Class 12 Mathematics (Maths) Official Sample Question Paper (2018-19) | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


1 
 
MATHEMATICS 
Time allowed : 3 hours                                                             Maximum Marks: 100 
General Instructions: 
1. All questions are compulsory. 
2. This question paper contains 29 questions. 
3. Questions 1 – 4 in Section A are very short-answer type questions carrying 1 mark each.  
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 marks each.  
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 marks each. 
6. Questions 24 – 29 in Section D are long-answer II type questions carrying 6 marks each. 
Section A 
1. If A and B are invertible matrices of order 3, 
1
1
| A | 2 and | (AB) | .
6
?
? ? ? Find |B|. 
2. Differentiate 
22
sin (x ) w.r.t x
2
. 
3. Write the order of the differential equation: 
                                     
3
2
2
d y dy
log x
dx
dx
??
??
??
?? ??
??
??
 
4. Find the acute angle which the line with direction cosines 
11
, , n
36
 makes with 
positive direction of z-axis.  
                                                     OR 
Find the direction cosines of the line:  
x 1 z 1
y
22
??
? ? ? 
Section B 
5. Let A = Z × Z and * be a binary operation on A defined by 
(a, b)*(c, d) (ad bc, bd). ?? 
Find the identity element for * in the set A. 
6. If 
3 1 1 0
A and I
1 2 0 1
? ? ? ?
??
? ? ? ?
?
? ? ? ?
, find k so that 
2
A 5A kI ?? . 
Page 2


1 
 
MATHEMATICS 
Time allowed : 3 hours                                                             Maximum Marks: 100 
General Instructions: 
1. All questions are compulsory. 
2. This question paper contains 29 questions. 
3. Questions 1 – 4 in Section A are very short-answer type questions carrying 1 mark each.  
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 marks each.  
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 marks each. 
6. Questions 24 – 29 in Section D are long-answer II type questions carrying 6 marks each. 
Section A 
1. If A and B are invertible matrices of order 3, 
1
1
| A | 2 and | (AB) | .
6
?
? ? ? Find |B|. 
2. Differentiate 
22
sin (x ) w.r.t x
2
. 
3. Write the order of the differential equation: 
                                     
3
2
2
d y dy
log x
dx
dx
??
??
??
?? ??
??
??
 
4. Find the acute angle which the line with direction cosines 
11
, , n
36
 makes with 
positive direction of z-axis.  
                                                     OR 
Find the direction cosines of the line:  
x 1 z 1
y
22
??
? ? ? 
Section B 
5. Let A = Z × Z and * be a binary operation on A defined by 
(a, b)*(c, d) (ad bc, bd). ?? 
Find the identity element for * in the set A. 
6. If 
3 1 1 0
A and I
1 2 0 1
? ? ? ?
??
? ? ? ?
?
? ? ? ?
, find k so that 
2
A 5A kI ?? . 
2 
 
7. Find : 
2 2 2
2
(x sin x)sec x
dx
1x
?
?
?
 
8. Find: 
x
3
e (x 3)
dx
(x 1)
?
?
?
 
 
                                     OR 
Find: 
1
4
4
5
(x x)
dx
x
?
?
 
9. Form the differential equation of all circles which touch the x-axis at the origin. 
10. Find the area of the parallelogram whose diagonals are represented by the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a 2i 3j 4k and b 2i j 2k ? ? ? ? ? ? 
OR 
Find the angle between the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a i j k a n d b i – j k ? ? ? ? ? 
11. If A and B are two independent events, prove that A ? and B are also independent. 
12. One bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. A 
ball is transferred from first bag to the second bag and then a ball is drawn from the 
second bag. Find the probability that the ball drawn is red. 
OR 
 If P(A) = 0.6, P(B) = 0.5 and P(A|B) = 0.3, then find P( ? ?? B). 
Section C 
13. Prove that the function f:[0, ?) ? R given by f(x) = 9x
2
 + 6x – 5 is not invertible. Modify 
the codomain of the function f to make it invertible, and hence find f
–1
. 
OR 
Check whether the relation R in the set R of real numbers, defined by 
R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive. 
14. Find the value of : 
11
1
sin 2 tan cos(tan 2 2)
4
??
??
?
??
??
 
 
 
Page 3


1 
 
MATHEMATICS 
Time allowed : 3 hours                                                             Maximum Marks: 100 
General Instructions: 
1. All questions are compulsory. 
2. This question paper contains 29 questions. 
3. Questions 1 – 4 in Section A are very short-answer type questions carrying 1 mark each.  
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 marks each.  
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 marks each. 
6. Questions 24 – 29 in Section D are long-answer II type questions carrying 6 marks each. 
Section A 
1. If A and B are invertible matrices of order 3, 
1
1
| A | 2 and | (AB) | .
6
?
? ? ? Find |B|. 
2. Differentiate 
22
sin (x ) w.r.t x
2
. 
3. Write the order of the differential equation: 
                                     
3
2
2
d y dy
log x
dx
dx
??
??
??
?? ??
??
??
 
4. Find the acute angle which the line with direction cosines 
11
, , n
36
 makes with 
positive direction of z-axis.  
                                                     OR 
Find the direction cosines of the line:  
x 1 z 1
y
22
??
? ? ? 
Section B 
5. Let A = Z × Z and * be a binary operation on A defined by 
(a, b)*(c, d) (ad bc, bd). ?? 
Find the identity element for * in the set A. 
6. If 
3 1 1 0
A and I
1 2 0 1
? ? ? ?
??
? ? ? ?
?
? ? ? ?
, find k so that 
2
A 5A kI ?? . 
2 
 
7. Find : 
2 2 2
2
(x sin x)sec x
dx
1x
?
?
?
 
8. Find: 
x
3
e (x 3)
dx
(x 1)
?
?
?
 
 
                                     OR 
Find: 
1
4
4
5
(x x)
dx
x
?
?
 
9. Form the differential equation of all circles which touch the x-axis at the origin. 
10. Find the area of the parallelogram whose diagonals are represented by the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a 2i 3j 4k and b 2i j 2k ? ? ? ? ? ? 
OR 
Find the angle between the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a i j k a n d b i – j k ? ? ? ? ? 
11. If A and B are two independent events, prove that A ? and B are also independent. 
12. One bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. A 
ball is transferred from first bag to the second bag and then a ball is drawn from the 
second bag. Find the probability that the ball drawn is red. 
OR 
 If P(A) = 0.6, P(B) = 0.5 and P(A|B) = 0.3, then find P( ? ?? B). 
Section C 
13. Prove that the function f:[0, ?) ? R given by f(x) = 9x
2
 + 6x – 5 is not invertible. Modify 
the codomain of the function f to make it invertible, and hence find f
–1
. 
OR 
Check whether the relation R in the set R of real numbers, defined by 
R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive. 
14. Find the value of : 
11
1
sin 2 tan cos(tan 2 2)
4
??
??
?
??
??
 
 
 
3 
 
15. Using properties of determinants, prove that: 
2 2 2
a b c c b
a c b c a (a b c) (a b c )
a b b a c
??
? ? ? ? ? ? ?
??
 
16. If 
sin x x
dy
y x sin(x ), find
dx
?? 
                                        OR
 
If 
2
2 4 1
2
dy
y log(1 2t t ), x tan t, find
dx
?
? ? ? ? 
17. If 
1
y cos(m cos x)
?
? 
Show that: 
2
22
2
d y dy
(1 x ) x m y
dx
dx
? ? ? = 0 
18. Find the equations of the normal to the curve 
3
y 4x 3x 5 ? ? ? which are perpendicular to 
the line 9x y 5 0 ? ? ? . 
19. Find: 
4
22
x1
dx
x(x 1)
?
?
?
 
20. Evaluate: 
1
2
1
x | x | 1
dx
x 2 | x | 1
?
??
??
?
 
21. Find the particular solution of the following differential equation. 
x
cos y dx (1 2e )sin y dy 0; y(0)
4
?
?
? ? ? ? 
                                             OR
 
Find the general solution of the differential equation: 
dx y tan y x tan y xy
dy y tan y
??
? 
22. If 
ˆ ˆ ˆ ˆ ˆ ˆ
p i j k a nd q i – 2 j k ? ? ? ? ? , find a vector of magnitude 53 units perpendicular to 
the vector q and coplanar with vectors p and q . 
23. Find the vector equation of the line joining (1, 2, 3) and ( –3, 4, 3) and show that it is 
perpendicular to the z-axis. 
 
Page 4


1 
 
MATHEMATICS 
Time allowed : 3 hours                                                             Maximum Marks: 100 
General Instructions: 
1. All questions are compulsory. 
2. This question paper contains 29 questions. 
3. Questions 1 – 4 in Section A are very short-answer type questions carrying 1 mark each.  
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 marks each.  
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 marks each. 
6. Questions 24 – 29 in Section D are long-answer II type questions carrying 6 marks each. 
Section A 
1. If A and B are invertible matrices of order 3, 
1
1
| A | 2 and | (AB) | .
6
?
? ? ? Find |B|. 
2. Differentiate 
22
sin (x ) w.r.t x
2
. 
3. Write the order of the differential equation: 
                                     
3
2
2
d y dy
log x
dx
dx
??
??
??
?? ??
??
??
 
4. Find the acute angle which the line with direction cosines 
11
, , n
36
 makes with 
positive direction of z-axis.  
                                                     OR 
Find the direction cosines of the line:  
x 1 z 1
y
22
??
? ? ? 
Section B 
5. Let A = Z × Z and * be a binary operation on A defined by 
(a, b)*(c, d) (ad bc, bd). ?? 
Find the identity element for * in the set A. 
6. If 
3 1 1 0
A and I
1 2 0 1
? ? ? ?
??
? ? ? ?
?
? ? ? ?
, find k so that 
2
A 5A kI ?? . 
2 
 
7. Find : 
2 2 2
2
(x sin x)sec x
dx
1x
?
?
?
 
8. Find: 
x
3
e (x 3)
dx
(x 1)
?
?
?
 
 
                                     OR 
Find: 
1
4
4
5
(x x)
dx
x
?
?
 
9. Form the differential equation of all circles which touch the x-axis at the origin. 
10. Find the area of the parallelogram whose diagonals are represented by the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a 2i 3j 4k and b 2i j 2k ? ? ? ? ? ? 
OR 
Find the angle between the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a i j k a n d b i – j k ? ? ? ? ? 
11. If A and B are two independent events, prove that A ? and B are also independent. 
12. One bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. A 
ball is transferred from first bag to the second bag and then a ball is drawn from the 
second bag. Find the probability that the ball drawn is red. 
OR 
 If P(A) = 0.6, P(B) = 0.5 and P(A|B) = 0.3, then find P( ? ?? B). 
Section C 
13. Prove that the function f:[0, ?) ? R given by f(x) = 9x
2
 + 6x – 5 is not invertible. Modify 
the codomain of the function f to make it invertible, and hence find f
–1
. 
OR 
Check whether the relation R in the set R of real numbers, defined by 
R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive. 
14. Find the value of : 
11
1
sin 2 tan cos(tan 2 2)
4
??
??
?
??
??
 
 
 
3 
 
15. Using properties of determinants, prove that: 
2 2 2
a b c c b
a c b c a (a b c) (a b c )
a b b a c
??
? ? ? ? ? ? ?
??
 
16. If 
sin x x
dy
y x sin(x ), find
dx
?? 
                                        OR
 
If 
2
2 4 1
2
dy
y log(1 2t t ), x tan t, find
dx
?
? ? ? ? 
17. If 
1
y cos(m cos x)
?
? 
Show that: 
2
22
2
d y dy
(1 x ) x m y
dx
dx
? ? ? = 0 
18. Find the equations of the normal to the curve 
3
y 4x 3x 5 ? ? ? which are perpendicular to 
the line 9x y 5 0 ? ? ? . 
19. Find: 
4
22
x1
dx
x(x 1)
?
?
?
 
20. Evaluate: 
1
2
1
x | x | 1
dx
x 2 | x | 1
?
??
??
?
 
21. Find the particular solution of the following differential equation. 
x
cos y dx (1 2e )sin y dy 0; y(0)
4
?
?
? ? ? ? 
                                             OR
 
Find the general solution of the differential equation: 
dx y tan y x tan y xy
dy y tan y
??
? 
22. If 
ˆ ˆ ˆ ˆ ˆ ˆ
p i j k a nd q i – 2 j k ? ? ? ? ? , find a vector of magnitude 53 units perpendicular to 
the vector q and coplanar with vectors p and q . 
23. Find the vector equation of the line joining (1, 2, 3) and ( –3, 4, 3) and show that it is 
perpendicular to the z-axis. 
 
4 
 
Section D 
24.  If A = 
3 1 2
3 2 3
2 0 1
??
??
?
??
?? ?
??
, find A
–1
. 
Hence, solve the system of equations: 
  3x + 3y+ 2z = 1 
  x + 2y = 4 
  2x – 3y –z = 5 
OR
 
Find the inverse of the following matrix using elementary transformations. 
2 1 3
5 3 1
3 2 3
? ??
??
?
??
?? ?
??
 
25. A cuboidal shaped godown with square base is to be constructed. Three times as much 
cost per square meter is incurred for constructing the roof as compared to the walls. Find 
the dimensions of the godown if it is to enclose a given volume and minimize the cost of 
constructing the roof and the walls. 
26. Find the area bounded by the curves y x, 2y 3 x and x axis ? ? ? ? . 
OR 
Find the area of the region. 
{(x, y) : x
2
 + y
2
 ? 8, x
2
 ? 2y} 
27. Find the equation of the plane through the line 
x 1 y 4 z 4
3 2 2
? ? ?
??
?
 and parallel to the 
line  
x 1 1 y z 2
2 4 1
? ? ?
?? . Hence, find the shortest distance between the lines. 
OR 
Show that the line of intersection of the planes x + 2y + 3z = 8 and 2x + 3y + 4z = 11 is 
coplanar with the line 
x 1 y 1 z 1
1 2 3
? ? ?
?? . Also find the equation of the plane containing 
them.  
 
 
Page 5


1 
 
MATHEMATICS 
Time allowed : 3 hours                                                             Maximum Marks: 100 
General Instructions: 
1. All questions are compulsory. 
2. This question paper contains 29 questions. 
3. Questions 1 – 4 in Section A are very short-answer type questions carrying 1 mark each.  
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 marks each.  
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 marks each. 
6. Questions 24 – 29 in Section D are long-answer II type questions carrying 6 marks each. 
Section A 
1. If A and B are invertible matrices of order 3, 
1
1
| A | 2 and | (AB) | .
6
?
? ? ? Find |B|. 
2. Differentiate 
22
sin (x ) w.r.t x
2
. 
3. Write the order of the differential equation: 
                                     
3
2
2
d y dy
log x
dx
dx
??
??
??
?? ??
??
??
 
4. Find the acute angle which the line with direction cosines 
11
, , n
36
 makes with 
positive direction of z-axis.  
                                                     OR 
Find the direction cosines of the line:  
x 1 z 1
y
22
??
? ? ? 
Section B 
5. Let A = Z × Z and * be a binary operation on A defined by 
(a, b)*(c, d) (ad bc, bd). ?? 
Find the identity element for * in the set A. 
6. If 
3 1 1 0
A and I
1 2 0 1
? ? ? ?
??
? ? ? ?
?
? ? ? ?
, find k so that 
2
A 5A kI ?? . 
2 
 
7. Find : 
2 2 2
2
(x sin x)sec x
dx
1x
?
?
?
 
8. Find: 
x
3
e (x 3)
dx
(x 1)
?
?
?
 
 
                                     OR 
Find: 
1
4
4
5
(x x)
dx
x
?
?
 
9. Form the differential equation of all circles which touch the x-axis at the origin. 
10. Find the area of the parallelogram whose diagonals are represented by the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a 2i 3j 4k and b 2i j 2k ? ? ? ? ? ? 
OR 
Find the angle between the vectors 
ˆ ˆ ˆ ˆ ˆ ˆ
a i j k a n d b i – j k ? ? ? ? ? 
11. If A and B are two independent events, prove that A ? and B are also independent. 
12. One bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. A 
ball is transferred from first bag to the second bag and then a ball is drawn from the 
second bag. Find the probability that the ball drawn is red. 
OR 
 If P(A) = 0.6, P(B) = 0.5 and P(A|B) = 0.3, then find P( ? ?? B). 
Section C 
13. Prove that the function f:[0, ?) ? R given by f(x) = 9x
2
 + 6x – 5 is not invertible. Modify 
the codomain of the function f to make it invertible, and hence find f
–1
. 
OR 
Check whether the relation R in the set R of real numbers, defined by 
R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive. 
14. Find the value of : 
11
1
sin 2 tan cos(tan 2 2)
4
??
??
?
??
??
 
 
 
3 
 
15. Using properties of determinants, prove that: 
2 2 2
a b c c b
a c b c a (a b c) (a b c )
a b b a c
??
? ? ? ? ? ? ?
??
 
16. If 
sin x x
dy
y x sin(x ), find
dx
?? 
                                        OR
 
If 
2
2 4 1
2
dy
y log(1 2t t ), x tan t, find
dx
?
? ? ? ? 
17. If 
1
y cos(m cos x)
?
? 
Show that: 
2
22
2
d y dy
(1 x ) x m y
dx
dx
? ? ? = 0 
18. Find the equations of the normal to the curve 
3
y 4x 3x 5 ? ? ? which are perpendicular to 
the line 9x y 5 0 ? ? ? . 
19. Find: 
4
22
x1
dx
x(x 1)
?
?
?
 
20. Evaluate: 
1
2
1
x | x | 1
dx
x 2 | x | 1
?
??
??
?
 
21. Find the particular solution of the following differential equation. 
x
cos y dx (1 2e )sin y dy 0; y(0)
4
?
?
? ? ? ? 
                                             OR
 
Find the general solution of the differential equation: 
dx y tan y x tan y xy
dy y tan y
??
? 
22. If 
ˆ ˆ ˆ ˆ ˆ ˆ
p i j k a nd q i – 2 j k ? ? ? ? ? , find a vector of magnitude 53 units perpendicular to 
the vector q and coplanar with vectors p and q . 
23. Find the vector equation of the line joining (1, 2, 3) and ( –3, 4, 3) and show that it is 
perpendicular to the z-axis. 
 
4 
 
Section D 
24.  If A = 
3 1 2
3 2 3
2 0 1
??
??
?
??
?? ?
??
, find A
–1
. 
Hence, solve the system of equations: 
  3x + 3y+ 2z = 1 
  x + 2y = 4 
  2x – 3y –z = 5 
OR
 
Find the inverse of the following matrix using elementary transformations. 
2 1 3
5 3 1
3 2 3
? ??
??
?
??
?? ?
??
 
25. A cuboidal shaped godown with square base is to be constructed. Three times as much 
cost per square meter is incurred for constructing the roof as compared to the walls. Find 
the dimensions of the godown if it is to enclose a given volume and minimize the cost of 
constructing the roof and the walls. 
26. Find the area bounded by the curves y x, 2y 3 x and x axis ? ? ? ? . 
OR 
Find the area of the region. 
{(x, y) : x
2
 + y
2
 ? 8, x
2
 ? 2y} 
27. Find the equation of the plane through the line 
x 1 y 4 z 4
3 2 2
? ? ?
??
?
 and parallel to the 
line  
x 1 1 y z 2
2 4 1
? ? ?
?? . Hence, find the shortest distance between the lines. 
OR 
Show that the line of intersection of the planes x + 2y + 3z = 8 and 2x + 3y + 4z = 11 is 
coplanar with the line 
x 1 y 1 z 1
1 2 3
? ? ?
?? . Also find the equation of the plane containing 
them.  
 
 
5 
 
28.  A manufacturer makes two types of toys A and B. Three machine are needed for this 
purpose and the time (in minutes) required for each toy on the machines is given below: 
 
 
Types of Toys 
Machines 
I II III 
A 20 10 10 
B 10 20 30 
 
 The machines I, II and III are available for a maximum of 3 hours, 2 hours and 2 hours 30 
minutes respectively. The profit on each toy of type A is ` 50 and that of type B is ` 60. 
Formulate the above problem as a L.P.P and solve it graphically to maximize profit. 
29. The members of a consulting firm rent cars from three rental agencies: 
50% from agency X, 30% from agency Y and 20% from agency Z. 
From past experience it is known that 9% of the cars from agency X need a service and 
tuning before renting, 12% of the cars from agency Y need a service and tuning before 
renting and 10% of the cars from agency Z need a service and tuning before renting. If 
the rental car delivered to the firm needs service and tuning, find the probability that 
agency Z is not to be blamed.