CBSE Class 12 Mathematics Question Paper 2020 Set 65-B

H$moS> Z§.
Code No. 65(B)
amob Z§. narjmWu H$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
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n¥ð> 19 h¢ & paper contains 19 printed pages.

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(III) H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| (III) Please check that this question
>36 àíZ h¢ & paper contains 36 questions.

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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.

J{UV
(Ho$db ZoÌhrZ narjm{W©`m| Ho$ {bE)
MATHEMATICS
(FOR BLIND CANDIDATES ONLY)
{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 80
Time allowed : 3 hours Maximum Marks : 80

.65(B) 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$s{OE :
(i) `h àíZ-nÌ Mma IÊS>m| _| {d^m{OV {H$`m J`m h¡ – H$, I, J Ed§ K & Bg àíZ-nÌ
_| 36 àíZ h¢ & g^r àíZ A{Zdm`© h¢ &
(ii) IÊS> H$ _| àíZ g§»`m 1 go 20 VH$ 20 àíZ h¢ Ed§ àË`oH$ àíZ 1 A§H$ H$m h¡ &
(iii) IÊS> I _| àíZ g§»`m 21 go 26 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 2 A§H$m| H$m h¡ &
(iv) IÊS> J _| àíZ g§»`m 27 go 32 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 4 A§H$m| H$m h¡ &
(v) IÊS> K _| àíZ g§»`m 33 go 36 VH$ 4 àíZ h¢ Ed§ àË`oH$ àíZ 6 A§H$m| H$m h¡ &
(vi) àíZ-nÌ _| g_J« na H$moB© {dH$ën Zht h¡ & VWm{n EH$-EH$ A§H$ dmbo VrZ àíZm| _|,
Xmo-Xmo A§H$m| dmbo Xmo àíZm| _|, Mma-Mma A§H$m| dmbo Xmo àíZm| _| Am¡a N :-N : A§H$m| dmbo
Xmo àíZm| _| Am§V[aH$ {dH$ën {XE JE h¢ & Eogo àíZm| _| go Ho$db EH$ hr {dH$ën H$m
CÎma {b{IE &
(vii) BgHo$ A{V[aº$, Amdí`H$VmZwgma, àË`oH$ IÊS> Am¡a àíZ Ho$ gmW `Wmo{MV {ZX}e {XE
JE h¢ &
(viii) Ho$bHw$boQ>am| Ho$ à`moJ H$s AZw_{V Zht h¡ &

IÊS> H$

àíZ g§»`m 1 go 20 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &
àíZ g§»`m 1 go 10 VH$ ~hþ{dH$ënr` àíZ h¢ & ghr {dH$ën Mw{ZE &

1. Eogo g^r H$mo{Q> 2  2 Ho$ g§^m{dV Amì`yhm| H$s Hw$b g§»`m, {OZH$m àË`oH$
Ad`d 2 `m 3 h¡, h¡
(A) 4
(B) 8
(C) 16
(D) 32
.65(B) 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. This
question paper carries 36 questions. All questions are compulsory.
(ii) Section A – Question numbers 1 to 20 comprises of 20 questions of
1 mark each.
(iii) Section B – Question numbers 21 to 26 comprises of 6 questions of
2 marks each.
(iv) Section C – Question numbers 27 to 32 comprises of 6 questions of
4 marks each.
(v) Section D – Question numbers 33 to 36 comprises of 4 questions of
6 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 six 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 calculators is not permitted.
SECTION A
Question numbers 1 to 20 carry 1 mark each.
Question numbers 1 to 10 are multiple choice type questions. Select the
correct option.
1. Total number of possible matrices of order 2  2 with each
entry 2 or 3 is
(A) 4
(B) 8
(C) 16
(D) 32

.65(B) 3 P.T.O.

2. erfm] (– 2, 0), (2, 0) VWm (0, k) dmbo EH$ {Ì^wO H$m joÌ\$b 4 dJ© BH$mB©
h¡ & k H$m _mZ h¡
(A) 4
(B) 2
(C) –4
(D) 6

3. ì`§OH$ 2 cosec–1 2 + cos–1  1  H$m _mZ h¡
2

(A)
3

(B) –
3
2
(C) –
3
2
(D)
3

dx
4.
 16  9x 2 ~am~a h¡
1 3x
(A) tan –1 c
4 4
1 3x
(B) tan –1 c
12 4
1 3x
(C) tan –1 c
3 4
1 9x
(D) tan –1 c
12 16
.65(B) 4

2. The area of a triangle with vertices (– 2, 0), (2, 0) and (0, k) is
4 sq. units. The value of k is
(A) 4
(B) 2
(C) –4
(D) 6

1
3. The value of the expression 2 cosec–1 2 + cos–1   is
2

(A)
3

(B) –
3
2
(C) –
3
2
(D)
3

dx
4.
 16  9x 2 is equal to
1 3x
(A) tan –1 c
4 4
1 3x
(B) tan –1 c
12 4
1 3x
(C) tan –1 c
3 4
1 9x
(D) tan –1 c
12 16
.65(B) 5 P.T.O.

5. EH$ {Ì^wO {OgH$s Xmo ^wOmE± g{Xem| ^i + ^k VWm 2 ^i + ^j + ^k Ûmam {Zê${nV
h¢, H$m joÌ\$b h¡
3
(A)
2
(B) 3
(C) 3
3
(D)
2

6. `{X EH$ aoIm Ho$ {XŠH$moÁ`m a, a, a h¢, Vmo
(A) a>0
(B) a = 1 `m a = –1

(C) 0<a<1
1
(D) a= `m a = – 1
3 3

7. g_Vb 3x – 2y + 6z + 11 = 0, x-Aj Ho$ gmW sin–1 () H$m H$moU ~ZmVm
h¡ &  H$m _mZ h¡
3
(A) –
7
3
(B)
7
2
(C)
7
3
(D)
2
.65(B) 6

5. The area of the triangle whose two sides are represented by
^ ^ ^ ^ ^
the vectors i + k and 2 i + j + k is

3
(A)
2
(B) 3
(C) 3
3
(D)
2

6. If the direction cosines of a line are a, a, a, then

(A) a>0

(B) a = 1 or a = –1

(C) 0<a<1

1 1
(D) a= or a = –
3 3

7. The plane 3x – 2y + 6z + 11 = 0, makes an angle sin–1 () with
x-axis. The value of  is
3
(A) –
7
3
(B)
7
2
(C)
7
3
(D)
2
.65(B) 7 P.T.O.

8. _mZm f :  , f(x) = 5x – 3 Ûmam n[a^m{fV h¡ & Vmo f –1(x) h¡
x3
(A)
5
x
(B) –5
3
x
(C) 3
5
x
(D) –3
5

9. `{X A VWm B Xmo Eogr KQ>ZmE± h¡§ {H$ P(A) = 0·2, P(B) = 0·4 VWm
P(A  B) = 0·08 h¢, Vmo P(A|B) ~am~a h¡
(A) 0·02
(B) 0·2
(C) 0·4
(D) 0·08

10. EH$ W¡bo _| 4 bmb VWm 3 H$mbr J§oX| h¢ & `{X W¡bo _| go {~Zm à{VñWmnZm Ho$
2 J|X| `mÑÀN>`m {ZH$mbr JB© h¢, Vmo R>rH$ EH$ bmb J|X Ho$ àmá hmoZo H$s àm{`H$Vm
h¡
1
(A)
7
2
(B)
7
4
(C)
7
3
(D)
14
.65(B) 8

8. Let f :  be defined by f(x) = 5x – 3. Then, f –1(x) is
given by
x3
(A)
5
x
(B) –5
3
x
(C) 3
5
x
(D) –3
5
9. If A and B be two events such that P(A) = 0·2, P(B) = 0·4 and
P(A  B) = 0·08, then P(A|B) is
(A) 0·02
(B) 0·2
(C) 0·4
(D) 0·08
10. A bag contains 4 red and 3 black balls. If 2 balls are drawn
from the bag at random without replacement, then the
probability of getting exactly one red ball is
1
(A)
7
2
(B)
7
4
(C)
7
3
(D)
14
.65(B) 9 P.T.O.

àíZ g§»`m 11 go 15 VH$ Ho$ g^r àíZm| Ho$ Imbr ñWmZ ^[aE &

11. Amì`yh A VWm B EH$-Xÿgao Ho$ ì`wËH«$_ hm|Jo Ho$db `{X __________ &

12. \$bZ f(x) = |x – 3|, x  , x = __________ na AdH$bZr` Zht h¡ &

13. \$bZ f(x) = loge (sin x),   ,   na {Za§Va __________ h¡ &
2 
AWdm
AdH$bOm| Ho$ à`moJ go 26 H$m g{ÞH$Q> _mZ, Xe_bd Ho$ Xmo ñWmZm| VH$
h¡ __________ &

14. EH$ a¡{IH$ àmoJ«m_Z g_ñ`m _|, Cg a¡{IH$ \$bZ, {OgH$m A{YH$V_ `m Ý`yZV_
_mZ kmV H$aZm hmoVm h¡, H$mo a¡{IH$ __________ \$bZ H$hVo h¢ &

15. g{Xe  ^ ^ ^
a = – 2 i + 3 j + 6 k H$s {Xem _| n[a_mU 14 dmbm g{Xe
h¡ __________ &

àíZ g§»`m 16 go 20 A{V g§{já CÎma dmbo àíZ h¢ &
bc 1 a ( b  c)
16. gma{UH$ ca 1 b(c  a) H$m _mZ kmV H$s{OE &
ab 1 c(a  b)

17. _mZ kmV H$s{OE :
/4

 (x3 + x cos x + tan5 x) dx
– /4

.65(B) 10

Fill in the blanks in question numbers 11 to 15.

11. Matrices A and B will be inverse of each other only if
__________ .

12. The function f(x) = |x – 3|, x  is not differentiable at
x = __________ .

 
13. The function f(x) = loge (sin x) is strictly __________ on  ,   .
2 
OR
The approximate value of 26 , using differentials, up to
2 places of decimal is __________ .

14. In an LPP, the linear function which has to be maximised or
minimised is called a linear __________ function.

15. A vector of magnitude 14 in the direction of the vector
 ^ ^ ^
a = – 2 i + 3 j + 6 k is __________ .

Question numbers 16 to 20 are very short answer type questions.

bc 1 a ( b  c)
16. Find the value of the determinant ca 1 b(c  a) .
ab 1 c(a  b)

17. Evaluate :
/4

 (x3 + x cos x + tan5 x) dx
– /4

.65(B) 11 P.T.O.

18. kmV H$s{OE :
6 cos x – 9 sin x
 6 cos x  4 sin x
dx

AWdm
kmV H$s{OE :
(x – 5) e x
 (x – 3)3 dx
19. kmV H$s{OE :

 tan 2 (3x  5) dx

20. dH«$m| Ho$ Hw$b y = a cos (x + b), {Og_| a, b ñdoÀN> AMa h¢, H$mo {Zê${nV
H$aZo dmbo AdH$b g_rH$aU H$mo kmV H$s{OE &
AWdm
AdH$b g_rH$aU y dx – (x + 2y2) dy = 0 Ho$ hb H$aZo Ho$ {bE g_mH$bZ
JwUH$ kmV H$s{OE &
IÊS> I
àíZ g§»`m 21 go 26 VH$ àË`oH$ àíZ 2 A§H$m| H$m h¡ &
21. Xem©BE {H$ g^r dmñV{dH$ YZ g§»`mAm| Ho$ g_wƒ` _| R = {(a, b) : a  b3}
Ûmam n[a^m{fV g§~§Y R, Z Vmo g_{_V h¡ Am¡a Z hr g§H«$m_H$ h¡ &
AWdm
{gÕ H$s{OE {H$ :
 12  3  56 
cos–1   + sin–1   = sin–1  
 13  5  65 

22. `{X (x2 + y2)2 = xy h¡, Vmo dy kmV H$s{OE &
dx
.65(B) 12

18. Find :
6 cos x – 9 sin x
 6 cos x  4 sin x
dx

OR
Find :
(x – 5) e x
 (x – 3)3 dx
19. Find :

 tan 2 (3x  5) dx

20. Form the differential equation representing the family of
curves y = a cos (x + b), where a, b are arbitrary constants.
OR
Find the integrating factor for the solution of the differential
equation y dx – (x + 2y2) dy = 0.

SECTION B

Question numbers 21 to 26 carry 2 marks each.
21. Show that the relation R in the set of all positive real numbers
defined by R = {(a, b) : a  b3} is neither symmetric nor
transitive.
OR
Prove that :
 12  3  56 
cos–1   + sin–1   = sin–1  
 13  5  65 

dy
22. If (x2 + y2)2 = xy, then find .
dx

.65(B) 13 P.T.O.

23. EH$ b§~-d¥Îmr` ~obZ H$s {ÌÁ`m 2 cm/s H$s Xa go ~‹T> ahr h¡ O~{H$ BgH$s
D±$MmB© 8 cm/s H$s Xa go KQ> ahr h¡ & Cg g_` O~ BgH$s {ÌÁ`m 3 cm VWm
D±$MmB© 6 cm h¡, BgHo$ Am`VZ Ho$ ~XbZo H$s Xa kmV H$s{OE &
24. Xem©BE {H$ q~Xþ {OZHo$ pñW{V g{Xe 2^i – ^j + ^k , ^i – 3^j – 5 ^k VWm
^ ^ ^
3 i – 4 j – 4 k h¢, EH$ g_H$moU {Ì^wO Ho$ erf© h¢ &
AWdm
g{Xem| Ho$ à`moJ go {Ì^wO ABC H$m joÌ\$b kmV H$s{OE {OgHo$ erf©
A(1, 1, 1), B(1, 2, 3) VWm C(2, 3, 1) h¢ &
 ^ ^ ^ ^ ^
25. aoImAm| r = (2 j – 3 k ) + ( i + 2 j + 2 k ) VWm
 ^ ^ ^ ^ ^ ^
r = (2 i + 6 j + 3 k ) + (2 i + 3 j – 6 k ) Ho$ ~rM H$m H$moU kmV
H$s{OE &
26. Xmo nmgm| H$mo EH$ ~ma CN>mbm J`m & {X`m J`m h¡ {H$ nmgm| na AmZo dmbr XmoZm|
g§»`mE± {^Þ h¢, Vmo KQ>Zm ‘nmgm| na AmB© g§»`mAm| H$m `moJ\$b 6 h¡,’ H$s
àm{`H$Vm kmV H$s{OE &
IÊS> J
àíZ g§»`m 27 go 32 VH$ àË`oH$ àíZ 4 A§H$m| H$m h¡ &
27. f(x) = 9x2 + 6x – 5 Ûmam àXÎm \$bZ f : +  [– 5, ) na {dMma H$s{OE &
y 6 –1
Xem©BE {H$ f ì`wËH«$_Ur` \$bZ h¡ VWm f –1(y) = h¡, Ohm±
3
+ g^r G$UoVa dmñV{dH$ g§»`mAm| H$m g_wƒ` h¡ &

28. `{X y = (cos x)x h¡, Vmo dy kmV H$s{OE &
dx
AWdm
2
b4
`{X x = a cos  VWm y = b sin  h¡, Vmo {gÕ H$s{OE {H$ d y2  – 2 3
.
dx a y
.65(B) 14

23. The radius of a right circular cylinder is increasing at the rate
of 2 cm/s and its height is decreasing at the rate of 8 cm/s. Find
the rate of change of its volume, when the radius is 3 cm and
height is 6 cm.
^ ^ ^
24. Show that the points with position vectors 2 i – j + k ,
^ ^ ^ ^ ^ ^
i – 3 j – 5 k and 3 i – 4 j – 4 k are the vertices of a right
angled triangle.
OR
Using vectors, find the area of triangle ABC with vertices
A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1).
25. Find the angle between the lines
 ^ ^ ^ ^ ^
r = (2 j – 3 k ) + ( i + 2 j + 2 k ) and
 ^ ^ ^ ^ ^ ^
r = (2 i + 6 j + 3 k ) + (2 i + 3 j – 6 k ).
26. Two dice are thrown once. Given that two numbers appearing
on the dice are different, find the probability of the event ‘the
sum of numbers on the dice is 6’.

SECTION C
Question numbers 27 to 32 carry 4 marks each.
2 + 6x – 5. Show
27. Consider f : +  [– 5, ) given by f(x) = 9x
y 6 –1
that f is invertible with f –1(y) = , where + is the
3
set of all non-negative real numbers.

dy
28. If y = (cos x)x, find .
dx
OR
d 2y b4
If x = a cos  and y = b sin , then prove that – .
2 2 3
dx a y

.65(B) 15 P.T.O.

29. _mZ kmV H$s{OE :
2
x dx
1 (x  1) (x  2)

30. AdH$b g_rH$aU dy + y cot x = 2x + x2 cot x, (x  0) H$m {d{eï> hb
dx
kmV H$s{OE, {X`m J`m h¡ {H$ y = 0 O~ x   .
2

31. EH$ \$ZuMa \$_© Hw${g©`m± VWm _oµO ~ZmVr h¡, {Og_| àË`oH$ Ho$ {bE VrZ _erZm|
A, B VWm C H$m à`moJ hmoVm h¡ & EH$ Hw$gu ~ZmZo Ho$ {bE _erZ A na 2 K§Q>o,
_erZ B na 1 K§Q>m VWm _erZ C na 1 K§Q>m bJVm h¡ & EH$ _oµO ~ZmZo Ho$ {bE
_erZ A VWm _erZ B àË`oH$ na 1 K§Q>m VWm _erZ C na 3 K§Q>o bJVo
h¢ & EH$ Hw$gu Ho$ ~oMZo na < 300 VWm EH$ _oµO Ho$ ~oMZo na < 600 H$m bm^
hmoVm h¡ & EH$ gámh _| _erZ A, 70 K§Q>m|, _erZ B, 40 K§Q>m| VWm _erZ C,
90 K§Q>m| Ho$ {bE CnbãY h¡ & EH$ a¡{IH$ àmoJ«m_Z g_ñ`m ~ZmBE `h OmZZo Ho$
{bE {H$ à{V gámh \$_© {H$VZr Hw${g©`m± d {H$VZr _oµO| V¡`ma H$ao {Oggo \$_© H$mo
A{YH$V_ bm^ hmo &
32. EH$ W¡bo _| 2 g\o$X, 3 bmb VWm 4 Zrbo a§J H$s J§oX§o h¢ & W¡bo _| go EH$-EH$
H$aHo$ {~Zm à{VñWmnZm Ho$ 2 J|X| {ZH$mbr OmVr h¢ & bmb J§oXm| Ho$ AmZo H$s
g§»`m H$m àm{`H$Vm ~§Q>Z kmV H$s{OE & bmb J|Xm| H$s g§»`m H$m _mÜ` ^r kmV
H$s{OE &
AWdm
52 nÎmm| H$s Vme H$s JÈ>r _| go EH$ nÎmm Imo OmVm h¡ & eof nÎmm| _§o go Xmo nÎmo
(EH$-EH$ H$aHo$ {~Zm à{VñWmnZm Ho$) {ZH$mbo OmVo h¢, Omo XmoZm| BªQ> Ho$ nmE OmVo
h¢ & ImoE hþE nÎmo Ho$ BªQ> Ho$ nÎmo Ho$ hmoZo H$s àm{`H$Vm kmV H$s{OE &

.65(B) 16

29. Evaluate :
2
x dx
1 (x  1) (x  2)

30. Find the particular solution of the differential equation
dy
+ y cot x = 2x + x2 cot x, (x  0), given that y = 0 when
dx

x .
2

31. A furniture firm manufactures chairs and tables, each
requiring the use of three machines A, B and C. Production of
one chair requires 2 hours on machine A, 1 hour on machine B
and 1 hour on machine C. Each table requires 1 hour each on
machines A and B and 3 hours on machine C. The profit
obtained by selling one chair is < 300; while by selling one
table, the profit is < 600. The total time available per week on
machine A is 70 hours, on machine B is 40 hours and on
machine C is 90 hours. Formulate an LPP to determine the
number of chairs and tables the firm should make per week in
order to get maximum profit.

32. A bag contains 2 white, 3 red and 4 blue balls. Two balls are
drawn one-by-one without replacement from the bag. Find the
probability distribution of the number of red balls. Also, find
the mean of the number of red balls.
OR
A card from a pack of 52 playing cards is lost. From the
remaining cards of the pack, two cards are drawn (one-by-one
without replacement) and both are found to be diamonds. Find
the probability of the lost card being a diamond card.
.65(B) 17 P.T.O.

IÊS> K
àíZ g§»`m 33 go 36 VH$ àË`oH$ àíZ 6 A§H$m| H$m h¡ &
 2 –1 1 
 
33. `{X A  – 1 2 – 1 h¡, Vmo Xem©BE {H$ A3 – 6A2 + 9A – 4I = O
 
1 –1 2 
Am¡a AV: A–1 kmV H$s{OE &
AWdm
gma{UH$m| Ho$ JwUY_m] Ho$ à`moJ go {gÕ H$s{OE {H$ :
3a –ab –ac
–ba 3b – b  c  3(a  b  c) (ab  bc  ca)
–ca –cb 3c

34. f(x) = 12x4/3 – 6x1/3, x [–1, 1] Ûmam àXÎm EH$ \$bZ f Ho$ {Zanoj CƒV_
Am¡a {Zanoj {ZåZV_ _mZ kmV H$s{OE &

35. g_mH$bZ Ho$ à`moJ go aoImAm| 2x + y = 4, 3x – 2y = 6 VWm x – 3y + 5 = 0
Ûmam {Kao joÌ H$m joÌ\$b kmV H$s{OE &

36. {gÕ H$s{OE {H$ q~XþAm| A(0, –1, –1) VWm B(4, 5, 1) go hmoH$a OmZo dmbr
aoIm, {~§XþAm| C(3, 9, 4) VWm D(– 4, 4, 4) go hmoH$a OmZo dmbr aoIm H$mo
à{VÀN>oX H$aVr h¡ &
AWdm
q~Xþ P(4, 3, 2) go g_Vb x + 2y + 3z = 2 na S>mbo JE b§~ Ho$ nmX Ho$
{ZX}em§H$ VWm b§~ H$s b§~mB© kmV H$s{OE & q~Xþ P H$m {XE JE g_Vb _|
à{Vq~~ ^r kmV H$s{OE &
.65(B) 18

SECTION D
Question numbers 33 to 36 carry 6 marks each.

 2 –1 1 
 
33. If A   – 1 2 – 1 , show that A3 – 6A2 + 9A – 4I = O and
 
1 –1 2 
hence find A–1.
OR
Using the properties of determinants, prove that
3a –ab –ac
–ba 3b – b  c  3(a  b  c) (ab  bc  ca)
–ca –cb 3c

34. Find the absolute maximum and absolute minimum values of
a function f given by f(x) = 12x4/3 – 6x1/3, x [–1, 1].

35. Using integration, find the area of the region bounded by the
lines 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0.

36. Prove that the line through A(0, –1, –1) and B(4, 5, 1)
intersects the line through C(3, 9, 4) and D(– 4, 4, 4).
OR
Find the coordinates of the foot of perpendicular and the
perpendicular distance from the point P(4, 3, 2) to the plane
x + 2y + 3z = 2. Also, find the image of P in the plane.

.65(B) 19 P.T.O.