|JEE MAIN 2019 | DATE : 9 APRIL 2019 (SHIFT-2) EVENING...

Main_09-04-2019_EVENING [ 1 ]

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|JEE MAIN 2019 | DATE : 9 APRIL 2019 (SHIFT-2) EVENING

MATHEMATICS This section contain 30 multiple choice questions. Each question has 4 choice (1), (2), (3) and (4) for its answer, out of which Only One is correct 1. If p (q r) is false, then the truth values of p, q, r are respectively:

(1) T, F, F (2) F, F, F (3) F, T, T (4) T, T, F

2. The total number of matrices

0 2y 1

A 2x y 1 , (x, y R, x y)2x y 1

for which T3A A 3 is :

(1) 6 (2) 4 (3) 3 (4) 2

3. Let z C be such that | z | 1. If 5 3z5(1 z)

, then :

(1) 4 m( ) 5 (2) 5 Re( ) 1 (3) 5 m( ) 1 (4) 5 Re( ) 4

4. The value of sin10° sin30° sin50° sin70° is :

(1) 118

(2) 116

(3) 136

(4) 132

5. If sec x 2e (sec x tanx f(x) (sec x tan x sec x))dx sec xe f(x) C, then a possible choice of f(x) is

:

(1) 1sec x tan x2

(2) 1sec x x tan x2

(3) 1x sec x t an x2

(4) 1sec x t an x2

6. If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively then a value of its th11 term is :

(1) 36 (2) 25 (3) 25 (4) 35

7. If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trivial

solution (x, y, z) then x y z ky z x is equal to :

(1) 34

(2) 12

(3) 4 (4) 14

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8. The domain of the definition of the function 3102

1f(x) log (x x)4 x

is :

(1) ( 1, 0) (1, 2) (2, ) (2) (1, 2) (2, )

(3) ( 2, 1) ( 1, 0) (2, ) (4) ( 1, 0) (1, 2) (3, )

9. The vertices B and C of a ABC lie on the line, x 2 y 1 z3 0 4

such that BC = 5 units. Then

the area (in sq. units) of this triangle, given that the point A(1, -1, 2), is :

(1) 34 (2) 2 34 (3) 6 (4) 5 17

10. A rectangle is inscribed in a circle with a diameter lying along, the line 3y = x + 7. If the two adjacent vertices of the rectangle are 8, 5 and (6, 5), then the area of the rectangle (in sq. units) is :

(1) 98 (2) 72 (3) 84 (4) 56

11. If m is chosen in the quadratic equation 2 2 2 2(m 1) x 3x (m 1) 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :

(1) 8 5 (2) 4 3 (3) 8 3 (4) 10 5

12. If some three consecutive coefficients in the binomial expansion of nx 1 in powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficients is :

(1) 625 (2) 227 (3) 964 (4) 232

13. The sum of the series th1 2 3 3 5 4 7 ......upto 11 terms is :

(1) 945 (2) 915 (3) 916 (4) 946

14. If dycos x y sin x 6x, (0 x )dx 2

and y 0,

3

then

y6

is equal to :

(1) 2

2

(2) 2

4 3

(3) 2

2 3 (4)

2

2 3

15. The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x,

42, 67, 70, y are 42 and 35 respectively, then yx

is equal to :

(1) 8/3 (2) 9/4 (3) 7/3 (4) 7/2

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16. The area (in sq. units) of the smaller of the two circles that touch the parabola, 2y 4x at the point (1, 2) and the x-axis is :

(1) 4 (2 2) (2) 4 (3 2) (3) 8 (3 2 2) (4) 8 (2 2)

17. Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisements is :

(1) 13.9 (2) 12.8 (3) 13 (4) 13.5

18. If a unit vector a

makes angles / 3 with i, / 4 with j and (0, ) with k , then a value of is ?

(1) 23 (2) 5

12 (3) 5

6 (4)

4

19. If the function a| x | 1, x 5

f(x)b | x | 3, x 5

is continuous at x = 5, then value of a – b is :

(1) 25

(2) 25

(3) 25

(4) 25

20. The area (in sq. units) of the region 2yA (x, y) : x y 4

2

is :

(1) 16 (2) 18 (3) 533

(4) 30

21. Let P be the plane, which contains the line of intersection of the planes, x + y + z – 6 = 0 and 2x + 3y + z + 5 = 0 and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to

(1) 11/ 5 (2) 205 5 (3) 63 5 (4) 17 / 5

22. Some identical balls are arranged in rows to form an equlateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is :

(1) 225 (2) 262 (3) 157 (4) 190 23. Two poles standing on a horizontal ground are of heights 5m and 10m respectively. The line

joining their tops makes an angle of 15° with the ground. Then the distance (in m) between the poles, is :

(1) 5 (2 3)2

(2) 5 (2 3) (3) 10 ( 3 1) (4) 5 ( 3 1)



24. The common tangent to the circle 2 2 2 2x y 4 and x y 6x 8y 24 0 also passes through the point :

(1) (4, 2) (2) (6, 2) (3) ( 4, 6) (4) ( 6, 4)

25. If f : R R is a differentiable function and f(2) 6, then f (x)

x 26

2t dtlim(x 2) is :

(1) 24 f '(2) (2) 12 f '(2) (3) 0 (4) 2 f '(2)

26. If f(x) = x[x] , x R,4

where [x] denotes the greatest integer function, then :

(1) Both x 4lim f(x)

and x 4lim f(x)

exist but are not equal.

(2) f is continuous at x = 4. (3)

x 4lim f(x)

exists but x 4lim f(x)

does not exist.

(4) x 4lim f(x)

exists but x 4lim f(x)

does not exist.

27. If the tangent to the parabola 2y x at a point ( , ), ( 0) is also a tangent to the ellipse, 2 2x 2y 1, then is equal to :

(1) 2 2 1 (2) 2 1 (3) 2 2 1 (4) 2 1

28. The value of the integral 1

1 2 4

0

xcot (1 x x )dx is :

(1) e1 log 2

4 2 (2) elog 2

2 (3) e

1 log 22 2 (4) elog 2

4

29. A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is

1 1tan .2

Water is poured into it at a constant rate of 5 cubic meter per minute. The the rate (in

m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m, is :

(1) 2 / (2) 1/ 5 (3) 1/ 15 (4) 1/ 10

30. If the two lines x + (a - 1) y = 1 and 2x + 2a y 1(a R {0, 1}) are perpendicular, then the distance of their point of intersection from the origin is :

(1) 25

(2) 25

(3) 25

(4) 25

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SOLUTION 1. (1) P (q r) is false only when P is true and q r is false.

i.e truth values of p, q, r are T, F, F respectively. 2. (2)

Given 0 2y 1

A 2x y 12x y 1

T3A A 3

0 2x 2x 0 2y 1 3 0 02y y y 2x y 1 0 3 01 1 1 2x y 1 0 0 3

2

2

8x 0 0 3 0 00 6y 0 0 3 00 0 3 0 0 0

3 1 3 1x & y (x, y) ,2 2 2 2 2 2

4 matrices are possible

3. (2)

5 3zw5(1 z)

5w 5wz 5 3z

5w 5z5w 3

| z | 1

5w 5 15w 3

35w 5 w5



1Re(w)5

5Re(w) 1

4. (2) sin10 sin30 sin50 sin70

1 sin10 sin50 sin702

1 1 sin302 4

1sin sin(60 ) sin(60 ) sin34

116

5. (4)

sec x 2 sec xe sec x tanx f(x) sec x tanx sec x) dx e f(x) c

Differentiating both sides w.r.t. x sec x 2 sec xe (sec x tanx f(x) (sec x tanx sec x)) e f '(x) f(x) sec x tan x

2f '(x) sec x sec x tan x

f(x) sec x tan x c

6. (2) Let 1t a d

2t a

3t a d

1 2 3t t t 33

3 , 05

3 , 05

1, 0

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a 11

1 2 3t t t 1155

211(121 d ) 1155

2d 121 105 d 4

11t a 9d 11 36 47, 25

7. (2) Given system of homogeneous equations are 2x 3y z 0 ………….(i)

x k y 2z 0 ………….(ii)

2x y z 0 ………….(iii)

Non –trivial solution

2 3 11 k 2 02 1 1

1(3 1) k (2 2) 2 ( 2 6) 0

4k 18

Or, n9k eq (ii) becomes2

9x y 2z 02

Solving (i) & (iii) 4x 2y 0 2x y

x 1y 2

Solving (i) & (ii) (3 2k) y 3z 0

y 16y 3zz 2

Solving (ii) & (iii)

510x z 02

5z 20x

z 4x

x y z 1ky z x 2

8. (1) Given

3102

1f(x) log (x x)4 x

f(x) is defined only when



2 2(i) 4 x 0 & (ii) x x 0

x 2, & x(x 1) (x 1) 0

x ( 1, 0) (1, 2) (2, )

9. (1)

x 2 y 1 z3 0 4

P (3 2, 1, 4 )

AP line

Dot product of direction ratio of line and AB

is zero.

3(3 3) 0(1 1) 4(4 2) 0

25 17

1725

1 68P , 1,25 25

2 224 18Length AP 4

25 25

2 144 625 8125

2 85025

Area of 1 2ABC 5 8502 25

_+ +_

-1 0 1

P

A

B5 C

(1, -1, 2)

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10. (3)

Slipe of AB = of CD = 0

8 62

C(6, ) & D( 8, )

1

Centre of circle is O( 1, 2) & M( 1, 5)

Area = 14 × 6 = 84 11. (1) Given quadratic eqn is 2 2 2 2(m 1)x 3x (m 1) 0

2

3m 1

Sum of roots is greatest when m = 0 2x 3x 1 0

3 & 1

3 3 2 2

2 24

9 4 (9 1)

8 5 12. (4) Let three consecutive Binomial coefficients are n n n

r 1 r r 1C , C , C

D(-8, )

7

3,

MA

C(6, )

B(6, )5

3y = x + 7O

(-8, )5



Given

n n nr 1 r r 1C C C

2 15 70(i) (ii) (iii)

From (i) & (ii) n

Crn

r 1

C 15 n r 1 152 r 2C

17r 2n 2 ………..(iv) From (ii) & (iii)

n

Cr 1n

r

C 7015C

n r 14r 1 3

17r 3n 14 ………..(v) Solving (iv) & (v) n = 16 & r = 2 16 16 16

1 2 3C , C , C , are three consecutive coefficient

Average = 16 16 16

1 2 3C C C3

16 120 5603

232 13. (4) thS 1 2 3 3 5 4 7 .......up to 11 then

thr term of sequence is rT r(2r 1) Required sum

r 11

rr 1

T

r 11 r 11

2

r 1 r 12 r r

11 12 23 11 1226 2

11 92 11 6 11 86 946 14. (C) Given differential equation is

dycos x y sin x 6x 0 xdx 2

dy y tan x 6x sec xdx

tanxdx n(cosx)F e e cos x

Main_09-04-2019_EVENING [ 11 ]


y cos x 6x dx 2y cos x 3x C

2

y 0 C3 3

2

2y sec x 3x3

2 2 22y

6 12 33 2 3

15. (3) Given numbers are in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y

Median = 35 34 x 35 x 362

Mean = 42 10 22 26 29 34 36 42 67 70 y 4210

336 y 420 y 84

y 7x 3

16. (3) Let required circle have centre C( , r) and radius = r

(Here 0, r 0 ) Equation of tangent at P is also tangent to circle x

2y 4x

uEq of tangent at (1, 2) on 2y 4x is

2y 2(x 1)

y x 1

Normal at P passes through centre of circle Equation normal is y = - x + 3

p(1, 2)

N T

( , r)C

(1,0)

y =4x2



r 3 …………..(i)

& | r 1| r2

………...(ii)

Solving (i) & (ii)

3 2r 1 r 2

r (2 2) 4

radius is minimum 4r 2 2 22 2

2A r 8 (3 2 2)

17. (1) Let population of the city is 100 n(A) 25 n(B) 20 n(A B) 8

n A B 17 & n A B 12

Percentage of the population who looks into odvertisements

30 40 8 5017 12100 100 100

51 48 40 13.910

18. (1)

Given, cos , cos , cos cos y3 4

2 2 2cos cos cos 1

21 1 cos 14 2

2 1cos4

1cos2

1cos cos2

2,3 3

19. (1) f(x) is continuous at x = 5

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x 5 x 5lim f(x) lim f(x) f(5)

a | 5 | 1 b | 5 | 3

5 (a b) 2

2a b5

20. 2y x y 4

2

2y 2x ………..(i)

y x 4 ……….(ii)

Solving (i) & (ii) 2(x 4) 2x

2x 10x 16 0

x 2, 8

A(2, 2) & B (8, 4)

Required area

4 2

2

yy 4 dy2

42 3

2

y y 32 44y 8 16 2 82 6 3 3

18

21. (1) Equation of plane P is

2x 3y z 5 x y z 6 0

y =4x2

A(0,-1)

y =2x2

Bx-y=4

(4,0)



i.e. x 2 y(3 ) z(1 ) 5 6 0 …………(i)

equation of xy-plane is z = 0 ………..(ii) plane (i) & (ii) are perpendicular 1 0 1

Equation of plane P is x 2y 11 0

Distance of point (0, 0, 256) from plane P is = 115

22. (4) Let r be the number of rows used to make equilateral triangle.

2r (r 1) 99 (r 2)2

2 2r r 198 2r 8r 8

2r 9r 190 0

r 10,19 r 0 r 19

Total number of balls = 19 20 1902

23. (2)

5tan(15 )x

113x 5 tan(75 ) 5 113

3 153 1

5

15°

5

M

A Bx

10

N

Main_09-04-2019_EVENING [ 15 ]


5 3 1 2 32

5 2 3

24. (2) 2 2

1S : x y 4

2 22S : x y 6x 8y 24

1 2r 2 r 7

1 2C (0, 0), C ( 3, 4)

1 2 1 2C C | r r |

1 2C C 5

1S touch 2S internally.

Equation Common tangent is 1 2S S 0

6x 8y 20

i.e. 3x + 4y = 10 which passes through point (6, -2) 25. (4) Given f(2) = 6

f (x)f (x) 66x 2 x 2

2 tdt2t dtlim limx 2 x 2

Using L-pt (denominator w.r.t x)

x 2

2 f(x) f(x)lim 2 f '(2) f '(2) 12f '(2)1

26. (2) Given

xf(x) x x R4

x 4 x 4

xlim f(x) lim x 3 0 34

x 4 x 4

xlim f(x) lim x 4 1 34

f(4) 4 1 3

f(x) is continuous at x = 4



27. (4) Equation of tangent at , on 2y x is

xy2

…………..(i)

( , ) lies on 2y x 2 ………..(ii)

Equation (i) is tangent to 2 2x 2y 1

Put = y from (i) & equate

2

22

(x )x 12

2 2 2 22 1 x 2x 2 0

Do

2 2 2 24 4 2 1 2

2

2 22 1 ( 2 )

( (2 1) ( 2) 0

2( 2 3 2) 0

2( 2 1) 0

1 2, 0

28. (1)

1

1 2 4

0

x cot (1 x x ) dx

Substitute 2x t

1

1 2

0

1 cot (1 t t ) dt2

1

1

0

1 t t2 tan dt1 t (1 t)

1 1

1 1

0 0

2 tan (t) dt tan (1 t) dt

1

1

0

tan (t) dt

1

1 2e

0

1t tan t log (1 t )2

elog 24 2

Main_09-04-2019_EVENING [ 17 ]


29. (2)

1 rtan2 x

x 2r

2 31 1v r x x3 12

2dv dx, 3xdt 12 dt

dx5 1004 dt

dx 1dt 5

meter / minute

30. (1)

1x (a 1) y 1 ma 1

22 2

22x a y 1 ma

1 2 2

2m m 1 1a a 1

3 2a a 2 0

2(a 1) (a 2a 2) 0 a 1

Equation lines are x 2y 1

& 2x y 1

Solving we get

x

A

M rP

1tan2



35x 3 x5

3 15x 3 x & y5 5

3 1P ,5 5

Distance of 3 1P ,5 5

from origin

1 2105 5

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|JEE MAIN 2019 | DATE : 9 APRIL 2019 (SHIFT-2) EVENING...

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