1

Sets
1. A and B are two sets such that n(A - B) = 14 + x, n(B - A) = 3x and n(A  B) = x. Draw a Venn
diagram to illustrate this information. If n(A) = n(B), find (i) the value of x (ii) n(A  B).
2. In class XI of a certain school, there are 20 students in a chemistry class and 30 students in a
physics class. Find the number of students which are either in chemistry class or in physics
class in the following cases: (i) the two classes meet at the same hour.

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(ii) the two classes meet at different hours and 10 students study both the subject.

3. In the adjacent Venn Diagram, if n() = 80, n(P) = 40,  P

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n(Q) = 28, n(P  Q) = 12 and n(P  R) = 10,
Q
R
(i) Mark the number of elements in each region
(ii) Determine the value of n(P  Q) and (n(Q  R)’).

A
4. A.T.V. survey gives the following data for T.V. watching: 60%, watch program A: 50%, watch
program B: 47% watch program C: 28% watch programs A and B: 23% watch programs A and
C: 18% watch programs B and C: 8% watch programs A, B and C. Draw a Venn diagram to
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illustrate this information and find (i) what percentage watch programs A and B but not C?
(ii) what percentage watch exactly two programs? (iii) what percentage do not watch any
program? Do you think that to some extent parents should monitor T.V. viewing habits of
children? If yes, then why?
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5. There are 240 students in class XI of a school, 130 play cricket, 100 pay football, 75 play
volleyball, 30 of these play cricket and football, 25 play volleyball and cricket, 15 play football
and volleyball. Also each student plays atleast one of three games. How many students play
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all the three games.

6. In a survey of 100 students, the number of students studying the various languages were
found to be: English only 18, English but not Hindi 23, English and Sanskrit 8, English 26,
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Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find (i) how many students were studying
Hindi? (ii) how many students were studying English and Hindi?
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7. A college awarded 38 medals for Honesty, 15 for punctuality and 20 for obedience. If these
medals were bagged by a total of 58 medals and only 3 students got medals for all three
values, how many students received medals for exactly two of the three values? Which value
you prefer to be awarded most and why?
8. From 50 students taking examination in Mathematics, Physics and Chemistry each of the
students has passed in atleast one of the subjects, 37 passed Mathematics, 24 Physics and 43
Chemistry.
 2

Almost 19 passed Mathematics and Physics, almost 29 Mathematics and Chemistry and
almost 20 Physics and Chemistry. What is the largest possible number that could have passed
in all the three subjects.
9. In class XI of a certain school, 50 students eat burger and 42 students eat noodles in lunch
time, If 24 students eat both burger and noodles, find the number of students who eat
(i) burger only (ii) noodles only (iii) any of the two food items
10. In a survey of 600 students in a school, 150 students were found to be drinking tea and 225

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drinking milk and 100 students were drinking both tea and milk. How many students were
drinking neither tea nor milk? What do you think which drink should a student prefer and
why?

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11. In a survey of 100 students regarding watching T.V., it was found that 28 watch action
movies, 30 watch comedy serials, 42 watch news channels, 8 watch action movies and
comedy serials, 10 watch action movies and news channels, 5 watch comedy serials and
news channels and 3 watch all the three program. Draw a Venn diagram to illustrate this

A
information and find (i) how many watch news channels only? (ii) how many do not watch
any of the three program? According to you which T.V. program is useful and why?
PT
12. For any sets A and B, show that (i) (A  B)  (A - B) = A, (ii) A  (B – A) = A  B.
13. For sets A, B and C, using properties of sets, prove that: (i) (A  B)  (A  B’) = A, (ii) (A 
B) - B = A – B, (iii) A – (A  B) = A – B, (iv) (A  B) – C = (A - C)  (B - C) (v) A – (B  C) = (A -
B)  (A - C), (vi) A  (B - C) = (A  B) – (A  C).
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14. For any sets A, B and C, using properties of sets, prove that:
(i) A – (A - B) = A  B , (ii) (A - B)  (A - C) = A – (B  C)
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15. If A  B’ =  then show that A  B.

16. By using properties of sets, prove that A  B if and only if B’  A’.
For any sets A and B, prove that P(A)  P(B)  P(A  B).
P

17.
18. A and B are two finite sets such that n(A)= m1 and n(B) = m2, then find the least and the
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greatest values of n(A  B).

 3

Relation and Functions

1. The adjoining diagram shows a relation between the sets P and Q. Write this relation (i) in
roster form (ii) in set builder form. What is its domain and range?
2. Find the domain and the range of the relation R defined by R = {x + 1, x + 3): x  {0, 1, 2, 3, 4,
5}}.
3. If R = {(x, y): x, y  W, 2x + y = 8}, then (i) find the domain and the range of R. (ii) write R as a
set of ordered pairs.

R
4. Find the domain and the range of the relation R given by R = {(x, y): y = x + 6/x, where x, y 
N and x < 6}.

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5. Find the linear relation between the components of the ordered pairs of the relation R where
= {(2, 1), (4, 7), (1, -2),……}.
6. If A = {1, 2, 3, 5}, B = {4, 6, 9} and a relation R from A to B is defined by R = {(x, y): the difference
between x and y is odd, x  A, y  B}. Then (i) write R in the roster form. (ii) represents R by
an arrow diagram.

A
7. Let A = {3, 5} and B = {7, 11}. Let R = {(a, b): a  A, b  B, a – b is odd}. Show that R is an empty
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relation from A to B.
8. If R = {(x, y): x, y  W, x2 + y2 = 25}, then find the domain and the range of R. Also write R in
Roster form.
9. If R = {(x, y): x, y  Z, x2 + y2 = 64}, then write R in Roster form.
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10. If R is the relation on N defined by R = {(x, y): y x + 12/x, x, y N}, then find (i) R in Roster
form, (ii) domain of R, (iii) range of R.
G

11. Write the domain and the range of the relation (x, y): x = 3y and x and y are natural numbers
less than 10.
12. Let R = {(x, y): x, y  Z, y = 2x - 4}. If (a, -2) and (4, b2) belongs to R, find the values of a and b.
P

13. Find the linear relation between the components of the ordered pairs of the relation R where
R

(i) R = {(-1, -1), (0, 2), (1, 5),….}, (ii) R = {(0, 2), (-1, 5), (2, -4),…}
14. A relation ‘f’ is defined by f: x → x2 – 2, where x  [-1, -2, 0, 2]. (i) List the elements of f. (ii) Is
f a function?
15. If A and B are finite sets such that n(A) = p and n(B) = q, then find the numbers of functions
from A to B.
16. A function ‘f’ be defined by f(x) = x2 + 1 where x  {-1, 0, 1, 3}. List the elements of f.
 4

17. If a function ‘f’ is defined by f(x) = 2x – 1 where x  {-2, 0, 3, 5}, the find its range.
18. If f(x) = ax + b, where a and b are integers, f(-1) = -5 and f(3) = 3, find a and b.
19. If (a, 8) and (2, b) are ordered pairs which belong to the mapping f: x → 3x + 4 where x  R,
find a and b.
20. If a function f from R to R is defined by f = {(x, 3x - 5): x  R}, find the values of a and b given
that (a, 4) and (1, b) belongs to f.
21. Let ‘f’ be a function defined by f: x → 5x2 + 2, x  R. Find (i) image of 3 under f. (ii) f(3) x f(2)

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(iii) x such that f(x) = 22.
x +7
22. Find the domain of the following functions: (i) f(x) = x2 + 2x + 1
2

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(ii) f(x) =
x − 8 x + 12 x − 8x + 4
2

x 2 − 3x + 1
23. If f(x) = , find f(-2) + f(1/3).
x −1

24. If f(x) = x2 – 3x + 1, find x  R such that f(2x) = f(x).

25. If f(x + 1) = 3x + 7, find (i) f(-2), (ii) f(3x).

A
26. Find the range of the function f(x) = 2 – 3x, x  R, x > 0.
PT
27. Find the domain and the range of the function f(x) = 3x 2 – 5. Also find (-3) and the numbers
which are associated with the numbers 43 in its range.
1
28. Find the domain and the range of the following functions: (i) f(x) = x − 1 (ii) f(x) = .
5− x
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29. Find the domain and the range of the following functions:
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x −3 2
1
(i) f(x) = (ii) f(x) = x 2 (iii) f(x) =
2x + 1 1+ x 1 − x2

(ii) f(x) = x2 − x + 1 .
2
x
30. Find the domain and the range of the following functions: (i) f(x) =
1+ x 2
x + x +1
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31. Find the domain and the range of the following functions:
R

1
(i) f(x) = x 2 − 4 (ii) f(x) = 9 − x 2 (iii) f(x) =
9 − x2

32. Find the domain and the range of the following functions: (i) f(x) = -|x| (ii) f(x) = 1 - |x - 2|.
x +2
33. Find the domain and the range of the functions f defined by f(x) = .
| x + 2|
 5

1 1
34. Find the domain of the following functions: (i) f(x) = (ii) f(x) = .
x +| x | x −| x |

1
35. Find the domain and the range of the functions f given by f(x) = .
[ x ]2 − [ x] − 6

36. Find the domain and the range of the following functions: (i) f(x) =|x -3|, (ii) f(x) = 3 - |x - 2|.
| x |− x
37. If a real function f is defined by f(x) = , then find its range.
2x

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38. Draw the graph of the following real functions and hence find their range:
x2 − 1
(i) f(x) = 2x – 1 (ii) f(x) =

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x −1

39. Draw the graphs of the following real functions and hence find their range:
f(x) = x2 (ii) f(x) = x3

A
40. Draw the graphs of the following real functions and hence find their range:
1 − x , x0
(i) f(x) =  1  x  R, x  0 (ii) f(x) = 1, x=0
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x  x + 1,
 x 0

 x + 1, -1  x  1
41. Draw the graphs of the real functions f defined by f(x) = 3 − x , 1  x  3 . Hence find its
6 - 2 x , 3 x 4

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range.
42. Draw the graphs of the real functions f defined by f(x) = |1 + x| + |1 - x|, x  [-2, 2]. Hence
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find its range.

1
43. Find the domain of the functions f defined by f(x) = 4 − x + .
x −1
2
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44. If f(x) = x − , prove that (f(x))3= f(x3) + 3f  1  .

1
x x
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x −1
45. If f(x) = , then show that (i) f  1  = − f (x) (ii) f  − 1  = − 1
x +1 x  x f (x)
 6

Trigonometric Functions
 3 5 7
1. cos2 + cos2 + cos2 + cos2 =2
8 8 8 8

  3      
2. 3 sin 4      sin 4  3      2 sin 6      sin 6  5      1
  2    2  
3. Find the values of: (i) cos 4950 (ii) sin 12300 (iii) cot (-3150) (iv) tan (-15900)

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4. Prove that the following: (i) cos700 cos100 + sin700 sin100 = 1/2
(ii) sin (400 + ) cos (100 + ) – cos (400 + ) sin (100 + ) = 1/2

SI
5. If x lies in the first quadrant and cosx = 8/17, then find the value of:

     2 
cos   x   cos   x   cos   x
6  4   3 


6. If x cos  = y cos   

2  
 = z cos   
3  
4 

A
 , then find the value of xy + yz+ zx.
3 
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cos 260  sin 260
7. Prove that: (i) tan 500 = tan 400 + 2 tan 100, (ii) tan 710 =
cos 260  sin 260
m 1 
(iii) If tan  = and tan  = , show that     .
m 1 2m  1 4
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sin( x  y )
8. If tan x = 2 tan y, prove that  3.
sin( x  y )
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   
9. Show that cot   x  cot   x  = 1.
4  4 

10. If x – y = , prove that (1 + tan x) (1 + tan y) = 2 tan x.
P

4
11. If sinx sin y – cos x cos y + 1 = 0, prove that 1 + cot x tan y = 0.
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n sin  cos 
12. If tan  = , show that the tan ( - ) = (1 - n)tan.
1  n sin 2 
sin   cos 
13. If tan x = , then show that the sin  + cos  = 2 cos x.
sin   cos 
14. Prove that: (i) cot 2x + tan x = cosec 2x
(ii) cos 2x cos 2y + cos2(x + y) - cos2(x - y) = cos (2x + 2y).
 7

k 1
15. If x + y = z and tan x = k tan y, then prove that sin z = sin( x  y ).
k 1
16. If tan  and tan  are distinct roots of atan  + b sec = c, then show that tan ( + ) =
2ac
.
a  c2
2

19  4 
17. Find the value of (i) sin (ii) tan   .
4  3 

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18. Express the following as functions of angles less than 450: (i) sin (-17850) (ii) cosec (-74980).
 2  29    5 
19. Prove that: (i) 3cos2 + sec + 5tan2 = (ii)  3cos sec  4sin tan  cos 2 = 1.

SI
4 3 3 2  3 3 6 4

20. Evaluate the following: (i) 2 sin 1350 cos2100 tan 2400 cot 3000 sec 3300
 3 5 7
(ii) sin 6900 cos 9300 + tan (-7650) cosec (-11700) (iii) sin2 + sin2 + sin2 + sin2

A
8 8 8 8
 3 5 7   5 7
(iv) sin2 + sin2 + sin2 + sin2 (v) tan + tan + tan + tan .
4 4 4 4 12 16 12 16
PT
 3   3 
21. Prove that: cos x + sin   x  - sin   x  + cos   x  = 0.
 2   2 

 
tan   x 
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cos x sin( x) 2 
22. Simplify the following: (i)  
  sin(  x) cot x
sin   x 
2 
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   3 
sin(  x) cos x   x  tan   x  cot  2  x 
(ii) 2   2 
 3 
sin(2  x) cos(2  x) cos ec( x)sin   x
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 2 

     
23. Find y from the following equation: cos ec   x   y cos x cot   x   sin   x  .
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2  2  2 
24. If 8x = , show that cos 7x + cos x = 0.

tan1550  tan1150 1  x 2
25. If tan250 = x prove that  .
1  tan1550 tan1150 2x

cos1350  cos1200
26. Prove that  3  2 2.
cos1350  cos1200
 8

A B C
27. If A, B and C are angles of a triangle, prove that (i) cos  sin
2 2
tan( B  C )  tan(C  A)  tan( A  B)
(ii)  1.
tan(  A)  tan(2  B)  tan(3  C )

 A B  CD
28. If A, B, C and D are angles of a quadrilateral, then prove that: sin    sin  .
 2   2 
29. If A, B, C and D are angles of a cyclic quadrilateral, prove that: sin A + sin B = sin C + sin D.

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30. Find tan 150 and hence show that tan 150 + cot 150 = 4.
11

SI
31. Evaluate : (i) cos 1950 (ii) sin
12

7  7  3 tan 690  tan 660

32. Show that: (i) sin cos - cos sin = (ii)  1.
12 4 12 4 2 1  tan 690 tan 660

A
     x   x 
33. Evaluate (i) cos2   x  - sin2   x  (ii) sin2    - sin2    .
4  4   8 2  8 2
PT
8 5
34. If ,  lies in the first quadrant and sin = and tan = , find the values of sin ( - ),
17 12
cos ( - ) and tan ( - ).
1 1
35. A positive acute angle is divided into two parts whose tangents are and . Show that the
2 3
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
angle is .
4
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36. Prove that tan220 + tan 230 + tan 220tan230 = 1.

37. If A + B = 2250, prove that tan A + tan B = 1 - tan A tan B.

cos110  sin110
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0 0 0 0
38. Prove that: (i) tan 70 = tan20 + 2tan 50 (ii) tan 56 =
cos110  sin110
39. Prove that: (i) sin (x + y) sin (x - y) + sin (y - z) + sin (y + z) sin (z + x) sin (z - x) = 0
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x x
(ii) 1 + tan x tan = sec x = tan x cot - 1.
2 2
40. Prove that: tan 13 x = tan 4x + tan 9x + tan 4x tan 9x tan 13x.
41. Prove that: cos 2x cos 2y + sin2(x - y) - sin2(x + y) = cos (2x + 2y).
 9

cos( x  y ) 1
42. If cot x cot y = 2, show that  .
cos( x  y ) 3

sin( A  B) m  1
43. If tan A = m tan B, prove that  .
sin( A  B) m  1

tan  x x y
44. If  +  =  and  , then prove that sin ( - ) = sin  .
tan  y x y

       

R
45. Prove that: tan x tan  x   + tan  x   tan  x   + tan  x   tan x = -3.
 3  3  3  3

 2   4 
46. Show that: (i) cos100 + cos110 + cos1300 = 0 (ii) sin  + sin   

SI
 + sin    = 0
 3   3 
 9 3 5
(iii) 2 cos cos cos cos = 0.
13 13 13 13

47. Prove that: (i) cos200 cos400 cos800 =

1
8

A(ii) sin100 sin500 sin 600 sin 700 =

3
16
.
PT
1 m
48. If cos (x + y) = m cos (x - y), then prove that tan x = cot y .
1 m
mn
49. If m sin x = n sin (x + 2), then prove that tan (x + ) cot  = .
mn
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  k 1   
50. If sin  = k sin , prove that  tan .
2 k 1 2
G

    1    
51. Prove that: (i) sin   x  sin   x  = cos 2 x (ii) sec   x  sec   x  =2 sec2x.
4  4  2 4  4 

 5   5 
(iii) sin   x  + sin   x  = cos x.
 6  6
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 
52. Prove that: cos520 + cos680 + cos1720 = 0 (ii) cos200 + cos1000 + cos1400 = 0.
R

1 1
53. Prove that: (i) sin100 sin500 sin700 = (ii) sin100sin300sin500sin700 =
8 16
3
(iii) sin200 sin400 sin600 sin800 =
16
1 3
54. Prove that: (i) cos200 cos400 cos600cos800 = (ii) cos100 cos300cos500cos700 = .
16 16
 10

55. Prove that: (i) tan200 tan400 tan800 = tan600 (ii) tan100 tan500 tan700 = tan300.

 2   2   3 5 7
56. Prove that: (i) cos x + cos   x  + cos   x  = 0 (ii) cos +cos +cos + cos = 0.
 3   3  8 8 8 8

   
57. Prove that: (i) 4 sin x sin x   x  sin   x  = sin 3x
3  3 

   
(ii) 4 cos x cos x   x  cos   x  = sin 3x = cos3x
3  3 

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sin 5x  2sin 8x  sin11x sin 8x
(iii) sin2x + sin2(x - y) - 2sinx cos y sin (x - y) = sin2y (iv) 
sin 8x  2sin11x  sin14 x sin11x

SI
     
58. Prove that: cos  + cos  + cos  + cos ( +  + ) = 4cos cos cos
2 2 2
1 1 x y 3
59. If cos x + cos y = and sin x + sin y = , prove that tan  .

A
3 4 2 4
x y
60. If sin x + sin y = a and cos x + cos y = b, find the value of tan .
PT
2
sin( x  y) a  b tan x a
61. If  , then show that tan  .
sin( x  y) a  b tan y b

1 m
62. If cos (x + 2y) = m cos x, prove that cot y = tan( x  y) .
U

1 m
 sec8x  1 tan 8x
63. Prove that: (i) 2  2  2cos 4 x  2cos x, 0  x  (ii) 
G

4 sec 4 x  1 tan 2 x
       
64. Prove that: (i) tan   x   tan   x   2sec 2 x (ii) tan x tan   x   tan   x   tan 3x
4  4  3  3 
P

65. Show that 2 sin2 + 4 cos ( + ) sin  sin  + cos2 ( + ) = cos 2.
66. If cos  + cos  = 0, sin  + sin  = 1, then prove that cos 2 + cos 2 + 2cos ( + ) = 0.
R

 2   4  3
67. Prove that: sin 3 x  sin3   x   sin3   x    sin 3x .
 3   3  4

 3 3 5 7
68. Find the value of: cos4  cos4  cos4  cos4 cos4 .
8 8 8 8 8
69. Prove that: cos5x = 16cos5x - 20 cos3x + 5 cosx.
 11

x m
70. If tan  , prove that m sin x + n cos x = n.
2 n
x  x  
71. If cos x = cos  cos , prove that tan tan  tan 2 .
2 2 2
72. If sin x + sin y = a and cos x + cos y = b, find (i) sin (x + y) (ii) cos (x + y).
 
73. Find the values of sin and cos .
10 5

R
74. Find the value of tan90 – tan 270 –tan630 + tan810.
1

SI
75. If sin x = , find the value of sin3x.
3
3 
76. If sin x = and 0 < x < , find the value of (i) sin2x (ii) cos2x (iii) tan2x (iv) sin 4x.
5 2

A
1  sin 2 x  cos 2 x cos3 x  sin3 x 1
77. Prove that: (i)  tan x (ii)  (2  sin 2 x)
1  sin 2 x  cos 2 x cos x  sin x 2
1  cos 2 x  sin x sin16 x
PT
(iii)  tan x (iv) cos x cos 2 x cos 4 x cos8x 
sin 2 x  cos x 16sin x

1  sin x  x 

(v) 1  cos2 2 x  2 cos4 x  sin 4 x  (vi)
1  sin x
 tan   
 4 2
U

 x    2 
(vii) tan     tan x  sec x (viii) tan x  tan   x   tan   x   3tan 3x
 4 2 3   3 
G

    1
(ix) cos x  cos   x  cos   x   cos3x
3  3  4
sin x cos x
78. If  , prove that a sin 2x + b cos 2x = b.
P

a b
79. If tan2 x = 2 tan2 y + 1, show that: cos2x + sin2y = 0.
R

1 1 3 1 
80. If 2 cos y = x + , prove that cos 3y =  x  3  .
x 2 x 
2sin 2 y
81. If tan y = 3 tan x, prove that tan (x + y) = .
1  2cos 2 y
82. Prove that: (i) cos 4x = 1 – 8cos2x + 8 cos4x (ii) sin 4x = 4 sinx cos3x - 4 cosx sin3x
(iii) sin 5x = 5 sin x – 20 sin3 x + 16 sin5x.
 12

2b
83. If a cos 2 + b sin 2 = c has  and  its roots, then prove that tan  + tan  = .
ac
 2   2  3
84. Prove that cos3x +cos3   x  + cos3   x  = cos3x
 3   3  4
0 0
1 1
85. Evaluate without using table: (i) 2 cos 22 sin 22 (ii) 2 cos2150 – 1
2 2
(iii) 8 cos3220 – 6cos200 (iv) 3 sin 400 – 4sin3400

R
   3  5  7  1
86. Prove that: 1  cos 1  cos 1  cos 1  cos  .
 8  8  8  8  8

SI
sin B
87. In a triangle ABC, if cos A = , then prove that the triangle is isosceles.
2sin C
2cos A cos B 2cos C a b
88. In a triangle ABC, if     , then prove that the triangle is right

A
a b c bc ca
angled.
PT
89. In a triangle ABC, if a cos A = b cos B, then prove that either the triangle is isosceles or right-
angled.
1 1 3
90. In a triangle ABC, prove that C = 600 if and only if   .
ac bc a bc
U

bc ca ab cos A cos B cos C

91. In a triangle ABC, if   , then prove that   .
11 12 13 7 19 25
G

a 2  b2 sin A sin B
92. In any triangle ABC, prove that  =  .
2 sin( A  B)
93. In any triangle ABC, prove that (i) a cos A + b cos B + c cos C = 2 asin B sin C
P

82
(ii) a cos A + b cos B + c cos C = .
abc
R

1
94. If in a ABC, a = 2 cm, b = 3 cm and sin A = , find B.
3

95. If the angles of ABC are in AP and b : c = 3 : 2 , then find A.

96. If the sides of a ABC are in ratio 4: 5: 6, prove that one angle is twice the other.

97. If the two sides of a triangle and the included angle are a = 3  1, b = 2, C = 600, find the
other two angles and the third side.
 13

 2 C A
98. Prove for any triangle: 2 a sin  c sin 2   c  a  b .
 2 2
99. In a triangle ABC, sin 2A +sin 2B = sin 2C, prove than either A = 900 or B = 900.
100. In a triangle ABC, sin2A +sin2B = sin2C, prove than either ABC is right angled.
101. If the angle of a triangle are in the ratio 1 : 2 : 3, prove that the corresponding sides of the
triangle are in the ratio 1: 3 : 2.

R
102. If A, B,C are the angles of a triangle and tan A = 1 and tan B = 2, prove that tan C = 3. If a,
a b c
b, c are the corresponding sides, then prove that   .
5 2 2 3

SI

103. In a triangle ABC, a = 1, b = 3 and C = . Find the other two angles and the third sides.
6

1 1 1 23  3 1 1 
Answer: 3. (i)  , (ii) , (iii) 1, (iv) 5.   6. 0

A
 
2 2 3 17  2 2 2 

1 3
PT
17. (i) , (ii)  3 18. (i) sin 150, (ii) sec 280 20. (i) 1, (ii) +1 (iii) 2, (iv) 2 (v) 1
2 4

3 1 3 1 3 1
22. (i) 3, (ii) 1 23. tan x 30. 31.(i)  , (ii)
3 1 2 2 2 2
U

0
1 1 21 220 21 a 3
33. (i) , (ii) sin x 34. , , 60. 68.
2 2 221 221 220 b 2
G

2ab b2  a 2 5 1 5  1 23
72. (i) 2 2 , (ii) 2 73. , 74. 4 75.
a b b  a2 4 4 27

24 7 24 336 1 3 3
P

76. (i) , (ii) , (iii) , (iv) 85. (i) , (ii) , (iii) 1, (iv)
25 25 7 625 2 2 2

94.  B = 300 or 1500 95. A = 750

R

97. A = 750,  B = 450, c = 6 103. A = 300,  B = 1200, c = 1

 14

SUMERMAL JAIN PUBLIC SCHOOL

B-2, JANKPURI NEW DELHI
CLASS-XI COMPLEX NUMBER
Q1. Evaluate the following:

R
( )
4 n +3
(i) i135 (ii) i19 (iii) i −999 (iv) − −1 ,n N

SI
Q2. Show that:

2 2 3
 19  1  25   17  1 34   18  1  24 
(i) i +    = −4 (ii) i −    = 2i (iii) i +    = 0
  i     i     i  

n+1 n+ 2 n+3
(iv) i + i + i + i = 0 , for all n  N
n

 (i
13
+ i n+1 ), where n  N . A
PT
n
Q3. Evaluate
n =1

Q4. Evaluate 1 + i 2 + i 4 + i 6 + ..... + i 2 n .

Q5. Express each of the following in the form a + ib :

U

(i) (5 − 3i)
3
(ii) (− 3 + −2) 2 3 − i ( )
G

Q6. Express each one of the following in the standard from a + ib.

(1 + i ) 2 1 1
(i) (ii) (iii)
3−i −2 + −3 1 − cos + 2i sin 
P

Q7. Prove that the following complex numbers are purely real:
R

 2 + 3i  2 − 3i   3 + 2i   3 − 2i 
(i)    (ii)  + 
 3 + 4i  3 − 4i   2 − 3i   2 + 3i 

Q8. Perform the indicated operation and find the result in the form a + ib.

2 − −25 3 − −16
(i) (ii)
1 − −16 1 − −9
 15

Q9. Find the real values of x and y, if

(i) ( 3x − 7 ) + 2iy = −5 y + ( 5 + x ) i (ii) (1 − i ) x + (1 + i ) y = 1 − 3i

x −1 y −1
(iii) ( x + iy )( 2 − 3i ) = 4 + i (iv) + =i
3 +1 3 − i

10. Find real values of x and y for which the following equalities hold:

(i) (1 + i ) y 2 + ( 6 + i ) = ( 2 + i ) x ( ) ( )

R
(ii) x 4 + 2 xi − 3x 2 + iy = ( 3 − 5i ) + (1 + 2iy )

c+i b 2c

SI
Q11. If a + ib = , where C is real, prove that: a 2 + b 2 = 1 and = 2
c −i a c −1

Q12. If ( x + iy ) = a + ib, x, y, a b  R. show that

1/3

+ = 4 ( a 2 − b2 )

A
− = −2 ( a 2 − b 2 )
x y x y
(i) (ii)
a b a b
PT
Q13. Multiply 3 − 2i by its conjugate.

1
Q14. Find the conjugate of .
3 + 4i
U

Q15. Express the following complex numbers in the standard form. Also, find their conjugate:

( 2 + 3i )
2
5 + 12i + 5 − 12i
(i) (ii)
G

2−i 5 + 12i − 5 − 12i

Q16. Find real values of x and y for which the complex numbers −3 + ix y and x + y + 4i are
2 2

conjugate of each other.

P

a + ib a − ib a 2 + b2
Q17. If = x + iy, prove that 2 = x − iy and 2 = x2 + y 2 .
R

c + id c − id c +d 2

(a + i ) 2 (a 2 + 1)2
Q18. If = p + iq, show that: p 2 + q 2 = .
(2a − i) (4a 2 + 1)

1+ i 
n

Q19. Find the least positive value of n , if   = 1.

 1− i 
 16

Q20. If x = −5 + 2 −4, find the value of x 4 + 9 x 3 + 35 x 2 − x + 4.

Q21. Find the value of x3 + 7 x 2 − x + 16, when x = 1 + 2i

z −1
Q22. If z is a complex number such that z = 1 , prove that is purely imaginary. What will be your
z +1
conclusion if z = 1?

Answer

R
1 x.(i) − i (ii) − i (iii) i (iv) i

SI
2. (i) -4 (ii) 2i (iii) 0 (iv) 0

3. (-1+i)

0, if n is odd

A
4.
1, if n is even

( ) ( )
PT
5. (i) -10-198i (ii) 2 −6 + 3+2 6 i

1 3 2 3
6. (i) − + i (ii) −
− i (iii)
5 5 7 7
 1 − cos    −2sin  
U

  + i 
 2 − 2cos  + 3sin    2 − 2cos  + 3sin  
2 2
G

13
7. (i) , which is purely real. (ii) 0, which is purely real.
25

22 3 3 1
8. (i) + i (ii) + i
P

17 17 2 2

5 14
9. (i) x = −1, y = 2. (ii) x = 2, y = −1 (iii) x = ,y= (iv) x = −4, y = 6.
R

13 13

10. (i) x = 5 and y = 2 or , x = 5 and y = −2 (ii) x = 2 and y = 3 or , x = −2 and y = 1/ 3.

13. 13

3 4
14. z = + i
25 25
 17

22 19 3
15. (i) z = − − i (ii) z = 0 + i
5 5 2

16. x = 1, y = −4 or , x = −1, y = −4

19. 4

20. −160

21. −17 + 24i

R
22. 0, which is purely real

SI
A
PT
U
G
P
R
 18

CLASS 11
MODULE-3

MCQ
1 1 x
Q1 If + = , then the value of x is
8! 9! 10!
(A) 100 (B)64 (C)81 (D)72
Q2 How many numbers are there between 99 and 1000 having 7 in the unit place?
(A) 100 (B)64 (C)90 (D)72
Q3 How many numbers are there between 99 and 1000 having at least one of their digits 7?
(A) 1000 (B)252 (C)810 (D)152
Q4 In how many ways can 5 children be arranged in a line such that two particular children of

R
them are always together
(A)48 (B)64 (C)90 (D)72
Q5 In how many ways can 5 children be arranged in a line such that two particular children of

SI
them are never together.
(A)100 (B)64 (C)90 (D)72
Q6 6 In how many ways 3 mathematics books, 4 history books, 3 chemistry books and 2 biology
books can be arranged on a shelf so that all books of the same subjects are together.
(A)48854 (B)64264 (C)41472 (D)47241
Q7 A
There are four bus routes between A and B; and three bus routes between B and C. A man can
travel round-trip in number of ways by bus from A to C via B. If he does not want to use same
bus route more than once, in how many ways can he make round trip?
PT
(A) 72 (B) 144 (C) 14 (D) 19
Q8 In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7 men
and 5 women?
(A) 45 (B) 350 (C) 4200 (D) 230
Q9 All the letters of the word ‘EAMCOT’ are arranged in different possible ways. The number of
U

such arrangements in which no two vowels are adjacent to each other is

(A) 360 (B) 144 (C) 72 (D) 54
Q10 Ten different letters of alphabet are given. Words with five letters are formed from these
G

given letters. Then the number of words which have at least one letter repeated is
(A) 69760 (B) 30240 (C) 99748 (D) 99784
Q11 The number of signals that can be sent by 6 flags of different colours taking one or more at a
time is
(A) 63 (B) 1956 (C) 720 (D) 21
P

Q12 In an examination there are three multiple choice questions and each question has 4 choices.
Number of ways in which a student can fail to get all answer correct is
R

(A) 11 (B) 12 (C) 27 (D) 63

Q13 If C(n,12) = C(n,8), then n is equal to
(A) 20 (B) 12 (C) 6 (D) 30
Q14 The number of possible outcomes when a coin is tossed 6 times is
(A) 36 (B) 64 (C) 12 (D) 32
Q15 The number of different four-digit numbers that can be formed with the digits 2, 3, 4, 7 and
using each digit only once is
(A) 120 (B) 96 (C) 24 (D) 100
Q16 The sum of the digits in unit place of all the numbers formed with the help of 3, 4, 5 and 6
taken all at a time is
(A) 432 (B) 108 (C) 36 (D) 18
 19

Q17 Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5
consonants is equal to
(A) 60 (B) 120 (C) 7200 (D) 720
Q18 A five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without
repetitions. The total number of ways this can be done is
(A) 216 (B) 600 (C) 240 (D) 3125
Q19 Everybody in a room shakes hands with everybody else. The total number of handshakes is 66.
The total number of persons in the room is
(A) 11 (B) 12 (C) 13 (D) 14
Q20 The number of triangles that are formed by choosing the vertices from a set of 12 points, seven
of which lie on the same line is
(A) 105 (B) 15 (C) 175 (D) 185

R
Q21 The number of parallelograms that can be formed from a set of four parallel lines intersecting
another set of three parallel lines is
(A) 6 (B) 18 (C) 12 (D) 9

SI
Q22 The number of ways in which a team of eleven players can be selected from 22 players always
including 2 of them and excluding 4 of them is
(A) 16C11 (B) 16C5 (C) 16C9 (D) 20C9
Q23 The number of 5-digit telephone numbers having atleast one of their digits repeated is
(A) 90,000 (B) 10,000 (C) 30,240 (D) 69,760
Q24

Q25
(B) 126 (C) 128
A
The number of ways in which we can choose a committee from four men and six women so
that the committee includes at least two men and exactly twice as many women as men is
(A) 94 (D) None
The total number of 9-digit numbers which have all different digits is
PT
(A) 10! (B) 9! (C) 9 × 9! (D) 10×10!
Q26 The number of words which can be formed out of the letters of the word ARTICLE, so that
vowels occupy the even place is
(A) 1440 (B) 144 (C) 7! (D) 4C4 × 3C3
Q27 Given 5 different green dyes, four different blue dyes and three different red dyes, the number
U

of combinations of dyes which can be chosen taking at least one green and one blue dye is
(A) 3600 (B) 3720 (C) 3800 (D) 3600
Q28 The total number of terms in the expansion of (x + a) 51 – (x – a)51 after simplification is
G

(a) 102 (b) 25 (c) 26 (d) None of these

Q29 If (1 − x + x ) = a0 + a1 x + a2 x + a3 x + ....... + a2 n x 2 n , then (1 − x + x 2 ) n = a0 + a2 + a4 ....... + a2 n
2 n 2 3

is equal to
P

Q30 The number of terms in the expansion of (a + b + c)n, where n ∈ N is

R

MCQ Answers
Q1 (A) Q2(C) Q3(B) Q4(A) Q5(D) Q6(C) Q7(A)
Q8(B) Q9(B) Q10(A) Q11(B) Q12(D) Q13(A) Q14(B)
Q15(C) Q16(B) Q17(C) Q18(A) Q19(B) Q20(D) Q21(B)
Q22(C) Q23(D) Q24(A) Q25(C) Q26(B) Q27(B) Q28(C)
Q29(A) Q30(A)
 20

CASE BASED QUESTIONS-1

Q31 Read the following and answer the questions given below:
There are 10 mathematics teachers and 20 students in the school mathematics magazine
editorial team. This year the team decided to form a committee of 2 teachers and 3 students
who will look after a special section of the magazine - Insight in Mathematical modelling
(IMM).

R
Help the team to find out:
(i) In how many ways committee can be formed from the Editorial Team?

SI
(ii) In how many ways a particular student from the Editorial Team can be included in the
committee?
(iii) In how many ways a particular teacher can be included in the committee?
OR
In how many ways a particular student can be excluded from IMM?

CASE BASED QUESTION-2

A
PT
Q32 Read the following and answer the questions given below:

Dyes are water-soluble colorants which are primarily used in textile industry. A manufacturer of
colorants customizes the colours and further treats in accordance with client's requirements
U
G
P

A colorant expert is using five different green dyes, four different blue dyes and three different
red dyes to meet the requirements of the client.
R

Answer the following questions based on the number of dyes the expert is using:
(i) How many choices does the expert have for choosing a particular dye?
(ii) What is number of ways in which the expert can choose 3 red dyes?
(iii) In how many ways at least one green dye can be chosen by the expert?
OR
What is the number of combinations of dyes which can be chosen taking at least one green
dye and 3 red dyes?
Q33 In a small village, there are 87 families, of which 52 families have at most 2 children. In a rural
development programme 20 families are to be chosen for assistance, of which at least 18
families must have at most 2 children. In how many ways can the choice be made?
 21

Q34 A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not
want to borrow Mathematics Part II, unless Mathematics Part I is also borrowed. In how many
ways can he choose the three books to be borrowed?
Q35 A convex polygon has 44 diagonals. Find the number of its sides.
Q36 In how many ways can a football team of 11 players be selected from 16 players? How many
of them will
(i) include 2 particular players?
(ii) exclude 2 particular players?
Q37 A sports team of 11 students is to be constituted, choosing at least 5 from Class XI and at least
5 from Class XII. If there are 20 students in each of these classes, in how many ways can the
team be constituted?
Q38 A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected

R
if the team has
(i) no girls
(ii) at least one boy and one girl (iii) at least three girls.

SI
Q39 If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent,
in how many points will they intersect each other?
Q40 Find the number of integers greater than 7000 that can be formed with the digits 3, 5, 7, 8 and
9 where no digits are repeated.
Q41 There are 10 lamps in a hall. Each one of them can be switched on independently. Find the

Q42

Q43
A
number of ways in which the hall can be illuminated
Find the number of different words that can be formed from the letters of the word
‘TRIANGLE’ so that no vowels are together.
If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary. Then
PT
what is the rank of the word RACHIT?
Q44 A candidate is required to answer 7 questions out of 12 questions, which are divided into two
groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from
either group. Find the number of different ways of doing questions.
Q45 Out of 18 points in a plane, no three are in the same line except five points which are collinear.
U

Find the number of lines that can be formed joining the point.
Q46 Find the number of permutations of the letters of the word ALLAHABAD.
Q47 Find the value of n such that:
G

Q48 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many
P

of these arrangements,
(i) do the words start with P
(ii) do all the vowels always occur together
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(iii) do the vowels never occur together (iv) do the words begin with I and end in P?
Q49 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many
of these
(i) four cards are of the same suit,
(ii) four cards belong to four different suits,
(iii) are face cards,
(iv) two are red cards and two are black cards,
(v) cards are of the same colour?
Q50 A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done
when the committee consists of:
(i) exactly 3 girls?
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(ii) at least 3 girls?

(iii) at most 3 girls?
Q51 From a class of 30 students, 15 are to be chosen for an excursion party. There are 3 students
who decide that either all of them will join or none of them will join. In how many ways can
the excursion party be chosen?
Q52 The English alphabet has 5 vowels and 21 consonants. How many words with three different
vowels and four different consonants can be formed from the alphabet?
Q53 Compute (98)5 with the help of binomial theorem.
Q54 Which is larger (1.01)1000000 or 10,000?
Q55 Find the expansion of (3x2 – 2ax + 3a2 )3 using binomial theorem.
Q56 If a and b are distinct integers, prove that a – b is a factor of (an – bn), whenever n is a positive
integer.

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Assignment-Binomial Theorem
Class XI(Mathematics)
1) Using binomial theorem , expand each of the following
4 4
 2  x 2
i ) 3 x 2   , x  0 ii )(1  x  x 2 ) 4 iii )1    , x  0
 x  2 x
2) Evaluate the following:

  
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i) 1  2 x  1  2 x  5
ii )(0.99) 5  (1.01) 5

3) Using binomial theorem, prove that 24n+4-15n-16 is divisible by 225, where nN.

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4) Using binomial theorem determine which number is smaller (1.2)4000 or 800?
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 3 x3 

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5) Find the 4 term from the end in the expansion of  2   .
th

x 6
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 1 
6) Find the 13 term in the expansion of  9 x 
th
 ,x  0.
 3 x

7)
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

1 
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 .
2 x3 
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8) Find x, if 21st and 22nd terms in the expansion of (1+x)44 are equal.
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 1
9) Find the coefficient of x and x
32 -17
in the expansion of  x 4  3  .
 x 
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10) Find n, if the ratio of seventh term from the beginning to the seventh term from the end in the
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 1 
expansion of  3 2   is 1:6 .
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
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3

11) If the 6th, 7th and 8th terms in the expansion of (x+a)n are respectively 112, 7 and 1/4, find x, a,
n.
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12) Find the sixth term in the expansion of y1/ 2  x1/ 3 , if the binomial coefficient of the third
term from the end is 45.
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Assignment –Sequences and Series

Class-11(Mathematics)

(One Markers)

1) Find the value of 61/ 2  61/ 4  61/ 8.......

1 1
2) Find the 9th term and the general term of the progression: ,  ,1,  2,...
4 2
3) Find the 4th term from the end of the GP 3,6,12,24, ….. ,3072.

Very Short Questions (2 markers)

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4) Insert 5 geometric means between 576 and 9.

5) If a,b,c, d are in GP. Prove that a2-b2, b2-c2, c2-d2 are also in GP.

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6) Find the minimum value of the expression 3x+31-x, xR.

Short Answer Question (3 marks)

7) Which term of the GP 5, 10, 20,40,…. is 5120?

8) A
The fourth, seventh and the last term of a GP are 10, 80 and 2560 respectively. Find the first
term and the number of terms in the GP.
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9) If a,b,c are respectively the pth, qth and rth terms of the GP,show that

(q-r)loga+(r-p)logb+(p-q)logc=0

10) If the product of three terms in GP is 216 and the sum of their product in pairs is 156, find the
numbers.
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11) The product of first three terms of a GP is 1000. if 6 is added to its second term and 7 added to
its third term , the terms become in AP. Find the GP.
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12) Let S be the sum , P be the product and R be the sum of the reciprocals of 3 terms of a GP. Find
P2R3:S3 .

13) In a GP of even number of terms, the sum of all terms is 5 times the sum of the odd terms. Find
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the common ratio of the GP.

14) If x,y,z are distinct positive numbers, then prove that (x+y)(y+z)(z+x)>8xyz
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15) If the AM of two positive numbers a and b (a>b) is twice their geometric mean . Find the ratio of
the numbers.

Long Answer Questions(5 markers)

a 2  ab  b 2 b  a
16) If a, b, c are in GP then prove that : 
bc  ca  ab c  b
17) If A is the arithmetic mean and G1 , G2 be two geometric means between any two numbers then
G12 G22
prove that 2 A  
G2 G1
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18) Case Study Question

The inventer of the chess board suggested a reward of one grain of wheat for the first square, 2
grains for the second, 4 grains for the third and soon, doubling the number of the grains for the
subsequent squares.

Based on the above information, answer the following questions :

How many grains of wheat is given for seventh square?
How many grains would have to be given to inventer?

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ASSIGNMENT ON STRAIGHT LINES

CLASS XI (2023-24)
MATHEMATICS

Q1-7 are MCQs (1 markers)

1.The sum of X and Y intercept of the line 3x – 4y = 6 is-
(a) 1/2 (b) 3/2 (c) 5/2 (d)7/2
2.A point equidistant from the line 4x+3y+10= 0 , 5x-12y +26=0, 7x +24y-50= 0 is
(a) (1,-1) (b) (1,1) (c)(0,0) (d)(0,1)
3.The angle between the lines 2x-y+3=0, and x+2y+3=0
(a) 90° (𝑏)60° (𝑐)45° (𝑑)30°
4.The centroid of the triangle is (2,7) and two of its vertices are (4,8) and (-2,6). The

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third vertex is
(a)(0,0) (b)(4,7) (c) (7,4) (d)(7,7)

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5.The value of m for which the lines 3x+4y=5,5x+4y=4 and mx+4y =6 are concurrent is-
(a) 2 (b)1 (c) 4 (d)3
6. The distance between the parallel lines 3x-4y+9=0 and 6x-8y-15=0 is
(a)33 (b)33/10 (c) 33.5 (d ) 30
7.The equation of two sides of square are 5x-12y-65=0 and 5x-12y+26=0. The area of the
Square is
(a)65/17 (b) 63/17
A (c) 4 (d) none of these
Q8-12 are Short Answer type questions (2 markers)
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8.What can be said of a line if its slope is
(a) zero (b) positive (c) negative ( d) undefined
9.(a) Find the equation of line equidistant from y = 10 and y = -2
(b) Find the equation of line equidistant from x=-2 and x= 6
10.Prove that the line y- x+2 =0 divides the join of points (3,-1) and (8,9) in the ratio 2:3
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11.Find the equation of the line whose intercepts on X axis and Y axis are respectively twice
and thrice of those made by the line 3x+4y =12
12.Find the equation of line parallel to y axis and drawn through the point of intersection of
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the lines x-7y+5=0 and 3x+y = 0

Q13- 16 are Long answer type questions (3 markers)

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13.The equation of base of an equilateral triangle x +y -2 = 0 and the opposite vertex has
coordinates (2,-1). Find the area of triangle .
14.Find the equation of line which passes through P(1,-7) and meets the axes A and B
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respectively so that 4AP-3BP= 0

15.Find the distance of the point (2,3) from the line 2x-3y+9=0 measured along the line
x-y+1= 0
16.Find the equation of the straight line passing through the point (2,1) and bisecting the
portion of the line 3x-5y=15 lying between the axes.
Q17-18 are Very Long answer type questions (5 markers)
17.Find the equation of the line passing through the intersection of the lines 5x-6y-1=0
and 3x+2y+5=0 and perpendicular to the line 3x-5y+11=0
18.Find the image of the point (-8,12) with respect to the line mirror 4x+7y+13=0
Also find the length of perpendicular from the point to the given line.
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ASSIGNMENT CLASS – XI

INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

Objective Type Questions

1. The length of the foot of perpendicular drawn from the point P (3,4,5) on y- axis is

a) √41 b) √34 c) 5 d) none of these

2. The perpendicular distance of the point P (6,7,8) from xy - plane is
a) 8 b) 7 c) 6 d) none of these
3. If L is the foot of perpendicular drawn from the point P (3,4,5) on xz – plane, then coordinates of

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a) (3,0,0) b) (0,4,5) c) (3,0,5) d) (3,4,0)

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4. If L is the foot of perpendicular drawn from the point P (6,7,8) on y- axis, then coordinates of the
point L are
a) (6,0,0) b) (0,7,0) c) (0,0,8) d) none of these

5. If the distance between the points (a,0,1) and (0,1,2) is√27 , then the value of a is
a) 5 b) ±5 c) -5 A d) none of these
6. The point on x- axis which is equidistant from the points (3,2,2) and (5,4,4) is
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a) ( 4 ,0,0) b) ( 2 ,0,0) c) (− ,0,0) d) (2,0,0)
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7. The ratio in which the line segment joining the points A(4,8,10) and B(6,10, -8) is divided by yz-
plane is
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a) 2:3 internally b) 2:3 externally c) 3:2 internally d) 3:2 externally

8. The coordinates of the point which is equidistant from the points O(0,0,0) , A(a,0,0) , B(0,b,0) and
C(0,0,c) are
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𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐
a) (a,b,c) b) (( 2 , 2 , 2) 𝑐) ( 3 , 3 , 3) 𝑑) (− 2 , − 2 , − 2)

9. If (-1,5,c) is the mid- point of the line segment joining the points (a,3,-5) and (-4,b,1), then the
value of a, b, c are……………………..respectively
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10. Name the octant in which the following points lie :

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(i) (4, -3, 5) (ii) (6, -2, -5) (iii) (-4, 2, -1) (iv) (-2, -1, 6)
Short Answer Type
11. Find the value of x if distance between the points (x, -8, 4) and (3, -5, 4) is 5 units.
12. Find the coordinates of the point P which is five- sixth of the way from A(-2, 0, 6) to B(10, -6, -12)
13. Find the ratio in which the line joining the points (4, 4, -10) and (-2, 2, 4) is divided by
(i) yz- plane (ii) x + y + z = 3.
Also find the coordinates of the point of the division.
14. Find the centroid of a triangle, the mid- points of whose sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4)
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15. If the points A(1, 0, -6) , B(-3, p, q) and C(-5, 9, 6) are collinear, find the value of p and q.
16. Determine the point in xy- plane which is equidistant from the points (2,0,3) , (0,3,2) and (0,0,1)
17. Find the point on y- axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
18. Find the locus of the point which is equidistant from the points A(0, 2, 3) and B(2, -2, 1).
19. Find the coordinates of the points which trisect the line segment joining the points (2, 1, -3) and
(5, -8, 3)
20. Two vertices of a parallelogram (2, 5, -3), (3, 7, -5) and its diagonals meet in(4, 3, 3), find the
remaining vertices of the parallelogram.

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MODULE V
CHAPTERS : CONIC SECTIONS & INTRODUCTION TO 3-D CO-ORDINATE
GEOMETRY
KEY POINTS

CONIC SECTIONS

• Circle: It is the set of all points in a plane that are equidistant from a fixed
point in that plane

Equation of circle: (x – h)2 + (y – k)2 = r2 where Centre (h, k), radius = r

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If centre is at origin then equation of circle is : x2 + y2 = r2
The general equation of a circle is also of form :
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x2 + y2 + 2gx + 2fy +c = 0
Its centre is (-g,-f ) and r =√𝑔2 + 𝑓 2 − 𝑐
Conditions for a circle :
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(i) It must be an equation of the second degree in x and y .
(ii) The coefficients of x2 and y2 in general equation must be equal and equal
To unity.
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❖ Parabola: It is the set of all points in a plane which are equidistant from a f
fixed point (focus) and a fixed line (directrix) in the plane. Fixed point does not lie on
the line .

Right handed Left handed Upward Downward

Parabola Parabola Parabola Parabola

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Note: In the standard equation of parabola, a > 0.
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In case of parabola , e = 1
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* Ellipse: It is the set of points in a plane the sum of whose distances from two
fixed points in the plane is a constant and is always greater than the distances
between the fixed points
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In case of an ellipse , e < 1 .

Note: In an ellipse then b = a and equation of ellipse will be converted in equation of

the circle. Its eq. will be x2+ y2 = a2. It becomes a circle. For this circle, diameter is
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equal to length of major axis and e = 0.

* Ellipse: It is the set of points in a plane the sum of whose distances from two
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fixed points in the plane is a constant and is always greater than the distances
between the fixed points.

* Hyperbola: It is the set of all points in a plane, the differences of whose distance
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from two fixed points in the plane is a constant.

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In case of hyperbola , e > 1 .

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INTRODUCTION TO 3-D

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LIMITS AND DERIVATIVES

CLASS: XI
1
EVALUATE

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MCQS
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DERIVATIVES

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 69 PAPER
SAMPLE
MATHEMATICS CLASS-XI
Time: 3 hr M.M: 80

General Instructions:

1. This Question paper contains – five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices
in some questions.

2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.

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3. Section B has 5 very short Answer (VSA)-type questions of 2 marks each.

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4. Section C has 6 short Answer (SA)-type question of 3 marks each

5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.

6. Section E has 3 source based/case based/passage based questions of 4 marks each.

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SECTION-A
.
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(Multiple Choice Questions)

Each question carries 1 mark.

Q1. If n  A  115, n  B   326 are two sets n  A  B   47, then n  A  B  is

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(a) 68 (b) 65 (c) 62 (d) 57

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Q2. Domain of a 2  x 2 (a  0) is

 a   a, a  b  a, a  c  0, a  d  (a,0]

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 
Q3. The value of  z  3 z  3 is equivalent to.
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a z 3 b z 3 c  z3  3  d  None of these If

2

Q4. a  ib  c  id , then

 a  a2  c2  0 b  b2  c2  0  c  b2  d 2  0  d  a 2  b2  c2  d 2

Q5. If x is a real number and x  3, then

a x  3 b  3  x  3  c  x  3 d   3  x  3

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Q6. If x  2  9, then
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 a  x   7,11  b  x   11, 7 
 c  x   , 7   11,    d  x   , 7   [11, )

Q7. If n C12  n C8 , then n is equal to

(a) 20 (b) 12 (c) 6 (d) 30

Q8. The number of possible outcomes when a coin is tossed 6 times is.

(a) 36 (b) 64 (c) 12 (d) 32

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Q9. The coefficient of x n in the expansion of 1  x  and 1  x 
2n 2 n 1
are in the ratio.

 a  1: 2 b 1: 3  c  3:1  d  2 :1

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Q10. The total number of terms in the expansion of  x  a    x  a
100 100
after simplification is.

(a) 50 (b) 202 (c) 51 (d) None of these

Q11. Slope of a line which cuts off intercepts of equal lengths on the axes is

 a  1 b 0 c 2 A d  3
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Q12. If the parabola y 2  4ax passes through the point (3, 2), then the length of its latusrectum is

2 4 1
a b c d  4
3 3 3
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Q13. The length of the latusrectum of the ellipse 3x 2  y 2  12 is

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4
a 4 b  3 c 8 d 
3
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Q14. X-axis is the intersection of two planes

 a  XY and XZ b  YZ and ZX  c  XY andYZ  d  Noneof these

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Q15. L is the foot of the perpendicular drawn from a point P(3, 4, 5) on the XY-plane. The coordinates of point L are

(a) (3, 0, 0) (b) (0, 4, 5) (c) (3, 0, 5) (d) (3, 4, 0)

1  cos 4
Q16. lim is equal to.
 0 1  cos 6

4 1 1
a b c  d  1
9 2 2

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Q17. While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of
different colours. 71
29 1 26 27
a b c d 
52 2 51 51

Q18. If seven persons are to be seated in a row. Then, the probability that two particular persons sit next to other is.

1 1 2 1
a b c d 
3 6 7 2

ASSERTION-REASON BASED QUESTIONS

In the following question a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct answer

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out of the following choices.

(a) Both A and R are true and R is the correct explanation of A.

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(b) Both A and R are true but R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

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Q19. Assertion (A): The value of sin150o cos120o  cos 330o sin 660o is
2
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Reason (R): sin  A  B   sin A cos B  cos A sin B

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Q20. Assertion (A): eccentricity of the ellipse 9 x 2  4 y 2  36 is
3
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Reason (R): For an ellipse , e = c/a

SECTION- B (This section comprises 5 very sort answers (VSA) questions of 2 marks each)
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Q21. In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games. Find
the number of students who play neither.

 
Q22. Is g  1,1 ,  2,3 ,  3,5 ,  4,7  , a function, justify. If this is described by the relation, g  x    x   , then what value
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should be assigned to  and  ?

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 1 i   1 i 
3 3

Q23. If      x  iy, then find (x, y).

 1 i   1 i 

tan 3 x  tan x
Q24. lim .
x  /4  
cos  x  
 4

OR

tan 2 x  sin 2 x
Evaluate: lim
x 0 x3
3
Q25.If the letters of the word ‘ALGORITHM’ are arranged at random in a row what is the probability the letters ‘GOR’ must
remain together as a unit? 72
SECTION- C (This section comprises 6 short answers (SA) questions of 3 marks each)

sin   cos 
Q26. tan   , then show that sin   cos   2 cos .
sin   cos 

Q27. A sports team of 11 students is to be constituted, choosing atleast 5 from class XI and atleast 5 from class XII. If there are
20 students in each of these classes, in how many ways can the team be constituted?

Q28. If A is the arithmetic mean and G1 , G2 be two geometric mean between any two numbers, then prove that
2 2

2A  G  G 1 2
.

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G2 G1

OR

SI
Find the numbers in G.P. whose sum is 13 and the sum of whose squares is 91.

Q29. Find the equation of the line passing through the point of intersection of 2 x  y  5 and x  3 y  8  0 and parallel to
the line 3x  4 y  7.

Q30. lim
y 0
 x  y  sec  x  y   x sec x .
y
A
PT
  
Q31. Prove that tan   tan 60o    tan 120o    3 tan 3 . 
OR

If sin      a and sin      b, provethat cos 2      4ab cos      1  2a2  2b2 .

U

SECTION- D (This section comprises 4 Long Answers (LA) questions of 5 marks each)
G

Q32.Find the equation of the hyperbola, having

3
(i) Vertices   5,0 , foci   7,0 . (ii) length of latus rectum =8 and e  .
5
P

 
(iii) Foci 0,  10 , passing through (2, 3).
R

Q33. (i) Find the derivative of sin x with respect to x by first principle method.

x3  3
(ii) Find the derivative of f  x   with respect to x .
x 1

Q34. Find mean, variance and standard deviation using short cut method.

Height (cm) 70-75 75-80 80-85 85-90 90-95 95-100 100-105 105-110 110-115
No. of children 3 4 7 7 15 9 6 6 3

4
 x3  6 x 2  11x  6
Q35. Evaluate: lim . 73
x 2 x2  6x  8

OR

Differentiate the following with respect to x .

sin x  cos x  sec x  1 

 i     ii   
 sin x  cos x   sec x  1 

SECTION-E

(This section comprises of 3 case-study/passage-based questions of 4 marks each with sub-parts. First two case study
questions have three sub-parts (i), (ii), (iii) of marks 1, 1, 2 respectively. The third case study question has two sub-parts of 2

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marks each.)

Q36. A parking lot in a company is triangular shaped. Its sides are given by the equations

SI
AB : 3 y  5 x  2, BC : x  y  6  0 and AC : 3 y  x  2  0.

A
PT
Based on the above information, answer the following questions.
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(i) The coordinates of the vertex A are (1 mark)

(a) (- 1, -1) (b) ( -1, 2) (c) (1, 2) (d) – 1, 1)

G

(ii) The coordinates of the vertex B are (1 mark)

(a) (- 2, 2) (b) ( 2, -2) (c) (2, 4) (d) (2, -4)

(iii) Equation of line passing through A and perpendicular to BC is (2 marks)

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a x  y  0 b x – y  0 c x  2y  0 d  x – 2 y  0

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Q37. The number of bacteria in a certain culture doubles every hours. Given that the number of bacteria present at the and of
the 4th hour was 160000.

5
(i) The number of bacteria present originally was (1 mark)
74
(a) 40000 (b) 20000 (c) 5000 (d) 10000

(ii) The number of bacteria present at the and of 7th hour was (1 mark)

(a) 640000 (b) 1280000 (c) 40000 (d) 128000

(iii) The number of bacteria present at the beginning of 3rd hour was (2 mark)

(a) 2550000 (b) 20000 (c) 40000 (d) 80000

Q38. Isha and her 5 friends went for a trip to shillong. They stayed in a hotel. There were 4 vacant rooms A, B, C and D. Out of
these 4 vacant rooms, two rooms A and B were double share rooms and two rooms C and D can contain one person each.

R
SI
A
(A) If A and B rooms each are filled already, then find the number of ways in which room C can be filled. 2 marks
PT
(B) Find the number of ways in which room B can be filled. 2 marks
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G
P
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