Class 11 Mathematics (Maths) Previous Year Paper - 1 | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


   
Date:                         Class: XI 
Mathematics (Set - A) 
Time: 3 hrs                                                M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
 
Section A 
1 Describe the set {-1, 1} in set builder form. 1 
2 Find the value of .
3
31
sin
p
 
1 
3 Write the range of a Signum function? 
1 
4 Solve 2 8 3 > + x , when x is an integer. 
1 
 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are   
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
1 
Section B 
 
7 Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
4 
 
8 Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
4 
 
 
9 If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
OR 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
4 
 
10 
Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
 
4 
 
13 
Prove that: 
x
x
x
x
2 tan
8 tan
1 4 sec
1 8 sec
=
- - 
 
 
4 
 
14 
Using Principle of Mathematical Induction, prove that 
n n
14 41 - is a multiple of 27,  
for . N n ? 
 
4 
Page 2


   
Date:                         Class: XI 
Mathematics (Set - A) 
Time: 3 hrs                                                M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
 
Section A 
1 Describe the set {-1, 1} in set builder form. 1 
2 Find the value of .
3
31
sin
p
 
1 
3 Write the range of a Signum function? 
1 
4 Solve 2 8 3 > + x , when x is an integer. 
1 
 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are   
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
1 
Section B 
 
7 Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
4 
 
8 Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
4 
 
 
9 If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
OR 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
4 
 
10 
Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
 
4 
 
13 
Prove that: 
x
x
x
x
2 tan
8 tan
1 4 sec
1 8 sec
=
- - 
 
 
4 
 
14 
Using Principle of Mathematical Induction, prove that 
n n
14 41 - is a multiple of 27,  
for . N n ? 
 
4 
 
15 
Solve the given inequality and represent the solution graphically on a number line: 
   ( ) ( ) 0 3 2 3 7 2 5 = + - - x x ; . 47 6 19 2 + = + x x 
 
4 
 
16 
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so 
that the resulting mixture will contain more than 25% but less than 30% acid content? 
OR 
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The 
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of 
the 8% solution, how many litres of the 2% solution will have to be added? 
4 
 
17 
Find the ratio in which the join of A(2, -1, 3) and B(-1, 2, 1) is divided by the plane             
x + y+z = 5. Also find the coordinates of point of division  
4 
 
18 
Find the number of different 8-letter arrangements that can be made from the letters of the 
word DAUGHTER so that 
(i) all vowels occur together     (ii) all vowels do not occur together 
OR 
In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not 
come together? 
4 
19 A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be 
selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls? 
 
4 
Section C 
20 In a class of 60 students, 23 play Hockey, 15 play basketball and 20 play cricket, 7 play 
Hockey and Basketball, 5 play cricket and basketball, 4 play hockey and cricket and 15 
students do not play any of these games. Find 
a. How many play hockey, basketball and cricket. 
b. How many play hockey but not cricket. 
c. How many play hockey and cricket but not basketball. 
 
6 
 
21 (i) Prove that: ? 8 2 2 2 Cos + + + = 2 Cos? . 
(ii) Prove that: ( ) ( )
1 2 cos 2
1 2 cos 2
60 tan 60 tan
- +
= - ° + °
?
?
? ? . 
3+3 
 
22 Prove that: .
2
5
sin 5 sin
2
9
cos 3 cos
2
cos 2 cos
x
x
x
x
x
x = - 
OR 
Prove that:  .
2
3
3
cos
3
cos cos
2 2 2
= ?
?
?
?
?
?
- + ?
?
?
?
?
?
+ +
p p
x x x 
 
6 
 
23 Using Principle of Mathematical Induction, prove that 5 5 . 3 7 . 2 - +
n n
is divisible by 24, for 
all . N n ? 
OR 
Using Principle of Mathematical Induction, prove that: 
 
1
2
... 3 2 1
1
.......
3 2 1
1
2 1
1
1
+
=
+ + + +
+ +
+ +
+
+
+
n
n
n
 , for all . N n ? 
 
6 
 
 
24 Solve the following system of inequalities graphically:  
   
6 
 
 
25 (i) The mid points of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Find its 
vertices. 
(ii) Find the coordinates of a point equidistant from the four points  
      O(0, 0, 0),  A(a, 0, 0), B(0, b, 0) and C(0, 0, c). 
 
3+3 
 
26 Find the number of words with or without meaning which can be made using all the letters 
of the word AGAIN. If these words are written as in a dictionary, what will be the 50
th
 
word? 
6 
 
Page 3


   
Date:                         Class: XI 
Mathematics (Set - A) 
Time: 3 hrs                                                M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
 
Section A 
1 Describe the set {-1, 1} in set builder form. 1 
2 Find the value of .
3
31
sin
p
 
1 
3 Write the range of a Signum function? 
1 
4 Solve 2 8 3 > + x , when x is an integer. 
1 
 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are   
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
1 
Section B 
 
7 Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
4 
 
8 Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
4 
 
 
9 If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
OR 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
4 
 
10 
Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
 
4 
 
13 
Prove that: 
x
x
x
x
2 tan
8 tan
1 4 sec
1 8 sec
=
- - 
 
 
4 
 
14 
Using Principle of Mathematical Induction, prove that 
n n
14 41 - is a multiple of 27,  
for . N n ? 
 
4 
 
15 
Solve the given inequality and represent the solution graphically on a number line: 
   ( ) ( ) 0 3 2 3 7 2 5 = + - - x x ; . 47 6 19 2 + = + x x 
 
4 
 
16 
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so 
that the resulting mixture will contain more than 25% but less than 30% acid content? 
OR 
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The 
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of 
the 8% solution, how many litres of the 2% solution will have to be added? 
4 
 
17 
Find the ratio in which the join of A(2, -1, 3) and B(-1, 2, 1) is divided by the plane             
x + y+z = 5. Also find the coordinates of point of division  
4 
 
18 
Find the number of different 8-letter arrangements that can be made from the letters of the 
word DAUGHTER so that 
(i) all vowels occur together     (ii) all vowels do not occur together 
OR 
In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not 
come together? 
4 
19 A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be 
selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls? 
 
4 
Section C 
20 In a class of 60 students, 23 play Hockey, 15 play basketball and 20 play cricket, 7 play 
Hockey and Basketball, 5 play cricket and basketball, 4 play hockey and cricket and 15 
students do not play any of these games. Find 
a. How many play hockey, basketball and cricket. 
b. How many play hockey but not cricket. 
c. How many play hockey and cricket but not basketball. 
 
6 
 
21 (i) Prove that: ? 8 2 2 2 Cos + + + = 2 Cos? . 
(ii) Prove that: ( ) ( )
1 2 cos 2
1 2 cos 2
60 tan 60 tan
- +
= - ° + °
?
?
? ? . 
3+3 
 
22 Prove that: .
2
5
sin 5 sin
2
9
cos 3 cos
2
cos 2 cos
x
x
x
x
x
x = - 
OR 
Prove that:  .
2
3
3
cos
3
cos cos
2 2 2
= ?
?
?
?
?
?
- + ?
?
?
?
?
?
+ +
p p
x x x 
 
6 
 
23 Using Principle of Mathematical Induction, prove that 5 5 . 3 7 . 2 - +
n n
is divisible by 24, for 
all . N n ? 
OR 
Using Principle of Mathematical Induction, prove that: 
 
1
2
... 3 2 1
1
.......
3 2 1
1
2 1
1
1
+
=
+ + + +
+ +
+ +
+
+
+
n
n
n
 , for all . N n ? 
 
6 
 
 
24 Solve the following system of inequalities graphically:  
   
6 
 
 
25 (i) The mid points of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Find its 
vertices. 
(ii) Find the coordinates of a point equidistant from the four points  
      O(0, 0, 0),  A(a, 0, 0), B(0, b, 0) and C(0, 0, c). 
 
3+3 
 
26 Find the number of words with or without meaning which can be made using all the letters 
of the word AGAIN. If these words are written as in a dictionary, what will be the 50
th
 
word? 
6 
 
Mathematics (Set - A) 
Time: 3 hours                                        M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
3. Use of calculators is not permitted. 
 
Section A 
1 
Describe the set {-1, 1} in set builder form. 
 
{x : x?Z, x² - 1 = 0} 
1 
2 
Find the value of .
3
31
sin
p
 
2
3
 
1 
3 
Write the range of a Signum function? 
 
{-1, 0, 1} 
1 
4 
Solve 2 8 3 > + x , when x is an integer. 
 
{-1, 0, 1, 2, 3,…} 
1 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
 
210 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are  
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
 
(1, 1, 2) 
1 
Section B 
 
7 
Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
 
Solve to get {1, 3, 5, 7}    (ii) Solve to get {3, 5, 6}                             2 + 2 
4 
 
8 
Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
 
(i) LHS = ( ) ( ) ( ) = = ? = ' n n = ' n ? n A U A B B A B A B A RHS  (U = universal set) 
(ii) LHS = ( ) ( ) ( ) ( ) = ? = n ? = ' n n ? = ' n ? B A U B A A A B A A B A RHS  
                          (using distributive property)                                  2 + 2 
4 
 
 
9 
If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
 
( ) ( )
x x
x
f x f f
1
1
1
- = ?
?
?
?
?
?
+
- =   
2 marks for ( ) ( ) x f f and 2 marks for final answer 
OR 
4 
Page 4


   
Date:                         Class: XI 
Mathematics (Set - A) 
Time: 3 hrs                                                M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
 
Section A 
1 Describe the set {-1, 1} in set builder form. 1 
2 Find the value of .
3
31
sin
p
 
1 
3 Write the range of a Signum function? 
1 
4 Solve 2 8 3 > + x , when x is an integer. 
1 
 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are   
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
1 
Section B 
 
7 Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
4 
 
8 Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
4 
 
 
9 If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
OR 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
4 
 
10 
Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
 
4 
 
13 
Prove that: 
x
x
x
x
2 tan
8 tan
1 4 sec
1 8 sec
=
- - 
 
 
4 
 
14 
Using Principle of Mathematical Induction, prove that 
n n
14 41 - is a multiple of 27,  
for . N n ? 
 
4 
 
15 
Solve the given inequality and represent the solution graphically on a number line: 
   ( ) ( ) 0 3 2 3 7 2 5 = + - - x x ; . 47 6 19 2 + = + x x 
 
4 
 
16 
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so 
that the resulting mixture will contain more than 25% but less than 30% acid content? 
OR 
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The 
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of 
the 8% solution, how many litres of the 2% solution will have to be added? 
4 
 
17 
Find the ratio in which the join of A(2, -1, 3) and B(-1, 2, 1) is divided by the plane             
x + y+z = 5. Also find the coordinates of point of division  
4 
 
18 
Find the number of different 8-letter arrangements that can be made from the letters of the 
word DAUGHTER so that 
(i) all vowels occur together     (ii) all vowels do not occur together 
OR 
In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not 
come together? 
4 
19 A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be 
selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls? 
 
4 
Section C 
20 In a class of 60 students, 23 play Hockey, 15 play basketball and 20 play cricket, 7 play 
Hockey and Basketball, 5 play cricket and basketball, 4 play hockey and cricket and 15 
students do not play any of these games. Find 
a. How many play hockey, basketball and cricket. 
b. How many play hockey but not cricket. 
c. How many play hockey and cricket but not basketball. 
 
6 
 
21 (i) Prove that: ? 8 2 2 2 Cos + + + = 2 Cos? . 
(ii) Prove that: ( ) ( )
1 2 cos 2
1 2 cos 2
60 tan 60 tan
- +
= - ° + °
?
?
? ? . 
3+3 
 
22 Prove that: .
2
5
sin 5 sin
2
9
cos 3 cos
2
cos 2 cos
x
x
x
x
x
x = - 
OR 
Prove that:  .
2
3
3
cos
3
cos cos
2 2 2
= ?
?
?
?
?
?
- + ?
?
?
?
?
?
+ +
p p
x x x 
 
6 
 
23 Using Principle of Mathematical Induction, prove that 5 5 . 3 7 . 2 - +
n n
is divisible by 24, for 
all . N n ? 
OR 
Using Principle of Mathematical Induction, prove that: 
 
1
2
... 3 2 1
1
.......
3 2 1
1
2 1
1
1
+
=
+ + + +
+ +
+ +
+
+
+
n
n
n
 , for all . N n ? 
 
6 
 
 
24 Solve the following system of inequalities graphically:  
   
6 
 
 
25 (i) The mid points of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Find its 
vertices. 
(ii) Find the coordinates of a point equidistant from the four points  
      O(0, 0, 0),  A(a, 0, 0), B(0, b, 0) and C(0, 0, c). 
 
3+3 
 
26 Find the number of words with or without meaning which can be made using all the letters 
of the word AGAIN. If these words are written as in a dictionary, what will be the 50
th
 
word? 
6 
 
Mathematics (Set - A) 
Time: 3 hours                                        M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
3. Use of calculators is not permitted. 
 
Section A 
1 
Describe the set {-1, 1} in set builder form. 
 
{x : x?Z, x² - 1 = 0} 
1 
2 
Find the value of .
3
31
sin
p
 
2
3
 
1 
3 
Write the range of a Signum function? 
 
{-1, 0, 1} 
1 
4 
Solve 2 8 3 > + x , when x is an integer. 
 
{-1, 0, 1, 2, 3,…} 
1 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
 
210 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are  
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
 
(1, 1, 2) 
1 
Section B 
 
7 
Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
 
Solve to get {1, 3, 5, 7}    (ii) Solve to get {3, 5, 6}                             2 + 2 
4 
 
8 
Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
 
(i) LHS = ( ) ( ) ( ) = = ? = ' n n = ' n ? n A U A B B A B A B A RHS  (U = universal set) 
(ii) LHS = ( ) ( ) ( ) ( ) = ? = n ? = ' n n ? = ' n ? B A U B A A A B A A B A RHS  
                          (using distributive property)                                  2 + 2 
4 
 
 
9 
If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
 
( ) ( )
x x
x
f x f f
1
1
1
- = ?
?
?
?
?
?
+
- =   
2 marks for ( ) ( ) x f f and 2 marks for final answer 
OR 
4 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
Solving LHS and RHS     2 + 2 
10 Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
(i) Domain = (5, 8 )    Solve to get Range = (0, 8 )          1 + 1 
(ii) Domain = R – {3}    Solve to get Range = R – {-1}    1 + 1 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 
 
Z n n x x x
Z n n x x
x x
x x
x x x
? = ? = ? = - ? + = ? =
= - =
= - = - , 2 0 cos cos 0 1 cos
,
4
) 1 2 ( 0 2 cos
0 1 cos , 0 2 cos
0 ) 1 (cos 2 cos 2
0 2 cos 2 cos 2 cos 2
p
p
2  
2 marks for final answer 
4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
 
By sine formulae, a/sinA = b/sinB = c/sinC = k (say) 
 
( )
( ) C B k
C B k
c b
c b
sin sin
sin sin
+
- =
+
-                1 
     = 
2
cos
2
sin 2
2
sin
2
cos 2
C B C B
C B C B
- +
+
- +
+
     1 
  = 
2
tan
2
cot
C B C B - +
             1 
  = 
2
tan )
2 2
cot(
C B A - - p
     
  =
2
cot
2
tan
A
C B -             1 
 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
Consider ] sin cos cos [sin ) sin( C B C B a C B a - = -    
By sine formulae, a/sinA = b/sinB = c/sinC = k (say)     1 
                                     
( )
2 2
2 2 2 2 2 2
2 2
c b k
ac
b c a
ck
ab
c b a
bk a
- =
?
?
?
?
?
?
?
?
?
?
?
?
?
? - +
- ?
?
?
?
?
?
?
? - +
=
            1 
Similarly, ( )
2 2
) sin( a c k A C b - = -               
4 
Page 5


   
Date:                         Class: XI 
Mathematics (Set - A) 
Time: 3 hrs                                                M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
 
Section A 
1 Describe the set {-1, 1} in set builder form. 1 
2 Find the value of .
3
31
sin
p
 
1 
3 Write the range of a Signum function? 
1 
4 Solve 2 8 3 > + x , when x is an integer. 
1 
 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are   
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
1 
Section B 
 
7 Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
4 
 
8 Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
4 
 
 
9 If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
OR 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
4 
 
10 
Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
 
4 
 
13 
Prove that: 
x
x
x
x
2 tan
8 tan
1 4 sec
1 8 sec
=
- - 
 
 
4 
 
14 
Using Principle of Mathematical Induction, prove that 
n n
14 41 - is a multiple of 27,  
for . N n ? 
 
4 
 
15 
Solve the given inequality and represent the solution graphically on a number line: 
   ( ) ( ) 0 3 2 3 7 2 5 = + - - x x ; . 47 6 19 2 + = + x x 
 
4 
 
16 
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so 
that the resulting mixture will contain more than 25% but less than 30% acid content? 
OR 
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The 
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of 
the 8% solution, how many litres of the 2% solution will have to be added? 
4 
 
17 
Find the ratio in which the join of A(2, -1, 3) and B(-1, 2, 1) is divided by the plane             
x + y+z = 5. Also find the coordinates of point of division  
4 
 
18 
Find the number of different 8-letter arrangements that can be made from the letters of the 
word DAUGHTER so that 
(i) all vowels occur together     (ii) all vowels do not occur together 
OR 
In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not 
come together? 
4 
19 A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be 
selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls? 
 
4 
Section C 
20 In a class of 60 students, 23 play Hockey, 15 play basketball and 20 play cricket, 7 play 
Hockey and Basketball, 5 play cricket and basketball, 4 play hockey and cricket and 15 
students do not play any of these games. Find 
a. How many play hockey, basketball and cricket. 
b. How many play hockey but not cricket. 
c. How many play hockey and cricket but not basketball. 
 
6 
 
21 (i) Prove that: ? 8 2 2 2 Cos + + + = 2 Cos? . 
(ii) Prove that: ( ) ( )
1 2 cos 2
1 2 cos 2
60 tan 60 tan
- +
= - ° + °
?
?
? ? . 
3+3 
 
22 Prove that: .
2
5
sin 5 sin
2
9
cos 3 cos
2
cos 2 cos
x
x
x
x
x
x = - 
OR 
Prove that:  .
2
3
3
cos
3
cos cos
2 2 2
= ?
?
?
?
?
?
- + ?
?
?
?
?
?
+ +
p p
x x x 
 
6 
 
23 Using Principle of Mathematical Induction, prove that 5 5 . 3 7 . 2 - +
n n
is divisible by 24, for 
all . N n ? 
OR 
Using Principle of Mathematical Induction, prove that: 
 
1
2
... 3 2 1
1
.......
3 2 1
1
2 1
1
1
+
=
+ + + +
+ +
+ +
+
+
+
n
n
n
 , for all . N n ? 
 
6 
 
 
24 Solve the following system of inequalities graphically:  
   
6 
 
 
25 (i) The mid points of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Find its 
vertices. 
(ii) Find the coordinates of a point equidistant from the four points  
      O(0, 0, 0),  A(a, 0, 0), B(0, b, 0) and C(0, 0, c). 
 
3+3 
 
26 Find the number of words with or without meaning which can be made using all the letters 
of the word AGAIN. If these words are written as in a dictionary, what will be the 50
th
 
word? 
6 
 
Mathematics (Set - A) 
Time: 3 hours                                        M. M: 100  
General Instructions: 
1. All questions are compulsory. 
2. The question paper consists of 26 questions divided into three sections A, B and C. Section A 
comprises of 6 questions of 1 mark each. Section B comprises of 13 questions of 4 marks each and 
Section C comprises of 7 questions of 6 marks each. 
3. Use of calculators is not permitted. 
 
Section A 
1 
Describe the set {-1, 1} in set builder form. 
 
{x : x?Z, x² - 1 = 0} 
1 
2 
Find the value of .
3
31
sin
p
 
2
3
 
1 
3 
Write the range of a Signum function? 
 
{-1, 0, 1} 
1 
4 
Solve 2 8 3 > + x , when x is an integer. 
 
{-1, 0, 1, 2, 3,…} 
1 
5 
Seven athletes are participating in a race. In how many ways can the first three prizes be 
won? 
 
210 
1 
 
6 
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are  
(3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of point C. 
 
(1, 1, 2) 
1 
Section B 
 
7 
Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}. 
Find: (i) ( ) C B A ' n ? '  (ii) ( ) ( ). C A A B - ? - 
 
Solve to get {1, 3, 5, 7}    (ii) Solve to get {3, 5, 6}                             2 + 2 
4 
 
8 
Show that for any two sets A and B,  
(i) ) ( ) ( B A B A A - ? n = (ii) ) ( ) ( B A A B A ? = - ? 
 
(i) LHS = ( ) ( ) ( ) = = ? = ' n n = ' n ? n A U A B B A B A B A RHS  (U = universal set) 
(ii) LHS = ( ) ( ) ( ) ( ) = ? = n ? = ' n n ? = ' n ? B A U B A A A B A A B A RHS  
                          (using distributive property)                                  2 + 2 
4 
 
 
9 
If ( )
1
1
+
- =
x
x
x f , 1 - ? x , then show that ( ) ( )
x
x f f
1
- = , provided that 0 ? x 
 
( ) ( )
x x
x
f x f f
1
1
1
- = ?
?
?
?
?
?
+
- =   
2 marks for ( ) ( ) x f f and 2 marks for final answer 
OR 
4 
Let A = {1,2,3}, B = {4} and C = {5}.  
(i) Verify that: ( ) ( ) ( ) C A B A C B A × - × = - ×       (ii)  Find ( ) ( ) C A B A × n × 
 
Solving LHS and RHS     2 + 2 
10 Find the domain and range of the following real functions: 
 (i) 
5
1
) (
- =
x
x f  (ii) 
x
x
x f
- - =
3
2
) ( 
 
(i) Domain = (5, 8 )    Solve to get Range = (0, 8 )          1 + 1 
(ii) Domain = R – {3}    Solve to get Range = R – {-1}    1 + 1 
2+2 
11 Find the general solution of the following equation: . 0 2 cos 2 cos 3 cos = - + x x x 
 
Z n n x x x
Z n n x x
x x
x x
x x x
? = ? = ? = - ? + = ? =
= - =
= - = - , 2 0 cos cos 0 1 cos
,
4
) 1 2 ( 0 2 cos
0 1 cos , 0 2 cos
0 ) 1 (cos 2 cos 2
0 2 cos 2 cos 2 cos 2
p
p
2  
2 marks for final answer 
4 
 
12 
In triangle ABC, prove that: .
2
cot
2
tan
A
c b
c b C B
+
- =
- 
 
By sine formulae, a/sinA = b/sinB = c/sinC = k (say) 
 
( )
( ) C B k
C B k
c b
c b
sin sin
sin sin
+
- =
+
-                1 
     = 
2
cos
2
sin 2
2
sin
2
cos 2
C B C B
C B C B
- +
+
- +
+
     1 
  = 
2
tan
2
cot
C B C B - +
             1 
  = 
2
tan )
2 2
cot(
C B A - - p
     
  =
2
cot
2
tan
A
C B -             1 
 
OR 
 
In any triangle ABC, prove that: . 0 ) sin( ) sin( ) sin( = - + - + - B A c A C b C B a 
 
Consider ] sin cos cos [sin ) sin( C B C B a C B a - = -    
By sine formulae, a/sinA = b/sinB = c/sinC = k (say)     1 
                                     
( )
2 2
2 2 2 2 2 2
2 2
c b k
ac
b c a
ck
ab
c b a
bk a
- =
?
?
?
?
?
?
?
?
?
?
?
?
?
? - +
- ?
?
?
?
?
?
?
? - +
=
            1 
Similarly, ( )
2 2
) sin( a c k A C b - = -               
4 
And ( )
2 2
) sin( b a k B A c - = -                       1 
Hence LHS = 0 = RHS                         1 
 
13 Prove that: 
x
x
x
x
2 tan
8 tan
1 4 sec
1 8 sec
=
- - 
 
Considering LHS  
= 
RHS
x
x
x
x
x
x x
x
x x
x
x
x
x x
x
x
x
x
x
x
x
x
=
=
=
=
=
- - =
2 sin
2 cos
.
8 cos
8 sin
2 sin 2
2 cos 2 sin 2
.
8 cos
4 cos 4 sin 2
2 sin 2
4 sin
.
8 cos
4 cos 4 sin 2
2 sin 2
4 cos
.
8 cos
4 sin 2
4 cos 1
4 cos
.
8 cos
8 cos 1
2
2
2
2
     2 
2 marks for final answer 
 
4 
 
14 
Using Principle of Mathematical Induction, prove that 
n n
14 41 - is a multiple of 27, for 
. N n ? 
 
Step I – Let the mathematical statement be P(n) 
Step II – Prove P(1) to be true               1 
Step III – Assume P(k) to be true      1 
Step IV – Prove P(k+1) to be true ( adding and subtracting ( )
k
14 41 ).    2 
Step V – Using PMI, hence proved.      
4 
 
15 
Solve the given inequality and represent the solution graphically on a number line: 
   ( ) ( ) 0 3 2 3 7 2 5 = + - - x x ; . 47 6 19 2 + = + x x 
 
Solve to get [ ] 11 , 7 ? x   3 
Represent solution on a number line    2 
4 
 
16 
How many litres of water will have to be added to 1125 litres of the 45% solution of acid 
so that the resulting mixture will contain more than 25% but less than 30% acid content? 
 
Let x litres of water is to be added 
Then, 25% of (1125 + x) < 0% of x + 45% of 1125 < 30% of (1125 + x)     2 
Solve to get 562.5 < x < 900 litres.    2 
 
OR 
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The 
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres 
of the 8% solution, how many litres of the 2% solution will have to be added? 
 
Let x litres of 2% boric acid solution is to be added. 
Then, 4% of (640 + x) < 2% of x + 8% of 640 < 6% of (640 + x)      2 
Solve to get 320 < x < 1280 litres.    2 
4 
 
17 
Find the ratio in which the join of A(2, -1, 3) and B(-1, 2, 1) is divided by the plane             
x + y+z = 5. Also find the coordinates of point of division. 
 
Let Point C divide AB in the ratio be k : 1. 
4