Mathematics- Model Test Paper -1, class 11, CBSE, NCERT Notes | EduRev

: Mathematics- Model Test Paper -1, class 11, CBSE, NCERT Notes | EduRev

 Page 1


101 XI – Mathematics
MODEL TEST PAPER – I
Time : 3 hours Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) Q. 1 to Q. 10 of Section A are of 1 mark each.
(iii) Q. 11 to Q. 22 of Section B are of 4 marks each.
(iv) Q. 23 to Q. 29 of Section C are of 6 marks each.
(v) There is no overall choice. However an internal choice has been provided
in some questions.
SECTION A
1. A = {1, 2, 3, 4, 5, 6}, B = {2, 3, 5, 7, 9}
U = {1, 2, 3, 4, .....10}, Write (A – B)´
2. Express (1 – 2i)
–2
 in the standard form a + ib.
3. Find 20
th
 term from end of the A.P. 3, 7, 11, .... 407.
4. Evaluate 5
2
 + 6
2
 + 7
2
 + .... + 20
2
5. Evaluate
0
lim
x x
x
e e
x
?
?
?
6. Evaluate
2
0
1 1
lim
x
x x
x
?
? ? ?
7. A bag contains 9 red, 7 white and 4 black balls. If two balls are drawn
at random, find the probability that both balls are red.
8. What is the probability that an ordinary year has 53 Sundays?
9. Write the contrapositive of the following statement :
“it two lines are parallel, then they do not intersect in the same plane.”
Page 2


101 XI – Mathematics
MODEL TEST PAPER – I
Time : 3 hours Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) Q. 1 to Q. 10 of Section A are of 1 mark each.
(iii) Q. 11 to Q. 22 of Section B are of 4 marks each.
(iv) Q. 23 to Q. 29 of Section C are of 6 marks each.
(v) There is no overall choice. However an internal choice has been provided
in some questions.
SECTION A
1. A = {1, 2, 3, 4, 5, 6}, B = {2, 3, 5, 7, 9}
U = {1, 2, 3, 4, .....10}, Write (A – B)´
2. Express (1 – 2i)
–2
 in the standard form a + ib.
3. Find 20
th
 term from end of the A.P. 3, 7, 11, .... 407.
4. Evaluate 5
2
 + 6
2
 + 7
2
 + .... + 20
2
5. Evaluate
0
lim
x x
x
e e
x
?
?
?
6. Evaluate
2
0
1 1
lim
x
x x
x
?
? ? ?
7. A bag contains 9 red, 7 white and 4 black balls. If two balls are drawn
at random, find the probability that both balls are red.
8. What is the probability that an ordinary year has 53 Sundays?
9. Write the contrapositive of the following statement :
“it two lines are parallel, then they do not intersect in the same plane.”
XI – Mathematics 102
10. Check the validity of the compound statement “80 is a multiple of 5
and 4.”
SECTION B
11. Find the derivative of
sin x
x
 with respect to x from first principle.
OR
Find the derivative of 
sin cos
sin cos
x x x
x x x
?
?
 with respect to x.
12. Two students Ajay and Aman appeared in an interview. The probability
that Ajay will qualify the interview is 0.16 and that Aman will quality the
interview is 0.12. The probability that both will qualify is 0.04. Find the
probability that—
(a) Both Ajay and Aman will not qualify.
(b) Only Aman qualifies.
13. Find domain and range of the real function ? ?
2
3
1
f
x
x
?
?
14. Let R be a relation in set A = {1, 2, 3, 4, 5, 6, 7} defined as R = {(a, b):
a divides b, a ? b}. Write R in Roster form and hence write its domain and
range.
OR
Draw graph of f(x) = 2 + |x – 1|.
15. Solve :
2
1
sin cos .
4
x x ? ?
16. Prove that
9 5
cos 2 . cos cos 3 cos sin 5 sin .
2 2 2
? ? ?
? ? ? ? ?
17. If x and y are any two distinct integers, then prove by mathematical
induction that x
n
 – y
n
 is divisible by (x – y) 
. n N ? ?
18. If x + iy = (a + ib)
1/3
, then show that
? ?
2 2
4
a b
x y
x y
? ?
?
Page 3


101 XI – Mathematics
MODEL TEST PAPER – I
Time : 3 hours Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) Q. 1 to Q. 10 of Section A are of 1 mark each.
(iii) Q. 11 to Q. 22 of Section B are of 4 marks each.
(iv) Q. 23 to Q. 29 of Section C are of 6 marks each.
(v) There is no overall choice. However an internal choice has been provided
in some questions.
SECTION A
1. A = {1, 2, 3, 4, 5, 6}, B = {2, 3, 5, 7, 9}
U = {1, 2, 3, 4, .....10}, Write (A – B)´
2. Express (1 – 2i)
–2
 in the standard form a + ib.
3. Find 20
th
 term from end of the A.P. 3, 7, 11, .... 407.
4. Evaluate 5
2
 + 6
2
 + 7
2
 + .... + 20
2
5. Evaluate
0
lim
x x
x
e e
x
?
?
?
6. Evaluate
2
0
1 1
lim
x
x x
x
?
? ? ?
7. A bag contains 9 red, 7 white and 4 black balls. If two balls are drawn
at random, find the probability that both balls are red.
8. What is the probability that an ordinary year has 53 Sundays?
9. Write the contrapositive of the following statement :
“it two lines are parallel, then they do not intersect in the same plane.”
XI – Mathematics 102
10. Check the validity of the compound statement “80 is a multiple of 5
and 4.”
SECTION B
11. Find the derivative of
sin x
x
 with respect to x from first principle.
OR
Find the derivative of 
sin cos
sin cos
x x x
x x x
?
?
 with respect to x.
12. Two students Ajay and Aman appeared in an interview. The probability
that Ajay will qualify the interview is 0.16 and that Aman will quality the
interview is 0.12. The probability that both will qualify is 0.04. Find the
probability that—
(a) Both Ajay and Aman will not qualify.
(b) Only Aman qualifies.
13. Find domain and range of the real function ? ?
2
3
1
f
x
x
?
?
14. Let R be a relation in set A = {1, 2, 3, 4, 5, 6, 7} defined as R = {(a, b):
a divides b, a ? b}. Write R in Roster form and hence write its domain and
range.
OR
Draw graph of f(x) = 2 + |x – 1|.
15. Solve :
2
1
sin cos .
4
x x ? ?
16. Prove that
9 5
cos 2 . cos cos 3 cos sin 5 sin .
2 2 2
? ? ?
? ? ? ? ?
17. If x and y are any two distinct integers, then prove by mathematical
induction that x
n
 – y
n
 is divisible by (x – y) 
. n N ? ?
18. If x + iy = (a + ib)
1/3
, then show that
? ?
2 2
4
a b
x y
x y
? ?
?
103 XI – Mathematics
OR
Find the square roots of the complex number 7 – 24i
19. Find the equation of the circle passing through points (1, –2) and (4,
–3) and has i ts centre on the l i ne 3x + 4y = 7.
OR
The foci of a hyperbola coincide with of the foci of the ellipse
2 2
1.
25 9
x y
? ? Find the equation of the hyperbola, if its eccentricity is 2.
20. Find the coordinates of the point, at which yz plane divides the line
segment joining points (4, 8, 10) and (6, 10, –8).
21. How many words can be made from the letters of the word ‘Mathematics’,
in which all vowels are never together.
22. From a class of 20 students, 8 are to be chosen for an excusion party.
There are two students who decide that either both of them will join or
none of the two will join. In how many ways can they be choosen?
SECTION C
23. In a survey of 25 students, it was found that 15 had taken mathematics,
12 had taken physics and 11 had taken chemistry, 5 had taken
mathematics and chemistry, 9 had taken mathematics and physics, 4
had taken physics and chemistry and 3 had taken all the three subjects.
Find the number of students who had taken
(i) atleast one of the three subjects,
(ii) only one of the three subjects.
24. Prove that
3 3 3
3 2 4
cos cos cos cos 3 .
4 3 3
A A A A
? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
25. Solve the following system of inequations graphically
x + 2y ? 40, 3x + y ? 30, 4x + 3y ? 60, x ? 0, y ? 0
OR
Page 4


101 XI – Mathematics
MODEL TEST PAPER – I
Time : 3 hours Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) Q. 1 to Q. 10 of Section A are of 1 mark each.
(iii) Q. 11 to Q. 22 of Section B are of 4 marks each.
(iv) Q. 23 to Q. 29 of Section C are of 6 marks each.
(v) There is no overall choice. However an internal choice has been provided
in some questions.
SECTION A
1. A = {1, 2, 3, 4, 5, 6}, B = {2, 3, 5, 7, 9}
U = {1, 2, 3, 4, .....10}, Write (A – B)´
2. Express (1 – 2i)
–2
 in the standard form a + ib.
3. Find 20
th
 term from end of the A.P. 3, 7, 11, .... 407.
4. Evaluate 5
2
 + 6
2
 + 7
2
 + .... + 20
2
5. Evaluate
0
lim
x x
x
e e
x
?
?
?
6. Evaluate
2
0
1 1
lim
x
x x
x
?
? ? ?
7. A bag contains 9 red, 7 white and 4 black balls. If two balls are drawn
at random, find the probability that both balls are red.
8. What is the probability that an ordinary year has 53 Sundays?
9. Write the contrapositive of the following statement :
“it two lines are parallel, then they do not intersect in the same plane.”
XI – Mathematics 102
10. Check the validity of the compound statement “80 is a multiple of 5
and 4.”
SECTION B
11. Find the derivative of
sin x
x
 with respect to x from first principle.
OR
Find the derivative of 
sin cos
sin cos
x x x
x x x
?
?
 with respect to x.
12. Two students Ajay and Aman appeared in an interview. The probability
that Ajay will qualify the interview is 0.16 and that Aman will quality the
interview is 0.12. The probability that both will qualify is 0.04. Find the
probability that—
(a) Both Ajay and Aman will not qualify.
(b) Only Aman qualifies.
13. Find domain and range of the real function ? ?
2
3
1
f
x
x
?
?
14. Let R be a relation in set A = {1, 2, 3, 4, 5, 6, 7} defined as R = {(a, b):
a divides b, a ? b}. Write R in Roster form and hence write its domain and
range.
OR
Draw graph of f(x) = 2 + |x – 1|.
15. Solve :
2
1
sin cos .
4
x x ? ?
16. Prove that
9 5
cos 2 . cos cos 3 cos sin 5 sin .
2 2 2
? ? ?
? ? ? ? ?
17. If x and y are any two distinct integers, then prove by mathematical
induction that x
n
 – y
n
 is divisible by (x – y) 
. n N ? ?
18. If x + iy = (a + ib)
1/3
, then show that
? ?
2 2
4
a b
x y
x y
? ?
?
103 XI – Mathematics
OR
Find the square roots of the complex number 7 – 24i
19. Find the equation of the circle passing through points (1, –2) and (4,
–3) and has i ts centre on the l i ne 3x + 4y = 7.
OR
The foci of a hyperbola coincide with of the foci of the ellipse
2 2
1.
25 9
x y
? ? Find the equation of the hyperbola, if its eccentricity is 2.
20. Find the coordinates of the point, at which yz plane divides the line
segment joining points (4, 8, 10) and (6, 10, –8).
21. How many words can be made from the letters of the word ‘Mathematics’,
in which all vowels are never together.
22. From a class of 20 students, 8 are to be chosen for an excusion party.
There are two students who decide that either both of them will join or
none of the two will join. In how many ways can they be choosen?
SECTION C
23. In a survey of 25 students, it was found that 15 had taken mathematics,
12 had taken physics and 11 had taken chemistry, 5 had taken
mathematics and chemistry, 9 had taken mathematics and physics, 4
had taken physics and chemistry and 3 had taken all the three subjects.
Find the number of students who had taken
(i) atleast one of the three subjects,
(ii) only one of the three subjects.
24. Prove that
3 3 3
3 2 4
cos cos cos cos 3 .
4 3 3
A A A A
? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
25. Solve the following system of inequations graphically
x + 2y ? 40, 3x + y ? 30, 4x + 3y ? 60, x ? 0, y ? 0
OR
XI – Mathematics 104
A manufacturer has 600 litres of a 12% solution of acid. How many litres
of a 30% acid solution must be added to it so that acid content in the
resulting mixture will be more than 15% but less than 18%?
26. Find n, it the ratio of the fifth term from the beginning to the fifth term from
the end in the expansion of 
4
4
1
2 is 6 : 1.
3
n
? ?
?
? ?
? ?
27. The sum of two numbers is 6 times their geometric mean. Show that the
numbers are in the ratio ? ? ? ? : .
3 2 2 3 2 2 ? ?
28. Find the image of the point (3, 8) with respect to the line x + 3y = 7
assuming the line to be a plane mirror.
29. Calculate mean and standard deviation for the following data
Age Number of persons
20 – 30 3
30 – 40 51
40 – 50 122
50 – 60 141
60 – 70 130
70 – 80 51
80 – 90 2
OR
The mean and standard deviation of 20 observations are found to be 10
and 2 respectively. On rechecking it was found that an observation 12
was misread as 8. Calculate correct mean and correct standard deviation.
Page 5


101 XI – Mathematics
MODEL TEST PAPER – I
Time : 3 hours Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) Q. 1 to Q. 10 of Section A are of 1 mark each.
(iii) Q. 11 to Q. 22 of Section B are of 4 marks each.
(iv) Q. 23 to Q. 29 of Section C are of 6 marks each.
(v) There is no overall choice. However an internal choice has been provided
in some questions.
SECTION A
1. A = {1, 2, 3, 4, 5, 6}, B = {2, 3, 5, 7, 9}
U = {1, 2, 3, 4, .....10}, Write (A – B)´
2. Express (1 – 2i)
–2
 in the standard form a + ib.
3. Find 20
th
 term from end of the A.P. 3, 7, 11, .... 407.
4. Evaluate 5
2
 + 6
2
 + 7
2
 + .... + 20
2
5. Evaluate
0
lim
x x
x
e e
x
?
?
?
6. Evaluate
2
0
1 1
lim
x
x x
x
?
? ? ?
7. A bag contains 9 red, 7 white and 4 black balls. If two balls are drawn
at random, find the probability that both balls are red.
8. What is the probability that an ordinary year has 53 Sundays?
9. Write the contrapositive of the following statement :
“it two lines are parallel, then they do not intersect in the same plane.”
XI – Mathematics 102
10. Check the validity of the compound statement “80 is a multiple of 5
and 4.”
SECTION B
11. Find the derivative of
sin x
x
 with respect to x from first principle.
OR
Find the derivative of 
sin cos
sin cos
x x x
x x x
?
?
 with respect to x.
12. Two students Ajay and Aman appeared in an interview. The probability
that Ajay will qualify the interview is 0.16 and that Aman will quality the
interview is 0.12. The probability that both will qualify is 0.04. Find the
probability that—
(a) Both Ajay and Aman will not qualify.
(b) Only Aman qualifies.
13. Find domain and range of the real function ? ?
2
3
1
f
x
x
?
?
14. Let R be a relation in set A = {1, 2, 3, 4, 5, 6, 7} defined as R = {(a, b):
a divides b, a ? b}. Write R in Roster form and hence write its domain and
range.
OR
Draw graph of f(x) = 2 + |x – 1|.
15. Solve :
2
1
sin cos .
4
x x ? ?
16. Prove that
9 5
cos 2 . cos cos 3 cos sin 5 sin .
2 2 2
? ? ?
? ? ? ? ?
17. If x and y are any two distinct integers, then prove by mathematical
induction that x
n
 – y
n
 is divisible by (x – y) 
. n N ? ?
18. If x + iy = (a + ib)
1/3
, then show that
? ?
2 2
4
a b
x y
x y
? ?
?
103 XI – Mathematics
OR
Find the square roots of the complex number 7 – 24i
19. Find the equation of the circle passing through points (1, –2) and (4,
–3) and has i ts centre on the l i ne 3x + 4y = 7.
OR
The foci of a hyperbola coincide with of the foci of the ellipse
2 2
1.
25 9
x y
? ? Find the equation of the hyperbola, if its eccentricity is 2.
20. Find the coordinates of the point, at which yz plane divides the line
segment joining points (4, 8, 10) and (6, 10, –8).
21. How many words can be made from the letters of the word ‘Mathematics’,
in which all vowels are never together.
22. From a class of 20 students, 8 are to be chosen for an excusion party.
There are two students who decide that either both of them will join or
none of the two will join. In how many ways can they be choosen?
SECTION C
23. In a survey of 25 students, it was found that 15 had taken mathematics,
12 had taken physics and 11 had taken chemistry, 5 had taken
mathematics and chemistry, 9 had taken mathematics and physics, 4
had taken physics and chemistry and 3 had taken all the three subjects.
Find the number of students who had taken
(i) atleast one of the three subjects,
(ii) only one of the three subjects.
24. Prove that
3 3 3
3 2 4
cos cos cos cos 3 .
4 3 3
A A A A
? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
25. Solve the following system of inequations graphically
x + 2y ? 40, 3x + y ? 30, 4x + 3y ? 60, x ? 0, y ? 0
OR
XI – Mathematics 104
A manufacturer has 600 litres of a 12% solution of acid. How many litres
of a 30% acid solution must be added to it so that acid content in the
resulting mixture will be more than 15% but less than 18%?
26. Find n, it the ratio of the fifth term from the beginning to the fifth term from
the end in the expansion of 
4
4
1
2 is 6 : 1.
3
n
? ?
?
? ?
? ?
27. The sum of two numbers is 6 times their geometric mean. Show that the
numbers are in the ratio ? ? ? ? : .
3 2 2 3 2 2 ? ?
28. Find the image of the point (3, 8) with respect to the line x + 3y = 7
assuming the line to be a plane mirror.
29. Calculate mean and standard deviation for the following data
Age Number of persons
20 – 30 3
30 – 40 51
40 – 50 122
50 – 60 141
60 – 70 130
70 – 80 51
80 – 90 2
OR
The mean and standard deviation of 20 observations are found to be 10
and 2 respectively. On rechecking it was found that an observation 12
was misread as 8. Calculate correct mean and correct standard deviation.
105 XI – Mathematics
MODEL TEST PAPER – I
Time : 3 hours Maximum Marks : 100
SECTION A
Note : For 1 mark questions in Section A, full marks are given if answer is
correct (i.e. the last step of the solution). Here, solution is given for your
help.
Marks
1. A – B = {1, 4, 6}
(A – B)
c
 = {2, 3, 5, 7, 8, 9, 10} ...(1)
2.
? ?
? ?
–2
2
1
1 2i
1– 2i
? ?
2
1 1 –3 4i
3 4i –3 4i 1 4i 4i
?
? ? ?
? ? ? ? ?
2
3 4i
9 16i
? ?
?
?
–3 4
i
25 25
? ?
...(1)
3. The given A.P. can be written in reverse order as 407, 403, 399, .....
Now 20th term = a + 19d
= 407 + 19 × (–4)
= 407 – 76
= 331 ...(1)
4. 5
2
 + 6
2
 + 7
2
 + ..... + 20
2
20 4
2 2
r 1 k 1
r k
? ?
? ?
? ?
? ? ? ?
2
n n 1 2n 1
n
6
? ?
? ? ?
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