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JEE-ADVANCED-OCTOBER 2021PAPER-2_DATE-03/10/2021

1

MATHEMATICS SECTION-1

· This section contains SIX (06) questions.

· Each question has FOUR options (A), (B), (C) and (D). 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,

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 unanswered;

Negative Marks : −2 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(s) (i.e. the question is unanswered) will get 0 marks and

choosing any other option(s) will get −2 marks.

1. Let S1 = {(i,j,k) : i, j, k Î{1,2,…,10}},

S2 = {(i,j) : 1≤ i < j + 2 ≤ 10,i , j Î {1,2,…,10}},

S3 = {(i, j, k, l) : 1≤i < j < k < l, i, j, k, l Î {1,2,…,10}} and

S4 = {(i, j, k, l) : i, j, k and l are distinct elements in {1,2,…,10}}.

If the total number of elements in the set Sr is nr, r = 1, 2, 3, 4, then which of the following

statements is (are) TRUE ?

(A) n1 = 1000 (B) n2 = 44 (C) n3 = 220 (D) 4n420

12=

Ans. (A,B,D)


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Sol. S1 : 103 S2 : i j 1 7 choice 2 6 choice

M

7 ,

S2 = 8

j 11

j 1

C+

=å =

8

j 1

( j 1)=

+å = 2 + 3 + ---- + 8 + 9 = 44

S3 = 104C 210=

S4 = 104C 4! 420´ =

2. Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE ?

(A) cos P ³ 1 – 2p

2pr (B)

q r p rcosR cos P cosQ

p q p q

æ ö æ ö- -³ +ç ÷ ç ÷+ +è ø è ø

(C) q r sin Q sin R

2p sin P+

< (D) If p < q and p < r, then cos Q > pr

and cos R > pq

Ans. (A,B)

Sol. (A) cos P = 2 2 2 2 2q r p 2qr p p

12qr 2qr 2qr

+ - -³ ³ - A is true

(B) p cos R + q cos R ³ q cosP – r cosP + p cosQ – rcosQ p cos R + r cos P + q cos R + r cos Q ³ q cos P + p cosQ q + p ³ r

(C) 2 qrq r 2 sin Qsin R

p p sin P+

³ ³

(D) option (D) in wrong when ÐR = 90°

3. Let : ,2 2p pé ù-ê úë û

→ ℝ be a continuous function such that f(0) = 1 and 3

0

f(t)dt

p

ò = 0. Then which of

the following statements is (are) TRUE ?

(A) The equation f(x) − 3cos 3x = 0 has at least one solution in 0,3pæ ö

ç ÷è ø

(B) The equation f(x) − 3sin 3x = 6

-p

has at least one solution in 0,3pæ ö

ç ÷è ø

(C) 2

x

0

xx 0

x f(t)dt

lim 11 e®

= --

ò

(D)

x

02x 0

sin x f(t)dt

lim 1x®

= -ò

Ans. (A,B,C )


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3

Sol. Option(A)

( ) ( )( )x

0

g x f x 3cos3x dx= -ò

( )g 0 0= , g 03pæ ö =ç ÷

è ø

Option (B)

( ) ( )x

0

6g x f x 3sin3x dxæ ö= - +ç ÷pè øò

( )g 0 0= , g 2 2 03pæ ö = - + =ç ÷

è ø

Option(C)

( )

( )( )

( )2

x x

0 0

x 0 x 0x

22

x f t dt f t dt

lim lim f 0 1x1 e

xx

® ®= - = - = -

æ ö-ç ÷ç ÷ç ÷è ø

ò ò

Option (D)

( ) ( )

x x

0 02x 0 x 0

sin x f t dt f t dt

lim lim 1xx® ®

= =ò ò

4. For any real numbers a and b, let ya,b(x), x Î ℝ , be the solution of the differential equation

xdyy xe , y(1) 1

dxb+ a = = . Let S = {ya,b(x) : a, b Î ℝ}. Then which of the following functions

belong(s) to the set S ?

(A) f(x) = 2

x xx 1e e e

2 2- -æ ö+ -ç ÷

è ø (B) f(x) =

2x xx 1

e e e2 2

- -æ ö- + +ç ÷è ø

(C) f(x) = x 2

xe 1 ex e e

2 2 4-æ öæ ö- + -ç ÷ç ÷

è ø è ø (D) f(x) =

x 2xe 1 e

x e e2 2 4

-æ öæ ö- + +ç ÷ç ÷è ø è ø

Ans. (A,C)

Sol. xdyy xe

dxb+ a =

I.F = eax

So ( )xxye xe dxa+ba = ò

If a + b = 0 2

x xye C

2a = +


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4

passing through (1,1), So 2

x x 1ye e

2 2a a= - + .

for a = 1, y = 2

x xx 1e e e

2 2- -æ ö+ -ç ÷

è ø

Option (A) is true

If 0a + b ¹ ( ) ( )x x

x xe e dxye

a+b a+ba = -

a + b a + bò

Þ ( ) ( )

( )

x xx

2

xe e dxye C

a+b a+ba = - +

a + b a + b curve passing through (1, 1) and If a = b = 1,

then y = x 2

xe 1 ex e e

2 2 4-æ öæ ö- + -ç ÷ç ÷

è ø è ø

Option (C) is true

5. Let O be the origin and ÔA 2i 2 j k= + +uuur

$ $ , ÔB i – 2 j 2k= +uuur

$ $ and ( )1OC OB OA

2= - l

uuur uuur uuur

for same

l > 0. If ( ) 9OB OC

2´ =

uuur uuur

, then which of the following statements is(are) TRUE ?

(A) Projection of OCuuur

on OAuuur

is 32

-

(B) Area of the triangle OAB is 92

(C) Area of the triangle ABC is 92

(D) The acute angle between the diagonals of the parallelogram with adjacent sides OAuuur

and

OCuuur

is 3p

Ans. (A,B,C)

Sol. 9

OB OC2

´ =uuur uuur

Þ ( )OB OB OA 9´ - l =uuur uuur uuur

Þ ( )OB OA 9l ´ =uuur uuur

Þ 1l = Þ 1l ±


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5

Hence, l = 1

Þ ( )1OC OB OA

2= -

uuur uuur uuur

Þ ( )1 ˆ ˆ ÔC i 4 j k2

= - - +uuur

Þ ˆ ˆ ÂB i 4 j k= - - +uuur

and 5 1ˆ ˆ ÂC i 4 j k

2 2-

= - -uuur

area of (DOAB)= 1 9

OA OB2 2

´ =uuur uuur

area of (DABC) = 1 9

AB AC2 2

´ =uuur uuur

Projection of OC on OA = OC.OA 3

–2OA

=

uuur uuur

uuur

Acute angle between diagonals is tan–1

11

31

1–3

æ ö+ç ÷ç ÷ç ÷è ø

= tan–12

6. Let E denote the parabola y2 = 8x. Let P = (−2,4), and let Q and Q′ be two distinct points on E

such that the lines PQ and PQ′ are tangents to E. Let F be the focus of E. Then which of the

following statements is (are) TRUE ?

(A) The triangle PFQ is a right-angled triangle

(B) The triangle QPQ′ is a right-angled triangle

(C) The distance between P and F is 5 2

(D) F lies on the line joining Q and Q′

Ans. (A,B,D)

Sol. Point P(–2, 4) lies on directrix

PF = 4 2

By property option A,B,D are true


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6

SECTION-2 · This section contains THREE (03) question stems. · There are TWO (02) questions corresponding to each question stem. · The answer to each question is a NUMERICAL VALUE. · For each question, enter the correct numerical value corresponding to the answer in the designated

place using the mouse and the on-screen virtual numeric keypad. · If the numerical value has more than two decimal places, truncate/round-off the value to TWO

decimal places. · Answer to each question will be evaluated according to the following marking scheme: Full Marks : +2 If ONLY the correct numerical value is entered at the designated place; Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 7 and 8 Question Stem Consider the region R = {(x, y) Î ℝ × ℝ ∶ x ³ 0 and y2 £ 4 − x}. Let f be the family of all circles

that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in f. Let (a,b) be a point where the circle C meets the curve y2 = 4 − x.

7. The radius of the circle C is ___ . Ans. (1.5) 8. The value of a is ___ . Ans. (2) Sol. (7 to 8)

P(4 – b2, b)

(r, 0)

equation of normal at P is y – b = 2b(x – 4 + b2) Passing through (r, 0) Þ –1 = 2(r – 4 + b2) ........ (i) or b = 0 ........ (ii) Also r2 = (r + b2 – 4)2 + b2 ........ (iii) from (i) and (iii)

r = 32

, –52

so radius of circle is 1.5 and a = 2 from (ii) and (iii) r = 2 but in this case circle intersect at three points


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Question Stem for Question Nos. 9 and 10

Question Stem

Let f1 : (0,¥) → ℝ and f2 : (0, ¥) → ℝ be defined by 9.

x 21

j

1j 10

f (x) (t j) dt, x 0=

= - >Õò

and f2(x) = 98 (x – 1)50 – 600(x – 1)49 + 2450, x > 0.

where, for any positive integer n and real numbers a1, a2,…,an, n

ii 1a=Õ denotes the product of

a1, a2,…,an. Let mi and ni, respectively, denote the number of points of local minima and the

number of points of local maxima of function fi, i = 1,2, in the interval (0, ¥).

9. The value of 2m1 + 3n1 + m1n1 is ___ .

Ans. (57)

10. The value of 6m2 + 4 2 + 8 2 2 is ___.

Ans. (6)

Sol. (9 to 10)

(a) ( ) ( )( ) ( )x

2 21

10

f x t 1 t 2 t 21 dt= - - - - - - - -ò

Þ f1'(x) = (x –1) (x–2)2 ---------(x–21)21

1 2 3 4 5 20 21

– + –+ – +

-----------------

+ – +

At all odd integers from 1 to 21 f(x) will have an extrema with 1,5,9, 13, 17,21 being points of

minima & 3, 7, 11, 15, 19 being points of maxima

So m1 = 6 & n1 = 5

Hence 2m1 + 3n1 + m1n1 = 57

(b) f2' (x) = 98 × 50 (x–1)49 – 600 × 49 (x–1)48

= 4900 (x –1)48 ((x–1) –6

= 4900 (x–1)48 (x–7)

So extrema is at x = 7 only . which is minima

m2 = 1, n2 = 0

Hence 6m2 + 4n2 + 8m2n2 = 6


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8

Question Stem for Question Nos. 11 and 12 Question Stem

Let gi : 3

,8 8p pé ù

ê úë û ® ℝ, i = 1, 2 and

3f : ,

8 8p pé ù

ê úë û® ℝ be functions such that

g1(x) = 1, g2(x) = |4x – p| and f(x) = sin2x , for all x Î 3

,8 8p pé ù

ê úë û

Define

38

i i

8

S f(x).g (x)dx, i 1,2

p

p

= =ò

11. The value of 116Sp

is _____.

Ans. (2)

12. The value of 22

48Sp

is _____.

Ans. (1.5) Sol. (11 to 12)

( ) ( )38

1 1

8

S f x g x

p

p

= ò

Þ

38

21

8

S sin xdx

p

p

= ò (1)

Þ

38

21

8

3S sin x dx

8 8

p

p

p pæ ö= + -ç ÷è øò

=

38

2

8

cos x dx

p

pò (2)

add (1) & (2) we get

38

1 1

8

22S 1dx S

8 8

p

p

p p= = Þ =ò

Þ 116S2=

p


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9

Now

38

22

8

S sin 4x dx

p

p

= - pò (3)

Þ

38

22

8

S cos 4 x dx2

p

p

pæ ö= -ç ÷è øò

Þ

38

22

8

S cos 4x dx

p

p

= - pò (4)

add (3) & (4) we get

328

2

8

12S 4x dx 2

2 8 2 16

p

p

p p p= - p = ´ ´ ´ =ò

22

48S 31.5

2= =

p

SECTION-3 · This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02)

questions. · Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the

correct answer. · For each question, choose the option corresponding to the correct answer. · Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct option is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : −1 In all other cases.

Paragraph for Question Nos. 13 to 14

Let M = {(x,y) Î ℝ × ℝ ∶ x2 + y2 £ r2}, where r > 0. Consider the geometric progression an = n 1

12 - ,

n = 1,2,3,… . Let S0 = 0 and, for n ³ 1, let Sn denote the sum of the first n terms of this progression. For n ³ 1, let Cn denote the circle with center (Sn–1,0) and radius an, and Dn denote the circle with center (Sn–1, Sn–1) and radius an.

13. Consider M with r = 1025513

. Let k be the number of all those circles Cn that are inside M. Let l be

the maximum possible number of circles among these k circles such that no two circles intersect. Then

(A) k + 2l = 22 (B)2k + l = 26

(C) 2k + 3l = 34 (D) 3k + 2l = 40

Ans. (D)


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10

Sol. Sn = 1 + 2 n–1

1 1 1.......

2 2 2+ + +

= 2 – n–1

1

2

Centre of Cn is n–2

12 – ,0

2æ öç ÷è ø

and radius of Cn is n–1

1

2

When r = 1025513

< 2

Cn will lie inside M when 2 – n–2 n–1

1 1 10255132 2

+ <

Þ k = 10 and l = 5

so, 3k + 2l = 40

14. Consider M with r = ( )199

198

2 1 2

2

-. The number of all those circles Dn that are inside M is

(A) 198 (B) 199 (C) 200 (D) 201

Ans. (B)

Sol. 2 Sn–1 + an < 199

198

2 –12

2

æ öç ÷è ø

199

n–2 n–1 198

1 1 2 –12 2 –

2 2 2

æ öæ ö + < ç ÷ç ÷è ø è ø

2

n–2 n–1 198

1 1 22 2 – 2 2 2 –

2 2 2+ <

n–2 198

1 1 2– 2 –

22 2æ ö <ç ÷è ø

n–2 198

1 (2 2 –1) 222 2

>

2n–2 < 1

2 –2

æ öç ÷è ø

2197

n – 2 £ 197

n £ 199

number of circle = 199


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11

Paragraph for Question Nos. 15 to 16

Let y1 : [0, ¥) → ℝ , y2 : [0, ¥) → ℝ , f : [0, ¥) → ℝ and g : [0, ¥) → ℝ be functions such that f(0) = g(0) = 0,

y1(x) = e–x + x, x ³ 0, y2(x) = x2 – 2x – 2e–x + 2, x ³ 0,

f(x) = 2x 2 t

x(| t | t )e dt, x 0-

-- >ò

and 2x t

0g(x) te dt, x 0-= >ò

15. Which of the following statements is TRUE ?

(A) 1

f( ln 3) g( ln3)3

+ =

(B) For every x > 1, there exists an a Î (1, x) such that y1(x) = 1 + ax (C) For every x > 0, there exists a b Î (0, x) such that y2(x) = 2x(y1(b) − 1)

(D) f is an increasing function on the interval 3

0,2

é ùê úë û

Ans. (C)

Sol. (A) f(x) = 2 2

x2 –t

0

(t t )e dt-ò ; x > 0

g(x) = 2x

– t

0

t e dtò ; x > 0

put t = u2

\ g(x) = 22 2

x x2 u 2 t

0 0

u e du 2 t e dt- -=ò ò

now, f(x) + g(x) = 2

xt

0

2te dt-ò = 1 – 2–xe

Þ f( ln 3) g( ln 3)+ = 1 – – ln3 2e

3=

(B) for a Î (1, x), Y1(x) – 1 – ax = 0 Þ e–x + x – 1 – ax = 0 Þ (e–x – 1) = x (a – 1) Which is not possible because LHS < 0 & RHS > 0 (C) Y'2(x) = 2Y1(x) – 2

from LMVT, 2 2(x) – (0)x 0

Y Y-

= Y'2(b) for atleast one b Î (0, x)

Þ Y2(x) = 2x(Y1(b) – 1)

(D) f'(x) = 2(x – x2)2–xe , x ³ 0

increasing in [0, 1] and decreasing in [1, ¥)


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12

16. Which of the following statements is TRUE ?

(A) y1(x) £ 1, for all x > 0

(B) y2(x) £ 0, for all x > 0

(C) f(x) ³ 1 – 2x 3 52 2

e x x3 5

- - + , for all x Î 1

0,2

æ öç ÷è ø

(D) 3 5 72 2 1g(x) x x x

3 3 7£ - + , for all x Î

10,

2æ öç ÷è ø

Ans. (D)

Sol. (A) e–x + x < 1 for x Î (0, ¥) is incorrect

LHS is increasing and unbounded function

(B) x2 – 2x – 2e–x + 2 < 1

for x Î (0, ¥) is incorrect because LHS ® ¥ when x ® ¥

(C) Now f(x) + g(x) = 1 – 2xe-

Þ f(x) = 1 – 2xe- – g(x) Þ f(x) £ 1 –

2x 3 52e x x

3 3- 2æ ö- -ç ÷

è ø

(D) g(x) = 2x

– t

0

te dtò

Þ g(x) £ 2x 2

0

tt 1 t dt

2

æ ö- +ç ÷

è øò Þ g(x) £

23

x3 – 25

x5 + 17

x7

g(x) ³ ( )2x

0

t 1 t dt-ò Þ g(x) ³ 3 52x x

3 52

-

SECTION 4

· This section contains THREE (03) questions.

· The answer to each question is a NON-NEGATIVE INTEGER.

· For each question, enter the correct integer corresponding to the answer using the mouse and the

on-screen virtual numeric keypad in the place designated to enter the answer.

· Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If ONLY the correct integer is entered;

Zero Marks : 0 In all other cases.

17. A number is chosen at random from the set {1,2,3,…,2000}. Let p be the probability that the

chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is ___ .

Ans. (214)


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13

Sol. Number multiple of 3 are 2000

3é ùê úë û

= 666

Number multiple of 7 are 2000

7é ùê úë û

= 285

Number multiple of 21 are 2000

21é ùê úë û

= 95

p = 666 285 95 856

2000 2000+ -

=

500 p = 214

18. Let E be the ellipse 2 2x y

116 9

+ = . For any three distinct points P, Q and Q¢ on E, let M(P,Q) be the

mid-point of the line segment joining P and Q, and M(P,Q') be the mid-point of the line segment

joining P and Q¢. Then the maximum possible value of the distance between M(P,Q) and M(P,Q'),

as P, Q and Q¢ vary on E, is ___ .

Ans. (4)

Sol. Q

M

Q'

M' (4, 0)

P

MM' = 12

QQ'

Maximum distance between QQ' is 8

Hence, maximum distance between M and M' is 4

19. For any real number x, let [x] denote the largest integer less than or equal to x. If

10

0

10xI dx

x 1

é ù= ê ú

+ë ûò ,

then the value of 9I is ___ .

Ans. (182)

Sol. y = 10x 10

10 –x 1 (x 1)

=+ +

2

dy 10dx (x 1)

=+


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14

y

x

10

3.01

O 10 –1

10x

1x 1

=+

Þ x = 19

10x

4x 1

=+

Þ x = 23

10x

9x 1

=+

Þ x = 9

10

0

10xdx

x 1

é ùê ú

+ê úë ûò =

1/9 2/3 9 10

0 1/9 2/3 9

0 dx dx 2 dx 3 dx+ + +ò ò ò ò

= 2 1 2

2 9 33 9 3

æ ö æ ö- + - +ç ÷ ç ÷è ø è ø

= 5 50

39 3

+ +

= 182

9

\ 9I = 182

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