ – 4 ñda d 4 ì`§OZ & VrZ Aja `mÑÀN>`m MwZo OmVo h¢ & BZ_| EH$
go A{YH$ ñda MwZo OmZo H$s àm{`H$Vm kmV H$s{OE & 2
AWdm
EH$ W¡bo _| 5 g\o$X, 7 bmb d 4 H$mbr J|X| h¢ & Bg W¡bo go 4 J|X| EH$-EH$ H$aHo$
(à{VñWm{nV H$aVo hþE) {ZH$mbr OmVr h¢ & BZ_| go {H$gr ^r J|X Ho$ g\o$X Z hmoZo H$s
àm{`H$Vm Š`m h¡ ? 2

23. `{X A Am¡a B Xmo 2  2 Ho$ dJ©g_ Amì`yh h¢, Vmo A Am¡a B Ho$ ~rM H$m dh gå~ÝY kmV
H$s{OE, Omo A + B H$mo ^r dJ©g_ Amì`yh ~ZmVm h¡ & 2
.364 8

Note : Answer the following question numbers 16 to 20.

16. Prove that : 1
pqpq

x –1
17. Evaluate lim , if it exists. 1
x1 | x – 1|

OR
 1
Evaluate lim x –  , if it exists, where [x] denotes greatest integer  x. 1
x 
1 2
2

18. Find the equation of the perpendicular bisector of the line segment
joining the points (10, – 5) and (2, 3). 1

19. Give an example of two mutually exclusive events. 1

20. Mark the region represented by the inequation, 2x – y  1. 1

SECTION B

21. Find the matrix X, if the following matrix equation is true : 2

 2 – 1 – 1 –8 – 10
   
 1 0  X  1 –2 –5
   
– 3 4   9 22 15 

22. A word consists of 8 letters, 4 vowels and 4 consonants. Three letters are
chosen at random. Find the probability that more than one vowel is
selected. 2
OR

A bag contains 5 white, 7 red and 4 black balls. 4 balls are drawn one by
one with replacement from this bag. What is the probability that none is
a white ball ? 2

23. If A and B are two 2  2 idempotent matrices, then find the relation
satisfied by A and B for which A + B is also an idempotent matrix. 2

.364 9 P.T.O.

24. Hw$b H«$` \$bZ C(x) = 3x3 + 2x2 + 4x + 7 Ho$ {bE {gÕ H$s{OE {H$ gr_m§V Am¡gVZ
H«$` \$bZ, 1 (MC – AC) Ho$ ~am~a h¡ & 2
x
AWdm
kmV H$s{OE : 2
x5  1
 x  1 dx
2

25. {ZåZ{b{IV Am±H$‹S>m| go, 2015 H$mo AmYma df© _mZVo hþE, gab gm_wXm{`H$ {d{Y go
df© 2018 H$m _yë` gyMH$m§H$ kmV H$s{OE & 2

2015 _| _yë` 2018 _| _yë`
dñVw
(< _|) (< _|)
A 65 110
B 40 90
C 15 25
D 50 80
E 30 45

IÊS> J
26. ZrMo Xr JB© a¡{IH$ àmoJ«m_Z g_ñ`m H$m, AmboIr` {d{Y go hb kmV H$s{OE :
{ZåZ ì`damoYm|
x1 + x 2  1
x1  2
x2  4
x1  0, x2  0
Ho$ A§VJ©V z = 4x1 + 6x2 H$m A{YH$V_rH$aU H$s{OE & 4

27. Xe_bd g§»`m 24·1875 H$mo {Û-AmYmar g§»`m _| n[ad{V©V H$s{OE & 4
28. do AÝVamb kmV H$s{OE, {OZ_| \$bZ f(x) = 2x3 – 24x + 107 (H$) {Za§Va dY©_mZ, Am¡a
(I) {Za§Va õmg_mZ h¡ & 4
AWdm
do AÝVamb kmV H$s{OE, {OZ_| \$bZ f(x) = 5x3/2 – 3x5/2 (x > 0) dY©_mZ `m õmg_mZ h¡ & 4
.364 10

1
24. Prove that marginal average cost function is (MC – AC) for the total
x
cost function C(x) = 3x3 + 2x2 + 4x + 7. 2
OR
Find : 2
x5  1
 x  1 dx
2

25. From the following data, find the price index number for the year 2018,
taking 2015 as the base year, using Simple Aggregative Method : 2

Price in 2015 Price in 2018
Commodity
(in <) (in <)

A 65 110
B 40 90
C 15 25
D 50 80
E 30 45

SECTION C
26. Solve the following LPP graphically : 4
Maximise : z = 4x1 + 6x2
subject to constraints :
x1 + x 2  1
x1  2
x2  4
x1  0, x2  0

27. Convert the decimal number 24·1875 into its binary equivalent. 4

28. Find the interval/s in which the function f(x) = 2x3 – 24x + 107 is
(a) strictly increasing, and (b) strictly decreasing. 4
OR
Find the interval in which the function f(x) = 5x3/2 – 3x5/2 (x > 0) is
increasing or decreasing. 4

.364 11 P.T.O.

29. EH$ {ÛnX ~§Q>Z _§o, _mÜ` VWm _mZH$ {dMbZ H«$_e: 12 Am¡a 2 h¢ & P(X  2) kmV
H$s{OE & 4
AWdm
`{X X EH$ Eogm ßdmgm| Ma h¡ {OgHo$ {bE P(X = 2) = 9P(X = 4) + 90P(X = 6) h¡,
Vmo X H$m _mÜ` kmV H$s{OE & 4

30. {ZåZ{b{IV g_`-loUr go, 3-dfu` J{V_mZ _mÜ` go CnZ{V _mZ kmV H$s{OE : 4

{~H«$s {~H«$s
df© df©
(hOma < _|) (hOma < _|)
2011 9 2015 18
2012 11 2016 17
2013 12 2017 19
2014 15

IÊS> K
31. EH$ ì`{º$ Zo < 50 _yë` Ho$ 500 gm_mÝ` eo`a, < 5 Ho$ àr{_`_ na IarXo, {OZH$m
bm^m§e 8% h¡ & Bg ì`{º$ H$m bm^ à{VeV Š`m h¡ ? 5

32. ZrMo Xr JB© a¡{IH$ àmoJ«m_Z g_ñ`m H$m, AmboIr` {d{Y go Bï>V_ hb kmV H$s{OE :
{ZåZ ì`damoYm|
7x1 + 8x2  168
14x1 + 8x2  224
2x1 + 4x2  60
x1  0, x2  0
Ho$ A§VJ©V C = 5x1 + 4x2 H$m Ý`yZV_rH$aU H$s{OE & 5
AWdm
EH$ OdmZ ì`{º$ H$mo 25 {H$_r à{V KÊQ>m H$s J{V go _moQ>a-gmB{H$b MbmZo Ho$ {bE,
< 2 à{V {H$_r H$s Xa go noQ´>mob na IM© H$aZm n‹S>Vm h¡ & `{X dh _moQ>a-gmB{H$b H$mo
40 {H$_r à{V KÊQ>m H$s J{V go MbmVm h¡, Vmo Cgo < 5 à{V {H$_r H$s Xa go noQ´>mob na
IM© H$aZm n‹S>Vm h¡ & `h ì`{º$ noQ´>mob na < 100 IM© H$aHo$ EH$ K§Q>o _| A{YH$V_ {H$VZr
Xÿar _moQ>a-gmB{H$b Mbm gH$Vm h¡ ? BgH$mo a¡{IH$ àmoJ«m_Z g_ñ`m _| ì`º$ H$s{OE Am¡a
hb H$s{OE & 5

.364 12

29. In a binomial distribution, the mean and the standard deviation are
12 and 2 respectively. Find P(X  2). 4
OR
If X is a Poisson variate such that P(X = 2) = 9P(X = 4) + 90P(X = 6),
then find the mean of X. 4

30. From the following time-series, determine the trend value by 3-yearly
moving averages : 4

Sales Sales
Year Year
(in < ’000) (in < ’000)

2011 9 2015 18
2012 11 2016 17
2013 12 2017 19
2014 15

SECTION D

31. What rate percent will a man get from his 500 common shares of par
value of < 50 each, bought at < 5 premium, the rate of dividend being
8% ? 5

32. Find the optimal solution to the LPP given below, using graphical
method : 5
Minimise : C = 5x1 + 4x2
subject to constraints : 7x1 + 8x2  168
14x1 + 8x2  224
2x1 + 4x2  60
x1  0, x2  0
OR
If a young man rides his motorcycle at 25 km per hour, he has to spend
< 2 per km on petrol. If he rides at a faster speed of 40 km per hour, the
petrol cost increases to < 5 per km. He has < 100 to spend on petrol and
wishes to find, what is the maximum distance that he can travel within
one hour. Express this as an LPP and then solve it. 5
.364 13 P.T.O.

33. Xem©BE {H$ Xr hþB© {V`©H$ D±$MmB© Am¡a _hÎm_ Am`VZ dmbo b§~-d¥Îmr` e§Hw$ H$m AY©-erf©
H$moU `{X  hmo, Vmo tan  = 2 . 5
AWdm
EH$ l bå~mB© Ho$ Vma H$mo Xmo Qw>H$‹S>m| _| {d^m{OV {H$`m OmVm h¡ & EH$ Qw>H$‹S>o go dJ© VWm Xÿgao
go d¥Îm ~Zm`m OmVm h¡ & XmoZm| Qw>H$‹S>m| H$s b§~mB© {H$VZr-{H$VZr hmoZr Mm{hE {Oggo dJ© Ed§
d¥Îm H$m Hw$b joÌ\$b Ý`yZV_ hmo ? 5

34. {ZåZ{b{IV dñVwAm|/godmAm| Ho$ {bE EH$ amÁ` Ho$ ^rVa H«$`-{dH«$` H$aZo go {~b H$s
YZam{e kmV H$s{OE & 5

A§{H$V _yë` (< _|) 16,000 10,000 12,000 7,500

~Å>m ({_{VH$mQ>m)% 25 30 20 35

gr.Or.Eg.Q>r.% 5 9 6 8

.364 14

33. If  is the semi-vertical angle of a right circular cone of maximum volume
whose slant height is given, then show that tan  = 2. 5

OR

A piece of wire of length l is cut into two parts, one of which is bent in the
shape of a circle and the other into the shape of a square. How should the
wire be cut so that the sum of the areas of the circle and the square is
minimum ? 5

34. Find the amount of bill for the following intra-state transaction of
goods/services. 5

MRP (in <) 16,000 10,000 12,000 7,500

Discount % 25 30 20 35

CGST % 5 9 6 8

.364 15 P.T.O.