Sample Question Paper 8 - Math, Class 11 | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
CBSE Board 
Class XI Mathematics 
Sample Paper – 8 
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
2
1
x
2
4x 1
lim
2x 1
?
?
?
. 
 
2. Write the negation of the statement: The number 2 is greater than 7. 
 
3. Find the value of 
1
i
?
. 
OR 
Find modulus of 1 – 3i. 
 
4. If variance of a distribution is 4 then find standard deviation of the distribution. 
 
                                                                     SECTION – B 
 
5. If A = {a, b, c, d}, B = {f, b, d, g} and n(A ? B) = total number of elements = 8 then find 
n(A’ ? B’). 
 
6. Let f : R ? R be a function given by f(x) = x
2
 + 1, find : f
-1
 (-5). 
OR 
If f(x) = 
x1
x1
?
?
 then show that f(1/x) = -f(x). 
 
7. Find the radian measures corresponding to the 5° 37’ 30’’. 
Page 2


  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
CBSE Board 
Class XI Mathematics 
Sample Paper – 8 
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
2
1
x
2
4x 1
lim
2x 1
?
?
?
. 
 
2. Write the negation of the statement: The number 2 is greater than 7. 
 
3. Find the value of 
1
i
?
. 
OR 
Find modulus of 1 – 3i. 
 
4. If variance of a distribution is 4 then find standard deviation of the distribution. 
 
                                                                     SECTION – B 
 
5. If A = {a, b, c, d}, B = {f, b, d, g} and n(A ? B) = total number of elements = 8 then find 
n(A’ ? B’). 
 
6. Let f : R ? R be a function given by f(x) = x
2
 + 1, find : f
-1
 (-5). 
OR 
If f(x) = 
x1
x1
?
?
 then show that f(1/x) = -f(x). 
 
7. Find the radian measures corresponding to the 5° 37’ 30’’. 
  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
OR 
Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 
15 °. 
 
8. If A = {1, 4}, B = {2, 3, 6} and C = {2, 3, 7} verify that A × (B – C) = A × B – A × C  
 
9. Solve sin
2 
x + sin x – 2 = 0 where 0° < ? < 360° 
OR 
Prove that tan tan 2sec
4 2 4 2
? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
10. Given below are two statements :  
p: 25 is a multiple of 5 
q: 25 is a multiple of 8 
Write the compound statement connecting these two statements with “OR” and check 
its validity. 
 
11. Find the domain for which the functions f(x) = 2x
2
 – 1 and g(x) = 1 – 3x are equal. 
 
12. In what ratio the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4. 
 
SECTION – C 
 
13. Prove that 
sec8 1 tan8
sec4 1 tan2
? ? ?
?
? ? ?
 
 
 
14. Which of the following relations are functions? 
1. A is the capital of b where b ? B and B is the set of all countries, a ? A and A is 
the set of capital cities of countries. 
2. y < x + 3 
3. y is a Maths pupil of x, where x represents any Maths teacher in a school. 
4. y = 3x + 2 
 
15. Let A be the set of first ten natural numbers and let R be a relation on A defined by                    
(x, y) ? R ? x + 2y = 10 i. e. R = {(x, y) : x ? A, y ? A and x + 2y = 10} Express R and R
-1
 
as sets of ordered pairs. Also determine domains of R and R
-1
 and ranges of R and R
-1
. 
 
16. If log10 2, log10 (2
x
 – 1) and log10 (2
x
 + 3) are in A. P. then find the value of x. 
 
17. If 1, ?, ?
2
 are the cube roots of unity, prove that  
(1 + ?)
3
 – (1 + ?
2
)
3
 = 0 and (x – y)(x ? – y)(x?
2
 – y) = x
3
 – y
3 
 
 
Page 3


  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
CBSE Board 
Class XI Mathematics 
Sample Paper – 8 
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
2
1
x
2
4x 1
lim
2x 1
?
?
?
. 
 
2. Write the negation of the statement: The number 2 is greater than 7. 
 
3. Find the value of 
1
i
?
. 
OR 
Find modulus of 1 – 3i. 
 
4. If variance of a distribution is 4 then find standard deviation of the distribution. 
 
                                                                     SECTION – B 
 
5. If A = {a, b, c, d}, B = {f, b, d, g} and n(A ? B) = total number of elements = 8 then find 
n(A’ ? B’). 
 
6. Let f : R ? R be a function given by f(x) = x
2
 + 1, find : f
-1
 (-5). 
OR 
If f(x) = 
x1
x1
?
?
 then show that f(1/x) = -f(x). 
 
7. Find the radian measures corresponding to the 5° 37’ 30’’. 
  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
OR 
Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 
15 °. 
 
8. If A = {1, 4}, B = {2, 3, 6} and C = {2, 3, 7} verify that A × (B – C) = A × B – A × C  
 
9. Solve sin
2 
x + sin x – 2 = 0 where 0° < ? < 360° 
OR 
Prove that tan tan 2sec
4 2 4 2
? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
10. Given below are two statements :  
p: 25 is a multiple of 5 
q: 25 is a multiple of 8 
Write the compound statement connecting these two statements with “OR” and check 
its validity. 
 
11. Find the domain for which the functions f(x) = 2x
2
 – 1 and g(x) = 1 – 3x are equal. 
 
12. In what ratio the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4. 
 
SECTION – C 
 
13. Prove that 
sec8 1 tan8
sec4 1 tan2
? ? ?
?
? ? ?
 
 
 
14. Which of the following relations are functions? 
1. A is the capital of b where b ? B and B is the set of all countries, a ? A and A is 
the set of capital cities of countries. 
2. y < x + 3 
3. y is a Maths pupil of x, where x represents any Maths teacher in a school. 
4. y = 3x + 2 
 
15. Let A be the set of first ten natural numbers and let R be a relation on A defined by                    
(x, y) ? R ? x + 2y = 10 i. e. R = {(x, y) : x ? A, y ? A and x + 2y = 10} Express R and R
-1
 
as sets of ordered pairs. Also determine domains of R and R
-1
 and ranges of R and R
-1
. 
 
16. If log10 2, log10 (2
x
 – 1) and log10 (2
x
 + 3) are in A. P. then find the value of x. 
 
17. If 1, ?, ?
2
 are the cube roots of unity, prove that  
(1 + ?)
3
 – (1 + ?
2
)
3
 = 0 and (x – y)(x ? – y)(x?
2
 – y) = x
3
 – y
3 
 
 
  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
18. An integer is chosen at random from the first two hundred positive integers. What is 
the probability that the integer chosen is divisible by 6 or 8? 
 
19. Evaluate 0.2345 
 
20. Determine the number of natural numbers smaller than 10
4
, in the decimal notation of 
which all the digits are distinct. 
OR 
 Find the sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at 
a time. 
 
21. Find the equation of the locus of all points such that the difference of their distances 
from (4, 0) and (-4, 0) is always equal to 2. 
OR 
         Find the lengths of the transverse and conjugate axes, co-ordinates of the foci, vertices 
and eccentricity for the hyperbola 9x
2
 – 16y
2
 = 144 
 
22. Evaluate the derivative of the following function at indicated points :  
1 sinx
1 cosx
?
?
at x
2
?
? 
OR 
 If y = 
23
x x x
1 ....
1! 2! 3!
? ? ? ? show that 
dy
y
dx
? 
 
23. Find the vertex, focus and directrix of the parabola 4y
2
 + 12x – 12y + 39 = 0. 
 
 
SECTION – D 
24. Prove that sin10°sin30°sin50°sin70° = 1/16 
 
OR 
 
Prove that cosA cos(60° – A) cos(60° + A) = 
1
cos3A
4
 
 
25. For a group of 200 candidates the mean and S. D. were found to be 40 and 15 
respectively. Later on it was found that the score 43 was misread as 34. Find the correct 
mean and correct S. D. 
26. If 
3
tan tan
4 2 4 2
? ? ? ? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 prove that 
3
2
3sin sin
sin
1 3sin
? ? ?
??
??
 
 
 
Page 4


  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
CBSE Board 
Class XI Mathematics 
Sample Paper – 8 
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
2
1
x
2
4x 1
lim
2x 1
?
?
?
. 
 
2. Write the negation of the statement: The number 2 is greater than 7. 
 
3. Find the value of 
1
i
?
. 
OR 
Find modulus of 1 – 3i. 
 
4. If variance of a distribution is 4 then find standard deviation of the distribution. 
 
                                                                     SECTION – B 
 
5. If A = {a, b, c, d}, B = {f, b, d, g} and n(A ? B) = total number of elements = 8 then find 
n(A’ ? B’). 
 
6. Let f : R ? R be a function given by f(x) = x
2
 + 1, find : f
-1
 (-5). 
OR 
If f(x) = 
x1
x1
?
?
 then show that f(1/x) = -f(x). 
 
7. Find the radian measures corresponding to the 5° 37’ 30’’. 
  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
OR 
Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 
15 °. 
 
8. If A = {1, 4}, B = {2, 3, 6} and C = {2, 3, 7} verify that A × (B – C) = A × B – A × C  
 
9. Solve sin
2 
x + sin x – 2 = 0 where 0° < ? < 360° 
OR 
Prove that tan tan 2sec
4 2 4 2
? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
10. Given below are two statements :  
p: 25 is a multiple of 5 
q: 25 is a multiple of 8 
Write the compound statement connecting these two statements with “OR” and check 
its validity. 
 
11. Find the domain for which the functions f(x) = 2x
2
 – 1 and g(x) = 1 – 3x are equal. 
 
12. In what ratio the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4. 
 
SECTION – C 
 
13. Prove that 
sec8 1 tan8
sec4 1 tan2
? ? ?
?
? ? ?
 
 
 
14. Which of the following relations are functions? 
1. A is the capital of b where b ? B and B is the set of all countries, a ? A and A is 
the set of capital cities of countries. 
2. y < x + 3 
3. y is a Maths pupil of x, where x represents any Maths teacher in a school. 
4. y = 3x + 2 
 
15. Let A be the set of first ten natural numbers and let R be a relation on A defined by                    
(x, y) ? R ? x + 2y = 10 i. e. R = {(x, y) : x ? A, y ? A and x + 2y = 10} Express R and R
-1
 
as sets of ordered pairs. Also determine domains of R and R
-1
 and ranges of R and R
-1
. 
 
16. If log10 2, log10 (2
x
 – 1) and log10 (2
x
 + 3) are in A. P. then find the value of x. 
 
17. If 1, ?, ?
2
 are the cube roots of unity, prove that  
(1 + ?)
3
 – (1 + ?
2
)
3
 = 0 and (x – y)(x ? – y)(x?
2
 – y) = x
3
 – y
3 
 
 
  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
18. An integer is chosen at random from the first two hundred positive integers. What is 
the probability that the integer chosen is divisible by 6 or 8? 
 
19. Evaluate 0.2345 
 
20. Determine the number of natural numbers smaller than 10
4
, in the decimal notation of 
which all the digits are distinct. 
OR 
 Find the sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at 
a time. 
 
21. Find the equation of the locus of all points such that the difference of their distances 
from (4, 0) and (-4, 0) is always equal to 2. 
OR 
         Find the lengths of the transverse and conjugate axes, co-ordinates of the foci, vertices 
and eccentricity for the hyperbola 9x
2
 – 16y
2
 = 144 
 
22. Evaluate the derivative of the following function at indicated points :  
1 sinx
1 cosx
?
?
at x
2
?
? 
OR 
 If y = 
23
x x x
1 ....
1! 2! 3!
? ? ? ? show that 
dy
y
dx
? 
 
23. Find the vertex, focus and directrix of the parabola 4y
2
 + 12x – 12y + 39 = 0. 
 
 
SECTION – D 
24. Prove that sin10°sin30°sin50°sin70° = 1/16 
 
OR 
 
Prove that cosA cos(60° – A) cos(60° + A) = 
1
cos3A
4
 
 
25. For a group of 200 candidates the mean and S. D. were found to be 40 and 15 
respectively. Later on it was found that the score 43 was misread as 34. Find the correct 
mean and correct S. D. 
26. If 
3
tan tan
4 2 4 2
? ? ? ? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 prove that 
3
2
3sin sin
sin
1 3sin
? ? ?
??
??
 
 
 
  
 
CBSE XI | Mathematics 
Sample Paper – 8 
 
     
27. Solve the following system of inequalities graphically:  
x + 2y ? 10; x + y ? 1; x - y ? 0; x ? 0; y ? 0 
 
OR 
       For the purpose of an experiment an acid solution between 4% and 6% is required.       
        640 liters of 8% acid solution and a 2% acid solution are available in a laboratory. How 
many liters of the 2% solution needs to be added to the 8% solution? 
28. 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
? ? ? ? ? ? ? ? ?
?
?
?
?
?
?
? ?
?
?
 
29. A student wants to buy a computer for Rs. 12,000. He has saved up to Rs. 6000 which he 
pays as cash. He is to pay the balance in annual installments of Rs. 500 plus an interest 
of 12% on the unpaid amount. How much will the computer cost him? 
OR 
          Find the value of
 
2 2 2
2 2 2
1 2 2 3 3 4 ....uptill the nth term
 
1 2 2 3 3 4 ....uptill the nth term
? ? ? ? ? ?
? ? ? ? ? ?
.