Sample Question Paper 3 - Math, Class 12 JEE Notes | EduRev

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JEE : Sample Question Paper 3 - Math, Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
Mathematics 
Class XII 
Sample Paper 3 
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and 
draw the graph, so that it represents a function. 
 
 
 
2. Find the value of ‘ ?’ for which ?(î + j + 
ˆ
k ) is a unit vector. 
 
 
3. Find the slope of the tangent to the curve y = x
3
 – x + 1 at the point where the curve cuts 
the y-axis. 
 
 
4. This 3 x 2 matrix gives information about the number of men and women workers in 
three factories I, II and III who lost their jobs in the last 2 months. What do you infer 
from the entry in the third row and second column of this matrix? 
 
                                          Men workers                    Women workers 
 Factory I                    40                                         15 
 
 Factory II                  35                                          40 
 
 Factory III                 72                                          64 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
Mathematics 
Class XII 
Sample Paper 3 
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and 
draw the graph, so that it represents a function. 
 
 
 
2. Find the value of ‘ ?’ for which ?(î + j + 
ˆ
k ) is a unit vector. 
 
 
3. Find the slope of the tangent to the curve y = x
3
 – x + 1 at the point where the curve cuts 
the y-axis. 
 
 
4. This 3 x 2 matrix gives information about the number of men and women workers in 
three factories I, II and III who lost their jobs in the last 2 months. What do you infer 
from the entry in the third row and second column of this matrix? 
 
                                          Men workers                    Women workers 
 Factory I                    40                                         15 
 
 Factory II                  35                                          40 
 
 Factory III                 72                                          64 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
OR 
If for any 2 x 2 square matrix A, A (adj A) = 
80
80
??
??
??
 , then write the value of 
     |A|. 
 
 
SECTION – B 
 
5. Evaluate: 
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
 
 
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
 
 
7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear? 
 
 
 
8. 
?
??
??
??
1
25
Write A for A = .
13
 
 
OR
 
 
If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
 
9. 
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
 
 
10.  
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
 
 
OR 
 
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area 
increasing when the length of an edge is 10 cm ? 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
Mathematics 
Class XII 
Sample Paper 3 
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and 
draw the graph, so that it represents a function. 
 
 
 
2. Find the value of ‘ ?’ for which ?(î + j + 
ˆ
k ) is a unit vector. 
 
 
3. Find the slope of the tangent to the curve y = x
3
 – x + 1 at the point where the curve cuts 
the y-axis. 
 
 
4. This 3 x 2 matrix gives information about the number of men and women workers in 
three factories I, II and III who lost their jobs in the last 2 months. What do you infer 
from the entry in the third row and second column of this matrix? 
 
                                          Men workers                    Women workers 
 Factory I                    40                                         15 
 
 Factory II                  35                                          40 
 
 Factory III                 72                                          64 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
OR 
If for any 2 x 2 square matrix A, A (adj A) = 
80
80
??
??
??
 , then write the value of 
     |A|. 
 
 
SECTION – B 
 
5. Evaluate: 
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
 
 
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
 
 
7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear? 
 
 
 
8. 
?
??
??
??
1
25
Write A for A = .
13
 
 
OR
 
 
If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
 
9. 
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
 
 
10.  
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
 
 
OR 
 
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area 
increasing when the length of an edge is 10 cm ? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
11. Obtain the differential equation of the family of circles passing through the points (a,0) and 
(-a,0). 
 
12. ?
2 1 1
If P(A)= , P(B)= ,    P(A B)= , then find P(A /B).
5 3 5
 
 
OR 
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the 
event “number obtained is even” and B be the event “Number obtained is red.” Find if A 
and B are independent events. 
 
 
 
SECTION – C 
 
13. 
? ?
?
??
3
5
3
x 5 x
Differentiate   ,  wrt  x
7 3x 8 5x
 
OR
 
 
? ? ?
2 '' '
If  y  =  acos(log x) + bsin(log x), prove that  x y xy y 0,
 
 
14.  
 
? ? ? ? Let  f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function
 
 
15. 
 
 
Find the distance between the parallel planes
r. 2i 1j 3k 4 and  r. 6i 3j 9k 13 0
 
 
16.  
 
2 2 2 2
A  plane is at a distance of p units from the origin. 
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?
 
  
 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
Mathematics 
Class XII 
Sample Paper 3 
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and 
draw the graph, so that it represents a function. 
 
 
 
2. Find the value of ‘ ?’ for which ?(î + j + 
ˆ
k ) is a unit vector. 
 
 
3. Find the slope of the tangent to the curve y = x
3
 – x + 1 at the point where the curve cuts 
the y-axis. 
 
 
4. This 3 x 2 matrix gives information about the number of men and women workers in 
three factories I, II and III who lost their jobs in the last 2 months. What do you infer 
from the entry in the third row and second column of this matrix? 
 
                                          Men workers                    Women workers 
 Factory I                    40                                         15 
 
 Factory II                  35                                          40 
 
 Factory III                 72                                          64 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
OR 
If for any 2 x 2 square matrix A, A (adj A) = 
80
80
??
??
??
 , then write the value of 
     |A|. 
 
 
SECTION – B 
 
5. Evaluate: 
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
 
 
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
 
 
7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear? 
 
 
 
8. 
?
??
??
??
1
25
Write A for A = .
13
 
 
OR
 
 
If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
 
9. 
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
 
 
10.  
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
 
 
OR 
 
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area 
increasing when the length of an edge is 10 cm ? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
11. Obtain the differential equation of the family of circles passing through the points (a,0) and 
(-a,0). 
 
12. ?
2 1 1
If P(A)= , P(B)= ,    P(A B)= , then find P(A /B).
5 3 5
 
 
OR 
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the 
event “number obtained is even” and B be the event “Number obtained is red.” Find if A 
and B are independent events. 
 
 
 
SECTION – C 
 
13. 
? ?
?
??
3
5
3
x 5 x
Differentiate   ,  wrt  x
7 3x 8 5x
 
OR
 
 
? ? ?
2 '' '
If  y  =  acos(log x) + bsin(log x), prove that  x y xy y 0,
 
 
14.  
 
? ? ? ? Let  f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function
 
 
15. 
 
 
Find the distance between the parallel planes
r. 2i 1j 3k 4 and  r. 6i 3j 9k 13 0
 
 
16.  
 
2 2 2 2
A  plane is at a distance of p units from the origin. 
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?
 
  
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
17. Four cards are drawn successively with replacement from a well shuffled deck of 52 
cards. What is the probability that  
(i) All the four cards are spades? 
(ii) Only 3 cards are spades?  
(iii) None is a spade? 
 
18. 
2
1 2 1 2
5
Solve the equation: (tan x) (cot x)
8
??
?
?? 
 
 
19. Find the equation of a tangent to  the curve given by 
33
x asin t , y bcos t ?? at  
        a point, where t
2
?
?
.  
20. Evaluate:
4
0
sinx cosx
dx
9 16sin2x
?
?
?
?
 
 
 
21. If 
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
 then prove that 
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??
  
 
OR 
 
 
?
??
? ? ? ?
??
2
2
2
If is one of the cube roots of unity, evaluate the given determinant
1
1
1
 
 
22. ? Show that the function f defined by f(x)  =  1-x + x , x   R is continuous. 
OR 
 Show that a logarithmic function is continuous at every point in its domain. 
 
23. 
? ? ?
?
? ? ? ?
? 2
(3 sin 2)cos
Evaluate: d
5 cos 4sin
 
 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
Mathematics 
Class XII 
Sample Paper 3 
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and 
draw the graph, so that it represents a function. 
 
 
 
2. Find the value of ‘ ?’ for which ?(î + j + 
ˆ
k ) is a unit vector. 
 
 
3. Find the slope of the tangent to the curve y = x
3
 – x + 1 at the point where the curve cuts 
the y-axis. 
 
 
4. This 3 x 2 matrix gives information about the number of men and women workers in 
three factories I, II and III who lost their jobs in the last 2 months. What do you infer 
from the entry in the third row and second column of this matrix? 
 
                                          Men workers                    Women workers 
 Factory I                    40                                         15 
 
 Factory II                  35                                          40 
 
 Factory III                 72                                          64 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
OR 
If for any 2 x 2 square matrix A, A (adj A) = 
80
80
??
??
??
 , then write the value of 
     |A|. 
 
 
SECTION – B 
 
5. Evaluate: 
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
 
 
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
 
 
7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear? 
 
 
 
8. 
?
??
??
??
1
25
Write A for A = .
13
 
 
OR
 
 
If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
 
9. 
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
 
 
10.  
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
 
 
OR 
 
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area 
increasing when the length of an edge is 10 cm ? 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
11. Obtain the differential equation of the family of circles passing through the points (a,0) and 
(-a,0). 
 
12. ?
2 1 1
If P(A)= , P(B)= ,    P(A B)= , then find P(A /B).
5 3 5
 
 
OR 
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the 
event “number obtained is even” and B be the event “Number obtained is red.” Find if A 
and B are independent events. 
 
 
 
SECTION – C 
 
13. 
? ?
?
??
3
5
3
x 5 x
Differentiate   ,  wrt  x
7 3x 8 5x
 
OR
 
 
? ? ?
2 '' '
If  y  =  acos(log x) + bsin(log x), prove that  x y xy y 0,
 
 
14.  
 
? ? ? ? Let  f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function
 
 
15. 
 
 
Find the distance between the parallel planes
r. 2i 1j 3k 4 and  r. 6i 3j 9k 13 0
 
 
16.  
 
2 2 2 2
A  plane is at a distance of p units from the origin. 
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?
 
  
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
17. Four cards are drawn successively with replacement from a well shuffled deck of 52 
cards. What is the probability that  
(i) All the four cards are spades? 
(ii) Only 3 cards are spades?  
(iii) None is a spade? 
 
18. 
2
1 2 1 2
5
Solve the equation: (tan x) (cot x)
8
??
?
?? 
 
 
19. Find the equation of a tangent to  the curve given by 
33
x asin t , y bcos t ?? at  
        a point, where t
2
?
?
.  
20. Evaluate:
4
0
sinx cosx
dx
9 16sin2x
?
?
?
?
 
 
 
21. If 
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
 then prove that 
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??
  
 
OR 
 
 
?
??
? ? ? ?
??
2
2
2
If is one of the cube roots of unity, evaluate the given determinant
1
1
1
 
 
22. ? Show that the function f defined by f(x)  =  1-x + x , x   R is continuous. 
OR 
 Show that a logarithmic function is continuous at every point in its domain. 
 
23. 
? ? ?
?
? ? ? ?
? 2
(3 sin 2)cos
Evaluate: d
5 cos 4sin
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 3  
 
     
SECTION - D 
 
24. Show that the right circular cone of least curved surface and given volume has an 
altitude equal to 2 times the radius of the base. 
 
OR 
 
     
43
Find the points at which the function f given by f(x) = (x - 2 ) (x 1) is minimum. ?
 
 
25. Obtain the inverse of the following matrix using elementary operations. 
0 1 2
A= 1 2 3
3 1 1
??
??
??
??
??
 
26.  
     
?
?
22
22
22
22
Calculate the area 
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a 
ab
xy
(ii) Triangle AOB is in the first quadrant  of the ellipse  + 1,where OA = a and OB = b.
ab
     Find the area enclosed between the chord AB and the arc AB of the ellipse 
(iii) Find the ratio of the two areas found.
 
OR
 
       
??
??
2 2 2
2
Find the smaller of the two areas in which the circle x y 2a
is divided by the parabola y ax, a 0
 
 
27. Find the equation of a plane that is parallel to the x-axis and passes through the line 
common to two intersectiing planes r. i+j+k 1 0 and r. 2i+3j-k 4 
 
28. Two trainee carpenters A and B earn Rs. 150 and Rs. 200 per day respectively. A can 
make 6 frames and 4 stools per day while B can make 10 frames and 4 stools per day. 
How many days shall each work, if it is desired to produce atleast 60 frames and 32 
stools at a minimum labour cost? Solve the problem graphically. 
 
 
 
 
 
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