Sample Question Paper 2 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XII Mathematics 
Sample Paper – 2 
Time: 3 hrs  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. Is ‘*’, defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. (a, b), a binary operation?  
         
   
2. If
2 3 1 0
A' and B
1 2 1 2
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
, then find (A + 2B)’.  
OR 
0   a   -3
If the matrix A = 2    0  -1 is skew symmetric, find the values of 'a' and 'b'
b     1   0
??
??
??
??
??
     
     
3. Find the projection of a = 
ˆ
i 3k ? on b = 
ˆ ˆ
3i j 4k ?? .   
     
4. Find the equation of a line parallel to the x-axis and passing through the origin.     
 
       
SECTION – B 
 
5. Find the value of x. 
11
3
sin sin cos x 1
5
??
??
??
??
??
 
 
 
6. 
?? ??
??
??
5 x x 1
For what value of x, the matrix  is singular?
24
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XII Mathematics 
Sample Paper – 2 
Time: 3 hrs  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. Is ‘*’, defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. (a, b), a binary operation?  
         
   
2. If
2 3 1 0
A' and B
1 2 1 2
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
, then find (A + 2B)’.  
OR 
0   a   -3
If the matrix A = 2    0  -1 is skew symmetric, find the values of 'a' and 'b'
b     1   0
??
??
??
??
??
     
     
3. Find the projection of a = 
ˆ
i 3k ? on b = 
ˆ ˆ
3i j 4k ?? .   
     
4. Find the equation of a line parallel to the x-axis and passing through the origin.     
 
       
SECTION – B 
 
5. Find the value of x. 
11
3
sin sin cos x 1
5
??
??
??
??
??
 
 
 
6. 
?? ??
??
??
5 x x 1
For what value of x, the matrix  is singular?
24
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7.  
??
32
The income (I) of a doctor is given by 
I = x 3x 5x. 
Can an insurance agent ensure the growth of his income?
 
 
8. ? ? ? Find the distance of the plane 3x 4y 12z 3  from the origin. 
 
OR 
 
Find the shortest distance between the lines
r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k)
? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
 
 
9. Without expanding, find the value of the following determinant 
 
0 q r r s
r q 0 p q
s r q r 0
??
??
??
 
 
10. Evaluate: 
1 cot x
dx
x log sin x
?
?
?
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
 
11. ? ? ? Write the direction cosines of the vectors 2i j 5k. 
 
 
12. If A = 
a 0 0
0 a 0
0 0 a
??
??
??
??
??
, where a is a non zero real number then without actually evaluating 
adjA, find the value of |adj A|. 
 
OR 
 
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XII Mathematics 
Sample Paper – 2 
Time: 3 hrs  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. Is ‘*’, defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. (a, b), a binary operation?  
         
   
2. If
2 3 1 0
A' and B
1 2 1 2
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
, then find (A + 2B)’.  
OR 
0   a   -3
If the matrix A = 2    0  -1 is skew symmetric, find the values of 'a' and 'b'
b     1   0
??
??
??
??
??
     
     
3. Find the projection of a = 
ˆ
i 3k ? on b = 
ˆ ˆ
3i j 4k ?? .   
     
4. Find the equation of a line parallel to the x-axis and passing through the origin.     
 
       
SECTION – B 
 
5. Find the value of x. 
11
3
sin sin cos x 1
5
??
??
??
??
??
 
 
 
6. 
?? ??
??
??
5 x x 1
For what value of x, the matrix  is singular?
24
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7.  
??
32
The income (I) of a doctor is given by 
I = x 3x 5x. 
Can an insurance agent ensure the growth of his income?
 
 
8. ? ? ? Find the distance of the plane 3x 4y 12z 3  from the origin. 
 
OR 
 
Find the shortest distance between the lines
r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k)
? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
 
 
9. Without expanding, find the value of the following determinant 
 
0 q r r s
r q 0 p q
s r q r 0
??
??
??
 
 
10. Evaluate: 
1 cot x
dx
x log sin x
?
?
?
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
 
11. ? ? ? Write the direction cosines of the vectors 2i j 5k. 
 
 
12. If A = 
a 0 0
0 a 0
0 0 a
??
??
??
??
??
, where a is a non zero real number then without actually evaluating 
adjA, find the value of |adj A|. 
 
OR 
 
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  3 
 
SECTION – C 
 
13. Show that the function f: N ? N defined by f(n) = n – (–1)
n
 for all (n ? N), is a bijection. 
 
OR 
 
   Show that relation R defined by (a, b) R (c, d) ? a + d = b + c on the set N x N is an 
equivalence relation.  
 
14. Find the value of 
2
11
22
1 2x 1 y
tan sin cos
2
1 x 1 y
??
?? ?? ??
? ??
?? ?
?? ??
??
? ? ? ?
?? ?? ?? ??
, |x|<1, y > 0, xy < 1. 
 
15. Using properties of determinants prove that 
 
2
2
2
a 1 ab ac
ab b 1 bc
ca cb c 1
?
?
?
 = 1 + a² + b² + c² 
OR 
 
 Find the equation of the line joining points A(1, 3) and B(0, 0) using determinants and 
find k if D(k, 0) is a point such that the area of ?ABD is 3 square units. 
 
16. If sin y = x sin(a + y), prove that 
? ?
2
sin a y
dy
dx sina
?
? 
 
OR 
 
   Differentiate w.r.t. x  
   y = (sin x)
tan x 
+ (cos x)
sec x 
 
17. Discuss the continuity of the function f(x) at x = 1 
 
31
x , x 1
22
3
Given , x 1
2
3
x , 1 x 2
2
?
? ? ?
?
?
?
?
?
?
?
? ? ?
?
?
 
 
18. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums. 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XII Mathematics 
Sample Paper – 2 
Time: 3 hrs  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. Is ‘*’, defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. (a, b), a binary operation?  
         
   
2. If
2 3 1 0
A' and B
1 2 1 2
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
, then find (A + 2B)’.  
OR 
0   a   -3
If the matrix A = 2    0  -1 is skew symmetric, find the values of 'a' and 'b'
b     1   0
??
??
??
??
??
     
     
3. Find the projection of a = 
ˆ
i 3k ? on b = 
ˆ ˆ
3i j 4k ?? .   
     
4. Find the equation of a line parallel to the x-axis and passing through the origin.     
 
       
SECTION – B 
 
5. Find the value of x. 
11
3
sin sin cos x 1
5
??
??
??
??
??
 
 
 
6. 
?? ??
??
??
5 x x 1
For what value of x, the matrix  is singular?
24
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7.  
??
32
The income (I) of a doctor is given by 
I = x 3x 5x. 
Can an insurance agent ensure the growth of his income?
 
 
8. ? ? ? Find the distance of the plane 3x 4y 12z 3  from the origin. 
 
OR 
 
Find the shortest distance between the lines
r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k)
? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
 
 
9. Without expanding, find the value of the following determinant 
 
0 q r r s
r q 0 p q
s r q r 0
??
??
??
 
 
10. Evaluate: 
1 cot x
dx
x log sin x
?
?
?
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
 
11. ? ? ? Write the direction cosines of the vectors 2i j 5k. 
 
 
12. If A = 
a 0 0
0 a 0
0 0 a
??
??
??
??
??
, where a is a non zero real number then without actually evaluating 
adjA, find the value of |adj A|. 
 
OR 
 
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  3 
 
SECTION – C 
 
13. Show that the function f: N ? N defined by f(n) = n – (–1)
n
 for all (n ? N), is a bijection. 
 
OR 
 
   Show that relation R defined by (a, b) R (c, d) ? a + d = b + c on the set N x N is an 
equivalence relation.  
 
14. Find the value of 
2
11
22
1 2x 1 y
tan sin cos
2
1 x 1 y
??
?? ?? ??
? ??
?? ?
?? ??
??
? ? ? ?
?? ?? ?? ??
, |x|<1, y > 0, xy < 1. 
 
15. Using properties of determinants prove that 
 
2
2
2
a 1 ab ac
ab b 1 bc
ca cb c 1
?
?
?
 = 1 + a² + b² + c² 
OR 
 
 Find the equation of the line joining points A(1, 3) and B(0, 0) using determinants and 
find k if D(k, 0) is a point such that the area of ?ABD is 3 square units. 
 
16. If sin y = x sin(a + y), prove that 
? ?
2
sin a y
dy
dx sina
?
? 
 
OR 
 
   Differentiate w.r.t. x  
   y = (sin x)
tan x 
+ (cos x)
sec x 
 
17. Discuss the continuity of the function f(x) at x = 1 
 
31
x , x 1
22
3
Given , x 1
2
3
x , 1 x 2
2
?
? ? ?
?
?
?
?
?
?
?
? ? ?
?
?
 
 
18. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums. 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  4 
 
19. Evaluate: 
3
2
x
dx
x 5 x
?
??
 
 
20. A company has two plants to manufacturing scooters. Plant I manufactures 70% of the 
scooters and plant II manufactures 30%. At plant I, 30% of the scooters are rated of 
standard quality and at plant II, 90% of the scooters are rated of standard quality. A 
scooter is chosen at random and is found to be of standard quality. Find the probability 
that it is manufactured by plant II. 
 
     
21. Find the co-ordinates of points on line 
x 1 y 2 z 3
2 3 6
? ? ?
?? , which are at a distance of 3 
units from the point (1, -2, 3). 
 
22. Give the intervals in which the function f(x) = 
4sin x 2x x cos x
2 cos x
??
?
 is increasing or 
decreasing. 
 
23.  If 
1
2
2x
u sin
1x
?
??
?
??
? ??
 and
1
2
2x
v tan
1x
?
??
?
??
? ??
, where -1 < x < 1, then write the value of  
du
dv
. 
 
SECTION – D 
 
24. Given two matrices, 
1 1 0 2 2 4
A 2 3 4 and B 4 2 4
0 1 2 2 1 5
?? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
, verify that BA = 6I, use the 
result to solve the system x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7. 
 
OR 
 Find the inverse of 
3 0 1
A 2 3 0
0 4 1
? ??
??
?
??
??
??
  by elementary row transformation. 
 
25.  Solve the differential equation 
  (x² - y²) dx + 2xy dy = 0 
  Given that y = 1 when x = 1. 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XII Mathematics 
Sample Paper – 2 
Time: 3 hrs  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. Is ‘*’, defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. (a, b), a binary operation?  
         
   
2. If
2 3 1 0
A' and B
1 2 1 2
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
, then find (A + 2B)’.  
OR 
0   a   -3
If the matrix A = 2    0  -1 is skew symmetric, find the values of 'a' and 'b'
b     1   0
??
??
??
??
??
     
     
3. Find the projection of a = 
ˆ
i 3k ? on b = 
ˆ ˆ
3i j 4k ?? .   
     
4. Find the equation of a line parallel to the x-axis and passing through the origin.     
 
       
SECTION – B 
 
5. Find the value of x. 
11
3
sin sin cos x 1
5
??
??
??
??
??
 
 
 
6. 
?? ??
??
??
5 x x 1
For what value of x, the matrix  is singular?
24
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7.  
??
32
The income (I) of a doctor is given by 
I = x 3x 5x. 
Can an insurance agent ensure the growth of his income?
 
 
8. ? ? ? Find the distance of the plane 3x 4y 12z 3  from the origin. 
 
OR 
 
Find the shortest distance between the lines
r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k)
? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
 
 
9. Without expanding, find the value of the following determinant 
 
0 q r r s
r q 0 p q
s r q r 0
??
??
??
 
 
10. Evaluate: 
1 cot x
dx
x log sin x
?
?
?
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
 
11. ? ? ? Write the direction cosines of the vectors 2i j 5k. 
 
 
12. If A = 
a 0 0
0 a 0
0 0 a
??
??
??
??
??
, where a is a non zero real number then without actually evaluating 
adjA, find the value of |adj A|. 
 
OR 
 
-1 -1
2   -3
Given A = , compute A and show that 2A 9I A
-4   7
??
??
??
??
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  3 
 
SECTION – C 
 
13. Show that the function f: N ? N defined by f(n) = n – (–1)
n
 for all (n ? N), is a bijection. 
 
OR 
 
   Show that relation R defined by (a, b) R (c, d) ? a + d = b + c on the set N x N is an 
equivalence relation.  
 
14. Find the value of 
2
11
22
1 2x 1 y
tan sin cos
2
1 x 1 y
??
?? ?? ??
? ??
?? ?
?? ??
??
? ? ? ?
?? ?? ?? ??
, |x|<1, y > 0, xy < 1. 
 
15. Using properties of determinants prove that 
 
2
2
2
a 1 ab ac
ab b 1 bc
ca cb c 1
?
?
?
 = 1 + a² + b² + c² 
OR 
 
 Find the equation of the line joining points A(1, 3) and B(0, 0) using determinants and 
find k if D(k, 0) is a point such that the area of ?ABD is 3 square units. 
 
16. If sin y = x sin(a + y), prove that 
? ?
2
sin a y
dy
dx sina
?
? 
 
OR 
 
   Differentiate w.r.t. x  
   y = (sin x)
tan x 
+ (cos x)
sec x 
 
17. Discuss the continuity of the function f(x) at x = 1 
 
31
x , x 1
22
3
Given , x 1
2
3
x , 1 x 2
2
?
? ? ?
?
?
?
?
?
?
?
? ? ?
?
?
 
 
18. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums. 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  4 
 
19. Evaluate: 
3
2
x
dx
x 5 x
?
??
 
 
20. A company has two plants to manufacturing scooters. Plant I manufactures 70% of the 
scooters and plant II manufactures 30%. At plant I, 30% of the scooters are rated of 
standard quality and at plant II, 90% of the scooters are rated of standard quality. A 
scooter is chosen at random and is found to be of standard quality. Find the probability 
that it is manufactured by plant II. 
 
     
21. Find the co-ordinates of points on line 
x 1 y 2 z 3
2 3 6
? ? ?
?? , which are at a distance of 3 
units from the point (1, -2, 3). 
 
22. Give the intervals in which the function f(x) = 
4sin x 2x x cos x
2 cos x
??
?
 is increasing or 
decreasing. 
 
23.  If 
1
2
2x
u sin
1x
?
??
?
??
? ??
 and
1
2
2x
v tan
1x
?
??
?
??
? ??
, where -1 < x < 1, then write the value of  
du
dv
. 
 
SECTION – D 
 
24. Given two matrices, 
1 1 0 2 2 4
A 2 3 4 and B 4 2 4
0 1 2 2 1 5
?? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
, verify that BA = 6I, use the 
result to solve the system x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7. 
 
OR 
 Find the inverse of 
3 0 1
A 2 3 0
0 4 1
? ??
??
?
??
??
??
  by elementary row transformation. 
 
25.  Solve the differential equation 
  (x² - y²) dx + 2xy dy = 0 
  Given that y = 1 when x = 1. 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  5 
26.  A manufacturing company makes two models A and B of a product. Each piece of Model 
A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of 
Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For 
fabricating and finishing, the maximum labour hours available are 180 and 30 
respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 
12000 on each piece of Model B. How many pieces of Model A and Model B should be 
manufactured per week to realise a maximum profit? What is the maximum profit per 
week? 
 
OR 
 
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. 
 
 
27. Find the volume of the largest cylinder which can be inscribed in a sphere of radius r. 
 
28. Prove that the curves y² = 4x and x² = 4y divide the area of the square bonded by x = 0,  
x = 4, y = 4, and y = 0 into three equal parts. 
 
29. The probability of a shooter hitting a target is
3
4
. How many minimum numbers of 
times must he fire so that the probability of hitting the target atleast once is more that 
0.99? 
 
OR 
 
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.