Trigonometry Topper’s Package Mathematics - XI 15 1. TRIGONOMETRICAL RATIO 1. The sum of the series n1 n! sin 720 is : (a) sin sin sin 180 360 540 (b) sin sin sin sin 6 30 120 360 (c) sin sin sin sin 6 30 120 360 + sin 720 (d) sin sin 180 360 2. The value of 2 2 2cot ( /6) 4tan ( /6) 3cosec /6 is : (a) 2 (b) 4 (c) 4/3 (d) 3 3. If sin A 6 cos A 7 cos A then cos A + 6 sin A is equal to : (a) 6 sin A (b) 7 sin A (c) 6 cos A (d) 7 cos A 4. If 3 cos 2 and 3 sin 2 , where does not lie and lies in the third quadrant, then 2 2tan 3 tan cot cos is equal to : (a) 7 22 (b) 5 22 (c) 9 22 (d) 22 5 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 2 cos x cos x 1 , then the value of 12 sin x 10 8 6 3sin x 3sin x sin x 1 is : (a) 2 (b) 1 (c) –1 (d) 0 7. If 0, 4 and tan 1 2 t (tan ) ,t = cot (tan ) , cot 3 t (cot ) and tan 4 t (cot ) , then : (a) 1 2 3 4 t t t t (b) 4 3 1 2 t t t t (c) 3 1 2 4 t t t t (d) 2 3 1 4 t t t t 8. The expression 10 8 cos cos 13 13 + 3 cos 13 + 5 cos 13 is equal to : (a) –1 (b) 0 (c) 1 (d) None of these 9. If sin cosec 2 , then the value of 10 10 sin cosec is : (a) 2 (b) 2 10 (c) 2 9 (d) 10 10. 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 k 1 k (b) 1 k 1 k (c) k 1 k 1 (d) k 1 k 1 12. The value of 2 4 tan 2tan 4cot 5 5 5 is : (a) cot 5 (b) 2 cot 5 (c) 4 cot 5 (d) 3 cot 5 13. In a right angled triangle, if the hypotenuse is 22 times the length of perpendicular drawn TRIGONOMETRY Unit 2
15
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TrigonometryTopper’s Package Mathematics - XI
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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
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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
= 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
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(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 :
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
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 _________