Class 12 Mathematics (Maths) Official Sample Question Paper (2017-18) | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


 
SAMPLE QUESTION PAPER 
MATHEMATICS (041) 
CLASS XII – 2017-18 
 
Time allowed: 3 hours        Maximum Marks: 100 
 
General Instructions: 
 
(i) All questions are compulsory. 
(ii) This question paper contains 29 questions. 
(iii) Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each. 
(iv) Questions 5-12 in Section B are short-answertype questions carrying 2 marks each. 
(v) Questions 13-23 in Section C are long-answer-I type questions carrying 4 marks each. 
(vi) Questions 24-29 in Section D are long-answer-II type questions carrying 6 marks each. 
 
 
Section A 
Questions 1 to 4 carry 1 mark each. 
 
1. 
Let A= 
? ? 1,2,3,4 . Let R be the equivalence relation on A A ? defined by 
? ? ? ? d c R b a , , iffa d b c ? ? ? . Find the equivalence class ? ? 1,3 . ??
??
 
 
2. If 
ij
Aa ?? ?
??
is a matrix of order 22 ? , such that 15 A ?? and  
ij
C represents the cofactor 
of ,
ij
a then find 
21 21 22 22
a c a c ? 
 
3. 
Give an example of vectors  
4. Determine whether the binary operation  ?  on the set N of natural numbers 
defined by 2
ab
ab ?? is associative or not. 
 
Section B 
Questions 5 to 12 carry 2 marks each 
5. 
If 
11
4sin cos , xx ?
??
?? then find the value of x. 
 
6. 
Find the inverse of the matrix
32
53
???
??
?
??
. Hence, find the matrix P satisfying the 
matrix equation 
3 2 1 2
.
5 3 2 1
P
? ? ? ? ?
?
? ? ? ?
??
? ? ? ?
 
Page 2


 
SAMPLE QUESTION PAPER 
MATHEMATICS (041) 
CLASS XII – 2017-18 
 
Time allowed: 3 hours        Maximum Marks: 100 
 
General Instructions: 
 
(i) All questions are compulsory. 
(ii) This question paper contains 29 questions. 
(iii) Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each. 
(iv) Questions 5-12 in Section B are short-answertype questions carrying 2 marks each. 
(v) Questions 13-23 in Section C are long-answer-I type questions carrying 4 marks each. 
(vi) Questions 24-29 in Section D are long-answer-II type questions carrying 6 marks each. 
 
 
Section A 
Questions 1 to 4 carry 1 mark each. 
 
1. 
Let A= 
? ? 1,2,3,4 . Let R be the equivalence relation on A A ? defined by 
? ? ? ? d c R b a , , iffa d b c ? ? ? . Find the equivalence class ? ? 1,3 . ??
??
 
 
2. If 
ij
Aa ?? ?
??
is a matrix of order 22 ? , such that 15 A ?? and  
ij
C represents the cofactor 
of ,
ij
a then find 
21 21 22 22
a c a c ? 
 
3. 
Give an example of vectors  
4. Determine whether the binary operation  ?  on the set N of natural numbers 
defined by 2
ab
ab ?? is associative or not. 
 
Section B 
Questions 5 to 12 carry 2 marks each 
5. 
If 
11
4sin cos , xx ?
??
?? then find the value of x. 
 
6. 
Find the inverse of the matrix
32
53
???
??
?
??
. Hence, find the matrix P satisfying the 
matrix equation 
3 2 1 2
.
5 3 2 1
P
? ? ? ? ?
?
? ? ? ?
??
? ? ? ?
 
 
7. Prove that if 
1
1
2
x ?? then 
2
11
33
cos cos
2 2 3
xx
x
?
??
??
?
? ? ? ??
??
??
 
 
8. 
Find the approximate change in the value of  
2
1
x
, when x changes from x = 2 to 
 x = 2.002 
 
9. 
Find 
1 sin 2
1 cos 2
x
x
e dx
x
?
?
?
 
10. 
Verify that 
22
1 ax by ?? is a solution of the differential equation 
2
2 1 1
() x yy y yy ?? 
11. 
Find the Projection (vector)  of 
ˆˆ ˆ ˆ ˆ ˆ
2 on 2 . i j k i j k ? ? ? ? 
12. 
If A and B are two events such that 
? ? ? ? 0.4, 0.8 P A P B ?? and
? ?
0.6 P B A ? , 
then find
? ?
P A B . 
 
Section C 
Questions 13 to 23 carry 4 marks each. 
 
13. 
        |
   
 
  
 
 
 
 
  
|     
                       |
 
 
      
 
    
 
 
 
  
   
 
 
 
   
|  
 
 
14. 
Find ‘ a’ and ‘ b’ , if the function given by 
2
, if 1
()
2 1, if 1
ax b x
fx
xx
? ??
?
?
??
?
 
 is differentiable at    1 x ? 
OR 
 Determine the values of ‘a ’ and ‘b ’ such that the following function is continuous 
at x = 0: 
sin
sin
, if 0
sin( 1)
( ) 2,if 0
1
2 ,if 0
bx
xx
x
ax
f x x
e
x
bx
?
? ?
? ? ?
?
?
?
?
??
?
?
?
?
?
?
?
 
Page 3


 
SAMPLE QUESTION PAPER 
MATHEMATICS (041) 
CLASS XII – 2017-18 
 
Time allowed: 3 hours        Maximum Marks: 100 
 
General Instructions: 
 
(i) All questions are compulsory. 
(ii) This question paper contains 29 questions. 
(iii) Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each. 
(iv) Questions 5-12 in Section B are short-answertype questions carrying 2 marks each. 
(v) Questions 13-23 in Section C are long-answer-I type questions carrying 4 marks each. 
(vi) Questions 24-29 in Section D are long-answer-II type questions carrying 6 marks each. 
 
 
Section A 
Questions 1 to 4 carry 1 mark each. 
 
1. 
Let A= 
? ? 1,2,3,4 . Let R be the equivalence relation on A A ? defined by 
? ? ? ? d c R b a , , iffa d b c ? ? ? . Find the equivalence class ? ? 1,3 . ??
??
 
 
2. If 
ij
Aa ?? ?
??
is a matrix of order 22 ? , such that 15 A ?? and  
ij
C represents the cofactor 
of ,
ij
a then find 
21 21 22 22
a c a c ? 
 
3. 
Give an example of vectors  
4. Determine whether the binary operation  ?  on the set N of natural numbers 
defined by 2
ab
ab ?? is associative or not. 
 
Section B 
Questions 5 to 12 carry 2 marks each 
5. 
If 
11
4sin cos , xx ?
??
?? then find the value of x. 
 
6. 
Find the inverse of the matrix
32
53
???
??
?
??
. Hence, find the matrix P satisfying the 
matrix equation 
3 2 1 2
.
5 3 2 1
P
? ? ? ? ?
?
? ? ? ?
??
? ? ? ?
 
 
7. Prove that if 
1
1
2
x ?? then 
2
11
33
cos cos
2 2 3
xx
x
?
??
??
?
? ? ? ??
??
??
 
 
8. 
Find the approximate change in the value of  
2
1
x
, when x changes from x = 2 to 
 x = 2.002 
 
9. 
Find 
1 sin 2
1 cos 2
x
x
e dx
x
?
?
?
 
10. 
Verify that 
22
1 ax by ?? is a solution of the differential equation 
2
2 1 1
() x yy y yy ?? 
11. 
Find the Projection (vector)  of 
ˆˆ ˆ ˆ ˆ ˆ
2 on 2 . i j k i j k ? ? ? ? 
12. 
If A and B are two events such that 
? ? ? ? 0.4, 0.8 P A P B ?? and
? ?
0.6 P B A ? , 
then find
? ?
P A B . 
 
Section C 
Questions 13 to 23 carry 4 marks each. 
 
13. 
        |
   
 
  
 
 
 
 
  
|     
                       |
 
 
      
 
    
 
 
 
  
   
 
 
 
   
|  
 
 
14. 
Find ‘ a’ and ‘ b’ , if the function given by 
2
, if 1
()
2 1, if 1
ax b x
fx
xx
? ??
?
?
??
?
 
 is differentiable at    1 x ? 
OR 
 Determine the values of ‘a ’ and ‘b ’ such that the following function is continuous 
at x = 0: 
sin
sin
, if 0
sin( 1)
( ) 2,if 0
1
2 ,if 0
bx
xx
x
ax
f x x
e
x
bx
?
? ?
? ? ?
?
?
?
?
??
?
?
?
?
?
?
?
 
 
15. 
If 
2
1
log( ) , yx
x
?? then prove that
22
21
( 1) ( 1) 2 x x y x y ? ? ? ? . 
16. 
Find the equation(s) of the tangent(s) to the curve 
3
( 1)( 2) y x x ? ? ? at the points 
where the curve intersects the x –axis. 
OR 
Find the intervals in which the function 
4
( ) 3log(1 ) 4log(2 )
2
f x x x
x
? ? ? ? ? ?
?
 
is strictly increasing or strictly decreasing. 
17. A person wants to plant some trees in his community park. The local nursery has 
to perform this task. It charges the cost of planting trees by the following formula: 
32
( ) 45 600 , C x x x x ? ? ? Where x is the number of trees and C(x) is the cost of 
planting x trees in rupees. The local authority has imposed a restriction that it can 
plant 10 to 20 trees in one community park for a fair distribution. For how many 
trees should the person place the order so that he has to spend the least amount? 
How much is the least amount? Use calculus to answer these questions. Which 
value is being exhibited by the person? 
 
 
18. 
Find 
sec
1 cos
x
dx
ecx ?
?
 
19. Find the particular solution of the differential equation  : 
3
( 2 ) , (0) 1
yy
ye dx y xe dy y ? ? ? 
OR 
Show that (   )   (    )   is a homogenous differential equation. Also, 
find the general solution of the given differential equation. 
20. 
If  are three vectors such that 0 a b c ? ? ? , then prove that 
, 0. a b b c c a andhenceshowthat a b c
??
? ? ? ? ? ?
??
 
21. Find the equation of the line which intersects the lines  
?? ? ? ? ?
? ? ? ?
32 2 1 1 3
 and 
1 2 4 2 3 4
yy x z x z
and passes through the point (1, 1, 1). 
 
Page 4


 
SAMPLE QUESTION PAPER 
MATHEMATICS (041) 
CLASS XII – 2017-18 
 
Time allowed: 3 hours        Maximum Marks: 100 
 
General Instructions: 
 
(i) All questions are compulsory. 
(ii) This question paper contains 29 questions. 
(iii) Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each. 
(iv) Questions 5-12 in Section B are short-answertype questions carrying 2 marks each. 
(v) Questions 13-23 in Section C are long-answer-I type questions carrying 4 marks each. 
(vi) Questions 24-29 in Section D are long-answer-II type questions carrying 6 marks each. 
 
 
Section A 
Questions 1 to 4 carry 1 mark each. 
 
1. 
Let A= 
? ? 1,2,3,4 . Let R be the equivalence relation on A A ? defined by 
? ? ? ? d c R b a , , iffa d b c ? ? ? . Find the equivalence class ? ? 1,3 . ??
??
 
 
2. If 
ij
Aa ?? ?
??
is a matrix of order 22 ? , such that 15 A ?? and  
ij
C represents the cofactor 
of ,
ij
a then find 
21 21 22 22
a c a c ? 
 
3. 
Give an example of vectors  
4. Determine whether the binary operation  ?  on the set N of natural numbers 
defined by 2
ab
ab ?? is associative or not. 
 
Section B 
Questions 5 to 12 carry 2 marks each 
5. 
If 
11
4sin cos , xx ?
??
?? then find the value of x. 
 
6. 
Find the inverse of the matrix
32
53
???
??
?
??
. Hence, find the matrix P satisfying the 
matrix equation 
3 2 1 2
.
5 3 2 1
P
? ? ? ? ?
?
? ? ? ?
??
? ? ? ?
 
 
7. Prove that if 
1
1
2
x ?? then 
2
11
33
cos cos
2 2 3
xx
x
?
??
??
?
? ? ? ??
??
??
 
 
8. 
Find the approximate change in the value of  
2
1
x
, when x changes from x = 2 to 
 x = 2.002 
 
9. 
Find 
1 sin 2
1 cos 2
x
x
e dx
x
?
?
?
 
10. 
Verify that 
22
1 ax by ?? is a solution of the differential equation 
2
2 1 1
() x yy y yy ?? 
11. 
Find the Projection (vector)  of 
ˆˆ ˆ ˆ ˆ ˆ
2 on 2 . i j k i j k ? ? ? ? 
12. 
If A and B are two events such that 
? ? ? ? 0.4, 0.8 P A P B ?? and
? ?
0.6 P B A ? , 
then find
? ?
P A B . 
 
Section C 
Questions 13 to 23 carry 4 marks each. 
 
13. 
        |
   
 
  
 
 
 
 
  
|     
                       |
 
 
      
 
    
 
 
 
  
   
 
 
 
   
|  
 
 
14. 
Find ‘ a’ and ‘ b’ , if the function given by 
2
, if 1
()
2 1, if 1
ax b x
fx
xx
? ??
?
?
??
?
 
 is differentiable at    1 x ? 
OR 
 Determine the values of ‘a ’ and ‘b ’ such that the following function is continuous 
at x = 0: 
sin
sin
, if 0
sin( 1)
( ) 2,if 0
1
2 ,if 0
bx
xx
x
ax
f x x
e
x
bx
?
? ?
? ? ?
?
?
?
?
??
?
?
?
?
?
?
?
 
 
15. 
If 
2
1
log( ) , yx
x
?? then prove that
22
21
( 1) ( 1) 2 x x y x y ? ? ? ? . 
16. 
Find the equation(s) of the tangent(s) to the curve 
3
( 1)( 2) y x x ? ? ? at the points 
where the curve intersects the x –axis. 
OR 
Find the intervals in which the function 
4
( ) 3log(1 ) 4log(2 )
2
f x x x
x
? ? ? ? ? ?
?
 
is strictly increasing or strictly decreasing. 
17. A person wants to plant some trees in his community park. The local nursery has 
to perform this task. It charges the cost of planting trees by the following formula: 
32
( ) 45 600 , C x x x x ? ? ? Where x is the number of trees and C(x) is the cost of 
planting x trees in rupees. The local authority has imposed a restriction that it can 
plant 10 to 20 trees in one community park for a fair distribution. For how many 
trees should the person place the order so that he has to spend the least amount? 
How much is the least amount? Use calculus to answer these questions. Which 
value is being exhibited by the person? 
 
 
18. 
Find 
sec
1 cos
x
dx
ecx ?
?
 
19. Find the particular solution of the differential equation  : 
3
( 2 ) , (0) 1
yy
ye dx y xe dy y ? ? ? 
OR 
Show that (   )   (    )   is a homogenous differential equation. Also, 
find the general solution of the given differential equation. 
20. 
If  are three vectors such that 0 a b c ? ? ? , then prove that 
, 0. a b b c c a andhenceshowthat a b c
??
? ? ? ? ? ?
??
 
21. Find the equation of the line which intersects the lines  
?? ? ? ? ?
? ? ? ?
32 2 1 1 3
 and 
1 2 4 2 3 4
yy x z x z
and passes through the point (1, 1, 1). 
 
22. Bag I contains 1 white, 2 black and 3 red balls; Bag II contains 2 white, 1 black and 
1 red balls; Bag III contains 4 white, 3 black and 2 red balls. A bag is chosen at 
random and two balls are drawn from it with replacement. They happen to be 
one white and one red. What is the probability that they came from Bag III. 
23. Four bad oranges are accidentally mixed with 16 good ones. Find the probability 
distribution of the number of bad oranges when two oranges are drawn at 
random from this lot. Find the mean and variance of the distribution. 
 
Section D 
Questions 24 to 29 carry 6 marks each. 
24. 
If the function : f ? be defined by 3 2 ) ( ? ? x x f and : g ? by     
3
( ) 5, g x x ?? then find g f ? and show that g f ? is invertible. Also, find    
 
? ?
1
, fg
?
 
hence find ? ? ? ? 9
1 ?
g f ? . 
OR 
A binary operation ? is defined on the set  of real numbers by 
, if 0
.
, if 0
ab
ab
a b b
? ?
??
?
??
?
 If at least one of a and b is 0, then prove that . a b b a ? ? ?
Check whether ? is commutative. Find the identity element for ? , if it exists. 
 
25. 
If 
3 2 1
4 1 2
7 3 3
A
??
??
??
??
?? ?
??
 , then find 
1
A
?
and hence solve the following system of   
    equations:3 4 7 14,2 3 4, 2 3 0 x y z x y z x y z ? ? ? ? ? ? ? ? ? 
OR 
If
2 1 1
1 0 1
0 2 1
A
??
??
?
??
?? ?
??
, find the inverse of A using elementary row transformations 
and hence solve the following matrix equation
? ? 1 0 1 XA ?
. 
26. Using integration, find the area in the first quadrant bounded by the curve 
y x x ? ,  the circle 
22
2 xy ?? and  the y-axis 
Page 5


 
SAMPLE QUESTION PAPER 
MATHEMATICS (041) 
CLASS XII – 2017-18 
 
Time allowed: 3 hours        Maximum Marks: 100 
 
General Instructions: 
 
(i) All questions are compulsory. 
(ii) This question paper contains 29 questions. 
(iii) Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each. 
(iv) Questions 5-12 in Section B are short-answertype questions carrying 2 marks each. 
(v) Questions 13-23 in Section C are long-answer-I type questions carrying 4 marks each. 
(vi) Questions 24-29 in Section D are long-answer-II type questions carrying 6 marks each. 
 
 
Section A 
Questions 1 to 4 carry 1 mark each. 
 
1. 
Let A= 
? ? 1,2,3,4 . Let R be the equivalence relation on A A ? defined by 
? ? ? ? d c R b a , , iffa d b c ? ? ? . Find the equivalence class ? ? 1,3 . ??
??
 
 
2. If 
ij
Aa ?? ?
??
is a matrix of order 22 ? , such that 15 A ?? and  
ij
C represents the cofactor 
of ,
ij
a then find 
21 21 22 22
a c a c ? 
 
3. 
Give an example of vectors  
4. Determine whether the binary operation  ?  on the set N of natural numbers 
defined by 2
ab
ab ?? is associative or not. 
 
Section B 
Questions 5 to 12 carry 2 marks each 
5. 
If 
11
4sin cos , xx ?
??
?? then find the value of x. 
 
6. 
Find the inverse of the matrix
32
53
???
??
?
??
. Hence, find the matrix P satisfying the 
matrix equation 
3 2 1 2
.
5 3 2 1
P
? ? ? ? ?
?
? ? ? ?
??
? ? ? ?
 
 
7. Prove that if 
1
1
2
x ?? then 
2
11
33
cos cos
2 2 3
xx
x
?
??
??
?
? ? ? ??
??
??
 
 
8. 
Find the approximate change in the value of  
2
1
x
, when x changes from x = 2 to 
 x = 2.002 
 
9. 
Find 
1 sin 2
1 cos 2
x
x
e dx
x
?
?
?
 
10. 
Verify that 
22
1 ax by ?? is a solution of the differential equation 
2
2 1 1
() x yy y yy ?? 
11. 
Find the Projection (vector)  of 
ˆˆ ˆ ˆ ˆ ˆ
2 on 2 . i j k i j k ? ? ? ? 
12. 
If A and B are two events such that 
? ? ? ? 0.4, 0.8 P A P B ?? and
? ?
0.6 P B A ? , 
then find
? ?
P A B . 
 
Section C 
Questions 13 to 23 carry 4 marks each. 
 
13. 
        |
   
 
  
 
 
 
 
  
|     
                       |
 
 
      
 
    
 
 
 
  
   
 
 
 
   
|  
 
 
14. 
Find ‘ a’ and ‘ b’ , if the function given by 
2
, if 1
()
2 1, if 1
ax b x
fx
xx
? ??
?
?
??
?
 
 is differentiable at    1 x ? 
OR 
 Determine the values of ‘a ’ and ‘b ’ such that the following function is continuous 
at x = 0: 
sin
sin
, if 0
sin( 1)
( ) 2,if 0
1
2 ,if 0
bx
xx
x
ax
f x x
e
x
bx
?
? ?
? ? ?
?
?
?
?
??
?
?
?
?
?
?
?
 
 
15. 
If 
2
1
log( ) , yx
x
?? then prove that
22
21
( 1) ( 1) 2 x x y x y ? ? ? ? . 
16. 
Find the equation(s) of the tangent(s) to the curve 
3
( 1)( 2) y x x ? ? ? at the points 
where the curve intersects the x –axis. 
OR 
Find the intervals in which the function 
4
( ) 3log(1 ) 4log(2 )
2
f x x x
x
? ? ? ? ? ?
?
 
is strictly increasing or strictly decreasing. 
17. A person wants to plant some trees in his community park. The local nursery has 
to perform this task. It charges the cost of planting trees by the following formula: 
32
( ) 45 600 , C x x x x ? ? ? Where x is the number of trees and C(x) is the cost of 
planting x trees in rupees. The local authority has imposed a restriction that it can 
plant 10 to 20 trees in one community park for a fair distribution. For how many 
trees should the person place the order so that he has to spend the least amount? 
How much is the least amount? Use calculus to answer these questions. Which 
value is being exhibited by the person? 
 
 
18. 
Find 
sec
1 cos
x
dx
ecx ?
?
 
19. Find the particular solution of the differential equation  : 
3
( 2 ) , (0) 1
yy
ye dx y xe dy y ? ? ? 
OR 
Show that (   )   (    )   is a homogenous differential equation. Also, 
find the general solution of the given differential equation. 
20. 
If  are three vectors such that 0 a b c ? ? ? , then prove that 
, 0. a b b c c a andhenceshowthat a b c
??
? ? ? ? ? ?
??
 
21. Find the equation of the line which intersects the lines  
?? ? ? ? ?
? ? ? ?
32 2 1 1 3
 and 
1 2 4 2 3 4
yy x z x z
and passes through the point (1, 1, 1). 
 
22. Bag I contains 1 white, 2 black and 3 red balls; Bag II contains 2 white, 1 black and 
1 red balls; Bag III contains 4 white, 3 black and 2 red balls. A bag is chosen at 
random and two balls are drawn from it with replacement. They happen to be 
one white and one red. What is the probability that they came from Bag III. 
23. Four bad oranges are accidentally mixed with 16 good ones. Find the probability 
distribution of the number of bad oranges when two oranges are drawn at 
random from this lot. Find the mean and variance of the distribution. 
 
Section D 
Questions 24 to 29 carry 6 marks each. 
24. 
If the function : f ? be defined by 3 2 ) ( ? ? x x f and : g ? by     
3
( ) 5, g x x ?? then find g f ? and show that g f ? is invertible. Also, find    
 
? ?
1
, fg
?
 
hence find ? ? ? ? 9
1 ?
g f ? . 
OR 
A binary operation ? is defined on the set  of real numbers by 
, if 0
.
, if 0
ab
ab
a b b
? ?
??
?
??
?
 If at least one of a and b is 0, then prove that . a b b a ? ? ?
Check whether ? is commutative. Find the identity element for ? , if it exists. 
 
25. 
If 
3 2 1
4 1 2
7 3 3
A
??
??
??
??
?? ?
??
 , then find 
1
A
?
and hence solve the following system of   
    equations:3 4 7 14,2 3 4, 2 3 0 x y z x y z x y z ? ? ? ? ? ? ? ? ? 
OR 
If
2 1 1
1 0 1
0 2 1
A
??
??
?
??
?? ?
??
, find the inverse of A using elementary row transformations 
and hence solve the following matrix equation
? ? 1 0 1 XA ?
. 
26. Using integration, find the area in the first quadrant bounded by the curve 
y x x ? ,  the circle 
22
2 xy ?? and  the y-axis 
 
27. 
Evaluate the following: 
4
4
4
2 cos 2
x
dx
x
?
?
?
?
?
?
?
 
OR 
 Evaluate 
2
2
2
(3 2 4) x x dx
?
??
?
 as the limit of a sum.  
 
28. 
                             ^   ^   
^
?   from the line  
ˆˆ ˆ ˆ ˆ ˆ
2 ( 3 9 ) r i j k i j k ? ? ? ? ? ? ?
 
measured parallel to the plane: x – y + 2z – 3 = 0.
 
29. 
A company produces two different products. One of them needs 1/4 of an hour of 
assembly work per unit, 1/8 of an hour in quality control work and Rs1.2 in raw 
materials. The other product requires 1/3 of an hour of assembly work per unit, 
1/3 of an hour in quality control work and Rs 0.9 in raw materials. Given the 
current availability of staff in the company, each day there is at most a total of 90 
hours available for assembly and 80 hours for quality control. The first product 
described has a market value (sale price) of Rs 9 per unit and the second product 
described has a market value (sale price) of Rs 8 per unit. In addition, the 
maximum amount of daily sales for the first product is estimated to be 200 units, 
without there being a maximum limit of daily sales for the second product. 
Formulate and solve graphically the LPP and find the maximum profit.