Sample Question Paper 7 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
Mathematics 
Class XII 
Sample Paper – 7 
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. If A =[aij], such that 
ij
i 2j
a
2
?
? , find the value of element at 3
rd
 column and 2
nd
 row. 
2. Find 
dy
dx
, if y + sin y = cos x 
3. Determine the order and degree of the following differential equation: 
2
dy 1
2
dy
dx
dx
??
??
??
??
 
4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the 
vector 
ˆ ˆˆ
3i 2j 8k ?? . 
OR 
 
Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6). 
SECTION – B  
 
5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is 
even}, is an equivalence relation. 
6. Find matrix A if  
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
. 
7. sin x sin 2x sin 3x dx 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
Mathematics 
Class XII 
Sample Paper – 7 
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. If A =[aij], such that 
ij
i 2j
a
2
?
? , find the value of element at 3
rd
 column and 2
nd
 row. 
2. Find 
dy
dx
, if y + sin y = cos x 
3. Determine the order and degree of the following differential equation: 
2
dy 1
2
dy
dx
dx
??
??
??
??
 
4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the 
vector 
ˆ ˆˆ
3i 2j 8k ?? . 
OR 
 
Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6). 
SECTION – B  
 
5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is 
even}, is an equivalence relation. 
6. Find matrix A if  
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
. 
7. sin x sin 2x sin 3x dx 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
8. Evaluate:
2
2
dx
1 x 1 x
 
OR 
Evaluate:  
sin x a
dx
sin x a
 
9. Form a differential equation corresponding to y
2 
= a(b – x)(b + x), by eliminating 
parameters a and b 
10. If a 5, b 13, a b 60, find a b ? ? ? ? ? 
OR 
Find the value of ? which makes the vectors a, b, and c coplanar, where 
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k . 
11. Find the probability distribution of the number of Kings drawn when two cards are 
drawn one by one, without replacement, from a pack of 52 playing cards 
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls 
respectively. One bag is selected at random and from the bag selected, a ball is 
drawn. Find the probability that the ball is red. 
OR 
A company has two factories to manufacture machinery. Factory I manufactures 
70% of the machinery and factory II manufactures 30% of the machinery. At factory 
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of 
the machinery are rated to be of a standard quality. A machine is chosen at random 
and is found to be of a standard quality. What is the probability that it came from 
factory II? 
SECTION – C  
 
13. Let A = R × R and * be a binary operation on A defined by 
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.  
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A. 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
Mathematics 
Class XII 
Sample Paper – 7 
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. If A =[aij], such that 
ij
i 2j
a
2
?
? , find the value of element at 3
rd
 column and 2
nd
 row. 
2. Find 
dy
dx
, if y + sin y = cos x 
3. Determine the order and degree of the following differential equation: 
2
dy 1
2
dy
dx
dx
??
??
??
??
 
4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the 
vector 
ˆ ˆˆ
3i 2j 8k ?? . 
OR 
 
Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6). 
SECTION – B  
 
5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is 
even}, is an equivalence relation. 
6. Find matrix A if  
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
. 
7. sin x sin 2x sin 3x dx 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
8. Evaluate:
2
2
dx
1 x 1 x
 
OR 
Evaluate:  
sin x a
dx
sin x a
 
9. Form a differential equation corresponding to y
2 
= a(b – x)(b + x), by eliminating 
parameters a and b 
10. If a 5, b 13, a b 60, find a b ? ? ? ? ? 
OR 
Find the value of ? which makes the vectors a, b, and c coplanar, where 
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k . 
11. Find the probability distribution of the number of Kings drawn when two cards are 
drawn one by one, without replacement, from a pack of 52 playing cards 
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls 
respectively. One bag is selected at random and from the bag selected, a ball is 
drawn. Find the probability that the ball is red. 
OR 
A company has two factories to manufacture machinery. Factory I manufactures 
70% of the machinery and factory II manufactures 30% of the machinery. At factory 
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of 
the machinery are rated to be of a standard quality. A machine is chosen at random 
and is found to be of a standard quality. What is the probability that it came from 
factory II? 
SECTION – C  
 
13. Let A = R × R and * be a binary operation on A defined by 
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.  
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A. 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
OR 
 
-1 -1
4 4 4x+3
Consider f : R - - R -  given by f(x) =  Show that f is bijective.
3 3 3x+4
Find the inverse of f and hence find f (0) and x such that f (x)=2
? ? ? ?
?
? ? ? ?
? ? ? ?
 
 
14. Prove that 
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
15. If a, b, and c are in A.P., find the value of the determinant 
? =
x 1 x 2 x a
x 2 x 3 x b
x 3 x 4 x c
? ? ?
? ? ?
? ? ?
 
16. If 
22
1
22
1 x 1 x dy
y tan , 1 x 1,x 0, find
dx
1 x 1 x
?
??
? ? ? ??
? ? ? ? ?
??
? ? ?
??
??
. 
OR 
 
If 
? ?
2 2 1
y dy x y
log x y 2tan ,show that
x dx x y
?
? ??
? ? ?
??
?
??
 
17. If 
22
11
22
x y dy y
cos tan a, prove that .
x y dx x
??
?? ?
??
??
?
??
 
18. A balloon, which always remains spherical on inflation, is being by inflated by 
pumping in 900 cubic centimeters of gas per second. Find the rate at which the 
radius of the balloon increases when the radius is 15 cm. 
19. Evaluate: 
2
x+2
dx
x +5x+6
?
 
20. Prove that, 
0
x
dx
1 sin x
?
?
??
?
 
 
21. Solve the differential equation: 
1
dy
sin x y
dx
?
??
??
??
??
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
Mathematics 
Class XII 
Sample Paper – 7 
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. If A =[aij], such that 
ij
i 2j
a
2
?
? , find the value of element at 3
rd
 column and 2
nd
 row. 
2. Find 
dy
dx
, if y + sin y = cos x 
3. Determine the order and degree of the following differential equation: 
2
dy 1
2
dy
dx
dx
??
??
??
??
 
4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the 
vector 
ˆ ˆˆ
3i 2j 8k ?? . 
OR 
 
Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6). 
SECTION – B  
 
5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is 
even}, is an equivalence relation. 
6. Find matrix A if  
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
. 
7. sin x sin 2x sin 3x dx 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
8. Evaluate:
2
2
dx
1 x 1 x
 
OR 
Evaluate:  
sin x a
dx
sin x a
 
9. Form a differential equation corresponding to y
2 
= a(b – x)(b + x), by eliminating 
parameters a and b 
10. If a 5, b 13, a b 60, find a b ? ? ? ? ? 
OR 
Find the value of ? which makes the vectors a, b, and c coplanar, where 
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k . 
11. Find the probability distribution of the number of Kings drawn when two cards are 
drawn one by one, without replacement, from a pack of 52 playing cards 
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls 
respectively. One bag is selected at random and from the bag selected, a ball is 
drawn. Find the probability that the ball is red. 
OR 
A company has two factories to manufacture machinery. Factory I manufactures 
70% of the machinery and factory II manufactures 30% of the machinery. At factory 
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of 
the machinery are rated to be of a standard quality. A machine is chosen at random 
and is found to be of a standard quality. What is the probability that it came from 
factory II? 
SECTION – C  
 
13. Let A = R × R and * be a binary operation on A defined by 
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.  
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A. 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
OR 
 
-1 -1
4 4 4x+3
Consider f : R - - R -  given by f(x) =  Show that f is bijective.
3 3 3x+4
Find the inverse of f and hence find f (0) and x such that f (x)=2
? ? ? ?
?
? ? ? ?
? ? ? ?
 
 
14. Prove that 
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
15. If a, b, and c are in A.P., find the value of the determinant 
? =
x 1 x 2 x a
x 2 x 3 x b
x 3 x 4 x c
? ? ?
? ? ?
? ? ?
 
16. If 
22
1
22
1 x 1 x dy
y tan , 1 x 1,x 0, find
dx
1 x 1 x
?
??
? ? ? ??
? ? ? ? ?
??
? ? ?
??
??
. 
OR 
 
If 
? ?
2 2 1
y dy x y
log x y 2tan ,show that
x dx x y
?
? ??
? ? ?
??
?
??
 
17. If 
22
11
22
x y dy y
cos tan a, prove that .
x y dx x
??
?? ?
??
??
?
??
 
18. A balloon, which always remains spherical on inflation, is being by inflated by 
pumping in 900 cubic centimeters of gas per second. Find the rate at which the 
radius of the balloon increases when the radius is 15 cm. 
19. Evaluate: 
2
x+2
dx
x +5x+6
?
 
20. Prove that, 
0
x
dx
1 sin x
?
?
??
?
 
 
21. Solve the differential equation: 
1
dy
sin x y
dx
?
??
??
??
??
 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
OR 
Solve the differential equation: 
? ?
2 dy
4x y 1
dx
? ? ? 
 
22. The scalar product of the vector a i j k
? ? ? ?
? ? ? with a unit vector along the sum of vectors  
b 2i 4 j 5k
? ? ? ?
? ? ? and c ? i 2 j 3 k
? ? ? ?
? ? ?  is equal to one. Find the value of ? and hence find 
the unit vector along b c .
??
? 
 
23. Find the shortest distance between the lines given by 
x 3 y 5 z 7
1 2 1
 and 
x 1 y 1 z 1
7 6 1
 
 
SECTION – D 
 
24. If 
111
A 1 1 1
111
??
??
?
??
??
??
,then prove that 
n 1 n 1 n 1
n n 1 n 1 n 1
n 1 n 1 n 1
333
A 3 3 3
333
???
???
???
??
??
?
??
??
??
 for every positive integer n. 
OR 
Let 
x
0 tan
2
A
x
tan 0
2
?? ??
?
?? ??
??
??
?
??
??
?? ??
????
 and I be the identity matrix of order 2.  
Show that: I + A = (I – A) 
cosx sinx
sinx cosx
? ??
??
??
 
25. A square piece of tin with side 18 cm is to be made into a box without a top by 
cutting a square piece from each corner and folding up the flaps. What should be the 
side of the square to be cut off, so that the volume of the box is the largest? Also find 
the maximum volume of the box. 
 
26. 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. 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
Mathematics 
Class XII 
Sample Paper – 7 
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. If A =[aij], such that 
ij
i 2j
a
2
?
? , find the value of element at 3
rd
 column and 2
nd
 row. 
2. Find 
dy
dx
, if y + sin y = cos x 
3. Determine the order and degree of the following differential equation: 
2
dy 1
2
dy
dx
dx
??
??
??
??
 
4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the 
vector 
ˆ ˆˆ
3i 2j 8k ?? . 
OR 
 
Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6). 
SECTION – B  
 
5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is 
even}, is an equivalence relation. 
6. Find matrix A if  
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
. 
7. sin x sin 2x sin 3x dx 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
8. Evaluate:
2
2
dx
1 x 1 x
 
OR 
Evaluate:  
sin x a
dx
sin x a
 
9. Form a differential equation corresponding to y
2 
= a(b – x)(b + x), by eliminating 
parameters a and b 
10. If a 5, b 13, a b 60, find a b ? ? ? ? ? 
OR 
Find the value of ? which makes the vectors a, b, and c coplanar, where 
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k . 
11. Find the probability distribution of the number of Kings drawn when two cards are 
drawn one by one, without replacement, from a pack of 52 playing cards 
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls 
respectively. One bag is selected at random and from the bag selected, a ball is 
drawn. Find the probability that the ball is red. 
OR 
A company has two factories to manufacture machinery. Factory I manufactures 
70% of the machinery and factory II manufactures 30% of the machinery. At factory 
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of 
the machinery are rated to be of a standard quality. A machine is chosen at random 
and is found to be of a standard quality. What is the probability that it came from 
factory II? 
SECTION – C  
 
13. Let A = R × R and * be a binary operation on A defined by 
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.  
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A. 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
OR 
 
-1 -1
4 4 4x+3
Consider f : R - - R -  given by f(x) =  Show that f is bijective.
3 3 3x+4
Find the inverse of f and hence find f (0) and x such that f (x)=2
? ? ? ?
?
? ? ? ?
? ? ? ?
 
 
14. Prove that 
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
15. If a, b, and c are in A.P., find the value of the determinant 
? =
x 1 x 2 x a
x 2 x 3 x b
x 3 x 4 x c
? ? ?
? ? ?
? ? ?
 
16. If 
22
1
22
1 x 1 x dy
y tan , 1 x 1,x 0, find
dx
1 x 1 x
?
??
? ? ? ??
? ? ? ? ?
??
? ? ?
??
??
. 
OR 
 
If 
? ?
2 2 1
y dy x y
log x y 2tan ,show that
x dx x y
?
? ??
? ? ?
??
?
??
 
17. If 
22
11
22
x y dy y
cos tan a, prove that .
x y dx x
??
?? ?
??
??
?
??
 
18. A balloon, which always remains spherical on inflation, is being by inflated by 
pumping in 900 cubic centimeters of gas per second. Find the rate at which the 
radius of the balloon increases when the radius is 15 cm. 
19. Evaluate: 
2
x+2
dx
x +5x+6
?
 
20. Prove that, 
0
x
dx
1 sin x
?
?
??
?
 
 
21. Solve the differential equation: 
1
dy
sin x y
dx
?
??
??
??
??
 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
OR 
Solve the differential equation: 
? ?
2 dy
4x y 1
dx
? ? ? 
 
22. The scalar product of the vector a i j k
? ? ? ?
? ? ? with a unit vector along the sum of vectors  
b 2i 4 j 5k
? ? ? ?
? ? ? and c ? i 2 j 3 k
? ? ? ?
? ? ?  is equal to one. Find the value of ? and hence find 
the unit vector along b c .
??
? 
 
23. Find the shortest distance between the lines given by 
x 3 y 5 z 7
1 2 1
 and 
x 1 y 1 z 1
7 6 1
 
 
SECTION – D 
 
24. If 
111
A 1 1 1
111
??
??
?
??
??
??
,then prove that 
n 1 n 1 n 1
n n 1 n 1 n 1
n 1 n 1 n 1
333
A 3 3 3
333
???
???
???
??
??
?
??
??
??
 for every positive integer n. 
OR 
Let 
x
0 tan
2
A
x
tan 0
2
?? ??
?
?? ??
??
??
?
??
??
?? ??
????
 and I be the identity matrix of order 2.  
Show that: I + A = (I – A) 
cosx sinx
sinx cosx
? ??
??
??
 
25. A square piece of tin with side 18 cm is to be made into a box without a top by 
cutting a square piece from each corner and folding up the flaps. What should be the 
side of the square to be cut off, so that the volume of the box is the largest? Also find 
the maximum volume of the box. 
 
26. 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. 
  
 
CBSE XII | Mathematics 
Sample Paper – 7  
 
     
OR 
Find the area of the smaller region bounded by the ellipse 
22
xy
1
94
?? and the 
straight line
xy
1
32
?? . 
27. Find the angle between the following pair of lines: 
x 2 y 1 z 3 x 2 2y 8 z 5
and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ? ?
? ? ?
 
And check whether the lines are parallel or perpendicular. 
 
OR 
Write the vector equations of the following lines and hence determine the distance 
between them: 
x 1 y 2 z 4 x 3 y 3 z 5
;
2 3 6 4 6 12
? ? ? ? ? ?
? ? ? ? 
 
28. A brick manufacturer has two depots, A and B, with stocks of 30,000 and 20,000 
bricks respectively. He receives orders from three builders P, Q and R for 15,000, 
20,000 and 15,000 bricks respectively. The cost in rupees for transporting 1000 
bricks to the builders from the depots are given below:  
            To 
From 
P Q R 
A 40 20 30 
B 20 60 40 
 
How should the manufacturer fulfil the orders so as to keep the cost of 
transportation minimum? 
 
29. A factory manufactures screws. Machines X, Y and Z manufacture respectively 1000, 
2000, and 3000 screws, of which 1%, 1.5% and 2 % of their outputs are respectively 
defective. A screw is drawn at random from the product and is found to be defective. 
What is the probability that it is manufactured by machine X?