Sample Solution Paper 10 - Math, Class 12 JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Sample Solution Paper 10 - Math, Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 10 Solution 
  
SECTION – A 
 
1. By observation we find that 
2 + x = 10 
x = 8. 
 
2. 
? ?
d
cos x
dx
 
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
 
 
3. DE: 
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ? 
It is linear, since y siny ? is product of two different functions, and their individual 
power is one. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 10 Solution 
  
SECTION – A 
 
1. By observation we find that 
2 + x = 10 
x = 8. 
 
2. 
? ?
d
cos x
dx
 
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
 
 
3. DE: 
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ? 
It is linear, since y siny ? is product of two different functions, and their individual 
power is one. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
4. Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
 
 
OR 
 
 
 
 
Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 10 Solution 
  
SECTION – A 
 
1. By observation we find that 
2 + x = 10 
x = 8. 
 
2. 
? ?
d
cos x
dx
 
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
 
 
3. DE: 
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ? 
It is linear, since y siny ? is product of two different functions, and their individual 
power is one. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
4. Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
 
 
OR 
 
 
 
 
Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
SECTION – B  
 
5. Let X be the non-empty set for which P(X) is the power set. 
ARB ? A ? B 
i.  ARA ? A ? A, every set is a subset of itself.  R is reflexive         
ii.  If A, B, C ? P(X)  
       ARB ? A ? B, BRC ? B ?  C 
       A ? B and B ? C ?A ? C 
       So ARC; Hence R is transitive. 
iii. ARB ? A ? B does not imply B ? A  
So B R A  
R is not symmetric 
R is reflexive, transitive but not symmetric ?R is not an equivalence relation   
 
6. We have, 
 2A – 3B + 5C = O 
2A = 3B – 5C 
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
 
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
 
8 3 5
A
13 1 9
???
?
??
? ? ?
??
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 10 Solution 
  
SECTION – A 
 
1. By observation we find that 
2 + x = 10 
x = 8. 
 
2. 
? ?
d
cos x
dx
 
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
 
 
3. DE: 
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ? 
It is linear, since y siny ? is product of two different functions, and their individual 
power is one. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
4. Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
 
 
OR 
 
 
 
 
Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
SECTION – B  
 
5. Let X be the non-empty set for which P(X) is the power set. 
ARB ? A ? B 
i.  ARA ? A ? A, every set is a subset of itself.  R is reflexive         
ii.  If A, B, C ? P(X)  
       ARB ? A ? B, BRC ? B ?  C 
       A ? B and B ? C ?A ? C 
       So ARC; Hence R is transitive. 
iii. ARB ? A ? B does not imply B ? A  
So B R A  
R is not symmetric 
R is reflexive, transitive but not symmetric ?R is not an equivalence relation   
 
6. We have, 
 2A – 3B + 5C = O 
2A = 3B – 5C 
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
 
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
 
8 3 5
A
13 1 9
???
?
??
? ? ?
??
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
7.  
?
?
22
sin x cos x
dx
sin xcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log |sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
8.  
?
??
2
dx
5 8x x
 
?
?
?
?
?
? ? ?
?
? ? ? ? ?
?
? ? ?
?
??
??
??
??
??
??
??
2
2
2
22
dx
(x 8x 5)
dx
(x 8x 16 16 5)
dx
[(x 4) 21]
dx
( 21) (x 4)
1 21 (x 4)
log C
2 21 (x 4) 21
1 x 4 21
log C
2 21 x 4 21
 
 
 
 
 
 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 10 Solution 
  
SECTION – A 
 
1. By observation we find that 
2 + x = 10 
x = 8. 
 
2. 
? ?
d
cos x
dx
 
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
 
 
3. DE: 
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ? 
It is linear, since y siny ? is product of two different functions, and their individual 
power is one. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
4. Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
 
 
OR 
 
 
 
 
Let ? be the angles between, the given two lines 
So, the angle between them given their direction cosines is given by 
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
SECTION – B  
 
5. Let X be the non-empty set for which P(X) is the power set. 
ARB ? A ? B 
i.  ARA ? A ? A, every set is a subset of itself.  R is reflexive         
ii.  If A, B, C ? P(X)  
       ARB ? A ? B, BRC ? B ?  C 
       A ? B and B ? C ?A ? C 
       So ARC; Hence R is transitive. 
iii. ARB ? A ? B does not imply B ? A  
So B R A  
R is not symmetric 
R is reflexive, transitive but not symmetric ?R is not an equivalence relation   
 
6. We have, 
 2A – 3B + 5C = O 
2A = 3B – 5C 
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
 
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
 
8 3 5
A
13 1 9
???
?
??
? ? ?
??
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
7.  
?
?
22
sin x cos x
dx
sin xcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log |sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
8.  
?
??
2
dx
5 8x x
 
?
?
?
?
?
? ? ?
?
? ? ? ? ?
?
? ? ?
?
??
??
??
??
??
??
??
2
2
2
22
dx
(x 8x 5)
dx
(x 8x 16 16 5)
dx
[(x 4) 21]
dx
( 21) (x 4)
1 21 (x 4)
log C
2 21 (x 4) 21
1 x 4 21
log C
2 21 x 4 21
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 10 Solution  
 
     
OR 
 
Let I = 
2
2
4
2
2
1
1
x1
x
dx dx
1
x1
x
x
??
?
?
?
?
?
 
(Dividing numerator and denominator by x
2
)                   
I
2
2
1
1
x
dx
1
x2
x
?
?
?
??
??
??
??
                                                    
Substituting x -
1
x
 = t, 
2
1
1 dx dt
x
??
? ? ?
??
??
 we get,      
? ?
22
2
1
dt dt
I
t2
t2
1t
tan
22
?
??
??
?
?
?
          
    
1
2
1
1
x
1
x
tan
22
1 x 1
tan c
2 2x
?
?
??
?
??
?
??
??
??
??
?
??
??
??
??
       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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