Sample Question Paper 6 - Math, Class 12 JEE Notes | EduRev

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JEE : Sample Question Paper 6 - Math, Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
Mathematics 
Class XII 
Sample Paper – 6 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
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. Use of calculators is not permitted. 
 
SECTION – A 
 
1. A matrix has 12 elements. What are the possible orders it can have? 
 
2. Differentiate sin(2x
2
) w.r.t. x 
 
3. Determine the order and degree of the following differential equation: 
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??
 
4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has 
direction ratios proportional to 1, 1, -2. 
OR 
Find the equation of a line passing through (1,-1, 0) and parallel to the line 
x 2 2y 1 5 z
3 2 1
? ? ?
?? 
 
SECTION – B  
 
5. (i) Is the binary operation *, defined on set N, given by  
      
ab
a * b for all a,b N,
2
commutative?  
(ii) Is the above binary operation * associative? 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
Mathematics 
Class XII 
Sample Paper – 6 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
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. Use of calculators is not permitted. 
 
SECTION – A 
 
1. A matrix has 12 elements. What are the possible orders it can have? 
 
2. Differentiate sin(2x
2
) w.r.t. x 
 
3. Determine the order and degree of the following differential equation: 
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??
 
4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has 
direction ratios proportional to 1, 1, -2. 
OR 
Find the equation of a line passing through (1,-1, 0) and parallel to the line 
x 2 2y 1 5 z
3 2 1
? ? ?
?? 
 
SECTION – B  
 
5. (i) Is the binary operation *, defined on set N, given by  
      
ab
a * b for all a,b N,
2
commutative?  
(ii) Is the above binary operation * associative? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b. 
 
7. Evaluate: 
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx 
OR 
 
Evaluate: 
? ?
2
1x
x 1 2x
?
?
?
dx 
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??
 
 
9. Form differential equations of the family of curves represented by c( y + c )
2 
= x
3
, 
where c is a parameter 
 
10. Find the angle between and ab.  
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ? 
 
OR 
 
Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?
 
 
11. A die is tossed thrice. Find the probability of getting an odd number at least once. 
OR 
A random variable X has the following probability distribution. Find
 
X 0 1 2 3 4 5 
P(X) 0.1 K 0.2 2K 0.3 K 
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3) 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
Mathematics 
Class XII 
Sample Paper – 6 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
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. Use of calculators is not permitted. 
 
SECTION – A 
 
1. A matrix has 12 elements. What are the possible orders it can have? 
 
2. Differentiate sin(2x
2
) w.r.t. x 
 
3. Determine the order and degree of the following differential equation: 
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??
 
4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has 
direction ratios proportional to 1, 1, -2. 
OR 
Find the equation of a line passing through (1,-1, 0) and parallel to the line 
x 2 2y 1 5 z
3 2 1
? ? ?
?? 
 
SECTION – B  
 
5. (i) Is the binary operation *, defined on set N, given by  
      
ab
a * b for all a,b N,
2
commutative?  
(ii) Is the above binary operation * associative? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b. 
 
7. Evaluate: 
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx 
OR 
 
Evaluate: 
? ?
2
1x
x 1 2x
?
?
?
dx 
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??
 
 
9. Form differential equations of the family of curves represented by c( y + c )
2 
= x
3
, 
where c is a parameter 
 
10. Find the angle between and ab.  
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ? 
 
OR 
 
Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?
 
 
11. A die is tossed thrice. Find the probability of getting an odd number at least once. 
OR 
A random variable X has the following probability distribution. Find
 
X 0 1 2 3 4 5 
P(X) 0.1 K 0.2 2K 0.3 K 
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3) 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
12. Two cards are drawn successively with replacement from a well shuffled pack of 
52 cards. Find the probability distribution of the number of aces.  
 
SECTION – C  
 
13. Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary 
operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then 
find 
(i) The identify element of * in A. 
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and 
??
??
??
1
,4
2
. 
OR 
 
Let f : W ? W 
be defined as 
n 1, if n is odd
f n 
n 1, if n is even
 
Show that f is invertible and find the inverse of f. Here, W is the set of all whole 
numbers. 
 
 
14. Solve the Equation:  
    
11
1x
1+x
x,(x 0)
1
  tan = tan
2
??
?
?
??
??
??
 
 
 
15. Show that x = 2 is a root of the equation formed by the following determinant  
      
x -6 -1
2 -3x x-3 0
-3 2x x+2
?
 
Hence, solve the equation. 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
Mathematics 
Class XII 
Sample Paper – 6 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
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. Use of calculators is not permitted. 
 
SECTION – A 
 
1. A matrix has 12 elements. What are the possible orders it can have? 
 
2. Differentiate sin(2x
2
) w.r.t. x 
 
3. Determine the order and degree of the following differential equation: 
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??
 
4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has 
direction ratios proportional to 1, 1, -2. 
OR 
Find the equation of a line passing through (1,-1, 0) and parallel to the line 
x 2 2y 1 5 z
3 2 1
? ? ?
?? 
 
SECTION – B  
 
5. (i) Is the binary operation *, defined on set N, given by  
      
ab
a * b for all a,b N,
2
commutative?  
(ii) Is the above binary operation * associative? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b. 
 
7. Evaluate: 
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx 
OR 
 
Evaluate: 
? ?
2
1x
x 1 2x
?
?
?
dx 
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??
 
 
9. Form differential equations of the family of curves represented by c( y + c )
2 
= x
3
, 
where c is a parameter 
 
10. Find the angle between and ab.  
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ? 
 
OR 
 
Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?
 
 
11. A die is tossed thrice. Find the probability of getting an odd number at least once. 
OR 
A random variable X has the following probability distribution. Find
 
X 0 1 2 3 4 5 
P(X) 0.1 K 0.2 2K 0.3 K 
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3) 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
12. Two cards are drawn successively with replacement from a well shuffled pack of 
52 cards. Find the probability distribution of the number of aces.  
 
SECTION – C  
 
13. Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary 
operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then 
find 
(i) The identify element of * in A. 
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and 
??
??
??
1
,4
2
. 
OR 
 
Let f : W ? W 
be defined as 
n 1, if n is odd
f n 
n 1, if n is even
 
Show that f is invertible and find the inverse of f. Here, W is the set of all whole 
numbers. 
 
 
14. Solve the Equation:  
    
11
1x
1+x
x,(x 0)
1
  tan = tan
2
??
?
?
??
??
??
 
 
 
15. Show that x = 2 is a root of the equation formed by the following determinant  
      
x -6 -1
2 -3x x-3 0
-3 2x x+2
?
 
Hence, solve the equation. 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
16. If 
? ?
2
1 x dy
y ,prove that 1 x y 0
1 x dx
?
? ? ? ?
?
 
 
OR 
Differentiate w.r.t. x 
log10x + logx10 + logx x + log1010 
17. Differentiate w.r.t. x  
12
cos 1 x
ye
?
?
? 
 
18. Find the equation of tangent and normal to the curve 
5x
y 3e ?? where it crosses 
the y-axis. 
 
19. Evaluate:  
2
5x 3
dx
x 4x 10
?
?
??
 
 
 
20. Evaluate: 
? ?
2
2
x +2x +1 dx
0
?
 as limit of sum. 
 
21. Solve the differential equation (1 + e
2x
 ) dy +( 1 + y
2
 )e
x
 dx = 0  
given that when x = 0, y = 1. 
OR 
 
Solve the differential equation x (1 + y
2
) dx - y (1 + x
2
)dy = 0,  
given that y = 0, when x = 1. 
 
22. If a i j k and b j k , find a vector c such that a c b and a.c 3 
 
23. A variable plane is at a constant distance p from the origin and meet the 
coordinate axes in A, B, C. Show that the locus of the centroid of the tetrahedron 
OABC is  
x
-2
+y
-2
+z
-2
=16p
-2
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
Mathematics 
Class XII 
Sample Paper – 6 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
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. Use of calculators is not permitted. 
 
SECTION – A 
 
1. A matrix has 12 elements. What are the possible orders it can have? 
 
2. Differentiate sin(2x
2
) w.r.t. x 
 
3. Determine the order and degree of the following differential equation: 
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??
 
4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has 
direction ratios proportional to 1, 1, -2. 
OR 
Find the equation of a line passing through (1,-1, 0) and parallel to the line 
x 2 2y 1 5 z
3 2 1
? ? ?
?? 
 
SECTION – B  
 
5. (i) Is the binary operation *, defined on set N, given by  
      
ab
a * b for all a,b N,
2
commutative?  
(ii) Is the above binary operation * associative? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b. 
 
7. Evaluate: 
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx 
OR 
 
Evaluate: 
? ?
2
1x
x 1 2x
?
?
?
dx 
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??
 
 
9. Form differential equations of the family of curves represented by c( y + c )
2 
= x
3
, 
where c is a parameter 
 
10. Find the angle between and ab.  
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ? 
 
OR 
 
Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?
 
 
11. A die is tossed thrice. Find the probability of getting an odd number at least once. 
OR 
A random variable X has the following probability distribution. Find
 
X 0 1 2 3 4 5 
P(X) 0.1 K 0.2 2K 0.3 K 
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3) 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
12. Two cards are drawn successively with replacement from a well shuffled pack of 
52 cards. Find the probability distribution of the number of aces.  
 
SECTION – C  
 
13. Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary 
operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then 
find 
(i) The identify element of * in A. 
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and 
??
??
??
1
,4
2
. 
OR 
 
Let f : W ? W 
be defined as 
n 1, if n is odd
f n 
n 1, if n is even
 
Show that f is invertible and find the inverse of f. Here, W is the set of all whole 
numbers. 
 
 
14. Solve the Equation:  
    
11
1x
1+x
x,(x 0)
1
  tan = tan
2
??
?
?
??
??
??
 
 
 
15. Show that x = 2 is a root of the equation formed by the following determinant  
      
x -6 -1
2 -3x x-3 0
-3 2x x+2
?
 
Hence, solve the equation. 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
16. If 
? ?
2
1 x dy
y ,prove that 1 x y 0
1 x dx
?
? ? ? ?
?
 
 
OR 
Differentiate w.r.t. x 
log10x + logx10 + logx x + log1010 
17. Differentiate w.r.t. x  
12
cos 1 x
ye
?
?
? 
 
18. Find the equation of tangent and normal to the curve 
5x
y 3e ?? where it crosses 
the y-axis. 
 
19. Evaluate:  
2
5x 3
dx
x 4x 10
?
?
??
 
 
 
20. Evaluate: 
? ?
2
2
x +2x +1 dx
0
?
 as limit of sum. 
 
21. Solve the differential equation (1 + e
2x
 ) dy +( 1 + y
2
 )e
x
 dx = 0  
given that when x = 0, y = 1. 
OR 
 
Solve the differential equation x (1 + y
2
) dx - y (1 + x
2
)dy = 0,  
given that y = 0, when x = 1. 
 
22. If a i j k and b j k , find a vector c such that a c b and a.c 3 
 
23. A variable plane is at a constant distance p from the origin and meet the 
coordinate axes in A, B, C. Show that the locus of the centroid of the tetrahedron 
OABC is  
x
-2
+y
-2
+z
-2
=16p
-2
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 6 
 
     
SECTION – D 
 
24. Let A = 
23
12
??
??
?
??
 and f(x) = x
2
 – 4x + 7. Show that f(A) = O. Use this result to find 
A
5
. 
OR 
Let f(x) = x
2
 – 5x + 6. Find f(A), if A = 
2 0 1
2 1 3
1 1 0
??
??
??
?? ?
??
 
 
25. A window is in the form of a rectangle surmounted by a semicircular opening. 
The total perimeter of the window is 10 m. Find the dimensions of the window 
to admit maximum light through the whole opening. 
       
26. Using integration, find the area of the circle x
2 
+ y
2 
= 16 which is exterior to the 
parabola y
2 
= 6x. 
OR 
Find the area of the smaller region bounded by the ellipse  
22
22
x y x y
1 and the line 1
ab
ab
 
  
27. Find the equation of the plane passing through the point (-1, 3, 2) and 
perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0. 
 
OR 
Find the equation of the plane which contains the line of intersection of the 
planes
? ?
r. i 2j 3k 4 0 ? ? ? ? , 
? ?
r. 2i j k 5 0 ? ? ? ? and which is perpendicular to 
the plane
? ?
ˆ
r. 5i 3j 6k 8 0 ? ? ? ? . 
 
28. A firm makes items A and B and the total number of items it can make in a day is 
24. It takes one hour to make item A and only half an hour to make item B. The 
maximum time available per day is 16 hours. The profit per item of A is Rs. 300 
and Rs. 160 on one item of B. How many items of each type should be produced 
to maximize the profit? Solve the problem graphically. 
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