CBSE Past Year Paper Session (2018), Math Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 
  
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. 
Section A comprises of 4 questions of one mark each, Section B comprises of 8 
questions of two marks each, Section C comprises of 11 questions of four marks each 
and Section D comprises of 6 questions of six marks each. 
3. All questions in section A are to be answered in one word, one sentence or as per the 
exact requirement of the question. 
4. There is no overall choice. However, internal choice has been provided in 3 questions 
of four marks each and 3 questions of six marks each. You have to attempt only one of 
the alternatives in all such questions. 
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. 
 
SECTION – A 
1.  
o
Find the magnitude of each of two vectors a and b, having the same magnitude 
9
such that the angle between them is 60 and their scalar product is .
2
 
 
2. 
-1 1
Find the value of tan 3 cot ( 3).
?
?? 
 
3.  
if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value 
of (5) o (10) where  and o are binary operations.
??
?
 
 
 
4. 
0 a 3
If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b'
b 1 0
? ??
??
?
??
??
??
 
 
 
 
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 
  
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. 
Section A comprises of 4 questions of one mark each, Section B comprises of 8 
questions of two marks each, Section C comprises of 11 questions of four marks each 
and Section D comprises of 6 questions of six marks each. 
3. All questions in section A are to be answered in one word, one sentence or as per the 
exact requirement of the question. 
4. There is no overall choice. However, internal choice has been provided in 3 questions 
of four marks each and 3 questions of six marks each. You have to attempt only one of 
the alternatives in all such questions. 
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. 
 
SECTION – A 
1.  
o
Find the magnitude of each of two vectors a and b, having the same magnitude 
9
such that the angle between them is 60 and their scalar product is .
2
 
 
2. 
-1 1
Find the value of tan 3 cot ( 3).
?
?? 
 
3.  
if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value 
of (5) o (10) where  and o are binary operations.
??
?
 
 
 
4. 
0 a 3
If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b'
b 1 0
? ??
??
?
??
??
??
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
 
SECTION B 
 
5. A black and red die are rolled together. Find the conditional probability of obtaining 
the sum 8, given that the red die resulted in a number less than 4. 
  
6. If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
7. Find the differential equation representing the family of curves y = ae
bx+5
, where a and 
b are arbitrary constants. 
  
8.  
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
9. The total cost C(x) associated with the production of x units of an item is given by  
C(x) = 0.005x
3
 – 0.02x
2
 + 30x + 5000. Find the marginal cost when 3 units are 
produced, where by marginal cost we mean the instantaneous rate of change of total 
cost at any level of output. 
 
10. 
-1
1 cos x
Differentiate tan with respect to x.
sin x
???
??
??
 
 
11.  
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
12. Prove that : 
-1 -1 3
11
3 sin x sin (3x 4x ),x ,
22
 
 
 
 
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 
  
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. 
Section A comprises of 4 questions of one mark each, Section B comprises of 8 
questions of two marks each, Section C comprises of 11 questions of four marks each 
and Section D comprises of 6 questions of six marks each. 
3. All questions in section A are to be answered in one word, one sentence or as per the 
exact requirement of the question. 
4. There is no overall choice. However, internal choice has been provided in 3 questions 
of four marks each and 3 questions of six marks each. You have to attempt only one of 
the alternatives in all such questions. 
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. 
 
SECTION – A 
1.  
o
Find the magnitude of each of two vectors a and b, having the same magnitude 
9
such that the angle between them is 60 and their scalar product is .
2
 
 
2. 
-1 1
Find the value of tan 3 cot ( 3).
?
?? 
 
3.  
if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value 
of (5) o (10) where  and o are binary operations.
??
?
 
 
 
4. 
0 a 3
If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b'
b 1 0
? ??
??
?
??
??
??
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
 
SECTION B 
 
5. A black and red die are rolled together. Find the conditional probability of obtaining 
the sum 8, given that the red die resulted in a number less than 4. 
  
6. If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
7. Find the differential equation representing the family of curves y = ae
bx+5
, where a and 
b are arbitrary constants. 
  
8.  
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
9. The total cost C(x) associated with the production of x units of an item is given by  
C(x) = 0.005x
3
 – 0.02x
2
 + 30x + 5000. Find the marginal cost when 3 units are 
produced, where by marginal cost we mean the instantaneous rate of change of total 
cost at any level of output. 
 
10. 
-1
1 cos x
Differentiate tan with respect to x.
sin x
???
??
??
 
 
11.  
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
12. Prove that : 
-1 -1 3
11
3 sin x sin (3x 4x ),x ,
22
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
SECTION C 
13. Two numbers are selected at random (without replacement) from the first five positive 
integers. Let X denote the larger of the two numbers obtained. Find the mean and 
variance of X. 
 
14. An open tank with a square base and vertical sides is to be constructed from a metal 
sheet so as to hold a given quantity of water. Show that the cost of material will be least 
when depth of the tank is half of its width. If the cost is to be borne by nearby settled 
lower income families, for whom water will be provided, what kind of value is hidden 
in this question? 
  
15. Find the equations of the tangent and the normal, to the curve 16x
2
 + 9y
2
 = 145 at the 
point (x1, y1), where x1 = 2 and y1 > 0. 
OR 
       
4
32
x
   Find the intervals in which the function f(x) = x 5x 24x 12 is 
4
(a) strictly increasing, (b) strictly decreasing.
? ? ? ?
  
 
16.   
2 2 2
dy
If (x +y ) = xy, find .
dx
                  OR
dy
If x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
?
? ? ? ?
 
 
17. 
2
2
2
d y dy
if y = sin (sin x), prove that tan x +y cos x 0
dx dx
??  
 
18.  
x x 2
Find the particular solution of the differential equation e tan y dx + (2 - e ) sec y dy = 0,
given that y =  when x = 0.
4
                                                         OR
Find the particula
?
dy
r solution of the differential equation 2y tan x = sin x, given that 
dx
y = 0 when x = 
3
?
?
 
 
 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 
  
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. 
Section A comprises of 4 questions of one mark each, Section B comprises of 8 
questions of two marks each, Section C comprises of 11 questions of four marks each 
and Section D comprises of 6 questions of six marks each. 
3. All questions in section A are to be answered in one word, one sentence or as per the 
exact requirement of the question. 
4. There is no overall choice. However, internal choice has been provided in 3 questions 
of four marks each and 3 questions of six marks each. You have to attempt only one of 
the alternatives in all such questions. 
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. 
 
SECTION – A 
1.  
o
Find the magnitude of each of two vectors a and b, having the same magnitude 
9
such that the angle between them is 60 and their scalar product is .
2
 
 
2. 
-1 1
Find the value of tan 3 cot ( 3).
?
?? 
 
3.  
if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value 
of (5) o (10) where  and o are binary operations.
??
?
 
 
 
4. 
0 a 3
If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b'
b 1 0
? ??
??
?
??
??
??
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
 
SECTION B 
 
5. A black and red die are rolled together. Find the conditional probability of obtaining 
the sum 8, given that the red die resulted in a number less than 4. 
  
6. If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
7. Find the differential equation representing the family of curves y = ae
bx+5
, where a and 
b are arbitrary constants. 
  
8.  
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
9. The total cost C(x) associated with the production of x units of an item is given by  
C(x) = 0.005x
3
 – 0.02x
2
 + 30x + 5000. Find the marginal cost when 3 units are 
produced, where by marginal cost we mean the instantaneous rate of change of total 
cost at any level of output. 
 
10. 
-1
1 cos x
Differentiate tan with respect to x.
sin x
???
??
??
 
 
11.  
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
12. Prove that : 
-1 -1 3
11
3 sin x sin (3x 4x ),x ,
22
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
SECTION C 
13. Two numbers are selected at random (without replacement) from the first five positive 
integers. Let X denote the larger of the two numbers obtained. Find the mean and 
variance of X. 
 
14. An open tank with a square base and vertical sides is to be constructed from a metal 
sheet so as to hold a given quantity of water. Show that the cost of material will be least 
when depth of the tank is half of its width. If the cost is to be borne by nearby settled 
lower income families, for whom water will be provided, what kind of value is hidden 
in this question? 
  
15. Find the equations of the tangent and the normal, to the curve 16x
2
 + 9y
2
 = 145 at the 
point (x1, y1), where x1 = 2 and y1 > 0. 
OR 
       
4
32
x
   Find the intervals in which the function f(x) = x 5x 24x 12 is 
4
(a) strictly increasing, (b) strictly decreasing.
? ? ? ?
  
 
16.   
2 2 2
dy
If (x +y ) = xy, find .
dx
                  OR
dy
If x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
?
? ? ? ?
 
 
17. 
2
2
2
d y dy
if y = sin (sin x), prove that tan x +y cos x 0
dx dx
??  
 
18.  
x x 2
Find the particular solution of the differential equation e tan y dx + (2 - e ) sec y dy = 0,
given that y =  when x = 0.
4
                                                         OR
Find the particula
?
dy
r solution of the differential equation 2y tan x = sin x, given that 
dx
y = 0 when x = 
3
?
?
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
19.  
Find the shortest distance between the lines
r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k)
? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
 
  
20.  
2
Find :
2 cos x
 dx
(1 sinx)(1 sin x) ??
?
  
 
  
21. Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes 
the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a 
‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’. What is the probability that 
she threw 3, 4, 5 sor 6 with the die? 
 
22.  
Let a 4i 5j k, b i - 4j 5k and c 3i + j k. Find a vector d which is perpendicular
to both c and b and d . a = 21.
? ? ?
? ? ? ? ? ? ?
  
 
23.  
Using properties of determinants, prove that
   1                1        1+3x
1+3y             1           1 9(3xyz+xy+yz+zx)
   1               1+3z      1
?
 
 
24. Using integration,  find the area of the region in the first quadrant enclosed by the x-
axis, the line y = x and the circle x
2
 + y
2
 = 32  
  
25.  
 
Let A = {x Z :0 x 12). Show that
R = {(a,b): a,b A, a b is divisible by 4} is an equivalence relation.
Find the set of all elements related to 1. Also write the equivalence class {2}
                    
? ? ?
??
2
                           OR
x
Show that function f : R R defined by f(x) = ,  x R is neither one-one nor onto. 
x1
Also, if g : R R is defined as g(x) = 2x - 1, find fog (x)
? ? ?
?
?
 
  
 
 
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 
  
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. 
Section A comprises of 4 questions of one mark each, Section B comprises of 8 
questions of two marks each, Section C comprises of 11 questions of four marks each 
and Section D comprises of 6 questions of six marks each. 
3. All questions in section A are to be answered in one word, one sentence or as per the 
exact requirement of the question. 
4. There is no overall choice. However, internal choice has been provided in 3 questions 
of four marks each and 3 questions of six marks each. You have to attempt only one of 
the alternatives in all such questions. 
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. 
 
SECTION – A 
1.  
o
Find the magnitude of each of two vectors a and b, having the same magnitude 
9
such that the angle between them is 60 and their scalar product is .
2
 
 
2. 
-1 1
Find the value of tan 3 cot ( 3).
?
?? 
 
3.  
if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value 
of (5) o (10) where  and o are binary operations.
??
?
 
 
 
4. 
0 a 3
If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b'
b 1 0
? ??
??
?
??
??
??
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
 
SECTION B 
 
5. A black and red die are rolled together. Find the conditional probability of obtaining 
the sum 8, given that the red die resulted in a number less than 4. 
  
6. If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
7. Find the differential equation representing the family of curves y = ae
bx+5
, where a and 
b are arbitrary constants. 
  
8.  
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
9. The total cost C(x) associated with the production of x units of an item is given by  
C(x) = 0.005x
3
 – 0.02x
2
 + 30x + 5000. Find the marginal cost when 3 units are 
produced, where by marginal cost we mean the instantaneous rate of change of total 
cost at any level of output. 
 
10. 
-1
1 cos x
Differentiate tan with respect to x.
sin x
???
??
??
 
 
11.  
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
12. Prove that : 
-1 -1 3
11
3 sin x sin (3x 4x ),x ,
22
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
SECTION C 
13. Two numbers are selected at random (without replacement) from the first five positive 
integers. Let X denote the larger of the two numbers obtained. Find the mean and 
variance of X. 
 
14. An open tank with a square base and vertical sides is to be constructed from a metal 
sheet so as to hold a given quantity of water. Show that the cost of material will be least 
when depth of the tank is half of its width. If the cost is to be borne by nearby settled 
lower income families, for whom water will be provided, what kind of value is hidden 
in this question? 
  
15. Find the equations of the tangent and the normal, to the curve 16x
2
 + 9y
2
 = 145 at the 
point (x1, y1), where x1 = 2 and y1 > 0. 
OR 
       
4
32
x
   Find the intervals in which the function f(x) = x 5x 24x 12 is 
4
(a) strictly increasing, (b) strictly decreasing.
? ? ? ?
  
 
16.   
2 2 2
dy
If (x +y ) = xy, find .
dx
                  OR
dy
If x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
?
? ? ? ?
 
 
17. 
2
2
2
d y dy
if y = sin (sin x), prove that tan x +y cos x 0
dx dx
??  
 
18.  
x x 2
Find the particular solution of the differential equation e tan y dx + (2 - e ) sec y dy = 0,
given that y =  when x = 0.
4
                                                         OR
Find the particula
?
dy
r solution of the differential equation 2y tan x = sin x, given that 
dx
y = 0 when x = 
3
?
?
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
19.  
Find the shortest distance between the lines
r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k)
? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
 
  
20.  
2
Find :
2 cos x
 dx
(1 sinx)(1 sin x) ??
?
  
 
  
21. Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes 
the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a 
‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’. What is the probability that 
she threw 3, 4, 5 sor 6 with the die? 
 
22.  
Let a 4i 5j k, b i - 4j 5k and c 3i + j k. Find a vector d which is perpendicular
to both c and b and d . a = 21.
? ? ?
? ? ? ? ? ? ?
  
 
23.  
Using properties of determinants, prove that
   1                1        1+3x
1+3y             1           1 9(3xyz+xy+yz+zx)
   1               1+3z      1
?
 
 
24. Using integration,  find the area of the region in the first quadrant enclosed by the x-
axis, the line y = x and the circle x
2
 + y
2
 = 32  
  
25.  
 
Let A = {x Z :0 x 12). Show that
R = {(a,b): a,b A, a b is divisible by 4} is an equivalence relation.
Find the set of all elements related to 1. Also write the equivalence class {2}
                    
? ? ?
??
2
                           OR
x
Show that function f : R R defined by f(x) = ,  x R is neither one-one nor onto. 
x1
Also, if g : R R is defined as g(x) = 2x - 1, find fog (x)
? ? ?
?
?
 
  
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Set 3 
 
     
26.   
Find the distance of the point (-1,-5,-10) from the point of intersection of the line 
r = 2i - j + 2k +  (3i 4j + 2k) and the plane r = (i - j + k) 5.
? ? ?
? ? ?
 
  
27. A factory manufactures two types of screws A and B, each type requiring the use of two 
machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 
minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it 
takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to 
manufacture packet of screws ‘B’. Each machine is available for at most 4 hours on any 
day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws 
‘B’ at a profit of 1. Assuming that he can sell all the screws he manufactures how many 
packets of each type should the factory owner produce in a day in order to maximize 
his profit? Formulate the above LPP and solve it graphically and find the maximum 
profit. 
 
  
28.  
/4
0
3
2x
1
Evaluate :
sin x + cos x
               dx
16 9 sin 2x
                     OR
Evaluate :
                (x 3x e ) dx
as the limit of the sum
?
?
??
?
?
  
 
29.  
-1
2 3 5
If A = 3 2 4 , find A Use it to solve the system of equations
1 1 2
         2x -3y +5z = 11
         3x + 2y -4z = -5
          x + y - 2z = -3
                                          OR
Using el
? ??
??
?
??
??
?
??
ementary row transformations, find the inverse of the matrix
1 2 3
    A = 2 5 7
2 4 5
??
??
??
??
? ? ?
??