Embed Size (px)
Citation preview
JEE (Advanced) 2019 Paper 2
Page 1
JEE (ADVANCED) 2019 PAPER 2 PART-I MATHS
SECTION 1 (Maximum Marks:32)
This section contains EIGHT (08) questions. Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct
answer(s). For each question, choose the option(s) corresponding to (all) the correct answer(s). Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If only (all) the correct option(s) is(are) chosen; Partial Marks : +3 If all the four options are correct but ONLY three options are chosen; Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of
which are correct; Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct
option; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : -1 In all other cases.
For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (B) will get +2 marks; choosing ONLY (A) and (D) will get +2 marks; choosing ONLY (B) and (D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY (B) will get +1 mark; choosing ONLY (D) will get +1 mark; choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -1 mark.
JEE (Advanced) 2019 Paper 2
Page 2
1. For , 1a a , let
3 3
7/32 2 2
1 2 ......lim 541 1 1....
1 2
n
n
nan an an n
Then the possible value(s) of a is/are 1. – 9 2. – 6 3. 7 4. 8 2. Let :f be a function. We say that f has
PROPERTY 1 if
0
0limh
f h fh
exists and is finite, and
PROPERTY 2 if
20
0limh
f h fh
exists and is finite
Then which of the following options is/are correct ? 1. f x x has PROPERTY 1 2. sinf x x has PROPERTY 2
3. f x x x has PROPERTY 2 4. 2/3f x x has PROPERTY 1 3. For non – negative integers n, let
0
2
0
1 2sin sin2 2
1sin2
n
kn
k
k kn nf n
kn
Assuming cos-1x takes values in 0, which of the following options is/are correct?
1. If 1tan cos 6f ,then 2 2 1 0
2. 1sin 7cos 5 0f
3. 342
f
4. 1lim2n
f n
4. Let :f be given by 1 2 5f x x x x . Define 0
, 0x
F x f t dt x
Then when of the following options is/are correct? 1. F has a local minimum at x = 1 2. 0F x for all 0,5x
3. F has a local maximum at x = 2 4. F has two local maxima and one local minimum in 0,
JEE (Advanced) 2019 Paper 2
Page 3
5. Let
1 2 3
1 0 0 1 0 0 0 1 00 1 0 , 0 0 1 , 1 0 00 0 1 0 1 0 0 0 1
P I P P
4 5 6
0 1 0 0 0 1 0 0 10 0 1 , 1 0 0 , 0 1 01 0 0 0 1 0 1 0 0
P P P
and 6
1
2 1 31 0 23 2 1
Tk k
k
X P P
Where TkP denotes the transpose of the matrix kP . Then which of the following options is/ are
correct? 1. X is a symmetric matrix 2. The sum of diagonal entries of X is 18
3. X – 30I is an invertible matrix 4. If
1 11 1 ,1 1
X
then 30
6. Let 2sin , 0xf x x
x
Let x1 < x2 < x3 < …….. < xn <……. be all the points of local maximum of f and y1 < y2 < y3 < ……. < yn < ….. be all the points of local minimum of f . Then which of the following options is/are correct ? 1. 1 1x y 2. 1 2n nx x
3. 12 ,22nx n n
for every n 4. 1n nx y for every n
7. Let x and let
1 1 1 20 2 2 , 0 4 00 0 3 6
x xP Q
x x
and 1R PQP
Then which of the following options is/are correct?
1. det
2det 0 4 0 8
5
x xR
x x
, for all x
2. For x = 1, there exists a unit vector i j k for which
000
R
3. There exists a real number x such that PQ = QP
4. For x = 0, if
1 16R a a
b b
, then a + b = 5
JEE (Advanced) 2019 Paper 2
Page 4
8. Three lines
1 : ,L r i
2 : ,L r k j
3 : ,L r i j vk v
are given . For which point(s) Q on L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?
1. k 2. 12
k j 3. k j 4. 12
k j
SECTION 2 (Maximum Marks:18)
This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value of the answer using the mouse and the on-screen
virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value of TWO decimal places.
Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct option is entered; Zero Marks : 0 In all other cases.
1. Suppose
2
0 0
0 0
det 03
n nn
kk k
n nn n k
k kk k
k C k
C k C
Holds for some positive integer n. The 0 1
nnk
k
Ck equals____
2. The value of
10
1
0
11 7 7sec sec sec4 12 2 12 2k
kk
In the interval 3,
4 4
equals ___
3. The value of the integral
/2
50
3 cos
cos sind
equals ____
4. Let X denote the number of elements in a set X. Let S= {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 B A , equals ____
JEE (Advanced) 2019 Paper 2
Page 5
5. Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ___
6. Let 2a i j k and 2b i j k
be two vectors. Consider a vector
, ,c a b
. If the projection of c
on the vector a b
is 3 2 , then the minimum
value of .c a b c
equals_____
SECTION 3 (Maximum Marks:12)
This section contains TWO (02) List-Match sets. Each List-Match set has TWO (02) Multiple Choice Questions. Each List-Match set has two lists: List-I and List-II. List-I has Four entries (I), (II), (III) and (IV) and List-II has Six entries (P), (Q), (R), (S), (T) and (U). FOUR options are given in each Multiple 1. Answer the following by appropriately matching the
lists based on the information given in the paragraph Let sin cosf x x and cos 2 sing x x be two functions defined for x > 0. Define
the following sets whose elements are written in the increasing order : : 0 ,X x f x 1: 0Y x f x
: 0 ,Z x g x 1: 0W x g x
List – I contains the sets X, Y, Z and W. List – II contains some information regarding these sets. List – I List – II
(I) X (P) 3, ,4 ,7
2 2
(II) Y (Q) an arithmetic progression (III) Z (R) NOT an arithmetic progression
(IV) W (S) 7 13, ,
6 6 6
(T) 2, ,
3 3
(U) 3,
6 4
Which of the following is the only CORRECT combination? 1. (IV), (P), (R), (S) 2. (III), (P), (Q), (U) 3. (III), (R), (U) 4. (IV), (Q), (T) 2. Answer the following by appropriately matching the lists based on the information given in the
paragraph Let sin cosf x x and cos 2 sing x x be two functions defined for x > 0. Define
the following sets whose elements are written in the increasing order : : 0 ,X x f x 1: 0Y x f x
JEE (Advanced) 2019 Paper 2
Page 6
: 0 ,Z x g x 1: 0W x g x
List – I contains the sets X, Y, Z and W. List – II contains some information regarding these sets. List – I List – II
(I) X (P) 3, ,4 ,7
2 2
(II) Y (Q) an arithmetic progression (III) Z (R) NOT an arithmetic progression
(IV) W (S) 7 13, ,
6 6 6
(T) 2, ,
3 3
(U) 3,
6 4
Which of the following is the only CORRECT combination? 1. (I), (P), (R) 2. (I), (Q), (U) 3. (II), (R), (S) 4. (II), (Q), (T) 3. Answer the following by appropriately matching the lists based on the information given in the
paragraph.
Let the circles 2 21 : 9C x y and 2 2
2 : 3 4 16C x y , intersect at the points X and
Y. Suppose that another circle 2 2 23 :C x h y k r satisfies the following conditions :
(i) Centre of 3C is collinear with the centres of 1C and 2C
(ii) 1C and 2C both lie inside 3C , and
(iii) 3C touches 1C at M and 2C at N Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a
tangent to the parabola 2 8x y . There are some expressions given in the List – I whose values are given in List – II below List – I List – II (I) 2h k (P) 6
(II) Length of ZWLength of XY
(Q) 6
(III) Area of triangle MZNArea of triangle ZMW
(R) 54
(IV) (S) 215
(T) 2 6
(U) 103
Which of the following is the only CORRECT combination? 1. (I), (U) 2. (II), (T) 3. (II), (Q) 4. (I), (S)
JEE (Advanced) 2019 Paper 2
Page 7
4. Answer the following by appropriately matching the lists based on the information given in the paragraph
Let the circles 2 21 : 9C x y and 2 2
2 : 3 4 16C x y , intersect at the points X and
Y. Suppose that another circle 2 2 23 :C x h y k r satisfies the following conditions :
(i) Centre of 3C is collinear with the centres of 1C and 2C
(ii) 1C and 2C both lie inside 3C , and
(iii) 3C touches 1C at M and 2C at N Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a
tangent to the parabola 2 8x y . There are some expressions given in the List – I whose values are given in List – II below List – I List – II (I) 2h k (P) 6
(II) Length of ZWLength of XY
(Q) 6
(III) Area of triangle MZNArea of triangle ZMW
(R) 54
(IV) (S) 215
(T) 2 6
(U) 103
Which of the following is the only INCORRECT combination? 1. (III), (R) 2. (IV), (U) 3. (IV), (S) 4. (I), (P)
Mathematics Key and Solutions
Section-I: 1) 1,4 2) 1,4 3) 1,2,3 4) 1,2,3 5) 1,2,4 6) 2,3,4 7) 1,4 8) 2,4
Section-II: 1) 6.20 2) 0.00 3) 0.50 4) 1523.00 5) 30.00 6) 18.00
Section-III: 1) 1 2) 4 3) 3 4) 3
JEE (Advanced) 2019 Paper 2
Page 8
SECTION – I
1.
1/33
11
7/32 2
1 1
1
lim lim1 1 1
nn
rrn nn n
r r
rr n n
nn ran r a
n
=
11/3 4/3 1
00
1102
0
3 34 4 541 11
1
x dx x l
ldx a x a aa x
1 8 9 9 8a a or 8 9a or
2. If 0 0
0lim lim 0h h
hf x x h
h
If 20
sinh 0sin lim .h
f x x D Eh
If 20
0lim .h
h hf x x x D E
h
If 2/3
1/32/31/30 0
0lim lim 0h h
hf x x hh
3. 0
0
2 3cos cos2 2
2 21 cos2
n
k
n
k
kn nf n
kn
1sin
2 31 cos cos2 2 2sin
21
sin2 21 cos
2 2sin2
nn nn
n n nn
nn nn
n nn
2 cos
2 cos2 2
nn
n n
.
26 cos 2 1 2 1 08 8
f Tan
1sin 7 cos 5 sin 7. 07
f
34 cos
6 2f
; lim 1n
f n
JEE (Advanced) 2019 Paper 2
Page 9
4. 1 1 2 5F x f x x x x At 1x and 5x F x has local minima At 2x , F x has local maxima and 0 0,5F x x
1 2 5
5. Let 6
1
2 1 31 0 2 ,3 2 1
Tk k
k
A X P AP
Here T Tk k k kP P P P I for all k =1,2,3,4,5,6.
1 &T Tk k kP P P A A
6 6
1 1
TT T Tk k k k
k k
X P AP P AP X
Let 6 6 6
1 1 1
1 2 2 2 30 11 2 2 2 30 30 1 301 2 2 2 30 1
TK K K K
K K KB XB P AP B P AB P AB AB B
30 and 30X I is singular
6
1
18Tk k
k
Tr P AP
6. 2
14
cos 2 sin , 0x x x xf x xx
4
2 cos tan2 0
xx x xcos x
x
y
0 1/23/2 3
1y 1x2y
7. 1 2 2
24 12 4 10 8 0 4 0 8
5
x xR PQP Q x x
x x
If 1 44 0 0x R
JEE (Advanced) 2019 Paper 2
Page 10
For 1 1 1 2 0 0 6 3 0
10 0 2 2 0 4 0 0 3 2
60 0 3 0 0 6 0 0 2
x R
12 6 4
1 0 24 86
0 0 36
24 1
346 0 23
0 0 0
R I
2 44 0, 2 03 3b ba a
3, 2 5b a a b If PQ = QP no value of x . 8. , 0,0 , 0, ,1 , 1,1,P Q R
If P, Q, R are collinear 11 1
1 11 1,01
Section – II
1. 2 1
1
11 2 2 02
2 4
n n
n n
n nn n n
n
2 1 2 3 21 2 1 2 2 0n n nn n n n 4 1 1 2 0n n n n 2 3 4 0 4 0,n n n n n N
4 54
0
2 1 31 6.201 5 5k
k
Ck
2. 10
0
7 7sec sec12 2 12 2 2k
k kS
10
0
72cos6k
ec k
10
1
0
2 1 cos6
k
k
ec
4 1 4
1 15sec sec 1 04
3.
/2 /2
5 50 0
3 cos 3 sin
cos sin sin cosI d d
JEE (Advanced) 2019 Paper 2
Page 11
/2 /2 2
4 40 0
3 3sec2sin cos 1
I d dTan
Put, 2 2sec 2Tan t d tdt
4 3 4
0 0
6 1 16
1 1 1t dt dtt t t
3 2
0
1 1 1 1 13 3 0 1 0 1 0.503 2 3 2 2t t
I l
4. If n(B) = 1, n(A) = 6 no. of ordered pairs (A, B) = 6 If n(B) = 2, n(A) = 3 no. of ordered pairs (A, B) = 180 If n(B) = 2, n(A) = 6 no. of ordered pairs (A, B) = 15 If n(B) = 3, n(A) = 4 no. of ordered pairs (A, B) = 180 If n(B) = 3, n(A) = 6 no. of ordered pairs (A, B) = 20 If n(B) = 4, n(A) = 6 no. of ordered pairs (A, B) = 15 If n(B) = 5, n(A) = 6 no. of ordered pairs (A, B) = 6 Total number of ordered pairs (A, B) = 422. 5. Total number of ways = 2 3 5 30 6. 3 3a b i j
.3 2 2
c a b
a b
2
18 . 18c c c a b
SECTION III 1, 2 For x , 0 sin cos 0 cosf x x x n cos 0 ( )1( ) 1x or or
,2
nx n N
For Y, 1 0 cos sin 0f x cos x x sin 0 ( ) cos cos 0x or x
, ( ) cos 2 1 ,2
x n n N or x n n N
1 1cos ,2 2
x
22 ,2 , , 03 3
x n n n I x
For Z, 0 cos 2 sin 0 2 sin 2 1 ,2
g x x n n N
2 1 1 1 3 3sin , , ,4 4 4 4 4
nx
For W, 1 0 2 sin 2 sin cos 0g x x
JEE (Advanced) 2019 Paper 2
Page 12
sin 2 sin 0 cos 0x or x
2 1 , 2 sin2
x n n N or x n
1sin 1, ,02 2nx
2 1 , ,2
x n n n N
, 1 , 16 6
n nn n n
3,4
WN
M ZO(0,0)
Y
X
(3,4)
2 21
2 2 2 22 2
2 2 23
9 0,0
3 4 16 6 8 9 0 3,4
C x y O
C x y x y x y C
C x h y k r
Let the centre of C3 be C.
29 12: 3: 2 ,5 5
OC CC C
2 6, 2 3 5 4 12 6h k r r Eq. of common chord of C1 and C2 is 3x+4y-9=0. Let P be the point of intersection of OC1 and common chord.
6 12 6 24, ,5 5 5
81 242 925 5
6
54
CP ZP WP MP
XY
ZWXYMZNZMW
Equation of the common chord of C1 and C3 is 103 4 15 03
x y .
