Sample Solution Paper 9 - Math, Class 12 JEE Notes | EduRev

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JEE : Sample Solution Paper 9 - Math, Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 9 Solution 
  
SECTION – A 
 
1. a42, means element at 3
rd
 row and 2
nd
 column 
So, 
a32 = 10 
 
2. Differentiating w.r.t. x, we get, 
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
 
 
3. DE: 
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
  
It is linear, since x is independent variable. 
 
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 9 Solution 
  
SECTION – A 
 
1. a42, means element at 3
rd
 row and 2
nd
 column 
So, 
a32 = 10 
 
2. Differentiating w.r.t. x, we get, 
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
 
 
3. DE: 
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
  
It is linear, since x is independent variable. 
 
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
 
 
SECTION – B  
 
5. f: R+ ? [4, 8)  defined by f (x) = x² + 4 
 f is 1 – 1  
 Let x1, x2, ? R+ such that f(x1) = f(x2) 
 
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
 
 ? f is 1 – 1       
 f is onto: Let y ? [4, 8) 
 f(x) = y ? x
2
 + 4 = y 
   ? x = y4 ? 
 Since y ? [4, 8) ? x ? R+ 
 For y ? [4, 8) there is a x ? R+ such that f (x) = y. 
 So f is onto.    
 So, f is bijective function and hence f is invertible.   
 The inverse of f is defined by  
 f: [4, 8) ? R+ 
f
-1
(y) = y4 ? 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 9 Solution 
  
SECTION – A 
 
1. a42, means element at 3
rd
 row and 2
nd
 column 
So, 
a32 = 10 
 
2. Differentiating w.r.t. x, we get, 
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
 
 
3. DE: 
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
  
It is linear, since x is independent variable. 
 
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
 
 
SECTION – B  
 
5. f: R+ ? [4, 8)  defined by f (x) = x² + 4 
 f is 1 – 1  
 Let x1, x2, ? R+ such that f(x1) = f(x2) 
 
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
 
 ? f is 1 – 1       
 f is onto: Let y ? [4, 8) 
 f(x) = y ? x
2
 + 4 = y 
   ? x = y4 ? 
 Since y ? [4, 8) ? x ? R+ 
 For y ? [4, 8) there is a x ? R+ such that f (x) = y. 
 So f is onto.    
 So, f is bijective function and hence f is invertible.   
 The inverse of f is defined by  
 f: [4, 8) ? R+ 
f
-1
(y) = y4 ? 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
6.  
We have, 
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2
 
 
7. 
x
x 2x
e
I dx
5 4e e
?
?
??
 
??
xx
Let e t e dx dt
 
Now integral I becomes,  
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?
 
2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??
 
1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 9 Solution 
  
SECTION – A 
 
1. a42, means element at 3
rd
 row and 2
nd
 column 
So, 
a32 = 10 
 
2. Differentiating w.r.t. x, we get, 
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
 
 
3. DE: 
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
  
It is linear, since x is independent variable. 
 
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
 
 
SECTION – B  
 
5. f: R+ ? [4, 8)  defined by f (x) = x² + 4 
 f is 1 – 1  
 Let x1, x2, ? R+ such that f(x1) = f(x2) 
 
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
 
 ? f is 1 – 1       
 f is onto: Let y ? [4, 8) 
 f(x) = y ? x
2
 + 4 = y 
   ? x = y4 ? 
 Since y ? [4, 8) ? x ? R+ 
 For y ? [4, 8) there is a x ? R+ such that f (x) = y. 
 So f is onto.    
 So, f is bijective function and hence f is invertible.   
 The inverse of f is defined by  
 f: [4, 8) ? R+ 
f
-1
(y) = y4 ? 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
6.  
We have, 
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2
 
 
7. 
x
x 2x
e
I dx
5 4e e
?
?
??
 
??
xx
Let e t e dx dt
 
Now integral I becomes,  
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?
 
2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??
 
1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
8. 
x
3
(x 4)e
I .dx
(x 2)
?
?
?
?
 
x
33
x
23
x'
23
xx
23
x
2
x 2 2
I e .dx
(x 2) (x 2)
12
I e .dx
(x 2) (x 2)
Thus the given integral is of the form, 
12
I e f(x) f '(x) dx where, f(x) ; f (x)
(x 2) (x 2)
e 2e
I dx dx
(x 2) (x 2)
1 d 1
e
dx
(x 2) (x
?
?
?
??
??
?
??
??
??
??
??
??
??
??
??
??
??
?
? ? ? ?
??
??
??
??
??
2
x
2
dx
2)
e
So,I C
(x 2)
?
?? ??
?? ??
??
??
?? ??
??
?
 
 
OR 
2
3
x
dx
1x
?
?
 
Let 1 + x
3
 = t 
? 0 + 3x
2
dx = dt 
?
2
dt
x dx
3
? 
2
3
3
dt
x
3
dx
t
1x
1 dt
       
3t
1
       log t c
3
1
       log 1 x c
3
??
?
??
??
??
??
?
??
?
??
? ? ?
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 9 Solution 
  
SECTION – A 
 
1. a42, means element at 3
rd
 row and 2
nd
 column 
So, 
a32 = 10 
 
2. Differentiating w.r.t. x, we get, 
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
 
 
3. DE: 
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
  
It is linear, since x is independent variable. 
 
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
 
 
SECTION – B  
 
5. f: R+ ? [4, 8)  defined by f (x) = x² + 4 
 f is 1 – 1  
 Let x1, x2, ? R+ such that f(x1) = f(x2) 
 
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
 
 ? f is 1 – 1       
 f is onto: Let y ? [4, 8) 
 f(x) = y ? x
2
 + 4 = y 
   ? x = y4 ? 
 Since y ? [4, 8) ? x ? R+ 
 For y ? [4, 8) there is a x ? R+ such that f (x) = y. 
 So f is onto.    
 So, f is bijective function and hence f is invertible.   
 The inverse of f is defined by  
 f: [4, 8) ? R+ 
f
-1
(y) = y4 ? 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
6.  
We have, 
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2
 
 
7. 
x
x 2x
e
I dx
5 4e e
?
?
??
 
??
xx
Let e t e dx dt
 
Now integral I becomes,  
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?
 
2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??
 
1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
8. 
x
3
(x 4)e
I .dx
(x 2)
?
?
?
?
 
x
33
x
23
x'
23
xx
23
x
2
x 2 2
I e .dx
(x 2) (x 2)
12
I e .dx
(x 2) (x 2)
Thus the given integral is of the form, 
12
I e f(x) f '(x) dx where, f(x) ; f (x)
(x 2) (x 2)
e 2e
I dx dx
(x 2) (x 2)
1 d 1
e
dx
(x 2) (x
?
?
?
??
??
?
??
??
??
??
??
??
??
??
??
??
??
?
? ? ? ?
??
??
??
??
??
2
x
2
dx
2)
e
So,I C
(x 2)
?
?? ??
?? ??
??
??
?? ??
??
?
 
 
OR 
2
3
x
dx
1x
?
?
 
Let 1 + x
3
 = t 
? 0 + 3x
2
dx = dt 
?
2
dt
x dx
3
? 
2
3
3
dt
x
3
dx
t
1x
1 dt
       
3t
1
       log t c
3
1
       log 1 x c
3
??
?
??
??
??
??
?
??
?
??
? ? ?
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 9 Solution  
 
     
9. We have to differentiate it w.r.t. x two times 
? ?
? ?
? ?
2
2
2
2
2
2
Differentiating
dy
2y m 2x
dx
dy
y mx............... 1
dx
differentiating again
d y dy
ym
dx dx
from 1
d y dy y dy
y
dx dx x dx
which is the required differential equation
??
??
??
? ? ?
??
??
??
??
??
??
 
 
10.  
r. 3i 4j 12k 13 0 
22
2 2 2 2
2
22
3x 4y 12z 13 0
Distance of point (1, 1, p) from the plane , is given by
3 1 4 1 12 p 13 20 12p
3 4 12 3 4 12
Distance of point ( 3, 0, 1) from the plane , is given by
3 3 4 0 12 1 13
8
3 4 12 3
2
22
4 12
 
  
22
2 2 2 2
The two distances are equal
20 12p 8
3 4 12 3 4 12
20 12p 8
20 12p 8
7
p 1,
3
 
 
 
 
 
 
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