Sample Question Paper 2 - Math, Class 11 | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XI Mathematics 
Sample Paper – 2 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consist of 29 questions. 
3. Questions 1 – 4 in Section A are very short answer type questions carrying 1 mark 
each. 
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 mark each. 
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 mark 
each. 
6. Questions 24 – 29 in Section D are long-answer type II questions carrying 6 mark 
each. 
 
SECTION – A 
 
1. Find the sum to infinity of the sequence: 
2 3 4
1 1 1 1
...
3
3 3 3
? ? ? ? 
OR 
Use geometric series to express 0.555…= 0.5 as a rational number. 
 
2. Write the truth value of the statement p: Intersection of two disjoint sets is an empty set. 
  
3. Find cos cos sin sin
4 4 4 4
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
. 
4. 
1
Find the argument of 
1- i
. 
 
SECTION – B 
 
5. What is the eccentricity of the curve 4 x
2
 + y
2
 = 100? 
 
6. What is the probability that two friends will have the same birthday? 
OR 
The probability that a person visiting a dentist will have his teeth cleaned is 0.44, the 
probability that he will have a cavity filled is 0.24. The probability that he will have his 
teeth cleaned or a cavity filled is 0.6. What is the probability that a person visiting a 
dentist will have his teeth cleaned and cavity filled? 
 
Page 2


  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XI Mathematics 
Sample Paper – 2 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consist of 29 questions. 
3. Questions 1 – 4 in Section A are very short answer type questions carrying 1 mark 
each. 
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 mark each. 
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 mark 
each. 
6. Questions 24 – 29 in Section D are long-answer type II questions carrying 6 mark 
each. 
 
SECTION – A 
 
1. Find the sum to infinity of the sequence: 
2 3 4
1 1 1 1
...
3
3 3 3
? ? ? ? 
OR 
Use geometric series to express 0.555…= 0.5 as a rational number. 
 
2. Write the truth value of the statement p: Intersection of two disjoint sets is an empty set. 
  
3. Find cos cos sin sin
4 4 4 4
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
. 
4. 
1
Find the argument of 
1- i
. 
 
SECTION – B 
 
5. What is the eccentricity of the curve 4 x
2
 + y
2
 = 100? 
 
6. What is the probability that two friends will have the same birthday? 
OR 
The probability that a person visiting a dentist will have his teeth cleaned is 0.44, the 
probability that he will have a cavity filled is 0.24. The probability that he will have his 
teeth cleaned or a cavity filled is 0.6. What is the probability that a person visiting a 
dentist will have his teeth cleaned and cavity filled? 
 
  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7. Divide 20 into 4 parts which form an A.P. such that ratio of the product of the I
st
 
and the 4
th
 term to the product of the 2
nd
 and the 3
rd
 is 2: 3. 
 
8. If the sum of n terms of an A.P is (pn + qn
2
) where p, q are constants, find the common 
difference. 
 
9. Let R be a relation from N to N defined by 
R = {(a, b): a, b ?N and a = b
2
}. 
Then, which of the following statement is true? 
(i)  (a, a) ?R, for all a ?N 
(ii) (a, b) ?R, implies (b, a) ?R 
 
10. Differentiate |2x – 1| w.r.t. x. 
OR 
Differentiate x sin x log x with respect to x. 
 
11. One end of diameter of the circle x
2
+y
2
-3x+5y-4=0 is (2, 1). Find the co-ordinates of 
other end. 
 
12. Find the equation of ellipse with e = ¾, foci on y axis, centre at the origin & passing 
through point (6, 4). 
OR 
Find the distance between the directrices the ellipse 
22
xy
1
36 20
?? 
 
SECTION – C 
 
13. A school gave out medals on its sports day. 38 medals were given for soccer, 15 for 
basketball, and 20 for cricket. These medals were given to 58 students in all. Only three 
students got medals in all three sports.  How many students received medals in exactly 
two of the three sports?  
 
14. Show that: 2cos6? = 64cos
6
? – 96cos
4
? + 36cos²? – 2 
 
OR 
Show that: 
sin3 cos3
2
sin cos
??
??
??
 
 
15. In how many ways can 5 children be arranged in a row such that 2 boys x and y, (i) are 
always together (ii) are never sit together? 
OR 
Page 3


  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XI Mathematics 
Sample Paper – 2 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consist of 29 questions. 
3. Questions 1 – 4 in Section A are very short answer type questions carrying 1 mark 
each. 
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 mark each. 
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 mark 
each. 
6. Questions 24 – 29 in Section D are long-answer type II questions carrying 6 mark 
each. 
 
SECTION – A 
 
1. Find the sum to infinity of the sequence: 
2 3 4
1 1 1 1
...
3
3 3 3
? ? ? ? 
OR 
Use geometric series to express 0.555…= 0.5 as a rational number. 
 
2. Write the truth value of the statement p: Intersection of two disjoint sets is an empty set. 
  
3. Find cos cos sin sin
4 4 4 4
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
. 
4. 
1
Find the argument of 
1- i
. 
 
SECTION – B 
 
5. What is the eccentricity of the curve 4 x
2
 + y
2
 = 100? 
 
6. What is the probability that two friends will have the same birthday? 
OR 
The probability that a person visiting a dentist will have his teeth cleaned is 0.44, the 
probability that he will have a cavity filled is 0.24. The probability that he will have his 
teeth cleaned or a cavity filled is 0.6. What is the probability that a person visiting a 
dentist will have his teeth cleaned and cavity filled? 
 
  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7. Divide 20 into 4 parts which form an A.P. such that ratio of the product of the I
st
 
and the 4
th
 term to the product of the 2
nd
 and the 3
rd
 is 2: 3. 
 
8. If the sum of n terms of an A.P is (pn + qn
2
) where p, q are constants, find the common 
difference. 
 
9. Let R be a relation from N to N defined by 
R = {(a, b): a, b ?N and a = b
2
}. 
Then, which of the following statement is true? 
(i)  (a, a) ?R, for all a ?N 
(ii) (a, b) ?R, implies (b, a) ?R 
 
10. Differentiate |2x – 1| w.r.t. x. 
OR 
Differentiate x sin x log x with respect to x. 
 
11. One end of diameter of the circle x
2
+y
2
-3x+5y-4=0 is (2, 1). Find the co-ordinates of 
other end. 
 
12. Find the equation of ellipse with e = ¾, foci on y axis, centre at the origin & passing 
through point (6, 4). 
OR 
Find the distance between the directrices the ellipse 
22
xy
1
36 20
?? 
 
SECTION – C 
 
13. A school gave out medals on its sports day. 38 medals were given for soccer, 15 for 
basketball, and 20 for cricket. These medals were given to 58 students in all. Only three 
students got medals in all three sports.  How many students received medals in exactly 
two of the three sports?  
 
14. Show that: 2cos6? = 64cos
6
? – 96cos
4
? + 36cos²? – 2 
 
OR 
Show that: 
sin3 cos3
2
sin cos
??
??
??
 
 
15. In how many ways can 5 children be arranged in a row such that 2 boys x and y, (i) are 
always together (ii) are never sit together? 
OR 
  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  3 
 In how many ways can 5 men and 4 women be seated in a row, so that the women 
occupy even places only? 
 
16. Find the equation of the set of points P, the sum of whose distances from A(4, 0, 0) and  
B(-4, 0, 0) is equal to 10 ? 
 
17. 
9 3 5
Prove that 2 cos cos cos cos 0
13 13 13 13
? ? ? ?
? ? ? 
 
18. Find the domain and range of f(x) = x5 ?
 
OR 
 Find the domain and range of f(x) =
2
3
(2 x ) ?
 
 
19. A ladder 12 m long leaning against a wall begins to slide down. Its one end always 
remains on the wall and the other on the floor. Find the equation of the locus of a point 
P which is 3 m from the end in contact with the floor. Identify the conic section 
represented by the equation. 
  
20. Prove that a
n
 – b
n
 is a multiple of (a - b), where a and b are natural numbers. 
  
21. Find the equation of a line, perpendicular to the line whose equation is 6x – 7y + 8 = 0 
and which passes through the point of intersection of the two lines whose equations 
are 2x – 3y – 4 = 0 and 3x + 4y – 5 = 0. 
 
22. An administration assistant is given three letters to be mailed to three different people. 
He is also given three addressed envelopes in which to put them and send to three 
people X, Y and Z. What is the probability that atleast one person out of X, Y and Z got 
the letter written to him? 
 
23. If O is the sum of odd terms and E of even terms in the expansion of  (x + a)
n
, prove 
that: 
(i)  O
2
 - E
2
 = (x
2
 - a
2
)
n
 
(ii) 4OE = (x + a)
2n
 - (x - a)
2n 
 
SECTION – D 
 
24. The sum of n terms of two A.P.s are in the ratio (7n + 1) : (4n + 27). Find the ratio of 
their 13
th
 term. 
OR 
The ratio of the sums m and n terms of an A. P. is m
2
 : n
2
. Show that the ratio of the m
th 
and n
th
 term is (2m – 1) : (2n – 1). 
Page 4


  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  1 
CBSE Board 
Class XI Mathematics 
Sample Paper – 2 
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consist of 29 questions. 
3. Questions 1 – 4 in Section A are very short answer type questions carrying 1 mark 
each. 
4. Questions 5 – 12 in Section B are short-answer type questions carrying 2 mark each. 
5. Questions 13 – 23 in Section C are long-answer I type questions carrying 4 mark 
each. 
6. Questions 24 – 29 in Section D are long-answer type II questions carrying 6 mark 
each. 
 
SECTION – A 
 
1. Find the sum to infinity of the sequence: 
2 3 4
1 1 1 1
...
3
3 3 3
? ? ? ? 
OR 
Use geometric series to express 0.555…= 0.5 as a rational number. 
 
2. Write the truth value of the statement p: Intersection of two disjoint sets is an empty set. 
  
3. Find cos cos sin sin
4 4 4 4
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
. 
4. 
1
Find the argument of 
1- i
. 
 
SECTION – B 
 
5. What is the eccentricity of the curve 4 x
2
 + y
2
 = 100? 
 
6. What is the probability that two friends will have the same birthday? 
OR 
The probability that a person visiting a dentist will have his teeth cleaned is 0.44, the 
probability that he will have a cavity filled is 0.24. The probability that he will have his 
teeth cleaned or a cavity filled is 0.6. What is the probability that a person visiting a 
dentist will have his teeth cleaned and cavity filled? 
 
  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  2 
7. Divide 20 into 4 parts which form an A.P. such that ratio of the product of the I
st
 
and the 4
th
 term to the product of the 2
nd
 and the 3
rd
 is 2: 3. 
 
8. If the sum of n terms of an A.P is (pn + qn
2
) where p, q are constants, find the common 
difference. 
 
9. Let R be a relation from N to N defined by 
R = {(a, b): a, b ?N and a = b
2
}. 
Then, which of the following statement is true? 
(i)  (a, a) ?R, for all a ?N 
(ii) (a, b) ?R, implies (b, a) ?R 
 
10. Differentiate |2x – 1| w.r.t. x. 
OR 
Differentiate x sin x log x with respect to x. 
 
11. One end of diameter of the circle x
2
+y
2
-3x+5y-4=0 is (2, 1). Find the co-ordinates of 
other end. 
 
12. Find the equation of ellipse with e = ¾, foci on y axis, centre at the origin & passing 
through point (6, 4). 
OR 
Find the distance between the directrices the ellipse 
22
xy
1
36 20
?? 
 
SECTION – C 
 
13. A school gave out medals on its sports day. 38 medals were given for soccer, 15 for 
basketball, and 20 for cricket. These medals were given to 58 students in all. Only three 
students got medals in all three sports.  How many students received medals in exactly 
two of the three sports?  
 
14. Show that: 2cos6? = 64cos
6
? – 96cos
4
? + 36cos²? – 2 
 
OR 
Show that: 
sin3 cos3
2
sin cos
??
??
??
 
 
15. In how many ways can 5 children be arranged in a row such that 2 boys x and y, (i) are 
always together (ii) are never sit together? 
OR 
  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  3 
 In how many ways can 5 men and 4 women be seated in a row, so that the women 
occupy even places only? 
 
16. Find the equation of the set of points P, the sum of whose distances from A(4, 0, 0) and  
B(-4, 0, 0) is equal to 10 ? 
 
17. 
9 3 5
Prove that 2 cos cos cos cos 0
13 13 13 13
? ? ? ?
? ? ? 
 
18. Find the domain and range of f(x) = x5 ?
 
OR 
 Find the domain and range of f(x) =
2
3
(2 x ) ?
 
 
19. A ladder 12 m long leaning against a wall begins to slide down. Its one end always 
remains on the wall and the other on the floor. Find the equation of the locus of a point 
P which is 3 m from the end in contact with the floor. Identify the conic section 
represented by the equation. 
  
20. Prove that a
n
 – b
n
 is a multiple of (a - b), where a and b are natural numbers. 
  
21. Find the equation of a line, perpendicular to the line whose equation is 6x – 7y + 8 = 0 
and which passes through the point of intersection of the two lines whose equations 
are 2x – 3y – 4 = 0 and 3x + 4y – 5 = 0. 
 
22. An administration assistant is given three letters to be mailed to three different people. 
He is also given three addressed envelopes in which to put them and send to three 
people X, Y and Z. What is the probability that atleast one person out of X, Y and Z got 
the letter written to him? 
 
23. If O is the sum of odd terms and E of even terms in the expansion of  (x + a)
n
, prove 
that: 
(i)  O
2
 - E
2
 = (x
2
 - a
2
)
n
 
(ii) 4OE = (x + a)
2n
 - (x - a)
2n 
 
SECTION – D 
 
24. The sum of n terms of two A.P.s are in the ratio (7n + 1) : (4n + 27). Find the ratio of 
their 13
th
 term. 
OR 
The ratio of the sums m and n terms of an A. P. is m
2
 : n
2
. Show that the ratio of the m
th 
and n
th
 term is (2m – 1) : (2n – 1). 
  
 
CBSE XI | Mathematics 
Sample Paper – 2 
 
  www.topperlearning.com  4 
25. 
? ? ?
? ? ? ? ?
b c c a a b cosA cosB cosC
If in a ABC, ,then prove that:  .
12 13 15 2 7 11
 
 
OR 
If in a triangle ABC, 
? ?
? ?
sin A B
sin A
sinC sin B C
?
?
?
 prove that a
2
, b
2
, c
2
 are in A. P. 
26. Show by mathematical induction that the sum to n terms of the series  
 
? ?
? ?
2 2 2 2 2 2
2
n
2
1 2 2 3 2 4 5 2 6 ....is
n n 1
, when n is even
2
S
n n 1
, when n is odd
2
? ? ? ? ? ? ? ? ?
?
?
?
?
?
?
? ?
?
? 
OR 
 
Prove by induction that the sum of the cubes of three consecutive numbers is divisible 
by 9.
 
 
27. Graph the given inequalities and shade the common solution region. 
 2x + y ? 40, x + 2y  ? 50, x + y = 35 
 
28. Given below is the frequency distribution of weekly study hours of a group of class 11 
students. Find the mean, variance and standard deviation of the distribution using the 
short cut method. 
 
Classes Frequency 
0 - 10 5 
10 - 20 8 
20 - 30 15 
30 - 40 16 
40 - 50 6 
 
29. (i) Find the derivative of ??
1
f(x) , using the first  principle.
x
 
   (ii) Evaluate: 
x x x
2
x0
6 3 2 1
lim
x
?
? ? ?