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TrigonometryTopper’s Package Mathematics - XI
15
1. TRIGONOMETRICAL RATIO
1. The sum of the series n 1
n!sin720
is :
(a) sin sin sin180 360 540
(b) sin sin sin sin6 30 120 360
(c) sin sin sin sin6 30 120 360
+
sin720
(d) sin sin180 360
2. The value of 2 22cot ( /6) 4tan ( /6)
3cosec /6 is :(a) 2 (b) 4(c) 4/3 (d) 3
3. If sinA 6 cos A 7 cos A then
cos A + 6 sinA is equal to :
(a) 6 sinA (b) 7 sinA
(c) 6 cos A (d) 7 cos A
4. If 3cos2
and 3sin2
, where does
not lie and lies in the third quadrant, then
22tan 3 tan
cot cos
is equal to :
(a)722 (b)
522
(c)922 (d)
225
5. If sec m and tan= n, then
1 1(m n)m (m n)
is equal to :
(a) 2 (b) 2m(c) 2n (d) mn
6. If 2cos x cos x 1 , then the value of12sin x 10 8 63sin x 3sin x sin x 1 is :
(a) 2 (b) 1(c) –1 (d) 0
7. If 0,4
and tan
1 2t (tan ) ,t = cot(tan ) ,
cot3t (cot ) and tan
4t (cot ) , then :
(a) 1 2 3 4t t t t (b) 4 3 1 2t t t t
(c) 3 1 2 4t t t t (d) 2 3 1 4t t t t
8. The expression 10 8cos cos13 13 +
3cos13
+
5cos13
is equal to :(a) –1 (b) 0(c) 1 (d) None of these
9. If sin cosec 2 , then the value of10 10sin cosec is :
(a) 2 (b) 210
(c) 29 (d) 1010. If sin 3sin( 2 ) , then the value of
tan( ) 2tan is :(a) 3 (b) 2(c) –1 (d) 0
11. If tan = k cot ,then cos( )cos( )
is equal to :
(a)1 k1 k
(b)1 k1 k
(c)k 1k 1
(d)k 1k 1
12. The value of 2 4tan 2tan 4cot
5 5 5 is :
(a) cot5
(b)2cot5
(c) 4cot5
(d)3cot5
13. In a right angled triangle, if the hypotenuse is
2 2 times the length of perpendicular drawn
TRIGONOMETRYUnit
2
TrigonometryTopper’s Package Mathematics - XI
16
from the opposite vertex on the hypotenuse,then the other two angles are :
(a) ,3 6
(b) ,4 4
(c)3,
8 8
(d)5,
12 8
14. If 3cos2 1cos23 cos2
, then tan is equal to :
(a) 2 tan (b) tan
(c) sin2 (d) 2cot
15. Ifn
3m
m 0sin xsin3x C cos mx
is an identity in
x, where 0, 1 nC C ,...C are constants and nC 0then teh value of n is(a) 2 (b) 4(c) 6 (d) 8
16. If tan1sin sin 2 ,5 tan
A BB A B then
A
(a)53
(b) 23
(c)32
(d) 35
17. The value of0 0 0
10 10 10log tan1 log tan 3 ... log (tan89 ) isgiven by(a) 45 (log10 tan 10) (b) 45 log10 (tan 890)(c) 1 (d) 0
18. The value of the expression
11
11
2
sincos
cossin
sincos
yy
yy
yy
is equal to
(a) 0 (b) 1(c) sin y (d) cos y
19. The expression
cos cos coscos cos cos6 6 4 15 2 10
5 5 3 10x x x
x x x+ + +
+ +is equal to
(a) cos 2x (b) 2cos x(c) cos2 x (d) 1+cos x
20. If Pnn n cos sin . ,then 2 3 16 4p p- + =
(a) 2 (b) 3(c) 0 (d) 1
21. If A-B =4
, then (1+tan A) (1-tan B) is equal to:
(a) 2 (b) 1(c) 0 (d) 3
22. cot = sin2 ( n ,n integer ) if equals(a) 45° and 60° (b) 45° and 90°(c) 45° only (d) 90°only
23. If A lies in the 2nd quadrant and 3 tan A + 4 = 0then the value of 2 cot A-5 cos A + sin A is equal to
(a) 5310 (b)
3710
(c)2310 (d)
710
.
24. If tan 17
110
and sin where
02
2
, , then is equal to
(a)
4 (b)
34
(c)
8 (d)
38 2 .
25. If sin 2 2 12
sin ,
cos cos , cos2 2 32
2 then b g
(a)38 (b)
54
(c)34 (d)
58
26. If sin sin ,x x 2 1 then the value of
cos cos cos cos12 10 8 63 3 1x x x x is equal to(a) 0 (b) 1(c) –1 (d) 2
27. If p is the product of the sine of angle of atriangle and q be the product of their cosine,then tangent of the angle are root of theequation :
(a) 3 2 (1 ) 0qx qx p x q
(b) 3 2 (1 ) 0px qx p x q
(c) 3 2 2(1 ) 0q x px px qx p (d) none of the above
28. If and are +ve acute angles and
TrigonometryTopper’s Package Mathematics - XI
17
tan( ) , tan ( ) , 1 13
equals
(a) 7 300 (b) 37 300
(c) 52 300 (d) 60 300 29. If A + B + C =180°, then the value of
cot cot cot cot cot cotB C C A A B b gb gb g will be(a) sec A sec B sec C(b) cosec A cosec B cosec C(c) tanA tan B tan C(d) 1
30. If A + B + C =1800, then the value of
cot cot cotA B C will be2 2 2
(a) cot cot cotA B C2 2 2 (b) 4
2 2 2cot cot cotA B C
(c) 22 2 2
cot cot cotA B C (d) 8
2 2 2cot cot cotA B C
.
31. The value of cos cos cos 7
27
47
is
(a) 18 (b)
14
(c) 1
16 (d) 3
1632. The least value of 3 sin x – 4 cos x + 7 is
(a) 0 (b) 2(c) 3 (d) 6
33. If tan m 1;tan ,thenm 1 2m 1
(a)2 (b)
3
(c)4 (d) none of these
34. Which of the following numbers is / arerational?
(a) sin150 (b) cos150
(c) sin cos15150 0 (d) sin cos15 750 0
35. If cos A B b g 35 and tan A tan B = 2 , then
(a) cos cosA B 15 (b) cos cosA B
15
(c) sin sinA B 25 (d) sin sinA B
15
36. If cos coscos
,
2 12 then the value of tan
2 is
equal to :
(a) 32
tan (b) tan 2
(c)13 2
tan (d) 32
tan
37. a b ccos sin , then a bsin cos b g2 -is
equal to(a) a2 + b2 (b) b2 + c2
(c) c2 + a2 (d) a2 + b2 – c2
38. If tan x x for all xb g b gtan ,1 then value of
must be
(a) 00 (b) 300
(c) 450 (d) 600
39. The value of tan81 tan63 tan27 tan9 is :(a) 1 (b) 2(c) 3 (d) 4
40. If tan x ba
then the value of , a cos 2x + b sin2x
is :(a) a (b) a – b(c) a + b (d) b
41. If sin cos ,30 600 0 d i d i then :
(a) 300 (b) 0
(c) 600 (d) 600
42. If sin cos 2,cos sin
then the value of is
(a) 900 (b) 600
(c) 450 (d) 300
43. If sin + cos =1, then the value of sin cos
is :
(a) 1 (b)12
(c) 0 (d) 244. The expression
3 32
34 4s in s in F
HGIKJ
LNM
OQPb g
TrigonometryTopper’s Package Mathematics - XI
18
FHG
IKJ
LNM
OQP2
256 6sin sin
b gis equal to :(a) 0 (b) 1
(c) 3 (d) sin cos4 6
45. If sin sin sin , 1 2 3 3 thencos cos cos 1 2 3 (a) 3 (b) 2(c) 1 (d) 0
46. If for real value of x, cos = x + 1/x ,then
(a) is an acute angle
(b) is a right angle
(c) is an obtuse angle
(d) no value of is possible.
47. The expression 1tan cotA A
simplifies to
(a) sec A cosec A (b) sin A cos A(c) tan 2A (d) sin 2 A
48. The value of 23
cos x FHG
IKJ
is
(a) cos sinx x 3 (b) 3 2cos sinx x
(c) cos sinx x 3 (d) 3 2cos sinx x
49. If A+ B+C= , then cos B + cos C is equal to
(a) 22 2
cos cosA B C(b) 2
2 2sin cosA B C
(c) 22 2
cos cosC A B(d) 2
2 2cos cos .B C
50. In a triangle PQR, R 2
. If tan P2 and tan
Q2
are the roots of the equation
ax bx c a o then2 0 ( ), :
(a) a + b = c (b) b + c = a(c) a + c = b (d) b = c
51. If tan = t, then tan2 + sec2 =
(a)11
tt (b)
11
tt
(c)2
1tt
(d)2
1tt
52. The number of integral values of K for whichthe equation 7cos x + 5 sin x = 2k +1 has asolution is(a) 4 (b) 8(c) 10 (d) 12
53. If sin and cos are the roots of theequation ax bx c2 0 , then a, b, c satisfythe relation
(a) b a ac2 2 2 (b) a b ac2 2 2
(c) a b c2 2 2 (d) b a ac2 2 2
54. If cos cos20 2 10 2 k and x k ,then thepossible values of x between 00 and 3600 are(a) 1400 (b) 400 and 1400
(c) 500 and 1300 (d) 400 and 3200
55. tan tan tan tan20 40 3 20 400 0 0 0 is equal to
(a)3
2(b)
34
(c) 3 (d) 1
56. The value of
sin sin sin sin2 2 2 2
838
58
78
(a) 1 (b) 2
(c) 118 (d) 2 1
8
57. The value of tan tan
434
LNM
OQP L
NMOQP is :
(a) – 2 (b) 2(c) 1 (d) – 1
58. The value of sin sin sin 14
314
514
is :
(a)1
16 (b) 18
(c)12 (d) 1
59. sec22
4
xyx yb g is true if :
(a) x y 0 (b) x y x , 0
(c) x y (d) x y 0 0,
TrigonometryTopper’s Package Mathematics - XI
19
60. If sin cosx ec x 2 , then sin cosn nx ec x isequal to
(a) 2 (b) 2n
(c) 2 1n (d) 2 2n
61. sin22
4
x yxy
b g where x R , y R gives real
if and only if(a) x + y = 0 (b) x = y (c) |x | = |y | 0 (d) none of these
62. If 0 1800 0 then
2 2 2 2 1 .... cosb g there being n
number of 2’s, is equal to
(a) 22
cos n (b) 22 1cos
n
(c) 22 1cos
n (d) none of these
63. tan .tan .tan3 3
is equal to
(a) tan2 (b) tan3
(c) tan3 (d) none of these
64. If tan tan tan
FHG
IKJ F
HGIKJ3 3
= ktan 3then k is equal to(a) 1 (b) 3(c) 1/3 (d) none of these
65. The value of tan tan tan20 2 50 700 0 0 is(a) 1 (b) 0
(c) tan500 (d) none of these
66. If 4n then cot . cot 2 . cot 3 ....
cot 2 1n b g is equal to
(a) 1 (b) -1(c) (d) none of these
67.r
n rn
1
12cos
is equal to :
(a)n2 (b)
n 12
(c)n2
1 (d) none of these
68. The value of sin n n n
ton sin sin ...3 5
terms is equal to :(a) 1 (b) 0
(c)n2 (d) none of these
69. The sum to 20 terms of the series
1 1 1sin .sin2 sin2 .sin3 sin3 .sin4
+ .......
(a) 1 tan tan21cos
(b) 1 cot cot21cos
(c) 1 tan tan21sin
(d) 1 cot cos21cos
70. The value of 3 cos 20 sec20ec is(a) 2 (b) –2(c) –4 (d) 4
71. The value of the expressioncos1 .cos2 ...cos179 equals :(a) 0 (b) 1
(c)12 (d) –1
72. If ,2
, then the value of tanequals:
(a) tan tan (b) 2(tan tan )
(c) tan 2tan (d) 2tan tan 73. The value of
2 2 2 2sin 5 sin 10 sin 15 ... sin 90 is :(a) 9.5 (b) 9(c) 10 (d) 8.
74. Angles A, B and Cof a triangle are in AP withcommon difference 15 degree, then angle A isequal to(a) 45° (b) 60°(c) 75° (d) 30°
2. MAXIMUM VALUE, MINIMUM VALUE
75. The maximum value of f(x) sinx(1 cos x) is:
TrigonometryTopper’s Package Mathematics - XI
20
(a)3 3
4(b)
3 32
(c) 3 3 (d) 3
76. The maximum value of 2cos x3
–
2cos x3
is :
(a) 32
(b)12
(c)32
(d)32
77. The maximum value of 5cos 3cos3
+3 is :(a) 5 (b) 11(c) 10 (d) –1
78. The maximum value of 24sin x 12sinx 7 is
(a) 25 (b) 4(c) does not exit (d) none of these
79. The maximum value of sin cosx xFHG
IKJ F
HGIKJ
6 6
in
the interval 02
, FHG
IKJ is attained at
(a)
12 (b)6
(c)3 (d)
2
80. Maximum value of 5 33
1cos cos
FHG
IKJ is
(a) 5 (b) 6(c) 7 (d) 8
81. Maximum value of f x x x isb g sin cos
(a) 1 (b) 2
(c)12 (d) 2
82. The smallest value of satisfing
3 4cot tan b g is :
(a)23
(b)3
(c)6 (d)
12
83. sin cos is maximum when is equal to
(a) 300 (b) 450
(c) 600 (d) 900
84. The maximum value ofcos . cos .... cos , 1 2b g b g b gn under the restrictions
021 2 , ,..... n and
(a)1
2 2n/ (b)1
2n
(c)12 (d) 1
85. If A sin cos20 48 , then for all valuesof :(a) A 1 (b) 0 1 A(c) 1 3 A (d) none of these
86. If e ex xsin sin 4 0 then the number f realvalues of x is :(a) 0 (b) 2(c) 1 (d) Infinite
87. If A B A B 0 03
, , and y = tan A. tan B
then(a) the maximum value of y is 3
(b) the minimum value of y is 13
(c) the maximum value of y is 13
(d) the minimum value of y is 0
3. TRIGONOMETRICAL EQUATION
88. The sum of the solutions in (0,2 ) of the
equation cos xcos x3
cos x
3
=
14 is :
(a) 4 (b)
(c) 2 (d) 3
89. Let f(x)= xsin x,x 0 . Then, for all natural
numbers n, f (x) vanishes at :
(a) a unique point in the interval 1n,n2
(b) a unique point in the interval 1n ,n 12
(c) a unique point in the interval (n +1, n + 2)(d) two points in the interval (n, n + 1)
90. If cos x sin xlog tanx log cot x 0 , then the mostgeneral solution of x is :
TrigonometryTopper’s Package Mathematics - XI
21
(a) 32n4
, n Z (b) 2n4
, n Z
(c) n4
, n Z (d) None of these
91. The number of real solutions of the equations3 2x x 4x 2sinx 0 in 0 x 2 is :
(a) four (b) two(c) one (d) 0
92. The positive values of n>3 satisfying the
equation 1 1 1
2 3sin sin sinn n n
is :
(a) 8 (b) 6(c) 5 (d) 7
93. The solution of trigonometric equation4 4cos x sin x = 2cos(2x )cos(2x ) is
(a)1n 1x sin
2 5
(b) n
1n ( 1) 2 2x sin4 4 3
(c) 1n 1x cos
2 5
(d)n
1n ( 1) 1x cos4 4 5
94. If
sinx cos x cos xcos x sinx cos xcos x cos x sinx
= 0, then the number of
distinct real roots of this equation in the
interval
,
2 2(a) 1 (b) 3(c) 2 (d) 4
95. Solution of the equation 3tan( 15 ) =tan( 15 ) is :
(a) = n 3
(b) n3
(c) n4
(d) n4
96. The solution of the equation1 sin2x[sinx cos x] =2, x is :
(a) 2
(b)
(c) 4
(d)34
97. General solution of sin x cos x =a Rmin
2{1,a 4a 6} is :
(a) nn ( 1)2 4 (b) n2n ( 1)
4
(c) n 1n ( 1)4
(d) nn ( 1)
4 4
98. The number of solutions of the equation
sinxcos3x = sin3x cos5x in 0,2
is :
(a) 3 (b) 4(c) 5 (d) 6
99. If 21 sin sin .... 4 2 3 ;0 2
then :
(a) 3
(b) 6
(c) or3 6
(d)2or
3 3
100. If tan cot4 4
then :
(a) 0 (b) 2n
(c) 2n 1
(d) 2(2n 1) for all n is an integer101. The number of solutions of the pair of equation
22sin cos2 0 and 22cos 3sin 0 in
the interval [0,2 ] is :(a) 0 (b) 1(c) 2 (d) 4
102. The set of values of satisfying the inequation22sin 5sin 2 0 wheren 0 2 is :
(a)50, ,2
6 6
(b)
50, ,26 6
(c)20, ,2
3 3
(d) None of these
103. The set of values of x for which
tan3 tan 2 11 tan3 tan 2
x x
x x is :
(a) (b) 4
TrigonometryTopper’s Package Mathematics - XI
22
(c) , 1,2,3,...4
n n
(d) 2 , 1,2,3,...4
n n
104. The equationa x b x csin cos where c a b 2 2 has :(a) a unique solution(b) infinite no. of solution(c) no solution(d) none of these
105. The solution of tan2 tan 1 is :
(a)3 (b) 6 1
6n b g
(c) 4 16
n b g (d) 2 16
n b g
106. The number of solutions of sin cos2 3 3
in , is
(a) 4 (b) 2(c) 0 (d) none of these
107. The number of all possible triplets
a a a1 2 3, ,b g such that a a x a x1 2 322 0 cos sin
for all x is :(a) 0 (b) 1
(c) 2 (d) infinite108. For n general solution of the equation
3 1 3 1 2 e j e jsin cos is :
(a)
24 12
n
(b)
n n14 12
b g
(c)
24 12
n
(d)
n n14 12
b g109. The smallest positive angle satisfying the
equation sin cos2 2 14
0 is :
(a)2 (b)
3
(c)4 (d)
6
110. If sin 6 + sin 4 + sin 2 = 0 ,then =
(a)n or n
4 3 (b)
n or n
4 6
(c)n or n
42
6 (d) none of these
111. If sin cos ,
0 02
and then is equal
to :
(a)2 (b)
4
(c)6 (d) 0
112. If 3 2cos sin , then the value of willbe :
(a) 26
n (b) 2
3n
(c) 212
n (d) 2
12n
113. If sin cos cos sin ,b g b g then which of thefollowing is correct ?
(a) cos 3
2 2 (b) cos
FHG
IKJ 2
12 2
(c) cos
FHG
IKJ 4
12 2 (d) cos .
F
HGIKJ 4
12 2
114. The number of solutions of sin cos2 3 3
in , is
(a) 4 (b) 2(c) 0 (d) none of these
115. The most general solutions of2 31 |cos x| cos x |cos x| ....to2 = 4 are given by
(a) n n Z
3
, (b) 23
n n Z
,
(c) 2 23
n n Z
, (d) none of these
116. The set of values of satifying the inequation22sin 5sin 2 0, where 0 2 , is:
(a) 50, ,26 6
(b)
50, ,26 6
TrigonometryTopper’s Package Mathematics - XI
23
(c) 20, ,23 3
(d) none of these
117. If exp 2 4 6(sin sin sin .... )log 2ex x x
satisfies the equation 2 9 8 0x x . Then
the value of cos ,0
cos sin 2x x
x x
.
(a) 3 1 (b) 3 1
(c)1
3 1 (d)1
3 1
118. The value of (s) of in the interval ,2 2
of the equation
22 tan(1 tan )(1 tan )sec 2 0 is
(a)3
(b)3
(c)4
(d)3
4. INVERSE TRIGONOMETRIC FUNCTION
119. 1 1cos [cos{2cot ( 2 1)} is equal to :
(a) 2 1 (b) 4
(c)34
(d) 0
120. Given, 10 x2
, then the value of :2
1 1x 1 xtan sin sin x2 2
is :
(a) 1 (b) 3
(c) –1 (d)13
121. The value of 23 n
1
n 1 k 1cot cos 1 2k
:
(a)2325 (b)
2523
(c)2324 (d)
2423
122. If 1 1asin x bcos x c , then 1asin x +1bcos x is equal to :
(a)ab c(a b)
a b
(b) 0
(c)ab c(a b)
a b
(d) 2
123. 1 2 1 2 1 2cot (2.1 ) cot (2.2 ) cot (2.3 ) ... upto
(a) 4
(b) 3
(c) 2
(d) 5
124. If 1 12 2
acot(cos x) sec tanb a
then x is
equal to :
(a) 2 2
b
2b a(b) 2 2
a
2b a
(c)2 22b aa (d)
2 22b ab
125. If 9 x 1 , then 21 x 1[{x cos(cot x) +
sin 1 2 1/2(cot x)} 1] is equal to :
(a) 2
x
1 x(b) x
(c) 2x 1 x (d) 21 x
126. If 2
1 12 2
1 x 2x5cos 7sin1 x 1 x
– 14tan
22x
1 x
1tan x 5 then x is equal to :
(a) 3 (b) 3
(c) 2 (d) 3
127. If 1 2 1sec 1 x cosec 2
11 y 1coty z
= then x + y + z is equal to :(a) xyz (b) 2xyz(c) xyz2 (d) x2yz
128. The value of 1 11sin sin sec (3)
3
+ cos
1 11tan tan (2)2
(a) 1 (b) 2(c) 3 (d) 4
129. If 1 1 1 1tan a tan b sin 1 tan c, then :
(a) a b c abc
TrigonometryTopper’s Package Mathematics - XI
24
(b) ab bc ca abc
(c)1 1 1 1a b c abc =0
(d) ab + bc + ca = a + b + c
130. If sin sin tan
LNM
OQP
LNM
OQP
12
12
121
21
2aa
bb
x , then
x is equal to
(a)a b
ab1 (b)
bab1
(c)bab1 (d)
a bab1
5. PROERTIES OF TRIANGLE131. In a triangle, the lengths of two larger sides
are 10 cm and 9 cm. If the angles of the triangleare in AP, then the length of the third side is :
(a) 5 6 (b) 5 6
(c) 5 6 (d) 5 6
132. In any ABC ,
2 2(a b c)(b c a)(c a b)(a b c)
4b c
equals
(a) 2sin B (b) 2cos A(c) 2cos B (d) 2sin A
133. In a ABC , a = 8 cm, b = 10 cm, c = 12 cm. Thenrelation between angles of the tirangle is :(a) C = A+B (b) C = 2B(c) C = 2A (d) C = 3A
134. If the angles A, B and C of a triangle are in anarithmetic progression and if a, b and c denotethe lengths of the sides opposite to A, B and Crespectively, then the value of the expressiona csin2c sin2Ac a
is :
(a)12 (b)
32
(c) 1 (d) 3
135. In ABC , If 2 2A Bsin ,sin2 2 and 2 Csin
2 are in
H.P.Then a,b, and c will be in :(a) AP (b) GP(c) HP (d) None of these
136. In a ABC , if a = 3, b = 4, c = 5 then the
distance between its incentre andcircumcentre is :
(a)12 (b)
32
(c)32 (d)
52
137. In an equilateral triangle of side 2 3 cm, thecircumradius is :
(a) 1 cm (b) 3cm(c) 2 cm (d) 2 3 cm
138. In ABC , A2
b= 4, c =3 then the value ofRr is :
(a)52 (b)
72
(c)92 (d)
3524
139. The circumradius of the triangle whose sidesare 13, 12 and 5 is :
(a) 15 (b)132
(c)152 (d) 6
140. In any ABC if 2 cos B ac
, then the triangle
is :(a) right angled (b) equilateral(c) isosceles (d) none of these
141. In a triangle ABC, sin sin sinA B C 1 2 andcos cos cosA B C 2 , then the triangle is(a) equilateral(b) isosceles(c) right angled(d) right angled isosceles
142. In a triangle ABC cos cos cosA B C 32
,
then the triangle is(a) isosceles (b) right angled(c) equilateral (d) none of these
143. In a triangle ABC, if 1 1 3
a c b c a b c
,
then C is equal to(a) 300 (b) 600
(c) 450 (d) 900
TrigonometryTopper’s Package Mathematics - XI
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144. If in a , r r r r1 2 3 , then the is
(a) obtuse angled(b) right angled(c) isosceles right angled(d) none of these
145.1 2 3
1 1 1r r r
(a)1r (b) r
(c)2r (d) none of these
146. In a C , r r r r r r1 2 2 3 3 1 (a) s (b) (c) s2 (d) 2
147. In a C , 1 2 3r.r .r .r is equal to
(a) 2 (b) 2
(c)abc
R4 (d) none of these
148. If in a triangle R and r are the circumradiusand in-radius respectively, then the H.M. of theex-radii of the triangle is(a) 3r (b) 2R(c) R r (d) none of these
149. In a C , 2ac sin A B C LNM
OQP 2
(a) a b c2 2 2 (b) c a b2 2 2 (c) b c a2 2 2 (d) c a b2 2 2
150. 4R sin A sin B sin C is equal to :
(a) a + b + c (b) a b c r b g(b) a b c R b g (d) a b c r
R b g
151. If in a triangle ABC2 2cos cos cos ,A
aB
bC
cabc
bca
then the
value of the angle A is
(a)3 (b)
4
(c)2 (d)
6
152. If in a C AC
A BB C
, sinsin
sinsin
,
b gb g then
(a) a, b, c are in A.P.
(b) a b c are in A P2 2 2, , . .(c) a,b,c are in H.P.
(d) a b c arein H P2 2 2, , . .153. In an equilateral triangle circumradius :
in-radius : ex-radius r1b g (a) 1 : 1 : 1 (b) 1 : 2 : 3(c) 2 : 1 : 3 (d) 3 : 2 : 4
154. In a triangle ABC, a b c b c a bc if b gb g
(a) 0 (b) 0
(c) 0 4 (d) 4
155. In a ABC c bc b
A, tan 2 is equal to
(a) tan A B2F
HGIKJ (b) cot A B
2F
HGIKJ
(c) tan A BF
HGIKJ2 (d) none of these
156. In a ABC c a b a b c ab, . b gb g The
measure of C is
(a)3 (b)
6
(c)23
(d) none of these
157. If in a triangle ABC,b c c a a b
11 12 13, then
cos A is equal to :
(a)15 (b)
57
(c)1935 (d) none of these
158. If a cos A=b cos B, then ABC is(a) isosceles only(b) right angled only(c) equilateral(d) right angled or isosceles.
159. If the angled of a triangle are in the ratio1 : 2 : 3, the corresponding sides are in theratio :(a) 2 : 3 : 1 (b) 3 2 1: :
TrigonometryTopper’s Package Mathematics - XI
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(c) 2 3 1: : (d) 1 3 2: :
160. In a ABC, if Aa
Bb
Cc
cos cos cos and side a
= 2, then area of the triangle is(a) 1 (b) 2
(c)3
2(d) 3
161. 3 cos( ) a B C
(a) 3 abc (b) 3 (a + b + c)(c) abc (a + b + c) (d) 0.
162. In a ABC, 2ac sin A B C FHG
IKJ 2
(a) a b c2 2 2 (b) c a b2 2 2
(c) b c a2 2 2 (d) c a b2 2 2
163. The ex-radii of a triangle r r r1 2 3, , , are in H.P.Then the three sides of the triangle a,b and care in(a) A.P. (b) H.P.(c) G.P. (d) none of these
164. The area of the circle and the area of a regularpolygon of n sides and of perimeter equal tothat of the circle are in the ratio :
(a) tan : n n
FHG
IKJ (b) cos :
n nFHG
IKJ
(c) sin : n n
FHG
IKJ (d) cot : .
n nFHG
IKJ
165. In a ABC,8 2 2 2 2R a b c , then the triangle is(a) equilateral (b) isoscles(c) right angled (d) obtuse angled.
166. The value of 1 1 1 12
12
22
32r r r r
is equal to :
(a)a b c2 2 2
(b)
a b c2 2 2
2
(c)2 2 2
22a b c
(d) none of these
167. If R is the radius of the circumcircle of ABCand is its area , then :
(a) R abc
4 (b) R abc
(c) R a b c
(d) R a b c
4
168. In a ABC A B A B,cot tan 2 2
is equal to :
(a)a ba b
(b)a ba b
(c)a a bb a b
b gb g (d) none of these
169. In a ABC, if tan A2
56
and tan B2
2037
then
(a) 2a = b + c (b) a > b > c(c) 2c = a + b (d) none of these
170. If in a ABC aA
bB
then,cos cos
,
(a) 2 sin A sin B sin C = 1
(b) sin sin sin2 2 2A B C (c) 2 sin cos sinA B C(d) none of these
171. If in a ABC , 2 cos A sin C = sin B then thetriangle is :(a) equilatereal (b) isosceles(c) right angled (d) none of these
172. If the sides of a triangle are proportional tothe cosines of the opposite angles then thetriangle is :(a) right angled (b) equilateral(c) obtuse angled (d) none of these
173. The sides of a triangle are in AP and its area
is 35 (area of an equilateral triangle of the
same perimeter). Then the ratio of the sidesis(a) 1 : 2 : 3 (b) 3 : 5 : 7(c) 1 : 3 : 5 (d) none of these
174. In a ABC R circumradius and r = inradius.
The value of a A b B C
a b ccos cos cos
is equal to:
(a)Rr (b)
Rr2
(c)rR (d)
2rR
175. In an equilateral triangle, (circumradius):(inradius): (exradius) is equal to
TrigonometryTopper’s Package Mathematics - XI
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(a) 1 : 1 : 1 (b) 1 : 2 : 3(c) 2 : 1 : 3 (d) 3 : 2 : 4
176. Inradius of a circle which is inscribed in aisosceles triangle one of whose angle is 2 /3,is 3 , then area of triangle is :
(a) 4 3 (b) 12 7 3
(c) 12 7 3 (d) none of these
177. If the radius of the circumcircle of an isoscelestriangle PQR is equal to PQ = PR, then the angleP is:
(a)6
(b)3
(c)2
(d)23
178. If sides a, b, c are in the ratio 19 : 16 : 5, then
cot : cot : cot2 2 2A B C
equals :
(a) 1:15 : 4 (b) 15 :1: 4(c) 4 :1:15 (d) 1: 4 :15
6. HEIGHT AND DISTANCE179. ABCD is a square plot. The angle of elevation
of the top of a pole standing at D from A or Cis 30° and that from B is , then tan isequal to:
(a) 6 (b) 1/ 6
(c) 3 /2 (d) 2 /3
180. A vertical pole PO is standing at the centre Oof a square ABCD. If AC subtends an 90 atthe top P of the pole then the angle subtendedby a side of the square at P is :(a) 30° (b) 45°(c) 60° (d) None of these
181. A ladder rests against a wall so that its toptouches the roof of the hourse. If the laddermakes an angle of 60° with the horizontal andheight of the house is 6 3 m, then the lengthof the ladder is :
(a) 12 3 m (b) 12 m
(c)12 m
3 (d) None of these
182. The angles of elevation of the top of a tower attwo points, which are at distance a and b fromthe foot in the same horizontal line and on thesame side of the tower, are complementary.The height of the tower is :
(a) ab (b) ab
(c) a /b (d) b/a
183. ABC is a triangular park with AB = AC = 100m. A clock tower is situated at the mid point ofBC. The angle of elevation, if the top of thetower at A and B are cot–13.2 and 1cosec 2.6
respectively. The height of the tower is :(a) 16 m (b) 25 m(c) 50 m (d) None of these
184. From the top of a hill h meteres high, theangles of depressions of the top and the bottomof a pillar are and , respectively. The height(in metres) of the pillar is :
(a)h(tan tan )
tan
(b)h(tan tan )
tan
(c)h(tan tan )
tan
(d)h(tan tan )
tan
185. A round balloon of radius r substends an angle
at the eye of the observer, while the angleof elevation of its centre is . The height ofthe centre of balloon is :
(a) rcosec sin2
(b) rsin cosec2
(c) rsin cosec2
(d) rcosec sin2
186. The angle of elevation of the top of a TV towerfrom three points A, B and C in a straight linethrough the foot of the tower are , 2 and3 respectively. If AB = a, then height of thetower is :
(a) a tan (b) a sin(c) a sin2 (d) a sin3
187. A flagpole stands on a building of height 450 ftand an observer on a level ground is 300 ft fromthe base of the building. The angle of elevationof the bottom of the flagpole is 30° and theheight of the flagpole is 50 ft. If is the angle ofelevation of the top of the flagpole, then tan is
(a) 4
3 3 (b)32
TrigonometryTopper’s Package Mathematics - XI
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(c)92 (d)
35
188. From an aeroplane flying vertically above ahorizontal road, the angles of depression of twoconsecutive stones on the same side of theaeroplane are observed to be 30° and 60°,respectively. The height (in km) at which theaeroplane is flying, is :
(a)43 (b)
32
(c)23 (d) 2
189. The elevation of an object on a hill is observedfrom a certain point in the horizontal plane
through its base, to be 30°. After walking 120m towards it on level ground, the elevation iffound to be 60°. Then, the height (in metres)of the object is :
(a) 120 (b) 60 3(c) 120 3 (d) 60
190. A house subtends a right angle at the windowof an opposite house and the angle of elevationof the window from the bottom of the first houseis 60°. If the distance between the two housesis 6 m, then the height of the first house is :
(a) 8 3 m (b) 6 3m
(c) 4 3 m (d) None of these
INTEGER TYPE QUESTIONS
1. tan20°tan40°tan60°tan80° =
2. If tan – cot = a and sin + cos = b, then(b2 – 1)2 (a2 + 4) is equal to ________
3. In a ABC, if b2 + c2 = 3a2, then cotB + cot C –cotA = ____________
4. The number of points of intersection of 2y = 1and y = sinx, in –2 x 2 is _______
5. The number of pairs (x, y) satisyfing theequations sinx + siny = sin(x + y) and |x| +|y| = 1 is __________
6. If 0 x 2, then the number of solutions ofthe equation sin8x + cos6x = 1 ________
7. The number of values of x in the interval[0, 3] satisfying the equation 2sin2x + 5sinx– 3 = 0 ____________
8. The number of values of x in the interval[0, 5] satisfying the equation 3sin2x – 7sinx +2 = 0 _______
9. The maximum value of the expression
2 21
sin 3sin cos 5cos is
10. The 3sinA + 5cosA = 5, then the value of(3cosA – 5sinA)2 is _________
TrigonometryTopper’s Package Mathematics - XI
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1. (C), 2. (C), 3. (B), 4. (B), 5. (A), 6. (D), 7. (B),8. (B), 9. (A), 10. (D), 11. (A), 12. (A), 13. (C), 14. (A),
15. (C), 16. (C), 17. (D), 18. (D), 19. (B), 20. (C), 21. (A),22. (B), 23. (C), 24. (A), 25. (D), 26. (A), 27. (A), 28. (A),29. (B), 30. (A), 31. (A), 32. (B), 33. (C), 34. (C), 35. (A),36. (A), 37. (D), 38. (C), 39. (D), 40. (A), 41. (B), 42. (C),43. (C), 44. (B), 45. (D), 46. (D), 47. (B), 48. (A), 49. (B),50. (A), 51. (A), 52. (A), 53. (A), 54. (D), 55. (C), 56. (B),57. (D), 58. (D), 59. (C), 60. (A), 61. (C), 62. (A), 63. (B),64. (B), 65. (B), 66. (A), 67. (C), 68. (B), 69. (B), 70. (C),71. (A), 72. (C), 73. (A), 74. (A), 75. (A), 76. (C), 77. (A),78. (D), 79. (A), 80. (B), 81. (D), 82. (C), 83. (B), 84. (A),85. (B), 86. (A), 87. (D), 88. (C), 89. (B), 90. (B), 91. (C),92. (D), 93. (B), 94. (C), 95. (D), 96. (C), 97. (D), 98. (C),99. (D), 100. (D), 101. (C), 102. (A), 103. (A) 104. (C), 105. (B),
106. (D), 107. (D), 108. (A), 109. (B), 110. (A), 111. (B), 112. (C),113. (C), 114. (D), 115. (B), 116. (A), 117. (C), 118. (D), 119. (C),120. (A), 121. (B), 122. (A), 123. (A), 124. (A), 125. (C), 126. (D),127. (A), 128. (A), 129. (C), 130. (D), 131. (D), 132. (D), 133. (C),134. (D), 135. (C), 136. (D), 137. (C), 138. (A), 139. (B), 140. (C),141. (D), 142. (C), 143. (B), 144. (B), 145. (A), 146. (A), 147. (B),148. (A), 149. (B), 150. (D), 151. (C), 152. (B), 153. (C), 154. (C),155. (A), 156. (C), 157. (A), 158. (D), 159. (D), 160. (D), 161. (A),162. (B), 163. (B), 164. (A), 165. (A), 166. (B), 167. (A), 168. (A),169. (B), 170. (A), 171. (B), 172. (B), 173. (B), 174. (C), 175. (C),176. (C), 177. (B), 178. (D), 179. (B), 180. (C), 181. (B), 182. (B),183. (B), 184. (A), 185. (D), 186. (C), 187. (A), 188. (B), 189. (B),190. (A),
1. (3) 2. (4) 3. (0) 4. (4) 5. (6) 6. (5) 7. (4)8. (6) 9. (2) 10. (9)
INTEGER TYPE QUESTIONS