[1] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

1. A value of for which2 3

1 2

i

i

sin

sin

purely imaginary

is

(1) sinFHGIKJ

1 1

3(2)3

(3)6

(4) sinFHGIKJ

1 3

4

Sol. [1]2 + 3isin

1 2

2 3 1 2

1 4 2

i

i i

sin

sin sin

sin

b gb g

2 6 7

1 4

2

2

sin sin

sin

e j i

2 6 02 sin sin . 1

3

2. The system of linear equations

x y z

x y z

x y z

0

0

0

has a non-trivial solution for

(1) exactly three values of

(2) infinitely many values of

(3) exactly one value of (4) exactly two values of

Sol. [1]

1 1

1 1

1 1

0

1 1 0b g b g

1 0 1, , .

3. A wire of length 2 units is cut into two parts which arebent respectively to form a square of side = x units anda circle of radius = r units. If the sum of the areas of thesquare and the circle so formed is minimum, then

(1) 2x r (2) 2 4x r b g(3) 4 b gx r (4) x = 2r

Mathematics

Sol. [4] Let length of pieces be , 2

4 2 2x r , x r

4

2

2,

Area A x r 2 22

2

2

16 42

b g

A' b g b g 2

16

2

42 0

8

4

x

r

FHG

IKJ

2 2

84

2 28

4

2b g .

4. A man is walking towards a vertical pillar in a straightpath, at a unitform speed. At a certain point A on thepath, he observed that the angle of elevation ofthe topof the pillar is 30°. AFter walking for 10 minutes from Ain the same direction, at a point B, he observes that theangle of elevation of the top of the pillar is 60°. Thenthe time taken (in minutes) by him, from B to reach thepillar, is(1) 5 (2) 6(3) 10 (4) 20

Sol. [1] Let speed be x m/ min.

h

y tan 60 3 h y 3

andh

x y1030

1

3 tan

3 10y x y

y x 5

time 5min.

[2] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

5. Let two fair six-faced dice A and B be thrown

simultaneously. If E1 is the event that die A shows up

four, E2 is the event that die B shows up two and E3

is the event that the sum of numbers on both dice isodd, then which of the following statements is NOTtrue?

(1) E E1 2, and E3 are indepedent

(2) E1 and E2 are independent

(3) E2 and E3 are indepedent

(4) E1 and E3 are indepedent

Sol. [1] P E P E P E1 2 31

6

1

6

18

36

1

2b g b g b g , ,

P E E1 21

6

1

6

1

36 b g

P E E P E E2 3 1 33

36

1

12 b g b g.

6. If the standard deviation of the numbers 2, 3, a and 11is 3.5, then which of the following is true?

(1) 3 23 44 02a a

(2) 3 26 55 02a a

(3) 3 32 84 02a a

(4) 3 34 91 02a a

Sol. [3] 22

2

235

FHGGIKJJ

xi

n

xi

n.b g

a a2 2

134

5

16

512 25

FHGIKJ .

3 32 84 02a a .

7. For x R , f x xb g log sin2 and g x f f xb g b gc h ,

then

(1) g is differentiable at x = 0 and g' sin log0 2b g b g

(2) g is not differentiable at x = 0

(3) g' cos log0 2b g b g

(4) g' cos log0 2b g b g

Sol. [3] g x xb g ln sin ln sin2 2 in nbd of 0

g x xb g b g ln sin ln sin2 2

g x x x' cos ln sin cosb g b gb g 2

g' cos ln .0 2b g b g

8. The distance of the point 1 5 9, ,b g from the plane

x y z 5 measured along the line x y z is

(1)20

3(2) 3 10

(3) 10 3 (4)10

3

Sol. [3] Line through P 1 5 9, ,b g parallel to given line

x y zr

1

1

5

1

9

1...(1)

Point of int. of (1) with plane be

Q r r r 1 5 9, ,b glies on the plane

r r r 1 5 9 5b g b g b g r = 10

Q 9 15 1, ,b g PQ 10 3.

9. The eccentricity of the hyperbola whose length of thelatus rectum is equal to 8 and the length of its conjugateaxis is equal to half of the distance between its foci, is

(1) 3 (2)4

3

(3)4

3(4)

2

3

Sol. [4] 2b c

4 2 2 2 2b c a b b

a

2

2

1

3

eb

a 1

2

3

2

2.

[3] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

10. Let P be the point on the parabola, y x2 8 which is at

a minimum distance from the centre C of the circle,

x y2 26 1 b g . Then the equation of the circle,

passing through C and having its centre at P is

(1) x y x y2 2 4 9 18 0

(2) x y x y2 2 4 8 12 0

(3) x y x y2 2 4 12 0

(4) x yx

y2 2

42 24 0

Sol. [2] Let normal through P be

y mx m m 4 2 3 ...(1)

it passes (0, – 6)

6 0 4 2 3m m m 1

Q m m 2 4 2 42 , ,e j b g Circle x y 2 4 8

2 2b g b g x y x y2 2 4 8 12 0 .

11. If Aa b

L

NMMOQPP

5

3 2 and A adjA A AT , then 5a + b is

equal to(1) 13 (2) –1(3) 5 (4) 4

Sol. [3] A adj A AAT A I AAT

10 31 0

0 1

5

3 2

5 3

2a b

a b a

bLNMMOQPP

LNMMOQPP LNMMOQPPb g

15 2 0a b 10 3 13a b

a b 2

53,

5 5a b .

12. Consider

f xx

xxb g

FHG

IKJ FHGIKJ

tansin

sin, ,1 1

10

2

.

A normal to y = f(x) at x 6

also passes through the

point

(1)4

0,FHGIKJ (2) (0, 0)

(3) 02

3,F

HGIKJ (4)

6

0,FHGIKJ

Sol. [3] y f xx x

FHGIKJ

FHG

IKJ b g tan tan1

4 2 4 2

when x y 6 3

, anddy

dx

1

2

Normal y x FHGIKJ

3

26

22

3x y

.

13. Two sides of a rhombus are along the lines,

x y 1 0 and 7 5 0x y . If its diagonals

intersect at (–1, –2), then which one of the following isa vertex of this rhombus?

(1) FHG

IKJ

10

3

7

3, (2) 3 9,b g

(3) 3 8,b g (4)1

3

8

3, FHGIKJ

Sol. [4] Let slope of angle bisectors of sides be m

2

1

7 1

1 72

m

m m 2

1

2,

Diagonals y x y x 2 2 1 21

21b g b g b g b g,

2 0 2 5 0x y x y ,

A B D1 27

3

4

3

1

3

8

3, , , , , .b g FHG

IKJ FHG

IKJ

[4] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

14. If a curve y f x b g passes through the point 1 1,b gand satisfies the differential equation,

y xy dx xdy1 b g , then f FHGIKJ

1

2 is equal to

(1)4

5(2)

2

5

(3) 4

5(4)

2

5

Sol. [1] ydx xdy xy dx 2 0

ydx xdy

yxdx

20 d

x

yd

xFHGIKJ FHGIKJ

2

20

x

y

xC

2

2...(1)

it pases (1, –1) 11

2C

C 1

2

x

y

x

2

2

1

2 x

y

x

2 1

2

e j

f x yx

xb g

2

12

f FHGIKJ

1

2

5

4.

15. If all the words (with or without meaning) having fiveletters, formed using the letters of the word SMALLand arranged as in a dictionary; then the position ofthe word SMALL is(1) 58th (2) 46th

(3) 59th (4) 52nd

Sol. [1] Starting with A 4

212

!

!

Starting with L 4 24!

Starting with M 2

212

!

!

Starting SA 3

23

!

!

Starting with SL 3 6!Next : SMALLRank = 58.

16. If the 2nd, 5th and 9th terms of a non-constant A.P. are inG.P., then the commonr atio of this G.P. is

(1)7

4(2)

8

5

(3)4

3(4) 1

Sol. [3] a d a d a d , ,4 8 in G.P.

a d a d a d 4 82b g b gb g

8 2d ad da

8

Now ra d

a d

aa

aa

4 2

8

4

3.

17. If the number of terms in the expansion of

12 4

02

FHGIKJ

x xx

n

, , is 28, then the sum of the

coefficients fo all the terms in this expansion, is(1) 729 (2) 64(3) 2187 (4) 243

Sol. [1] Number of term = n rrC

11

n C22

Now n C 22 28 n = 6

sum of coefficient FHGIKJ 1

2

1

4

13

66 .

18. If the sum of the first ten terms of the series

13

52

2

53

1

54 4

4

5

2 2 22

2FHGIKJ FHGIKJ FHGIKJ FHG

IKJ ..., is

16

5m ,

then m is equal to(1) 99 (2) 102(3) 101 (4) 100

Sol. [3] 13

52

2

53

1

54 4

4

5

2 2 22

2FHGIKJ FHGIKJ FHGIKJ FHG

IKJ ...

8 12 16 20 10

5

16

5

2 2 2 2

2

... termsm

e j

[5] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

2 3 4 5 52 2 2 2 ... m

11 12 23

61 5

m m = 101.

19. If the line,x y z

3

2

2

1

4

3 lies in the plane,

x my z 9 , then 2 2m is equal to

(1) 2 (2) 26(3) 18 (4) 5

Sol. [1] 2 1 1 3 0 mb g b g 2 3 m

and 3 2 2 4 9b g b gb g m

3 2 5 m 1 1, .m

20. The Boolean Expression p q q p q ~ ~b g b g is

equivalent to

(1) p q ~ (2) ~ p q

(3) p q (4) p q

Sol. [4]p q p q p q p q p q q p q p q

T T F F F F T T

T F F T T F T T

F T T F F T T T

F F T T F F F F

~ ~ ~ ~ ~ ~ b g b g b g b g b g

21. The integral2 5

1

12 9

5 3 3

x x

x xdx

z e j

is equal to

(1)

x

x xC

10

5 3 22 1e j

(2)

x

x xC

5

5 3 21e j

(3)x

x xC

10

5 32 1

e j(4)

x

x xC

5

5 3 22 1

e j

where C is an arbitrary constant.

Sol. [3]2 5

1

12 9

5 3 3

x x

x xdx

z e j

z 2 5

1

3 6

2 5 3

x x

x xdx

e j

z dt

t3 let 1 2 5 x x t

t

Cx

x xC

2 10

5 32 2 1e j.

22. If one of the diameters of the circle, given by the

equation, x y x y2 2 4 6 12 0 , is a chord of a

circle S, whose centre is at (–3, 2), then the radius of Sis

(1) 10 (2) 5 2

(3) 5 3 (4) 5

Sol. [3]

r n 25 50 r 5 3.

23. lim...

/

n n

nn n n

n

FHG

IKJ

1 2 32

1b gb g is equal to

(1) 3 3 2log (2)18

4e

(3)27

2e(4)

92e

Sol. [3] Ln n n n

nn n

LNM

OQP

lim...

/1 2 2

2

1 2b gb g b g

ln lim ln ln ... lnLn

n

n

n

n

n n

nn

FHGIKJ

FHGIKJ

FHGIKJ

LNM

OQP

1 1 2 2

z ln 10

2

x dxb g

ln lnL 3 3 2 Le

272

.

[6] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

24. The centre of those circles which touch the circle,

x y x y2 2 8 8 4 0 , externally and also touch

the x-axis, lie on(1) a parabola(2) a circle(3) an ellipse which is not a circle(4) a hyperbola

Sol. [1]

h k k 4 4 42 2b g b g .

25. Let a b, and c be the three unit vectors such that

a b c b c e j e j3

2. Ifb is not parallel to c , then

the angle between a andb is

(1)5

6

(2)

3

4

(3)2

(4)2

3

Sol. [1] a b c a c b a b c e j b g e j

3

2

b ce j

a b

3

2 cos

3

2.

26. Let p xx

x lim tan

/

0

2 1 21e j then log p is equal to

(1)1

4(2) 2

(3) 1 (4)1

2

Sol. [4] P xx

x

lim tan/

0

2 1 21e j

P e x

x

x lim

tan0

2

2

P e 1 2/ .

27. If 0 2 x , then the number of real values of x, whichsatisfy the equation

cos cos cos cosx x x x 2 3 4 0 is

(1) 9 (2) 3(3) 5 (4) 7

Sol. [4] cos cos cos cosx x x x 2 3 4 0 2 2 2 3 0cos cos cos cosx x x x

25

2 20cos cos cosx

x x

x

2

3

2 5

3

5

7

5

9

5, , , , , , .

28. The sum of all real values of x satisfying the equation

x xx x2 4 60

5 5 12

e j is

(1) 5 (2) 3(3) –4 (4) 6

Sol. [2] x xx x2 4 60

5 5 12

e j

x x2 4 60 0 x x 10 6 0b gb g x 10 6,

x x2 5 5 1 x x 4 1 0b gb g x 1 4,

x x2 5 5 1 x x2 5 6 0

x x 2 3 0b gb g

x

rejected

B

2 3,

( )

Solution set = 10 6 1 4 2, , , ,l q .

[7] JEE Main Test 2016

"Guru-Kripa" Anasagar Circular Road, Ajmer Ph.No.(0145) 2631190, 2629236

RG ACADEMY

RG ACADEMY

30. If f x fx

x xb g FHGIKJ 2

13 0, , and

S x f x f x R : b g b gm r ; then S

(1) contains more than two elements(2) is an empty set(3) contains exactly one element(4) contains exatly two elements

Sol. [4] f x fx

xb g FHGIKJ 2

13 ...(1)

fx

f xx

12

3FHGIKJ b g

f xx

xb g 2

Now f x f xb g b g 2 2

xx

xx

x2 1

2 x

1

2.

29. The area (in sq. units) of the region

x y y x and x y x x y, : , ,b go t2 2 22 4 0 0 is

(1)2

2 3

3 (2)

4

3

(3) 8

3(4)

4 2

3

Sol. [3]

Req. Area : 2 20

2

z x dx

FHGIKJ

2 2 2

32

8

3.