CBSE Past Year Paper Session (2019) Set- 1, Math Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
i. All questions are compulsory. 
ii. The question paper consists of 29 questions divided into four 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. 
iii. All questions in Section A are to be answered in one word, one sentence or as 
per the exact requirement of the question. 
iv. There is no overall choice. However, internal choice has been provided in 1 
question of Section A, 3 questions of Section B, 3 questions of Section C and 3 
questions of Section D. 
You have to attempt only one of the alternatives in all such questions. 
v. Use of calculators is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION A 
Questions number 1 to 4 carry 1 mark each. 
 
1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 
 
2. If y = sin
-1
x + cos
-1
x, find 
dy
dx
. 
 
3. Write the order and the degree of the differential equation 
??
??
??
????
??
??
?? ?? ??
??
3
2
2
4
4
d y dy
x
dx dx
. 
 
4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? 
 
OR 
 
Find the cartesian equation of the line which passes through the point (-2, 4, -5) 
and is parallel to the line 
? ? ?
??
x 3 4 y z 8
.
3 5 6
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
i. All questions are compulsory. 
ii. The question paper consists of 29 questions divided into four 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. 
iii. All questions in Section A are to be answered in one word, one sentence or as 
per the exact requirement of the question. 
iv. There is no overall choice. However, internal choice has been provided in 1 
question of Section A, 3 questions of Section B, 3 questions of Section C and 3 
questions of Section D. 
You have to attempt only one of the alternatives in all such questions. 
v. Use of calculators is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION A 
Questions number 1 to 4 carry 1 mark each. 
 
1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 
 
2. If y = sin
-1
x + cos
-1
x, find 
dy
dx
. 
 
3. Write the order and the degree of the differential equation 
??
??
??
????
??
??
?? ?? ??
??
3
2
2
4
4
d y dy
x
dx dx
. 
 
4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? 
 
OR 
 
Find the cartesian equation of the line which passes through the point (-2, 4, -5) 
and is parallel to the line 
? ? ?
??
x 3 4 y z 8
.
3 5 6
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
  
Questions number 5 to 12 carry 2 marks each. 
  
5. If * is defined on the set R of all real numbers by * : a * b = ?
22
 a b , find the 
identity element, if it exists in R with respect to *. 
 
 
 
6. 
? ? ? ?
? ? ? ?
? ? ? ?
0     2 0        3a
If A = and kA = , then find the values of k, a and b.
3     -4 2b      24
 
 
7. Find: 
?
?
?
sinx - cosx
dx, 0 < x < /2
1 sin2x
 
  
8. Find: 
? ?
? ?
?
?
?
sin x a
dx
sin x a
 
 
OR 
 
Find: 
?(logx)
2
 dx 
 
 
9. Form the differential equation representing the family of curves y
2
 = m(a
2
 – x
2
) 
by eliminating the arbitrary constants ‘m’ and ‘a’. 
 
10. Find a unit vector perpendicular to both the vectors a and b, where 
?? a i - 7j + 7k and b 3i - 2j + 2k . 
 
OR 
 
Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 
 
11. Mother, father and son line up at random for a family photo. If A and B are two 
events given by A = Son on one end, B = Father in the middle, find P(B/A).
  
 
12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 
2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). 
Find the probability distribution of X. 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
i. All questions are compulsory. 
ii. The question paper consists of 29 questions divided into four 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. 
iii. All questions in Section A are to be answered in one word, one sentence or as 
per the exact requirement of the question. 
iv. There is no overall choice. However, internal choice has been provided in 1 
question of Section A, 3 questions of Section B, 3 questions of Section C and 3 
questions of Section D. 
You have to attempt only one of the alternatives in all such questions. 
v. Use of calculators is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION A 
Questions number 1 to 4 carry 1 mark each. 
 
1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 
 
2. If y = sin
-1
x + cos
-1
x, find 
dy
dx
. 
 
3. Write the order and the degree of the differential equation 
??
??
??
????
??
??
?? ?? ??
??
3
2
2
4
4
d y dy
x
dx dx
. 
 
4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? 
 
OR 
 
Find the cartesian equation of the line which passes through the point (-2, 4, -5) 
and is parallel to the line 
? ? ?
??
x 3 4 y z 8
.
3 5 6
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
  
Questions number 5 to 12 carry 2 marks each. 
  
5. If * is defined on the set R of all real numbers by * : a * b = ?
22
 a b , find the 
identity element, if it exists in R with respect to *. 
 
 
 
6. 
? ? ? ?
? ? ? ?
? ? ? ?
0     2 0        3a
If A = and kA = , then find the values of k, a and b.
3     -4 2b      24
 
 
7. Find: 
?
?
?
sinx - cosx
dx, 0 < x < /2
1 sin2x
 
  
8. Find: 
? ?
? ?
?
?
?
sin x a
dx
sin x a
 
 
OR 
 
Find: 
?(logx)
2
 dx 
 
 
9. Form the differential equation representing the family of curves y
2
 = m(a
2
 – x
2
) 
by eliminating the arbitrary constants ‘m’ and ‘a’. 
 
10. Find a unit vector perpendicular to both the vectors a and b, where 
?? a i - 7j + 7k and b 3i - 2j + 2k . 
 
OR 
 
Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 
 
11. Mother, father and son line up at random for a family photo. If A and B are two 
events given by A = Son on one end, B = Father in the middle, find P(B/A).
  
 
12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 
2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). 
Find the probability distribution of X. 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
OR 
 
A coin is tossed 5 times, Find the probability of getting (i) at least 4 heads, and 
(ii) at most 4 heads. 
 
 
SECTION C 
 
Questions number 13 to 23 carry 4 marks each. 
 
13. Show that the relation R on the set Z of all integers, given by  
R = [(a,b) : 2 divides (a - b)] is an equivalence relation. 
 
OR 
If f(x) = 
?
?
?
4x 3 2
, x 
6x 4 3
, show that fof(x) = x for all ?
2
x 
3
. Also, find the 
inverse of f. 
 
14. If tan
-1
x – cot
-1
x = tan
-1
??
??
??
1
3
, x > 0, find the value of x and hence find the  
value of sec
-1
2
x
??
??
??
.
 
  
15. Using properties of determinants, prove that  
b c a a
b c a b = 4abc
c c a b
?
?
?
 
 
16. If sin y = x sin (a + y), prove that 
2
dy sin (a + y)
dx sina
? 
 
OR 
If (sin x)
y
 = x + y, find 
dy
dx
. 
 
17. If y = (sec
-1
 x)
2
, x > 0, show that 
? ? ? ?
2
2 2 3
2
d y dy
x x 1 2x x 2 0
dx dx
? ? ? ? ? 
 
18. Find the equation of the tangent and the normal to the curve 
? ? ? ?
x - 7
y
x 2 x 3
?
??
 
at the point where it cuts the x-axis. 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
i. All questions are compulsory. 
ii. The question paper consists of 29 questions divided into four 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. 
iii. All questions in Section A are to be answered in one word, one sentence or as 
per the exact requirement of the question. 
iv. There is no overall choice. However, internal choice has been provided in 1 
question of Section A, 3 questions of Section B, 3 questions of Section C and 3 
questions of Section D. 
You have to attempt only one of the alternatives in all such questions. 
v. Use of calculators is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION A 
Questions number 1 to 4 carry 1 mark each. 
 
1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 
 
2. If y = sin
-1
x + cos
-1
x, find 
dy
dx
. 
 
3. Write the order and the degree of the differential equation 
??
??
??
????
??
??
?? ?? ??
??
3
2
2
4
4
d y dy
x
dx dx
. 
 
4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? 
 
OR 
 
Find the cartesian equation of the line which passes through the point (-2, 4, -5) 
and is parallel to the line 
? ? ?
??
x 3 4 y z 8
.
3 5 6
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
  
Questions number 5 to 12 carry 2 marks each. 
  
5. If * is defined on the set R of all real numbers by * : a * b = ?
22
 a b , find the 
identity element, if it exists in R with respect to *. 
 
 
 
6. 
? ? ? ?
? ? ? ?
? ? ? ?
0     2 0        3a
If A = and kA = , then find the values of k, a and b.
3     -4 2b      24
 
 
7. Find: 
?
?
?
sinx - cosx
dx, 0 < x < /2
1 sin2x
 
  
8. Find: 
? ?
? ?
?
?
?
sin x a
dx
sin x a
 
 
OR 
 
Find: 
?(logx)
2
 dx 
 
 
9. Form the differential equation representing the family of curves y
2
 = m(a
2
 – x
2
) 
by eliminating the arbitrary constants ‘m’ and ‘a’. 
 
10. Find a unit vector perpendicular to both the vectors a and b, where 
?? a i - 7j + 7k and b 3i - 2j + 2k . 
 
OR 
 
Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 
 
11. Mother, father and son line up at random for a family photo. If A and B are two 
events given by A = Son on one end, B = Father in the middle, find P(B/A).
  
 
12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 
2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). 
Find the probability distribution of X. 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
OR 
 
A coin is tossed 5 times, Find the probability of getting (i) at least 4 heads, and 
(ii) at most 4 heads. 
 
 
SECTION C 
 
Questions number 13 to 23 carry 4 marks each. 
 
13. Show that the relation R on the set Z of all integers, given by  
R = [(a,b) : 2 divides (a - b)] is an equivalence relation. 
 
OR 
If f(x) = 
?
?
?
4x 3 2
, x 
6x 4 3
, show that fof(x) = x for all ?
2
x 
3
. Also, find the 
inverse of f. 
 
14. If tan
-1
x – cot
-1
x = tan
-1
??
??
??
1
3
, x > 0, find the value of x and hence find the  
value of sec
-1
2
x
??
??
??
.
 
  
15. Using properties of determinants, prove that  
b c a a
b c a b = 4abc
c c a b
?
?
?
 
 
16. If sin y = x sin (a + y), prove that 
2
dy sin (a + y)
dx sina
? 
 
OR 
If (sin x)
y
 = x + y, find 
dy
dx
. 
 
17. If y = (sec
-1
 x)
2
, x > 0, show that 
? ? ? ?
2
2 2 3
2
d y dy
x x 1 2x x 2 0
dx dx
? ? ? ? ? 
 
18. Find the equation of the tangent and the normal to the curve 
? ? ? ?
x - 7
y
x 2 x 3
?
??
 
at the point where it cuts the x-axis. 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
 
19. Find: 
? ? ? ?
22
sin2x
dx
sin x 1 sin x 3 ??
?
 
 
20. Prove that 
? ? ? ?
bb
aa
/3
/6
f x dx = f a b x dx and hence evaluate
dx
.
1 tanx
?
?
??
?
??
?
 
 
 
21. Solve the differential equation: 
dy x y
dx x y
?
?
?
 
 
OR 
 
Solve the differential equation: 
(1 + x
2
)dy + 2xy dx = cot x dx 
 
22. Let a, b and c be three vectors such that  a  1,  b 2 and  c 3. ? ? ? 
If the projection of b along a is equal to the projection of c along a; and b, c
are perpendicular to each other, then find  3a - 2b + 2c .
 
 
23. Find the value of ? for which the following lines are perpendicular to each other: 
1
y
x 5 2 y 1 z x z 1
2
; 
5 2 5 1 1 2 3
?
? ? ? ?
? ? ? ?
? ? ? ?
 
Hence, find whether the lines intersect or not. 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
i. All questions are compulsory. 
ii. The question paper consists of 29 questions divided into four 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. 
iii. All questions in Section A are to be answered in one word, one sentence or as 
per the exact requirement of the question. 
iv. There is no overall choice. However, internal choice has been provided in 1 
question of Section A, 3 questions of Section B, 3 questions of Section C and 3 
questions of Section D. 
You have to attempt only one of the alternatives in all such questions. 
v. Use of calculators is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION A 
Questions number 1 to 4 carry 1 mark each. 
 
1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 
 
2. If y = sin
-1
x + cos
-1
x, find 
dy
dx
. 
 
3. Write the order and the degree of the differential equation 
??
??
??
????
??
??
?? ?? ??
??
3
2
2
4
4
d y dy
x
dx dx
. 
 
4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? 
 
OR 
 
Find the cartesian equation of the line which passes through the point (-2, 4, -5) 
and is parallel to the line 
? ? ?
??
x 3 4 y z 8
.
3 5 6
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
  
Questions number 5 to 12 carry 2 marks each. 
  
5. If * is defined on the set R of all real numbers by * : a * b = ?
22
 a b , find the 
identity element, if it exists in R with respect to *. 
 
 
 
6. 
? ? ? ?
? ? ? ?
? ? ? ?
0     2 0        3a
If A = and kA = , then find the values of k, a and b.
3     -4 2b      24
 
 
7. Find: 
?
?
?
sinx - cosx
dx, 0 < x < /2
1 sin2x
 
  
8. Find: 
? ?
? ?
?
?
?
sin x a
dx
sin x a
 
 
OR 
 
Find: 
?(logx)
2
 dx 
 
 
9. Form the differential equation representing the family of curves y
2
 = m(a
2
 – x
2
) 
by eliminating the arbitrary constants ‘m’ and ‘a’. 
 
10. Find a unit vector perpendicular to both the vectors a and b, where 
?? a i - 7j + 7k and b 3i - 2j + 2k . 
 
OR 
 
Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 
 
11. Mother, father and son line up at random for a family photo. If A and B are two 
events given by A = Son on one end, B = Father in the middle, find P(B/A).
  
 
12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 
2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). 
Find the probability distribution of X. 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
OR 
 
A coin is tossed 5 times, Find the probability of getting (i) at least 4 heads, and 
(ii) at most 4 heads. 
 
 
SECTION C 
 
Questions number 13 to 23 carry 4 marks each. 
 
13. Show that the relation R on the set Z of all integers, given by  
R = [(a,b) : 2 divides (a - b)] is an equivalence relation. 
 
OR 
If f(x) = 
?
?
?
4x 3 2
, x 
6x 4 3
, show that fof(x) = x for all ?
2
x 
3
. Also, find the 
inverse of f. 
 
14. If tan
-1
x – cot
-1
x = tan
-1
??
??
??
1
3
, x > 0, find the value of x and hence find the  
value of sec
-1
2
x
??
??
??
.
 
  
15. Using properties of determinants, prove that  
b c a a
b c a b = 4abc
c c a b
?
?
?
 
 
16. If sin y = x sin (a + y), prove that 
2
dy sin (a + y)
dx sina
? 
 
OR 
If (sin x)
y
 = x + y, find 
dy
dx
. 
 
17. If y = (sec
-1
 x)
2
, x > 0, show that 
? ? ? ?
2
2 2 3
2
d y dy
x x 1 2x x 2 0
dx dx
? ? ? ? ? 
 
18. Find the equation of the tangent and the normal to the curve 
? ? ? ?
x - 7
y
x 2 x 3
?
??
 
at the point where it cuts the x-axis. 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
 
19. Find: 
? ? ? ?
22
sin2x
dx
sin x 1 sin x 3 ??
?
 
 
20. Prove that 
? ? ? ?
bb
aa
/3
/6
f x dx = f a b x dx and hence evaluate
dx
.
1 tanx
?
?
??
?
??
?
 
 
 
21. Solve the differential equation: 
dy x y
dx x y
?
?
?
 
 
OR 
 
Solve the differential equation: 
(1 + x
2
)dy + 2xy dx = cot x dx 
 
22. Let a, b and c be three vectors such that  a  1,  b 2 and  c 3. ? ? ? 
If the projection of b along a is equal to the projection of c along a; and b, c
are perpendicular to each other, then find  3a - 2b + 2c .
 
 
23. Find the value of ? for which the following lines are perpendicular to each other: 
1
y
x 5 2 y 1 z x z 1
2
; 
5 2 5 1 1 2 3
?
? ? ? ?
? ? ? ?
? ? ? ?
 
Hence, find whether the lines intersect or not. 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION D 
 
Questions number 24 to 29 carry 6 marks each. 
 
24. If A = 
1
1 1 1
0 1 3 ' find A
1 2 1
?
??
??
??
??
?
??
 
Hence, solve the following system of equations: 
x + y + z = 6, 
y + 3z = 11 
and x – 2y + z = 0 
OR 
 
Find the inverse of the following matrix, using elementary transformations: 
2 3 1
A = 2 4 1
3 7 2
??
??
??
??
??
 
 
25. Show that the height of a cylinder, which is open at the top, having a given 
surface area and greatest volume, is equal to the radius of its base. 
 
26. Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using 
intergration. 
OR 
 
Find the area of the region bounded by the curves (x - 1)
2
 + y
2
 = 1 and x
2
 + y
2
 
= 1, using integration. 
 
27. Find the vector and cartesian equations of the plane passing through the points 
(2, 5, -3), (-2, -3, 5) and (5, 3, -3). Also, find the point of intersection of this 
plane with the line passing through points (3, 1, 5) and (-1, -3, -1). 
 
OR 
 
Find the equation of the plane passing through the intersection of the planes 
? ? ? ?
r.  i + j + k 1 and r.  2i + 3j - k + 4 0 ?? and parallel to x-axis. Hence, 
find the distance of the plane from x-axis.