Sample Solution Paper 6 - Math, Class 12 JEE Notes | EduRev

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JEE : Sample Solution Paper 6 - Math, Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 6 Solution 
 
SECTION – A  
1. A matrix is of order m × n, then it has mn elements. 
So, mn = 12 
Which means we have to find values of m and n, such that it will satisfy the 
above condition 
So all the possible cases are 
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1). 
 
2.  
Let f(x) = sin(2x
2
) 
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
 
 
3.  
Sol: 
Order: 3 
Degree: 1  
 
4.  
The Cartesian form of a line is given by 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2 
We get the equation as 
x 2 y 1 z 4
1 1 2
? ? ?
??
?
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 6 Solution 
 
SECTION – A  
1. A matrix is of order m × n, then it has mn elements. 
So, mn = 12 
Which means we have to find values of m and n, such that it will satisfy the 
above condition 
So all the possible cases are 
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1). 
 
2.  
Let f(x) = sin(2x
2
) 
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
 
 
3.  
Sol: 
Order: 3 
Degree: 1  
 
4.  
The Cartesian form of a line is given by 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2 
We get the equation as 
x 2 y 1 z 4
1 1 2
? ? ?
??
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
OR 
The equation can be written as 
1
y
x 2 z 5
2
3 1 1
?
??
??
?
 
D.R.S. of this line is proportional to 3, 1,-1. 
Also the line passes through (1,-1, 0) 
x 1 y 1 z 0
3 1 1
? ? ?
??
?
 
 
 
SECTION – B 
 
5.   
(i) For all 
ab
a,b N,a * b
2
 
Now, 
b a a b
b * a a * b
22
 
Thus, the binary operation * is commutative. 
 
(ii) Let a,b,c  N 
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
 
Thus, the binary operation * is not associative. 
 
 
 
 
 
 
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 6 Solution 
 
SECTION – A  
1. A matrix is of order m × n, then it has mn elements. 
So, mn = 12 
Which means we have to find values of m and n, such that it will satisfy the 
above condition 
So all the possible cases are 
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1). 
 
2.  
Let f(x) = sin(2x
2
) 
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
 
 
3.  
Sol: 
Order: 3 
Degree: 1  
 
4.  
The Cartesian form of a line is given by 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2 
We get the equation as 
x 2 y 1 z 4
1 1 2
? ? ?
??
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
OR 
The equation can be written as 
1
y
x 2 z 5
2
3 1 1
?
??
??
?
 
D.R.S. of this line is proportional to 3, 1,-1. 
Also the line passes through (1,-1, 0) 
x 1 y 1 z 0
3 1 1
? ? ?
??
?
 
 
 
SECTION – B 
 
5.   
(i) For all 
ab
a,b N,a * b
2
 
Now, 
b a a b
b * a a * b
22
 
Thus, the binary operation * is commutative. 
 
(ii) Let a,b,c  N 
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
 
Thus, the binary operation * is not associative. 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
6.  
The corresponding values of two equal matrices are equal 
a + b = 6 and ab = 8 
Therefore, 
8
b
a
? 
Substituting in first condition we get, 
? ? ? ?
2
8
a6
a
a 6a 8 0
a 4 a 2 0
a 4 or a 2 and
b 2 or b 4 respectively.
??
? ? ?
? ? ?
??
??
 
 
 
7.  
x
sin4x 4
Let I e dx
1 cos4x
?
? ??
?
??
?
??
 
x
x2
2
sin2(2x) 4
e dx
1 cos2(2x)
2sin2xcos2x 4
e dx [Using,sin2x 2sin x.cosxand2sin x 1 cos(2x)
2sin (2x)
?
?
?? ?
?
??
?
??
??
?
? ? ? ?
??
??
??
 
? ?
xx
2 2 2
x2
2(sin(2x)cos(2x) 4 sin(2x)cos(2x) 2
e dx e dx
2sin 2x sin 2x sin 2x
e cot(2x) 2cosec 2x dx
??
?
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
 
Now, let f(x) = cot(2x) then f’(x) = -2cosec
2
2x 
I ? ?
x
e f(x) f '(x) dx
?
?? 
So, I = e
x
f(x) + C = e
x 
cot 2x + C , where C is a constant 
Therefore, 
? ?
xx
sin4x 4
e dx e cot 2x + C
1 cos4x
?
? ??
?
??
?
??
 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 6 Solution 
 
SECTION – A  
1. A matrix is of order m × n, then it has mn elements. 
So, mn = 12 
Which means we have to find values of m and n, such that it will satisfy the 
above condition 
So all the possible cases are 
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1). 
 
2.  
Let f(x) = sin(2x
2
) 
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
 
 
3.  
Sol: 
Order: 3 
Degree: 1  
 
4.  
The Cartesian form of a line is given by 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2 
We get the equation as 
x 2 y 1 z 4
1 1 2
? ? ?
??
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
OR 
The equation can be written as 
1
y
x 2 z 5
2
3 1 1
?
??
??
?
 
D.R.S. of this line is proportional to 3, 1,-1. 
Also the line passes through (1,-1, 0) 
x 1 y 1 z 0
3 1 1
? ? ?
??
?
 
 
 
SECTION – B 
 
5.   
(i) For all 
ab
a,b N,a * b
2
 
Now, 
b a a b
b * a a * b
22
 
Thus, the binary operation * is commutative. 
 
(ii) Let a,b,c  N 
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
 
Thus, the binary operation * is not associative. 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
6.  
The corresponding values of two equal matrices are equal 
a + b = 6 and ab = 8 
Therefore, 
8
b
a
? 
Substituting in first condition we get, 
? ? ? ?
2
8
a6
a
a 6a 8 0
a 4 a 2 0
a 4 or a 2 and
b 2 or b 4 respectively.
??
? ? ?
? ? ?
??
??
 
 
 
7.  
x
sin4x 4
Let I e dx
1 cos4x
?
? ??
?
??
?
??
 
x
x2
2
sin2(2x) 4
e dx
1 cos2(2x)
2sin2xcos2x 4
e dx [Using,sin2x 2sin x.cosxand2sin x 1 cos(2x)
2sin (2x)
?
?
?? ?
?
??
?
??
??
?
? ? ? ?
??
??
??
 
? ?
xx
2 2 2
x2
2(sin(2x)cos(2x) 4 sin(2x)cos(2x) 2
e dx e dx
2sin 2x sin 2x sin 2x
e cot(2x) 2cosec 2x dx
??
?
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
 
Now, let f(x) = cot(2x) then f’(x) = -2cosec
2
2x 
I ? ?
x
e f(x) f '(x) dx
?
?? 
So, I = e
x
f(x) + C = e
x 
cot 2x + C , where C is a constant 
Therefore, 
? ?
xx
sin4x 4
e dx e cot 2x + C
1 cos4x
?
? ??
?
??
?
??
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
OR 
? ?
2
1x
dx
x 1 2x
?
?
?
 
Here 
? ?
2
1x
x 1 2x
?
?
 is an improper rational fraction. 
Reducing it to proper rational fraction gives 
? ? ? ?
2
1 x 1 1 2 x
x 1 2x 2 2 x 1 2x
??
??
??
??
??
??
??
…………(1) 
Now, let 
? ?
2 x A B
x 1 2x x (1 2x)
?
??
??
 
? ? ? ?
? ?
2 x A(1 2x) Bx
2 x A x(2A B)
x 1 2x x 1 2x
Equating the coefficientsweget,A 2 and B 3
2 x 2 3
So,
x 1 2x x (1 2x)
? ? ?
? ? ? ? ? ? ?
??
??
?
??
??
 
Substituting in equation (1), we get 
? ?
2
1 x 1 1 2 3
x 1 2x 2 2 x (1 2x)
?? ?
? ? ?
??
??
??
 
? ?
2
1 x 1 1 2 3
i.e dx dx
x 1 2x 2 2 x (1 2x)
??
?? ?? ?
? ? ?
?? ??
??
?? ??
 
dx dx 3 dx x 3 1
log x log 1 2x C
2 x 2 (1 2x) 2 2 ( 2)
x3
log x log 1 2x C
24
? ? ?
? ? ? ? ? ? ? ? ?
??
? ? ? ? ?
 
 
 
 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 6 Solution 
 
SECTION – A  
1. A matrix is of order m × n, then it has mn elements. 
So, mn = 12 
Which means we have to find values of m and n, such that it will satisfy the 
above condition 
So all the possible cases are 
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1). 
 
2.  
Let f(x) = sin(2x
2
) 
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
 
 
3.  
Sol: 
Order: 3 
Degree: 1  
 
4.  
The Cartesian form of a line is given by 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2 
We get the equation as 
x 2 y 1 z 4
1 1 2
? ? ?
??
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
OR 
The equation can be written as 
1
y
x 2 z 5
2
3 1 1
?
??
??
?
 
D.R.S. of this line is proportional to 3, 1,-1. 
Also the line passes through (1,-1, 0) 
x 1 y 1 z 0
3 1 1
? ? ?
??
?
 
 
 
SECTION – B 
 
5.   
(i) For all 
ab
a,b N,a * b
2
 
Now, 
b a a b
b * a a * b
22
 
Thus, the binary operation * is commutative. 
 
(ii) Let a,b,c  N 
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
 
Thus, the binary operation * is not associative. 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
6.  
The corresponding values of two equal matrices are equal 
a + b = 6 and ab = 8 
Therefore, 
8
b
a
? 
Substituting in first condition we get, 
? ? ? ?
2
8
a6
a
a 6a 8 0
a 4 a 2 0
a 4 or a 2 and
b 2 or b 4 respectively.
??
? ? ?
? ? ?
??
??
 
 
 
7.  
x
sin4x 4
Let I e dx
1 cos4x
?
? ??
?
??
?
??
 
x
x2
2
sin2(2x) 4
e dx
1 cos2(2x)
2sin2xcos2x 4
e dx [Using,sin2x 2sin x.cosxand2sin x 1 cos(2x)
2sin (2x)
?
?
?? ?
?
??
?
??
??
?
? ? ? ?
??
??
??
 
? ?
xx
2 2 2
x2
2(sin(2x)cos(2x) 4 sin(2x)cos(2x) 2
e dx e dx
2sin 2x sin 2x sin 2x
e cot(2x) 2cosec 2x dx
??
?
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
 
Now, let f(x) = cot(2x) then f’(x) = -2cosec
2
2x 
I ? ?
x
e f(x) f '(x) dx
?
?? 
So, I = e
x
f(x) + C = e
x 
cot 2x + C , where C is a constant 
Therefore, 
? ?
xx
sin4x 4
e dx e cot 2x + C
1 cos4x
?
? ??
?
??
?
??
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
OR 
? ?
2
1x
dx
x 1 2x
?
?
?
 
Here 
? ?
2
1x
x 1 2x
?
?
 is an improper rational fraction. 
Reducing it to proper rational fraction gives 
? ? ? ?
2
1 x 1 1 2 x
x 1 2x 2 2 x 1 2x
??
??
??
??
??
??
??
…………(1) 
Now, let 
? ?
2 x A B
x 1 2x x (1 2x)
?
??
??
 
? ? ? ?
? ?
2 x A(1 2x) Bx
2 x A x(2A B)
x 1 2x x 1 2x
Equating the coefficientsweget,A 2 and B 3
2 x 2 3
So,
x 1 2x x (1 2x)
? ? ?
? ? ? ? ? ? ?
??
??
?
??
??
 
Substituting in equation (1), we get 
? ?
2
1 x 1 1 2 3
x 1 2x 2 2 x (1 2x)
?? ?
? ? ?
??
??
??
 
? ?
2
1 x 1 1 2 3
i.e dx dx
x 1 2x 2 2 x (1 2x)
??
?? ?? ?
? ? ?
?? ??
??
?? ??
 
dx dx 3 dx x 3 1
log x log 1 2x C
2 x 2 (1 2x) 2 2 ( 2)
x3
log x log 1 2x C
24
? ? ?
? ? ? ? ? ? ? ? ?
??
? ? ? ? ?
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 Solution  
 
     
8.  
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
22
2
2x
I dx
x 1 x 3
Let x z
2xdx dz
dz
I
z 1 z 3
By partialfraction,
1 A B
z 1 z 3 z 1 z 3
1 A z 3 B z 1
?
?
?
??
?
??
??
??
??
? ? ? ?
? ? ? ? ?
 
? ? ? ?
? ? ? ?
? ? ? ?
22
22
2
2
Putting z 3, we obtain :
1 2B
1
B
2
1
A
2
1
1
1 2
2
z 1 z 3 z 1 z 3
dz 1 dz dz
z 1 z 3 2 z 1 z 3
11
log z 1 log z 3 C
22
2xdx 1 1
log x 1 log x 3 C
22
x 1 x 3
1 x 1
log C
2
x3
? ? ?
?
??
??
??
??
??
?
??
??
? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
??
?
??
?
 
 
 
 
 
 
 
 
 
 
 
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