Sample Question Paper 8 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
Mathematics 
Class XII 
Sample Paper – 8 
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. Write the position of the element 6 in the given matrix, and denote it as aij. 
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
 
 
2. Find 
dy
dx
, if 2x + 3y = cos x 
3. Is the differential equation given by 
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give 
reason. 
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for 
the line. 
OR 
 
Find the angle between following pairs of line 
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ? 
 
 
SECTION – B  
 
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. 
Write the operation table of the operation *. 
6.  Solve the matrix equation 
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
 
 
7. Evaluate:  
2
5x 2
dx
1 2x 3x
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
Mathematics 
Class XII 
Sample Paper – 8 
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. Write the position of the element 6 in the given matrix, and denote it as aij. 
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
 
 
2. Find 
dy
dx
, if 2x + 3y = cos x 
3. Is the differential equation given by 
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give 
reason. 
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for 
the line. 
OR 
 
Find the angle between following pairs of line 
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ? 
 
 
SECTION – B  
 
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. 
Write the operation table of the operation *. 
6.  Solve the matrix equation 
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
 
 
7. Evaluate:  
2
5x 2
dx
1 2x 3x
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
8. Evaluate:  
2
22
x
dx
x 4 x 9
 
 
OR 
Evaluate:  
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being 
parameters. 
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
 
OR 
 
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k 
and c 2i j 4k . 
 
 
11. 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? 
 
12. A random variable X  has the following probability  distribution : 
X 0 1 2 3 4 5 6 7 
P(X) 0 k 2k 2k 3k k
2
 2k
2 
7k
2
 +k
 
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ?  X < 3) 
OR 
 
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find 
their respective probabilities of winning, if A starts first. 
 
 
SECTION – C  
 
13. Let A = Q x Q and let * be a binary operation on A defined by                                                              
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative 
and associative. Then, with respect to * on A 
(i) Find the identify element in A. 
(ii) Find the invertible elements of A. 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
Mathematics 
Class XII 
Sample Paper – 8 
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. Write the position of the element 6 in the given matrix, and denote it as aij. 
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
 
 
2. Find 
dy
dx
, if 2x + 3y = cos x 
3. Is the differential equation given by 
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give 
reason. 
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for 
the line. 
OR 
 
Find the angle between following pairs of line 
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ? 
 
 
SECTION – B  
 
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. 
Write the operation table of the operation *. 
6.  Solve the matrix equation 
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
 
 
7. Evaluate:  
2
5x 2
dx
1 2x 3x
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
8. Evaluate:  
2
22
x
dx
x 4 x 9
 
 
OR 
Evaluate:  
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being 
parameters. 
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
 
OR 
 
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k 
and c 2i j 4k . 
 
 
11. 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? 
 
12. A random variable X  has the following probability  distribution : 
X 0 1 2 3 4 5 6 7 
P(X) 0 k 2k 2k 3k k
2
 2k
2 
7k
2
 +k
 
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ?  X < 3) 
OR 
 
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find 
their respective probabilities of winning, if A starts first. 
 
 
SECTION – C  
 
13. Let A = Q x Q and let * be a binary operation on A defined by                                                              
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative 
and associative. Then, with respect to * on A 
(i) Find the identify element in A. 
(ii) Find the invertible elements of A. 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
OR 
 
Let A = Q × Q, Q being the set of rational. Let ‘*’ be a binary operation on A, defined 
by (a, b) * (c, d) = (ac, ad + b). Show that 
(i) ‘*’ is not commutative 
(ii) ‘*’ is associative  
(iii) The identity element with respect to ‘*’ is (1, 0) 
 
14. Write in the simplest form: 
12
y cot 1 x x
?
? ? ?
??
??
??
 
 
 
15. Let a, b, and c be positive numbers , but not equal and not all are zero.
a b c
Show that the value of the determinant b c a is negative 
c a b
 
16. If sin y = x sin (a + y), prove that 
? ?
2
sin a y
dy
dx sina
?
? 
 
OR 
 
If
11
x y dy
y b tan tan , find
a x dx
??
??
??
??
??
 
 
17. If 
1
3
cos3x dy 3
y cos , then show that
cos x dx cosx cos3x
?
?? 
 
18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post 
which is 6 m high. Find the rate at which the length of his shadow increases. 
 
19. Evaluate: 
4 2 2 4
1
dx
sin x+sin xcos x+cos x
?
 
 
20. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums. 
 
21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0. 
 
OR 
Solve: ? ?
2
2
dy
x y a
dx
?? 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
Mathematics 
Class XII 
Sample Paper – 8 
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. Write the position of the element 6 in the given matrix, and denote it as aij. 
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
 
 
2. Find 
dy
dx
, if 2x + 3y = cos x 
3. Is the differential equation given by 
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give 
reason. 
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for 
the line. 
OR 
 
Find the angle between following pairs of line 
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ? 
 
 
SECTION – B  
 
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. 
Write the operation table of the operation *. 
6.  Solve the matrix equation 
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
 
 
7. Evaluate:  
2
5x 2
dx
1 2x 3x
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
8. Evaluate:  
2
22
x
dx
x 4 x 9
 
 
OR 
Evaluate:  
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being 
parameters. 
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
 
OR 
 
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k 
and c 2i j 4k . 
 
 
11. 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? 
 
12. A random variable X  has the following probability  distribution : 
X 0 1 2 3 4 5 6 7 
P(X) 0 k 2k 2k 3k k
2
 2k
2 
7k
2
 +k
 
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ?  X < 3) 
OR 
 
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find 
their respective probabilities of winning, if A starts first. 
 
 
SECTION – C  
 
13. Let A = Q x Q and let * be a binary operation on A defined by                                                              
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative 
and associative. Then, with respect to * on A 
(i) Find the identify element in A. 
(ii) Find the invertible elements of A. 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
OR 
 
Let A = Q × Q, Q being the set of rational. Let ‘*’ be a binary operation on A, defined 
by (a, b) * (c, d) = (ac, ad + b). Show that 
(i) ‘*’ is not commutative 
(ii) ‘*’ is associative  
(iii) The identity element with respect to ‘*’ is (1, 0) 
 
14. Write in the simplest form: 
12
y cot 1 x x
?
? ? ?
??
??
??
 
 
 
15. Let a, b, and c be positive numbers , but not equal and not all are zero.
a b c
Show that the value of the determinant b c a is negative 
c a b
 
16. If sin y = x sin (a + y), prove that 
? ?
2
sin a y
dy
dx sina
?
? 
 
OR 
 
If
11
x y dy
y b tan tan , find
a x dx
??
??
??
??
??
 
 
17. If 
1
3
cos3x dy 3
y cos , then show that
cos x dx cosx cos3x
?
?? 
 
18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post 
which is 6 m high. Find the rate at which the length of his shadow increases. 
 
19. Evaluate: 
4 2 2 4
1
dx
sin x+sin xcos x+cos x
?
 
 
20. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums. 
 
21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0. 
 
OR 
Solve: ? ?
2
2
dy
x y a
dx
?? 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
22.       
a) If 
ˆ ˆˆ
i . j.k represents the right handed system of mutually perpendicular vectors 
and
ˆ ˆ ˆ ˆ ˆ
3i j; 2i j 3k ? ? ? ? ? ? ? , then express ? in the form of 
12
? ? ? ? ? where 
1
? is 
parallel to 
2
and ?? is perpendicular to ? . 
b) Let a, b and c be three vectors of magnitude 3, 4 and 5 units respectively. If 
each of these is perpendicular to the sum of the other two vectors, find a b c ?? . 
 
23.   
Find the coordinates of the point, where the line 
x 2 y 1 z 2
3 4 2
 intersects the 
plane x – y + z – 5 = 0. Also find the angle between the line and the plane. 
 
 
SECTION – D 
 
24. If 
1 0 2
A 0 2 2
203
??
??
?
??
??
??
, then show that A is a root of polynomials f(x) = x
3
 – 6x
2
 + 7x + 2 
OR 
If 
1 1 0
A 0 1 1
2 3 4
?? ??
??
??
??
??
??
and 
1 2 3
B 0 1 0
1 1 0
??
??
?
??
??
??
, show that AB?BA 
 
25. Find the volume of the largest cylinder which can be inscribed in a sphere of radius 
r. 
 
26. Find the area of the region 
{(x, y):0 = y = x² + 1, 0 = y = x + 1, 0 = x = 2} 
 
OR 
 
Calculate the area  
?
22
22
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a 
ab
22
22
xy
(ii) Triangle AOB is in the first quadrant  of the ellipse  + 1,
ab
Where OA = a and OB = b.
Find the area enclosed between the chord AB and the arc AB of the ellipse
?
(iii) Find the ratio of the two areas found. 
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
Mathematics 
Class XII 
Sample Paper – 8 
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. Write the position of the element 6 in the given matrix, and denote it as aij. 
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
 
 
2. Find 
dy
dx
, if 2x + 3y = cos x 
3. Is the differential equation given by 
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give 
reason. 
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for 
the line. 
OR 
 
Find the angle between following pairs of line 
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ? 
 
 
SECTION – B  
 
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. 
Write the operation table of the operation *. 
6.  Solve the matrix equation 
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
 
 
7. Evaluate:  
2
5x 2
dx
1 2x 3x
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
8. Evaluate:  
2
22
x
dx
x 4 x 9
 
 
OR 
Evaluate:  
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being 
parameters. 
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
 
OR 
 
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k 
and c 2i j 4k . 
 
 
11. 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? 
 
12. A random variable X  has the following probability  distribution : 
X 0 1 2 3 4 5 6 7 
P(X) 0 k 2k 2k 3k k
2
 2k
2 
7k
2
 +k
 
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ?  X < 3) 
OR 
 
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find 
their respective probabilities of winning, if A starts first. 
 
 
SECTION – C  
 
13. Let A = Q x Q and let * be a binary operation on A defined by                                                              
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative 
and associative. Then, with respect to * on A 
(i) Find the identify element in A. 
(ii) Find the invertible elements of A. 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
OR 
 
Let A = Q × Q, Q being the set of rational. Let ‘*’ be a binary operation on A, defined 
by (a, b) * (c, d) = (ac, ad + b). Show that 
(i) ‘*’ is not commutative 
(ii) ‘*’ is associative  
(iii) The identity element with respect to ‘*’ is (1, 0) 
 
14. Write in the simplest form: 
12
y cot 1 x x
?
? ? ?
??
??
??
 
 
 
15. Let a, b, and c be positive numbers , but not equal and not all are zero.
a b c
Show that the value of the determinant b c a is negative 
c a b
 
16. If sin y = x sin (a + y), prove that 
? ?
2
sin a y
dy
dx sina
?
? 
 
OR 
 
If
11
x y dy
y b tan tan , find
a x dx
??
??
??
??
??
 
 
17. If 
1
3
cos3x dy 3
y cos , then show that
cos x dx cosx cos3x
?
?? 
 
18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post 
which is 6 m high. Find the rate at which the length of his shadow increases. 
 
19. Evaluate: 
4 2 2 4
1
dx
sin x+sin xcos x+cos x
?
 
 
20. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums. 
 
21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0. 
 
OR 
Solve: ? ?
2
2
dy
x y a
dx
?? 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
22.       
a) If 
ˆ ˆˆ
i . j.k represents the right handed system of mutually perpendicular vectors 
and
ˆ ˆ ˆ ˆ ˆ
3i j; 2i j 3k ? ? ? ? ? ? ? , then express ? in the form of 
12
? ? ? ? ? where 
1
? is 
parallel to 
2
and ?? is perpendicular to ? . 
b) Let a, b and c be three vectors of magnitude 3, 4 and 5 units respectively. If 
each of these is perpendicular to the sum of the other two vectors, find a b c ?? . 
 
23.   
Find the coordinates of the point, where the line 
x 2 y 1 z 2
3 4 2
 intersects the 
plane x – y + z – 5 = 0. Also find the angle between the line and the plane. 
 
 
SECTION – D 
 
24. If 
1 0 2
A 0 2 2
203
??
??
?
??
??
??
, then show that A is a root of polynomials f(x) = x
3
 – 6x
2
 + 7x + 2 
OR 
If 
1 1 0
A 0 1 1
2 3 4
?? ??
??
??
??
??
??
and 
1 2 3
B 0 1 0
1 1 0
??
??
?
??
??
??
, show that AB?BA 
 
25. Find the volume of the largest cylinder which can be inscribed in a sphere of radius 
r. 
 
26. Find the area of the region 
{(x, y):0 = y = x² + 1, 0 = y = x + 1, 0 = x = 2} 
 
OR 
 
Calculate the area  
?
22
22
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a 
ab
22
22
xy
(ii) Triangle AOB is in the first quadrant  of the ellipse  + 1,
ab
Where OA = a and OB = b.
Find the area enclosed between the chord AB and the arc AB of the ellipse
?
(iii) Find the ratio of the two areas found. 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 8  
 
     
27. Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and 
B(3, -1, 2) and parallel to the line 
x 4 y 3 z 1
1 4 7
? ? ?
??
?
 
OR 
Find the equation of the line passing through the point (-1,3,-2) and perpendicular 
to the lines
x y z x 2 y 1 z 1
and
1 2 3 3 2 5
. 
 
28. 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 realize a maximum profit? What is the 
maximum profit per week? 
 
29. Two bags A and B contain 3 red and 4 black balls, and 4 red and 5 black balls 
respectively. From bag A, one ball is transferred to bag B and then a ball is drawn from 
bag B. The ball is found to be red in colour. Find the probability that 
(a)The transferred ball is black ? 
(b) The transferred ball is red ?