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CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 8 Solution 
 
SECTION – A  
1. The element ‘6’ lies on 3
rd
 row and 3
rd
 column 
So, 
a33 = 6 
 
2. 2x + 3y = cos x  
Differentiating w.r.t. x, we get, 
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
 
 
3. DE: 
2
2
2
d y dy
s sy s
dx dx
?? 
It is nonlinear, since we have product of dependent variable and differential 
coefficient 
dy
y
dx
 
 
4. 
? ? y4
x 5 z 6
3 7 2
??
??
?? 
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2. 
So its vector equation is  
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ? 
 
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 8 Solution 
 
SECTION – A  
1. The element ‘6’ lies on 3
rd
 row and 3
rd
 column 
So, 
a33 = 6 
 
2. 2x + 3y = cos x  
Differentiating w.r.t. x, we get, 
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
 
 
3. DE: 
2
2
2
d y dy
s sy s
dx dx
?? 
It is nonlinear, since we have product of dependent variable and differential 
coefficient 
dy
y
dx
 
 
4. 
? ? y4
x 5 z 6
3 7 2
??
??
?? 
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2. 
So its vector equation is  
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ? 
 
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
SECTION – B  
 
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b} 
The operation table for the given operation * on the given set is as follows 
 
* 1 2 3 4 5 
1 1 1 1 1 1 
2 1 2 2 2 2 
3 1 2 3 3 3 
4 1 2 3 4 4 
5 1 2 3 4 5 
 
 
6. We have, 
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 8 Solution 
 
SECTION – A  
1. The element ‘6’ lies on 3
rd
 row and 3
rd
 column 
So, 
a33 = 6 
 
2. 2x + 3y = cos x  
Differentiating w.r.t. x, we get, 
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
 
 
3. DE: 
2
2
2
d y dy
s sy s
dx dx
?? 
It is nonlinear, since we have product of dependent variable and differential 
coefficient 
dy
y
dx
 
 
4. 
? ? y4
x 5 z 6
3 7 2
??
??
?? 
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2. 
So its vector equation is  
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ? 
 
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
SECTION – B  
 
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b} 
The operation table for the given operation * on the given set is as follows 
 
* 1 2 3 4 5 
1 1 1 1 1 1 
2 1 2 2 2 2 
3 1 2 3 3 3 
4 1 2 3 4 4 
5 1 2 3 4 5 
 
 
6. We have, 
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
7. 
2
5x 2
dx
1 2x 3x
 
2
2
2
2
2
x
5
5 dx
1 2x 3x
12
6x
5
5
dx
6 1 2x 3x
12
6x 2 2
5
5
dx
6 1 2x 3x
22
6x 2
5
5
dx
6 1 2x 3x
 
2 2
2
2
5 6x 2 5 22 1
dx dx
6 6 5 1 2x 3x
12
3x
39
5 11 1
log 1 2x 3x dx
69
12
x
39
 
21
21
1
x
5 11 3 3
log 1 2x 3x tan C
69
22
3
5 11 3x 1
log 1 2x 3x tan C
6
3 2 2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 8 Solution 
 
SECTION – A  
1. The element ‘6’ lies on 3
rd
 row and 3
rd
 column 
So, 
a33 = 6 
 
2. 2x + 3y = cos x  
Differentiating w.r.t. x, we get, 
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
 
 
3. DE: 
2
2
2
d y dy
s sy s
dx dx
?? 
It is nonlinear, since we have product of dependent variable and differential 
coefficient 
dy
y
dx
 
 
4. 
? ? y4
x 5 z 6
3 7 2
??
??
?? 
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2. 
So its vector equation is  
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ? 
 
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
SECTION – B  
 
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b} 
The operation table for the given operation * on the given set is as follows 
 
* 1 2 3 4 5 
1 1 1 1 1 1 
2 1 2 2 2 2 
3 1 2 3 3 3 
4 1 2 3 4 4 
5 1 2 3 4 5 
 
 
6. We have, 
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
7. 
2
5x 2
dx
1 2x 3x
 
2
2
2
2
2
x
5
5 dx
1 2x 3x
12
6x
5
5
dx
6 1 2x 3x
12
6x 2 2
5
5
dx
6 1 2x 3x
22
6x 2
5
5
dx
6 1 2x 3x
 
2 2
2
2
5 6x 2 5 22 1
dx dx
6 6 5 1 2x 3x
12
3x
39
5 11 1
log 1 2x 3x dx
69
12
x
39
 
21
21
1
x
5 11 3 3
log 1 2x 3x tan C
69
22
3
5 11 3x 1
log 1 2x 3x tan C
6
3 2 2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
8. 
2
Let x y
 
2
22
x y A B
y 4 y 9 y 4 y 9
x 4 x 9
y A(y 9) B(y 4)
Comparing both sides,
A B 1 and 9A 4B 0
49
Solving, we get A and B
55
 
22
11
11
49
I dx
5 x 4 5 x 9
4 1 x 9 1 x
tan tan C
5 2 2 5 3 3
2 x 3 x
tan tan C
5 2 5 3
 
 
OR 
 
x
3
x
3
x
33
x
23
(x 3)e
dx
(x 5)
(x 5 2)e
dx
(x 5)
(x 5) 2
e . dx
(x 5) (x 5)
12
e . dx
(x 5) (x 5)
?
?
??
?
?
?? ?
??
??
??
??
??
??
??
??
??
?
?
?
?
 
? ? ?
??
??
??
??
??
??
?
?
?
xx
x
23
x
2
This is of the form 
e [f(x) f '(x)]dx e f(x) C
12
e .dx
(x 5) (x 5)
e
C
(x 5)
 
 
 
 
 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
Mathematics 
Class XII 
Sample Paper – 8 Solution 
 
SECTION – A  
1. The element ‘6’ lies on 3
rd
 row and 3
rd
 column 
So, 
a33 = 6 
 
2. 2x + 3y = cos x  
Differentiating w.r.t. x, we get, 
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
 
 
3. DE: 
2
2
2
d y dy
s sy s
dx dx
?? 
It is nonlinear, since we have product of dependent variable and differential 
coefficient 
dy
y
dx
 
 
4. 
? ? y4
x 5 z 6
3 7 2
??
??
?? 
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2. 
So its vector equation is  
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ? 
 
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
SECTION – B  
 
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b} 
The operation table for the given operation * on the given set is as follows 
 
* 1 2 3 4 5 
1 1 1 1 1 1 
2 1 2 2 2 2 
3 1 2 3 3 3 
4 1 2 3 4 4 
5 1 2 3 4 5 
 
 
6. We have, 
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
7. 
2
5x 2
dx
1 2x 3x
 
2
2
2
2
2
x
5
5 dx
1 2x 3x
12
6x
5
5
dx
6 1 2x 3x
12
6x 2 2
5
5
dx
6 1 2x 3x
22
6x 2
5
5
dx
6 1 2x 3x
 
2 2
2
2
5 6x 2 5 22 1
dx dx
6 6 5 1 2x 3x
12
3x
39
5 11 1
log 1 2x 3x dx
69
12
x
39
 
21
21
1
x
5 11 3 3
log 1 2x 3x tan C
69
22
3
5 11 3x 1
log 1 2x 3x tan C
6
3 2 2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
8. 
2
Let x y
 
2
22
x y A B
y 4 y 9 y 4 y 9
x 4 x 9
y A(y 9) B(y 4)
Comparing both sides,
A B 1 and 9A 4B 0
49
Solving, we get A and B
55
 
22
11
11
49
I dx
5 x 4 5 x 9
4 1 x 9 1 x
tan tan C
5 2 2 5 3 3
2 x 3 x
tan tan C
5 2 5 3
 
 
OR 
 
x
3
x
3
x
33
x
23
(x 3)e
dx
(x 5)
(x 5 2)e
dx
(x 5)
(x 5) 2
e . dx
(x 5) (x 5)
12
e . dx
(x 5) (x 5)
?
?
??
?
?
?? ?
??
??
??
??
??
??
??
??
??
?
?
?
?
 
? ? ?
??
??
??
??
??
??
?
?
?
xx
x
23
x
2
This is of the form 
e [f(x) f '(x)]dx e f(x) C
12
e .dx
(x 5) (x 5)
e
C
(x 5)
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8 Solution  
 
     
9. We have to differentiate it w.r.t. x two times 
? ?
? ?
? ? ? ?
2
2
2
2
2
2
differentiating
dy
abcos bx c .......(1)
dx
differentiating again
dy
ab sin bx c .........(2)
dx
dy
b y............. y asin bx c
dx
which is the required differential equation
??
? ? ?
? ? ? ?
 
 
 
10. ABCD is a parallelogram with,  
AB 2i 4j 5k;AD i 2j 3k
Using the parallelogramlaw of vector addition, diagonal is given by
AC AB AD 2i 4j 5k i 2j 3k 3i 6j 2k
 
2 2 2
Unit vector parallel to diagonal AC
AC 3i 6j 2k 3i 6j 2k
                                                
3i 6j 2k AC 3 ( 6) (2)
3i 6j 2k 1
3i 6j 2k
7 49
 Area of the parallelogram 
2 2 2
ABCD= AB AD
i j k
2 4 5                                                                                       
1 2 3
i(12 10) j( 6 5) k( 4 4) i(22) j ( 11) k(0) i(22) j (11)
(22) (11) 0 11 5 sq units
 
 
 
 
 
 
 
 
 
 
 
 
 
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FAQs on Sample Solution Paper 8 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

1. What are the important topics that I should focus on for the Class 12 Math exam?
Ans. The important topics that you should focus on for the Class 12 Math exam include calculus, algebra, probability, trigonometry, and coordinate geometry. These topics are usually given more weightage in the exam and have a higher chance of appearing in the question paper.
2. How can I improve my problem-solving skills for the Class 12 Math exam?
Ans. To improve your problem-solving skills for the Class 12 Math exam, you can practice solving a variety of math problems from different sources such as textbooks, previous year question papers, and online resources. Additionally, you can also seek help from your teachers or join study groups to discuss and solve problems together.
3. Are there any specific tips to remember formulas and theorems for the Class 12 Math exam?
Ans. Yes, there are specific tips to remember formulas and theorems for the Class 12 Math exam. You can create flashcards or cheat sheets with the formulas and theorems written on them and revise them regularly. Additionally, you can also try to understand the derivations and proofs behind the formulas and theorems, as it will help you remember them better.
4. How should I manage my time during the Class 12 Math exam?
Ans. To manage your time during the Class 12 Math exam, it is important to practice solving previous year question papers or sample papers within the given time limit. This will help you get an idea of how much time you should allocate to each section or question. Additionally, you can also prioritize the questions based on their weightage or difficulty level to ensure that you allocate sufficient time to the important ones.
5. Are there any recommended resources or books for the Class 12 Math exam preparation?
Ans. Yes, there are several recommended resources and books for the Class 12 Math exam preparation. Some popular ones include NCERT textbooks, RD Sharma's Mathematics for Class 12, and RS Aggarwal's Senior Secondary School Mathematics for Class 12. Additionally, you can also refer to online platforms or apps that provide study materials, practice questions, and video tutorials for Class 12 Math.
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