IITJEE Main 2017 Question Paper with Solutions

RBS

Test Booklet Code

D

PAPER - 1 : MATHEMATICS, PHYSICS & CHEMISTRY

Do not open this Test Booklet until you are asked to do so. Read carefully the Instructions on the Back Cover of this Test Booklet.

Important Instructions:

1. Immediately fill in the particulars on this page of the Test Booklet with only Black Ball Point Pen provided in the examination hall.

2. The Answer Sheet is kept inside this Test Booklet. When you are directed to open the Test Booklet, take out the Answer Sheet and fill in the particulars carefully.

3. The test is of 3 hours duration.

4. The Test Booklet consists of 90 questions. The maximum marks are 360.

5. There are three parts in the question paper A, B, C consisting of, Mathematics, Physics and Chemistry having 30 questions in each part of equal weightage. Each question is allotted 4 (four) marks for each correct response.

6. Candidates will be awarded marks as stated above in instruction No. 5 for correct response of each question. 1/4 (one fourth) marks of the total marks allotted to the question (i.e., 1 mark) will be deducted for indicating incorrect response of each question. No deduction from the total score will be made if no response is indicated for an item in the answer sheet.

7. There is only one correct response for each question. Filling up more than one response in any question will be treated as wrong response and marks for wrong response will be deducted accordingly as per instruction 6 above.

8. For writing particulars/ marking responses on Side-1 and Side-2 of the Answer Sheet use only Black Ball Point Pen provided in the examination hall.

9. No candidates is allowed to carry any textual material, printed or written, bits of papers, pager, mobile phone, any electronic device, etc., except the Admit Card inside the examination room/hall.

10. Rough work is to be done on the space provided for this purpose in the Test Booklet only. This space is given at the bottom of each page and in 4 pages (Pages 2023) at the end of the booklet.

11. On completion of the test, the candidate must hand over the Answer Sheet to the Invigilator on duty in the Room / Hall. However, the candidates are allowed to take away this Test Booklet with them.

12. The CODE for this Booklet is D. Make sure that the CODE printed on Side2 of the Answer Sheet and also tally the serial number of the Test Booklet and Answer Sheet are the same as that on this booklet. In case of discrepancy, the candidate should immediately report the matter to the invigilator for replacement of both the Test Booklet and the Answer Sheet.

13. Do not fold or make any stray marks on the Answer Sheet.

Name of the Candidate (in Capital letters) : ______________________________________________

Roll Number : in figures

: in words ______________________________________________________________

Examination Centre Number : Name of Examination Centre (in Capital letters) : ____________________________________________

Candidate's Signature : ______________________

1. Invigilator's Signature : _________________ 2. Invigilator's Signature : _________________

(Pg. 1)

JEE-MAIN 2017 : Question Paper and Solution (2) Read the following instructions carefully :

1. The candidates should fill in the required particulars on the Test Booklet and Answer Sheet (Side-1) with Black Ball Point Pen.

2. For writing/marking particulars on Side-2 of the Answer Sheet, use Black Ball Point Pen only. 3. The candidates should not write their Roll Numbers anywhere else (except in the specified space) on the

Test Booklet/Answer Sheet. 4. Out of the four options given for each question, only one option is the correct answer. 5. For each incorrect response, ? (one-fourth) of the total marks allotted to the question (i.e., 1 mark) will

be deducted from the total score. No deduction from the total score, however, will be made if no response is indicated for an item in the Answer Sheet. 6. Handle the Test Booklet and Answer Sheet with care, as under no circumstances (except for discrepancy in Test Booklet Code and Answer Sheet Code), another set will be provided. 7. The candidates are not allowed to do any rough work or writing work on the Answer Sheet. All calculations/writing works are to be done in the space provided for this purpose in the Test Booklet itself, marked `Space for Rough Work'. This space is given at the bottom of each page and in 4 pages (Pages 20 ? 23) at the end of the booklet. 8. On completion of the test, the candidates must hand over the Answer Sheet to the Invigilator on duty in the Room/Hall. However, the candidates are allowed to take away this Test Booklet with them. 9. Each candidate must show on demand his/her Admit Card to the Invigilator. 10. No candidate, without special permission of the Superintendent or Invigilator, should leave his/her seat. 11. The candidates should not leave the Examination Hall without handing over their Answer Sheet to the Invigilator on duty and sign the Attendance Sheet again. Cases where a candidate has not signed the Attendance Sheet second time will be deemed not to have handed over the Answer Sheet and dealt with as an unfair means case. The candidates are also required to put their left hand THUMB impression in the space provided in the Attendance Sheet. 12. Use of Electronic/Manual Calculator and any Electronic Item like mobile phone, pager etc. is prohibited. 13. The candidates are governed by all Rules and Regulations of the Examination body with regard to their conduct in the Examination Hall. All cases of unfair means will be dealt with as per Rules and Regulations of the Examination body. 14. No part of the Test Booklet and Answer Sheet shall be detached under any circumstances. 15. Candidates are not allowed to carry any textual material, printed or written, bits of papers, pager, mobile phone, electronic device or any other material except the Admit Card inside the examination room/hall.

(Pg. 2)

(3) VIDYALANKAR : JEE-MAIN 2017 : Question Paper and Solution

.Questions and Solutions.

PART- A : MATHEMATICS

1. If S is the set of distinct values of `b' for which the following system of linear equations x + y + z = 1, x + ay + z = 1, ax + by + z = 0 has no solution, then S is : (1) an empty set (2) an infinite set (3) a finite set containing two or more elements (4) a singleton

1. (4)

1 11

= 1 a 1 = (a 1)2

a b1

x = (a 1) y = 0 z = a(a 1) If a = 1 = x = y = z = 0

Infinite solution For b = 1 & a = 1 x + y + z = 1

x + y + z = 0 no solution Only one value of b.

2. The following statement (p q) [(~ p q) q] is :

(1) a tautology

(2) equivalent to ~ p q

(3) equivalent to p ~ q

(4) a fallacy

2. (1)

(p q) [(p q) q] is

12 3

4

5

6

7

p q p p q p q [(p q)q] 4 6

TT F

T

T

T

T

TF F

F

T

F

T

FT T

T

T

T

T

FF T

T

F

T

T

3. If 5 tan2 xcos2 x = 2 cos 2x + 9, then the value of cos 4x is :

(1) 53

(2)

1 3

3. (4)

5 tan2 x cos2 x = 2 cos 2x + 9

(3)

2 9

5 sec2 x1cos2 x = 2 (2cos2x 1) + 9

5

1 cos2

x

1

cos2

x

=

4

cos2

x

2

+

9

(4) 79

(Pg. 3)

JEE-MAIN 2017 : Question Paper and Solution (4)

5

1

cos2 x cos4 cos2 x

x

=

4

cos2

x

+

7

5 5 cos2 x 5 cos4 x = 4 cos4 x + 7 cos2 x

9 cos4x + 12 cos2x 5 = 0

Put, cos2x = m

9m2 + 12m 5 = 0

9m2 + 15m 3m 5 = 0

3m (3m + 5) 1 (3m + 5) = 0

(3m + 5) (3m 1) = 0

m

=

1 3

or

m = 53

cos2

x

=

1 3

or

cos2x = 53

But, cos2x 53

cos2x

=

1 3

Now, cos 4x = 2 cos2 2x 1

= 2 (cos 2x)2 1

= 2 (2 cos2 x 1)2 1

=

2213

12

1

=

2 2312

1=

213

2

1

=

921 =

2

9

9

=

79

4. For three events A, B and C, P (Exactly one of A or B occurs)

= P (Exactly one of B or C occurs) = P (Exactly one of C or A occurs)

=

1 4

and P (All the three events occur simultaneously =

1 16

.

Then the probability that at least one of the events occurs is :

(1)

7 32

(2)

7 16

(3)

7 64

(4)

3 16

4. (2)

P(exactly one of A or B occurs) =

1 4

P(A) + P(B) 2P(A B) =

1 4

P(exactly one of B & C occurs) =

1 4

P(B) + P(C) 2P(B C) =

1 4

P(exactly one of C or A occurs) =

1 4

P(C) + P(A) 2P(A C) =

1 4

P(A

B

C)

=

1 16

Ad.

[P(A)

+ P(B)

+ (C)

P(A

B)

P(B

C)

P(C

A)]

=

3 8

.... (1)

Probability of at least one event

= P(A B C) = P(A) + P(B) + P(C) P(A B) P(B C) P(C A) + P(A B C)

= 83116 = 6161176

(Pg. 4)

(5) VIDYALANKAR : JEE-MAIN 2017 : Question Paper and Solution

11

5. Let be a complex number such that 2 + 1 = z, where z = 3 . If 1 2 1

1 2

then k is equal to :

(1) z

(2) z

5. (1)

(3) 1

(4) 1

w =

3i1 2

w

=

1 2

3i ,w212

3i

1 + w + w2 = 0, w3 = 1

11

1 11 1

3k = 1 w2 1 w2 1 w w2

1 w2 w7 1 w2 w

= 1(w2 w4) 1 (w w2) + 1 (w2 w) = w2 w4 w + w2 + w2 w = 3w2 2w w4 = 3w2 2w w = 3(w2 w)

k = w2 w

=

1 2

3i

1 2

3i

= 1

3i 1 2

3i22 3i

3i z

(2w + 1 = z = 3 i)

1 2 = 3k, 7

6. Let k be an integer such that the triangle with vertices (k, 3k), (5, k) and (k, 2) has area 28 sq. units. Then the orthocenter of this triangle is at the point :

(1)

2,

1 2

(2)

1,34

(3)

1,34

6. (4)

=

1 2

(k2

10 3k2)(15kk22k) 28

(4)

2,12

5k2 13k10 56

5k2 + 13k + 10 = 56 5k2 + 13k + 10 = 56

5k2 + 13k 46 = 0 5k2 + 23k 10k 46 = 0

k (5k + 23) 2(5k + 23) = 0

k

=

2,

23 5

k = 2 as integer

A (2, 6), B (5, 2) , C (2, 2).

As BC perpendicular to xaxis

eqn. of altitude AD is x = 2

For

BE,

y

2

=

1 2

(x

5)

Solving

orthocentre

2,12

C(2, 2)

D

B (5,2)

E A(2, 6)

(Pg. 5)

JEE-MAIN 2017 : Question Paper and Solution (6)

7. Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector.

Then the maximum area (in sq. m) of the flowerbed is :

(1) 12.5

(2) 10

(3) 25

(4) 30

7. (3)

2r + s = 20 (length of wire)

Now s = r

2r + r = 20

=

202r r

A =

1 2

r

2

=

1 2

r

2

202r r

A = 10 r r2

r

s

r

dA dr

= 10 2r = 0 r = 5

d2A dr2

=

2

<

0

Area

is

maximum

Maximum Area =

1 2

r21225

202r r

= 1220510

=

1 2

5

10

=

25

sq.m.

8. The area (in sq. units) of the region x,y:x0,x y 3,x2 4yandy1 x is :

(1)

59 12

(2)

3 2

(3)

7 3

(4)

5 2

8. (4)

x2 = 4y

1

1

Area = xdy ydxPQR

(0,3) Q(1,2)

y x1

0

0

=

1

2

0

1

ydy

0

x dx 12

(0,1) S R P(2,1)

0

(3,0)

=

2

2323

1 2

=

2

1 2

=

5 2

9. If the image of the point P (1, 2, 3) in the plane, 2x + 3y 4z + 22 = 0, measured parallel to the

line,

x 1

y 4

z 5

is Q, then PQ is

equal to :

(1) 3 5

(2) 2 42

(3) 42

9. (2)

Line PM (Parallel to given line) is

x 1 1

y

4

2

z

3 5

r(Let)

M (r + 1, 4r 2, 5r + 3)

M satisfy plane, 2x + 3y 4z + 22 = 0

2r + 2 + 12r 6 20r 12 + 22 = 0

6r + 6 = 0

r = 1

So, M = (2, 2, 8)

(4) 6 5

P(1,2,3) M

x 1

y 4

z 5

PM = (2 1)2 (2 2)2 (8 3)2 = 116 25 42

So, PQ = 2 42

(Pg. 6)

(7) VIDYALANKAR : JEE-MAIN 2017 : Question Paper and Solution

10.

If

for

x

0,14

,

the

derivative

of

tan

1

6x x 1 9x3

is

x.g x , then g(x) equals :

(1)

9 1 9x3

10. (1)

(2)

3x x 1 9x3

(3)

3x 1 9x3

(4)

3 1 9x3

y =

tan 1

2

1

3x 3x

x x

2

.

Let 3x x tan

= tan1 tan2 = 2 = 2tan1 3x x

dy dx

=

2

1

1 9x3

32x

1 2

=

9x 1 9x3

=

g(x) =

9 1 9x3

x

1

9 9x3

11.

If

2 sinxddxyy 1cosx

0 and

y(0)

=

1,

then

y

2

is

equal

to

:

(1)

1 3

(2)

2 3

11. (1)

ddxy2ycsoisnxx2cossinx x0

ddxy2

cos x sin

x

y

cos x 2sin

x

(3) 13

(4)

4 3

I.F. =

e

cos x 2sin x

dx

2sin

x

y . (2 + sin x) = c + ( cos x)dx

y (2 + sin x) = c sin x Given y (0) = 1

1(2 + 0) = c 0 c = 2

Soln. y(2 sin x) = 2 sin x

y(/2) =

2 2

1113

12. Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P

be a point on the ground such that AP = 2AB. If BPC = , then tan is equal to :

(1)

6 7

(2)

1 4

(3)

2 9

(4)

4 9

12. (3)

AP = 2AB

tan = AABP12

B

AB tan = AACP2A2B14

C

A

P

(Pg. 7)

JEE-MAIN 2017 : Question Paper and Solution (8)

tan

(

+

)

=

tantan 1tan tan

1 2

=

1 4

tan

114 .tan

12

1 4 tan 4 tan

2 + 8 tan = 4 tan

9 tan = 2

tan

=

2 9

13.

If

A

=

2 4

3 1

,

then

adj

(3A2

+

12A)

is

equal

to

:

(1)

72 63

84 51

(2)

51 84

63 72

(3)

51 63

84 72

13. (2)

A2

=

2 4

1324

131162

9 13

3A2

+

12A

=

48 36

27 39

+

24 48

36 12

=

72 84

63 51

Adj.

(3A2 +

12A) =

51 84

63 72

(4)

72 84

63 51

14. For any three positive real numbers a, b and c, 9 (25a2 + b2) + 25 (c2 3ac) = 15b (3a + c). Then :

(1) b, c and a are in G.P.

(2) b, c and a are in A.P.

(3) a, b and c are in A.P.

(4) a, b and c are in G.P.

14. (2)

225a2 + 9b2 + 25c2 75ac 45ab 15bc = 0

450a2 + 18b2 + 50c2 150ac 90ab 30bc = 0

(15a 5c)2 + (15a 3b)2 + (3b 5c)2 = 0

15a = 5c 15a = 3b 3b = 5c

15a = 3b = 5c

a 1

b 5

c 3

b, c, a in A.P.

15. The distance of the point (1, 3, 7) from the plane passing through the point (1, 1, 1) having

normal perpendicular to both the lines

x

1 1

y2 2

z4 3

and

x2 2

y 1 1

z7 1

is :

(1) 20 74

(2) 10 83

(3) 5 83

(4) 10 74

15. (2)

Plane passing (1, 1,1) is

a(x 1) + b(y + 1) + c(z + 1) = 0

(Pg. 8)

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