JEE ADVANCE MATHEMATICS SAMPLE PAPER – VI MATHEMATICS

JEE ADVANCE MATHEMATICS SAMPLE PAPER – VI
MATHEMATICS
SECTION – I
Straight Objective Type
This section contains 9 multiple choice questions numbered 1 to 9. Each question has
4 choices (a), (b), (c) and (d), out of which ONLY ONE choice is correct.
1.
The minimum value of
(a) 2 2
2.
The value of
4.
3
(c) 3 3
(d)
3
{sin{x}} (where {x} denotes fractional part of x) is
3
(b) 6 cos 1
(c) 6 (1 – cos 1)
(d) none of these
If A( z1 ) and B( z2 ) are two points on circle |z| = r then the tangents to the circle at A and B
will intersect at
z1 z 2
2 z1 z 2
z12 z22
z12 z22
(a)
(b)
(c)
(d)
z1 z 2
z1 z 2
z1 z2
2( z1 z2 )
The value of
(a)
5.
y4 z2
if x, y, z are positive numbers is
xyz
(b) 3 2
(a) 0
3.
x4
1
27
2
27
2
log5 16
2 log5 9
(b)
equals to
2 2
27
(c)
4
27
(d)
4 2
27
If f (x) = cos (4 {x}) + sin (4 {x}) where {x} denotes fractional part of x then which of the
following is true?
(a) f (x) is non periodic
(b) period of f (x) is 1/2
Space for rough work
6.
(c) period of f (x) is 1
(d) period of f (x) is 2
Vertices of a triangle are (1, 3 ) , (2 cos , 2 sin ) and (2 sin , – 2 cos ). Then locus of
orthocentre of the triangle is
(a) ( x 1)2 ( y
(b) ( x 2)2 ( y
3) 2 8
3) 2 4
(c) ( x 1)2 ( y
7.
3)2
(d) none of these
4
x
x3
3!
1
x2
2!
Solution of the differential equation
(a) 2 ye2 x
(c) ye2 x
c · e2 x 1
x5
.....
dx dy
5!
is
4
dx dy
x
.....
4!
(b) 2 ye2 x
c · e2 x 2
2c · e2 x 1
(d) none of these
 

  
w and w u v then value of [u v w] is
(d) none of these
8.


  
If u and v are two unit vectors such that u v u
(a) 1
(b) – 1
(c) 0
9.
The value of (1 tan1º )(1 tan 2º )(1 tan 3º )..........(1 tan 44º )(1 tan 45º ) is
(a) 221
(b) 224
(c) 223
(d) 222
SECTION II
Reasoning Type
This section contains 4 reasoning type questions numbered 10 to 13. Each question contains
Statement-1 and Statement-2. Each question has 4 choices (a), (b), (c) and (d), out of which
ONLY ONE is correct.
Directions: Read the following questions and choose
(A) If both the statements are true and statement-2 is the correct explanation of
statement-1.
(B) If both the statements are true but statement-2 is not the correct explanation of
statement-1.
Space for rough work
(C) If statement-1 is True and statement-2 is False.
(D) If statement-1 is False and statement-2 is True.
10.
Statement-1 :
Statement-2 :
(a) A
The differential equation of family of hyperbola whose assymptote lines
dy
are x + y – 1 = 0 and x – y – 1 = 0 is x – 1 = y .
dx
Eccentricity of rectangular hyperbola is 2 .
(b) B
(c) C
(d) D
(a) A
y = [[x]] – [x – 1] is always continuous (where [.] denotes greatest integer
function).
y = [x] is discontinuous at all integral values of x (where [.] denotes
greatest integer function).
(b) B
(c) C
(d) D
12.
Statement-1 :
Statement-2 :
(a) A
If y = f (x) is an even differentiable function, then f ' (0) 0 .
If f (x) is even then f ' ( x) is odd.
(b) B
(c) C
(d) D
13.
Statmenet-1 :
The direction cosines of a pair of lines satisfy (l + m) = n and
mn + nl + lm = 0, where l, m, n are direction cosines. The two lines are
perpendicular if = 2
Lines with direction cosines l1 , m1 , n1 and l 2 , m 2 , n 2 are perpendicular if
l1 l2 m1 m2 n1 n2 0 .
11.
Statement-1 :
Statement-2 :
Statmenet-2 :
(a) A
Space for rough work
(b) B
(c) C
(d) D
SECTION III
Linked Comprehension Type
This section contains 2 paragraphs M14 M19. Based upon these paragraphs, 6 multiple choice
questions have to be answered. Each question has 4 choices (a), (b), (c) and (d), out of which
ONLY ONE choice is correct.
Passage – I
To find point of contact Q( x1, y1 ) of a tangent to the graph of y = f (x) passing through origin
f ( x1 )
‘O’, we solve the equation f ' ( x1 )
x1
14.
The equation log mx
(a) 0 < p <
15.
The equation log mx
(a) p
16.
m
e
m
e
px where m is a positive constant has a single root if
e
e
m
(b) p
(c) 0 p
(d) p
m
m
e
px has exactly two roots (m is a positive constant) if
e
e
(b) p
(c) 0 p
(d) 0 p
m
m
m
e
Find the condition that the equation has exactly three roots (m is a positive constant) if
m
m
e
e
(a) p
(b) 0 p
(c) 0 p
(d) p
e
e
m
m
Passage – II
x R 0 and y = g (x) differentiable
x R 0 are two
If y = f (x) differentiable
curves such that they pass through (1, 1) and (2, 3) respectively. Also tangents to the two
curves where their abscissa are same intersect on y-axis and normals to the curves at the point
where their abscissa are equal intersect on x-axis.
17.
The curve f (x) is
Space for rough work
2
x
x
The curve g (x) is
1
(a) x
x
(a)
18.
19.
1
x
(b) 2 x 2
(b) x
(c)
2
x
2
x2
(c) x 2
(d) none of these
x
1
(d) none of these
x2
The number of positive integral solutions of f (x) = g (x) are
(a) 4
(b) 5
(c) 6
(d) none of these
SECTION IV
Matrix Match Type
This section contains 3 questions. Each question contains statements given in two columns
which have to be matched. Statements (A), (B), (C), (D) in Column I have to be matched
with statements (1, 2, 3, 4) in Column II. One statement in first column has one or more than
one match with the statements in second column. The answers to these questions have to be
appropriately bubbled as illustrated in the following example.
If the correct matches are A 1,3, B 3, C 2,3 and D 2,4, then the correctly bubbled
4 × 4 matrix should be as follows:
1 2 3 4
A
B
C
D
1.
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Match the functions in column–I with corresponding period in column–II
Column–I
Column–II
(A) sin 2 x + cos x
1. 2
(B) sin 1(cos x)
2.
Space for rough work
2
(C)
sin x
cos x
3.
(D)
tan x
cot x
4. 2
2.
Match the items of column–I with that of column–II
Column–I
Column–II
(A) If x, y
[0, 2 ] then total number of ordered pair
1. 4
(x, y) satisfying sin x cos y = 1 is
(B) If f (x) = sin x – cos x – kx + b decreases for all real values of
x then 2 2 k may be
2. 6
(C) The number of solution of the equation
sin 1 (| x 2 1 |) cos 1 (| 2 x 2 5 |)
2
3. 2
is
(D) The number of ordered pair (x, y) satisfying the equation sin x
4. 3
+ sin y = sin (x + y) and x
y 1 is
3.
Match the items of column–I with that of column–II
Column–I
Column–II
(A) If A, B, C are angles of acute angled triangle, then minimum value
1. 16
of tan 4 A tan 4 B tan 4 C is
(B) If any ABC the minimum value of cot
A
B
C
cot cot
2
2
2
is
2. 3 3
(C) If A, B, C are angles of a triangle such that tan A tan C = 2 and 3. 4
Space for rough work
tan B tan C = 18, then value of tan 2 C is
(D) The maximum value of 1 sin
is
Space for rough work
4
2 cos
4
, where
R
4. 27