CBSE Past Year Paper Session (2017), Math Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2017  
All India  
Time: 3 hours                                                              Maximum Marks: 100 
  
General Instructions:                                                                                                                     
(i) All questions are compulsory.  
(ii) There are 29 questions in all is 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, 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, an internal choice has been provided in 3 
questions of four marks each, 3 questions of six marks each. You have to 
attempt only one of the alternatives in all such questions.  
(v) Use of calculator is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION – A 
 
1. If for any 2 x 2 square matrix A, A (adj A) = 
??
??
??
80
08
 , then write the value of 
     l A l. 
 
2. Determine the value of ‘k’ for which the following function is continuous at x = 
3: 
? ?
?
?
?
?
?
2
x+3 -36
,x¹3
f(x)=
x-3
k , x=3
 
 
3. Find : 
?
22
sin x-cos x
dx
sin x cos x
 
 
4. Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. 
 
 
  
Page 2


  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2017  
All India  
Time: 3 hours                                                              Maximum Marks: 100 
  
General Instructions:                                                                                                                     
(i) All questions are compulsory.  
(ii) There are 29 questions in all is 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, 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, an internal choice has been provided in 3 
questions of four marks each, 3 questions of six marks each. You have to 
attempt only one of the alternatives in all such questions.  
(v) Use of calculator is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION – A 
 
1. If for any 2 x 2 square matrix A, A (adj A) = 
??
??
??
80
08
 , then write the value of 
     l A l. 
 
2. Determine the value of ‘k’ for which the following function is continuous at x = 
3: 
? ?
?
?
?
?
?
2
x+3 -36
,x¹3
f(x)=
x-3
k , x=3
 
 
3. Find : 
?
22
sin x-cos x
dx
sin x cos x
 
 
4. Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. 
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
Section B 
 
5. If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
6. Find the value of c in Rolle’s theorem for the function f(x)=x
3
 – 3x in 
??
??
- 3,0 . 
 
7. 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 ? 
 
8. Show that the function f(x) = x
3
-3x
2
+6x-100 is increasing on R. 
 
9. The x-coordinate of a point on the line joining the points P(2,2,1) and Q (5,1,-2) 
is 4. Find its z-coordinate. 
 
10. 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. 
 
11. Two tailors, A and B earn 300 and 400 per day respectively. A can stitch 6 
shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of 
trousers per day. To find how many days should each of them work and if it is 
desired to produce at least 60 shirts and 32 pairs of trousers at a minimum 
labour cost, formulate this as an LPP. 
 
12. 
? 2
find
dx
5 - 8x - x
 
 
  
Page 3


  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2017  
All India  
Time: 3 hours                                                              Maximum Marks: 100 
  
General Instructions:                                                                                                                     
(i) All questions are compulsory.  
(ii) There are 29 questions in all is 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, 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, an internal choice has been provided in 3 
questions of four marks each, 3 questions of six marks each. You have to 
attempt only one of the alternatives in all such questions.  
(v) Use of calculator is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION – A 
 
1. If for any 2 x 2 square matrix A, A (adj A) = 
??
??
??
80
08
 , then write the value of 
     l A l. 
 
2. Determine the value of ‘k’ for which the following function is continuous at x = 
3: 
? ?
?
?
?
?
?
2
x+3 -36
,x¹3
f(x)=
x-3
k , x=3
 
 
3. Find : 
?
22
sin x-cos x
dx
sin x cos x
 
 
4. Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. 
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
Section B 
 
5. If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
6. Find the value of c in Rolle’s theorem for the function f(x)=x
3
 – 3x in 
??
??
- 3,0 . 
 
7. 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 ? 
 
8. Show that the function f(x) = x
3
-3x
2
+6x-100 is increasing on R. 
 
9. The x-coordinate of a point on the line joining the points P(2,2,1) and Q (5,1,-2) 
is 4. Find its z-coordinate. 
 
10. 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. 
 
11. Two tailors, A and B earn 300 and 400 per day respectively. A can stitch 6 
shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of 
trousers per day. To find how many days should each of them work and if it is 
desired to produce at least 60 shirts and 32 pairs of trousers at a minimum 
labour cost, formulate this as an LPP. 
 
12. 
? 2
find
dx
5 - 8x - x
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
SECTION C 
 
13. 
?
-1 -1
x-3 x+3
If tan + tan = ,thenfindthevalueof x
x-4 x+4 4
 
 
14.  
 
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2
3
Using properties of determinants, provethat
a +2a 2a+1 1
2a+1 a+2 1 =(a-1)
3 3 1
Find matrix A such that
2 -1 -1 -8
1 0 A= 1 -2
-3 4 - 22
OR 
 
15.  
??
??
??
y x b
2
2
y
2
dy
If x +y =a ,then find
dx
OR
d y dy
If e (x+1)=1,then show that =
dx dx
 
 
16. Find 
? 22
cos ?
d ?
(4 + sin ? ) ( 5 - 4 c o s ? )
 
 
17. Find 
? ?
?
?
p
0
4
1
xtanx
dx
secx+tanx
OR
Evaluate
 x - 1 + x - 2 + x - 4 dx
 
 
18. Solve the differential equation (tan
-1
 x – y) dx  = (1+x
2
) dy. 
 
19.  
Showthat the pointsA,B,Cwith positionvectors 2 i -j + k, i -3 j -5 k and 3 i -4 j -4 k
respectively are the vertices of a right-angledtriangle, Hence find the areaof thetriangle
 
Page 4


  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2017  
All India  
Time: 3 hours                                                              Maximum Marks: 100 
  
General Instructions:                                                                                                                     
(i) All questions are compulsory.  
(ii) There are 29 questions in all is 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, 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, an internal choice has been provided in 3 
questions of four marks each, 3 questions of six marks each. You have to 
attempt only one of the alternatives in all such questions.  
(v) Use of calculator is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION – A 
 
1. If for any 2 x 2 square matrix A, A (adj A) = 
??
??
??
80
08
 , then write the value of 
     l A l. 
 
2. Determine the value of ‘k’ for which the following function is continuous at x = 
3: 
? ?
?
?
?
?
?
2
x+3 -36
,x¹3
f(x)=
x-3
k , x=3
 
 
3. Find : 
?
22
sin x-cos x
dx
sin x cos x
 
 
4. Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. 
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
Section B 
 
5. If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
6. Find the value of c in Rolle’s theorem for the function f(x)=x
3
 – 3x in 
??
??
- 3,0 . 
 
7. 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 ? 
 
8. Show that the function f(x) = x
3
-3x
2
+6x-100 is increasing on R. 
 
9. The x-coordinate of a point on the line joining the points P(2,2,1) and Q (5,1,-2) 
is 4. Find its z-coordinate. 
 
10. 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. 
 
11. Two tailors, A and B earn 300 and 400 per day respectively. A can stitch 6 
shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of 
trousers per day. To find how many days should each of them work and if it is 
desired to produce at least 60 shirts and 32 pairs of trousers at a minimum 
labour cost, formulate this as an LPP. 
 
12. 
? 2
find
dx
5 - 8x - x
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
SECTION C 
 
13. 
?
-1 -1
x-3 x+3
If tan + tan = ,thenfindthevalueof x
x-4 x+4 4
 
 
14.  
 
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2
3
Using properties of determinants, provethat
a +2a 2a+1 1
2a+1 a+2 1 =(a-1)
3 3 1
Find matrix A such that
2 -1 -1 -8
1 0 A= 1 -2
-3 4 - 22
OR 
 
15.  
??
??
??
y x b
2
2
y
2
dy
If x +y =a ,then find
dx
OR
d y dy
If e (x+1)=1,then show that =
dx dx
 
 
16. Find 
? 22
cos ?
d ?
(4 + sin ? ) ( 5 - 4 c o s ? )
 
 
17. Find 
? ?
?
?
p
0
4
1
xtanx
dx
secx+tanx
OR
Evaluate
 x - 1 + x - 2 + x - 4 dx
 
 
18. Solve the differential equation (tan
-1
 x – y) dx  = (1+x
2
) dy. 
 
19.  
Showthat the pointsA,B,Cwith positionvectors 2 i -j + k, i -3 j -5 k and 3 i -4 j -4 k
respectively are the vertices of a right-angledtriangle, Hence find the areaof thetriangle
 
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
 
20.  
Findthevalueof ? , if fo u r p o in t s w it h p o s it io n vectors 3i+6 j+9k,i+2 j+3k,2i+3j+k
and4i+6 j + ? k a r e c o p la n e r .
 
 
21. There are 4 cards numbered 1, 3, 5 and 7 one number on one card. Two cards 
are drawn at random without replacement. Let X denotes the sum of the 
numbers on the two drawn cards. Find the mean and variance of X. 
 
22. Of the students in a school, it is known that 30% have 100% attendance and 
70% students are irregular. Previous year results report that 70% of all 
students who have 100% attendance attain A grade and 10% irregular 
students attain A grade in their annual examination. At the end of the year one 
student is chosen at random from the school and he was found to have an A 
grade. What is the probability that the student has 100% attendance? Is 
regularity required only in school? Justify your answer. 
 
23.  
Maximisez=x+2y
Subject tothecontraints
x+2y 100
2x-y 0
2x+y 200
x,y 0
Solvethe above LPPgraphically
?
?
?
?
 
 
  
Page 5


  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2017  
All India  
Time: 3 hours                                                              Maximum Marks: 100 
  
General Instructions:                                                                                                                     
(i) All questions are compulsory.  
(ii) There are 29 questions in all is 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, 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, an internal choice has been provided in 3 
questions of four marks each, 3 questions of six marks each. You have to 
attempt only one of the alternatives in all such questions.  
(v) Use of calculator is not permitted. You may ask for logarithmic tables, if 
required. 
 
SECTION – A 
 
1. If for any 2 x 2 square matrix A, A (adj A) = 
??
??
??
80
08
 , then write the value of 
     l A l. 
 
2. Determine the value of ‘k’ for which the following function is continuous at x = 
3: 
? ?
?
?
?
?
?
2
x+3 -36
,x¹3
f(x)=
x-3
k , x=3
 
 
3. Find : 
?
22
sin x-cos x
dx
sin x cos x
 
 
4. Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. 
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
Section B 
 
5. If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
 
6. Find the value of c in Rolle’s theorem for the function f(x)=x
3
 – 3x in 
??
??
- 3,0 . 
 
7. 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 ? 
 
8. Show that the function f(x) = x
3
-3x
2
+6x-100 is increasing on R. 
 
9. The x-coordinate of a point on the line joining the points P(2,2,1) and Q (5,1,-2) 
is 4. Find its z-coordinate. 
 
10. 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. 
 
11. Two tailors, A and B earn 300 and 400 per day respectively. A can stitch 6 
shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of 
trousers per day. To find how many days should each of them work and if it is 
desired to produce at least 60 shirts and 32 pairs of trousers at a minimum 
labour cost, formulate this as an LPP. 
 
12. 
? 2
find
dx
5 - 8x - x
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
SECTION C 
 
13. 
?
-1 -1
x-3 x+3
If tan + tan = ,thenfindthevalueof x
x-4 x+4 4
 
 
14.  
 
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2
3
Using properties of determinants, provethat
a +2a 2a+1 1
2a+1 a+2 1 =(a-1)
3 3 1
Find matrix A such that
2 -1 -1 -8
1 0 A= 1 -2
-3 4 - 22
OR 
 
15.  
??
??
??
y x b
2
2
y
2
dy
If x +y =a ,then find
dx
OR
d y dy
If e (x+1)=1,then show that =
dx dx
 
 
16. Find 
? 22
cos ?
d ?
(4 + sin ? ) ( 5 - 4 c o s ? )
 
 
17. Find 
? ?
?
?
p
0
4
1
xtanx
dx
secx+tanx
OR
Evaluate
 x - 1 + x - 2 + x - 4 dx
 
 
18. Solve the differential equation (tan
-1
 x – y) dx  = (1+x
2
) dy. 
 
19.  
Showthat the pointsA,B,Cwith positionvectors 2 i -j + k, i -3 j -5 k and 3 i -4 j -4 k
respectively are the vertices of a right-angledtriangle, Hence find the areaof thetriangle
 
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
 
20.  
Findthevalueof ? , if fo u r p o in t s w it h p o s it io n vectors 3i+6 j+9k,i+2 j+3k,2i+3j+k
and4i+6 j + ? k a r e c o p la n e r .
 
 
21. There are 4 cards numbered 1, 3, 5 and 7 one number on one card. Two cards 
are drawn at random without replacement. Let X denotes the sum of the 
numbers on the two drawn cards. Find the mean and variance of X. 
 
22. Of the students in a school, it is known that 30% have 100% attendance and 
70% students are irregular. Previous year results report that 70% of all 
students who have 100% attendance attain A grade and 10% irregular 
students attain A grade in their annual examination. At the end of the year one 
student is chosen at random from the school and he was found to have an A 
grade. What is the probability that the student has 100% attendance? Is 
regularity required only in school? Justify your answer. 
 
23.  
Maximisez=x+2y
Subject tothecontraints
x+2y 100
2x-y 0
2x+y 200
x,y 0
Solvethe above LPPgraphically
?
?
?
?
 
 
  
  
 
CBSE XII  | Mathematics 
Board Paper – 2017 
 
     
SECTION D 
 
24.  
-4 4 4 1 -1 1
Determine the product -7 1 3 1 -2 -2 and use it to Solve the system of equations
5 -3 -1 2 1 3
x- y+z=4,x-2y-2x=9,2x+y+3z=1
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
 
25.  
-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
? ? ? ?
?
? ? ? ?
? ? ? ?
OR 
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  
Commucative and associative. Then, with respect to * on A 
(i) Find the identify element in A. 
(ii) Find the invertible elements of A. 
 
 
26. Show that the surface area of a closed cuboid with square base and given 
volume is minimum, when it is a cube. 
 
27. Using the method of integration, find the area of the triangle ABC, coordinates 
of whose vertices are A(4,1), B(6,6) and C(8,4). 
OR 
       Find the area enclosed between the parabola 4y = 3x
2
 and the straight line 
       3x - 2y + 12 = 0. 
 
28.  
? ? ? ?
dy
Find the particular solution of the diffrential equation x-y = x+2y ,
dx
given that y = 0 when x=1
 
 
29. Find the coordinates of the point where the line through the points (3,-4,-5) 
and (2,-3,1) crosses the plane determined by the points (1,2,3), (4,2,-3) and 
(0,4,3) 
OR 
      A variable plane which remains at a constant distance 3p from the origin cuts   
the coordinate axes at A,B,C. Show that the locus of the centroid of triangle 
ABC is 
2 2 2 2
1 1 1 1
+ + =
x y z p