[1] jee main test 2016 mathematics y[1] jee main test 2016 "guru-kripa" anasagar circular...
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[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.