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Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 1
TRIGONOMETRY
Sy l labus :
Trigonometrical identities and equations.Trigonometric functions. Inverse trigonometricalfunctions and their properties. Height and Distances.
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 2
C O N C E P T S
Trogonometric Identities & Ratios
C1 Basic Trigonometric Identities :
(a) sin2 + cos2 = 1; –1 sin 1; –1 cos 1 R
(b) sec2 – tan2 = 1; |sec | 1 R –
In,2
)1n2(
(c) cosec2 – cot2 = 1 ; |cosec | 1 R – {n, n I}
Practice Problems :
1. The value of cos2 50 + cos2 100 + ...... cos2 850 + cos2 900 will be
(a)2
17(b)
2
19(c) 1 (d) 0
(a)
2. If sec – tan = k then the value of cos will be
(a)1k
k22
(b)1k
k22
(c)k2
1k 2 (d)
k2
1k 2
(a)
3. If cosec A + cot A = 2
11 then tan A equals
(a)22
21(b)
16
15(c)
117
44(d)
43
117
(c)
[Answers : (1) a (2) a (3) c]
C2 Trigonometric Functions Of Allied Angles :
If is any angle, then –, 90 ± , 180 ± , 270 ± , 360 ± etc. are called Allied Angles.
(a) sin (–) = –sin ; cos (–) = cos
(b) sin (900 – ) = cos ; cos (900 – ) = sin
(c) sin (900 + ) = cos ; cos (900 + ) = – sin
(d) sin (1800 – ) = sin ; cos (1800 – ) = – cos
(e) sin (1800 + ) = – sin ; cos (1800 + ) = – cos
(f) sin (2700 – ) = – cos ; cos (2700 – ) = – sin
(g) sin (2700 + ) = – cos ; cos (2700 + ) = sin
(h) tan (900 – ) = cot ; cot (900 – ) = tan
Practice Problems :
1. If 2
1sin and is obtuse then cot equals
(a)3
1(b)
3
1 (c) 3 (d) 3
2. The value of sin 100 + sin 200 + sin 300......sin 3600 is
(a) 0 (b) 1 (c) –1 (d) none of these
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 3
3. The value of cos 10 . cos 20 . cos 30.......cos 1800 will be
(a) 0 (b) 1
(c) 100 (d) cannot be found
4. The value of cos 10 + cos 20 + cos 30.......cos 1800 will be
(a) 0 (b) –1 (c) 1 (d) none of these
5. The value of log [cos 10 + cos 20 + cos 30 ..... cos 1800] will be
(a) 0 (b) 1 (c) –1 (d) not defined
6. The value of log [cot 10 . cot 20 . cot 30 ..... cot 890] will be
(a) 0 (b) 1 (c) –1 (d) not defined
[Answers : (1) d (2) a (3) a (4) b (5) d (6) a]
C3 Trigonometric Functions of Sum of Difference of Two Angles :
(a) sin (A ± B) = sinA cosB ± cosA sin B
(b) cos (A ± B) = cosA cosB sinA sin B
(c) sin2A – sin2B = cos2B – cos2A = sin (A + B) . sin (A – B)
(d) cos2A – sin2B = cos2B – sin2A = cos (A + B) . cos (A – B)
(e)BtanAtan1
BtanAtan)BAtan(
(f)AcotBcot
1BcotAcot)BAcot(
(g)AtanCtanCtanBtanBtanAtan1
CtanBtanAtan–CtanBtanAtan)CBAtan(
Practice Problems :
1. If sin A + cos B = a and sin B + cos A = b then the value of sin (A + B) will be
(a) a2 + b2 (b)2
ba 22 (c)
2
2ba 22 (d)
2
2ba 22
2. If tan ( + ) = 2
1 and tan ( – ) =
3
1 then tan 2 will be
(a) 0 (b) 1 (c) –1 (d) none of these
3. If tan = 2
1 and tan =
3
1 then will be
(a) 0 (b)4
(c)
2
(d)
6
4. The value of cos2 480 – sin2 120 equals to
(a)4
15 (b)
4
15 (c)
8
15 (d)
8
15
5. The value of 3 cosec 200 – sec 200 is equal to
(a) 2 (b) 0
0
40sin
20sin2(c) 4 (d) 0
0
40sin
20sin4
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 4
6. The value of tan 5x . tan 3x . tan 2x is equal to
(a) tan 5x – tan 3x – tan 2x (b)x2cosx3cosx5cos
x2sinx3sinx5sin
(c) 0 (d) none of these
7. The value of 00
00
9sin9cos
9sin9cos
equal to
(a) tan 540 (b) tan 360 (c) tan 180 (d) none of these
8. If A + B + C = (A, B, C > 0) and the angle C is obtuse then
(a) tan A . tan B > 1 (b) tan A . tan B < 1
(c) tan A . tan B = 1 (d) none of these
[Answers : (1) c (2) b (3) b (4) c (5) c (6) a (7) a (8) b]
C4A Factorisation of the Sum of Difference of Two Sines or Cosines :
(a)2
DCcos
2
DCsin2DsinCsin
(b)2
DCsin
2
DCcos2DsinCsin
(c)2
DCcos
2
DCcos2DcosCcos
(d)2
DCsin
2
DCsin2DcosCcos
C4B Transformation of Products into Sum of Difference of Sines & Cosines :
(a) 2sinA cosB = sin(A + B) + sin(A – B)
(b) 2cosA sinB = sin (A + B) – sin(A – B)
(c) 2 cos A cos B = cos (A + B) + cos (A – B)
(d) 2 sin A sin B = cos (A – B) – cos (A + B)
C4C Multiple and Sub-multiple Angles :
(a) sin2A = 2sinA cosA; sin = 2
cos2
sin2
(b) cos2A = cos2A – sin2A = 2cos2A – 1 = 1 – 2 sin2A; 2 cos2
2
= 1 + cos ,
2 sin2
2
= 1 – cos .
(c)
2tan1
2tan2
tan;Atan1
Atan2A2tan
22
(d)Atan1
Atan1A2cos;
Atan1
Atan2A2sin
2
2
2
(e) sin 3A = 3 sin A – 4sin3A
(f) cos 3A = 4 cos3A – 3 cosA
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 5
(g)Atan31
AtanAtan3A3tan
2
3
C4D Important Trigonometric Ratios and Formula :
4
15
5cosor36cos&
4
1518sinor
10sin 00
sin2
2sin2cos......2cos.2cos.cos
n
n1n2
Practice Problems :
1. If sin A + sin B = a and cos A + cos B = b then the value of sin (A + B) will be
(a)22 ba
ab2
(b)
22 ba
ab2
(c)
ab2
ba 22 (d)
ab2
ba 22
2. The value of 00
00
40sin70cos
40cos70sin
equal to
(a) 1 (b)3
1(c) 3 (d)
2
1
3. If BcotAcosCcos
CsinAsin
, then A, B, C are in
(a) A.P. (b) G.P. (c) H.P. (d) none of these
4. The value of
nn
BcosAcos
BsinAsin
BsinAsin
BcosAcos
equals
(a)2
BAtan2 n
(b)2
BAcot2 n
(c) 0 (d) both (b) and (c) are correct
5. The value of sin 120 . sin 240 . sin 480 . sin 840 will be
(a)8
1(b)
16
1(c)
32
1(d)
64
1
6. The value of sin 360 . sin 720 . sin 1080 . sin 1440 will be
(a)4
1(b)
16
1(c)
4
3(d)
16
5
7. The value of 8
7cos
8
5cos
8
3cos
8cos 4444
(a)2
1(b)
4
1(c)
2
3(d)
4
3
8. The value of
8
7cos1
8
5cos1
8
3cos1
8cos1 will be
(a)2
1(b)
4
1(c)
8
1(d)
16
1
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 6
9. The value of
10
3sin
10sin equals to
(a)2
1(b)
2
1 (c)
4
1(d) 1
10. The value of 5
8cos.
5
4cos.
5
2cos.
5cos
will be
(a)16
1(b) 0 (c)
8
1 (d)
16
1
11. The value of 18
7sin
18
5sin
18sin
equal to
(a)2
1(b)
4
1(c)
8
1(d)
16
1
[Answers : (1) a (2) c (3) a (4) d (5) b (6) d (7) c (8) c (9) c (10) d (11) c]
C5 General Solutions of Trigonometrical Equations :
(i) If sin = sin then = n + (–1)n (n I)
(ii) If cos = cos then = 2n ± (n I)
(iii) If tan = tan then = n + (n I)
(iv) If sin2 = sin2
cos2 = cos2
tan2 = tan2
then = n ± (n I)
Practice Problems :
1. The most general value of satisfying the equation tan = –1 and cos = 2
1 is
(a)4
7n
(b)
4
7)1(n n
(c)4
7n2
(d) none of these
2. The number of solution of the given equation tan + sec = 3 where 0 < < 2 is
(a) 0 (b) 1 (c) 2 (d) 3
3. If 3(sec2 + tan2) = 5 then the general solution of is
(a)6
n2
(b)6
n
(c)3
n2
(d)3
n
4. If sin 3 = sin then the general value of is
(a)3
)1n2(,n2
(b)4
)1n2(,n
(c)3
)1n2(,n
(d) none of these
5. If tan m = tan n then then the consecutive value of will be in
(a) A.P. (b) G.P. (c) H.P. (d) none of these
6. The number of solutions of the equation tan x + sec x = 2 cos x lying in the interval [0, 2] is
(a) 0 (b) 1 (c) 2 (d) 3
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 7
7. The complete solution of the equation 7cos2x + sinxcosx – 3 = 0 is given by
(a) )In(2
n
(b) )In(4
n
(c) )In(3
4tann 1
(d)
3
4tank,
4
3n 1 (k, n I)
8. If sin + cos = 2 cos then the general value of is
(a)
4
n2 (b)
4
n2 (c)
4
n (d)
4
n
9. The general solution of the equation sin100x – cos100x = 1 is
(a) In,3
n2
(b) In,2
n2
(c) In,4
n
(d) In,3
n2
[Answers : (1) c (2) c (3) b (4) b (5) a (6) d (7) d (8) b (9) b]
Inverse Trigonometric Function :
C6 Principal Values & Domains of Inverse Trigonometric/Circular Functions :
Function Domain Range
(i) y = sin–1x where –1 x 12
y2
(ii) y = cos–1x where –1 x 1 0 y
(iii) y = tan–1x where x R2
y2
(iv) y = cosec–1x where x –1 or x 1 0y,2
y2
(v) y = sec–1x where x –1 or x 12
y;y0
(vi) y = cot–1x where x R 0 < y <
C7 Properties of Inverse Trigonometric Functions :
Property - A
(i) sin (sin–1x) = x, –1 x 1
(ii) cos (cos–1x) = x, –1 x 1
(iii) tan (tan–1x) = x, x R
(iv) cot (cot–1x) = x, x R
(v) sec (sec–1x) = x, x –1, x 1
(vi) cosec (cosec–1x) = x, x –1, x 1
Property - B
(i) sin–1 (sin x) = x,2
x2
(ii) cos–1 (cos x) = x; 0 x
(iii) tan–1 (tan x) = x,2
x2
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 8
(iv) cot–1 (cot x) = x; 0 x
(v) sec–1 (sec x) = x; 0 x 2
x
(vi) cosec–1 (cosec x) = x; x 0,2
x2
Property - E
(i) 1x1,2
xcosxsin 11
(ii) Rx,2
xcotxtan 11
(iii) 1|x|,2
xsecxeccos 11
Practice Problems :
1. The value of
3
2sinsin 1 is equal to
(a)3
2(b)
3
(c)
3
2 (d)
3
(d)
2. The value of )5(sinsin 1 is equal to
(a) 2 – 5 (b) 5 – 2 (c) 5 – (d) none of these
(b)
3. The value of
2
3coscos 1 is equal to
(a)2
(b) 0 (c)
2
(d)
2
3
(a)
4. The value of
7
5tantan 1 is equal to
(a)7
2(b)
7
5(c)
7
2 (d)
7
(c)
[Answers : (1) d (2) b (3) a (4) c]
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 9
INITIAL STEP EXERCISE
1.
x2
3tan.
2xcos
x2
7sinx
2
3cos.
2xtan 3
when simplified reduced to :
(a) sin x cos x (b) – sin2x
(c) – sin x cos x (d) sin2x
2. If cosA + cosB + cosC = 0 = sinA + sinB + sinC, thenthe value of cos (A – B) + cos (B – C) + cos (C – A)is :
(a) 3/2 (b) –(3/2)
(c) 3/4 (d) –(3/4)
3. The exact value of 000
000
110sin50sin20sin
35sin65sin80sin96
is equal to
(a) 12 (b) 24
(c) –12 (d) 48
4. If A + B + C = , then sin A + sin B + sin C =
(a)2
Ccos
2
Bcos
2
Acos4
(b)2
Ccos
2
Bcos
2
Asin4
(c)2
Ccos
2
Bsin
2
Acos4
(d)2
Csin
2
Bcos
2
Acos4
5. The value of x satisfying the equation
2
xsin2xsin1xsin1 are wheree
n I
(a) 2n x 2(n + 1)
(b) 2n x 2n +
(c) n x n +
(d) (2n – 1) x (2n + 1)
6. If A + B + C = , then cosA + cosB + cosC =
(a)2
Csin
2
Bsin
2
Asin41
(b)2
Ccos
2
Bsin
2
Asin1
(c)2
Csin
2
Bcos
2
Asin41
(d)2
Csin
2
Bsin
2
Acos1
7. If A and B are complimentary angles, then :
(a) 22
Btan1
2
Atan1
(b) 22
Bcot1
2
Acot1
(c) 22
Beccos1
2
Asec1
(d) 22
Btan1
2
Atan1
8. If = 3 and sin = 22 ba
a
. The value of the
expression a cosec – b sec is
(a) 22 ba
1
(b) 22 ba2
(c) a + b (d) none
9. If sin2x – sin x = 1, then the value of cos2x + cos4x is
(a) 1 (b) 2
(c) 3 (d) none of these
10. If cosA = tanB, cosB = tanC and cosC = tanA, then
(a) A = B = C
(b)2
15Asin
(c) sinC = 3 sin180
(d)4
15Bcos
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 10
FINAL STEP EXERCISE
1. If A = cos10x + sin6x, then
(a) A 1 (b) 1/2 A 3/2
(c) 0 < A 1 (d) None of these
2. Number of solutions of the simultaneous equationy = cos x and [y] = [sin y] is
(a) 2
(b) 4
(c) 6
(d) Infinitely many
3. 1cottan2cot1
cos
cossin
cossin2
33
if :
(a)
2,0 (b)
,
2
(c)
2
3, (d)
2,
2
3
4. The number of solutions of the equation1 + sin x sin2(x/2) = 0 is
(a) 0 (b) 1
(c) 2 (d) 3
5. The number of solutions of the equation2cos x = |sin x| in [–2, 2] is
(a) 1 (b) 2
(c) 4 (d) 8
6. If < tan–1x + cot–1x + cos–1x < , then possiblevalues of and are
(a) = 0, = (b) = /2, =
(c) = /2, = 3/2 (d) None of these
7. If x11 tan x = x11 , then
sin 4x =
(a) 4x (b) x
(c) 2x (d) none of these
8. , , and are the smallest positive angles inascending order of magnitude which have theirsines equal to the positive quantity k. The value of
2sin
2sin2
2sin3
2sin4
is equal to :
(a) k12 (b) k12
(c) k2 (d) 2k
9. The value of
14
7sin
14
5sin
14
3sin
14sin
14
13sin
14
11sin
14
9sin
is
(a) 1/4 (b) 1/8
(c) 1/64 (d) 1/128
11. If x = sec – tan and y = cosec + cot then :
(a)1y
1yx
(b)
x1
x1y
(c)1y
1yx
(d) xy + x + y + 1 = 0
12. The values of x satisfying the equation
1x2tanx4tan1
x2tanx4tan
aree
(a) /4 (b) n + /4
(c) 2n + /4 (d) no value
13. The set of values ‘a’ for which the equation,cos2x + a sin x = 2a – 7 possess a solution is :
(a) (–, 2) (b) [2, 6]
(c) (6, ) (d) (–, )
14. The value of
0000
2
17tan
2
167cot
2
167tan
2
17cot is
(a) a rational number
(b) 2(3 – 23)
(c) 2(3 + 3)
(d) none of these
15. If cos(sin x) = 1/2, then x may be in the interval
(a) [/4, /2] (b) [–/4, 0]
(c) [, 5/4] (d) [3/4, ]
16. The value of cos [tan–1{sin(cot–1x)}] is
(a)1x
x2
2
(b)
1x
1x2
2
(c)2x
1x2
2
(d)
1x
2x2
2
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 11
ANSWERS (INITIAL STEPEXERCISE)
1. d
2. b
3. b
4. a
5. c
6. a
7. a
8. b
9. a
10. a
11. c
12. d
13. b
14. a
15. a
16. c
ANSWERS (FINAL STEP EXERCISE)
1. c
2. c
3. b
4. a
5. d
6. d
7. b
8. b
9. c
10. c
11. d
12. d
13. b
14. b
15. b
16. b
10. If |2a sin2 + 2b sin cos + 2c cos2 – a – c | k areequality holds for some value of , then k2 =
(a) a2 + (b – c)2 (b) c2 + (a – b)2
(c) b2 + (a – c)2 (a) none of these
11. If 20 sin2 + 21 cos – 24 = 0 &
24
7 then
the values of cot 2
is :
(a) 3 (b)3
15
(c)3
15 (d) – 3
12. If + + + = , then sin2 + sin2 + sin2 +sin2 =
(a) cossinsin4
(b) sincossin4
(c) sinsincos4
(d) sinsinsin4
13. Two vertical poles 20 m and 80 m high stand aparton a horizontal plane. The height of the point ofintersection of the lines joining the top of each poleto the foot of the other is
(a) 15 m (b) 16 m
(c) 18 m (d) 50 m
14. A pole stands vertically, inside a triangular parkABC. If the angle of elevation of the top of the polefrom each corner of the park is same, then in ABC,the foot of the pole is at the
(a) centroid (b) circumcentre
(c) incentre (d) orthocentre
15. A vertical pole subtends an angle tan–1(1/2) at apoint P on the ground. The angle subtends by theupper half of the pole at the point P is
(a) tan–1(1/4) (b) tan–1(2/9)
(c) tan–1(1/8) (d) tan–1(2/3)
16. A pole of height h stands at one corner of a park inthe shape of an equilateral triangle. If is the anglewhich the pole subtends at the midpoint of theopposite side, the length of each side of the park is
(a) (3/2) h cot (b) (2/3) h cot
(c) (3/2) h tan (d) (2/3) h tan
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 12
AIEEE ANALYSIS [2002]
1. In a triangle ABC, 2 ca 2
CBAsin
is equal to
(a) a2 + b2 – c2 (b) c2 – a2 – b2
(c) c2 + a2 – b–2 (d) b2 – c2 – a2
2.
9
2tan
4
1tan 11 equal to
(a)
5
3tan
2
1 1 (b)
2
1tan 1
(c)
5
3cos
2
1 1 (d)
5
3sin
2
1 1
3. The equation
a sin x + b cos x = c, where |c| > 22 ba has
(a) infinite no. of solutions
(b) no solutions
(c) a unique solution
(d) none of these
4. If tan = 3
4 then sin is
(a)5
4notbut
5
4
(b)5
4notbut
5
4
(c)5
4or
5
4
(d) none of these
5. In a ABC, 5
2
2
Ctan,
6
5
2
Atan then
(a) a, c, b are in A.P.
(b) a, b, c are in A.P.
(c) b, a, c are in A.P.
(d) a, b, c are in G.P.
6. cos sin ( – ) + cos sin ( – ) + cos sin( – ) =
(a) 0
(b) 4 cos cos cos
(c) 1
(d) 1/2
7. If is a root of 25cos2 + 5cos – 12 = 0
2 then sin 2 is equal to
(a)25
24(b)
18
13
(c)18
13(d)
25
24
8. The value of 15tan1
15tan12
2
is
(a) 1 (b) 2
(c) 3 (d)2
3
9. If sin ( + ) = 1, sin ( – ) = 2
1then
tan ( + 2) tan (2 + ) is equal to
(a) zero (b) –1
(c) 1 (d) none of these
10. In a triangle ABC,
a = 4, b = 3, A = 600, then c is the root of theequation
(a) c2 – 3c + 7 = 0 (b) c2 – 3c – 7 = 0
(c) c2 + 3c – 7 = 0 (d) c2 + 3c + 7 = 0
11. xcostancoscot 11 then sin x =
(a)
2tan2 (b)
2cot
(c) tan (d)
2cot2
12.2
2
)yx(
xy4sin
is true if and only if
(a) x + y = 0 (b) x = y, x 0
(c) x 0, y 0 (d) x – y 0
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 13
AIEEE ANALYSIS [2003]
13. The trigonometric equation
sin–1x = 2sin–1a, has a solution for
(a) all real values of a
(b)2
1|a|
(c)2
1|a|
(d)2
1|a|
2
1
14. The upper 3/4 th portion of a vertical pole subtendsan angle tan–1(3/5) at a point in the horizontal planethrough its foot and at a distance 40 m from thefoot. A possible height of the vertical pole is
(a) 40 m (b) 60 m
(c) 80 m (d) 20 m
15. In a triangle ABC, medians AD and BE are drawn.
If AD = 4, DAB = 6
and ABE =
3
, then the
area of the ABC is
(a) 16/3 (b) 32/3
(c) 64/3 (d) 32/33
16. If in a triangle ABC
2
b3
2
Acosc
2
Ccosa 22
, then the sides a,
b and c
(a) are in G.P.
(b) are in H.P.
(c) satisfy a + b = c
(d) are in A.P.
17. The sum of the radii of inscribed and circumscribedcircles for an n sided regular polygon of side a, is
(a)
n2cot
2
a(b)
n2cota
(c)
n2cot
4
a(d)
ncota
AIEEE ANALYSIS [2004/2005]
18. Let , be such that < – < 3.If sin + sin = –21/65, and cos + cos = –27/65,
then the value of cos 2
is
(a)65
6(b)
130
3
(c)130
3 (d)
65
6
[2004]
19. The sides of a triangle are sin , cos and
cossin1 for some 0 < < 2
. The the
greatest angle of the triangle is
(a) 1200 (b) 900
(c) 600 (d) 1500
[2004]
20. A person standing on the bank of a river observesthat the angle of elevation of the top of the tree onthe opposite bank of the river is 600 and when theretires 40 meters away from the tree, the angle ofelevation becomes 300. The breadth of the river is
(a) 40 m (b) 30 m
(c) 20 m (d) 60 m
[2004]
21. In a triangle ABC, let C = 2
. If r is the inradius
and R is the circumradius of the triangle ABC, then2(r + R) equals
(a) a + b (b) b + c
(c) c + a (d) a + b + c
[2005]
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 14
22. If cos–1x – cos–1
2
y = , then 4x2 – 4xy cos + y2 is
equal to
(a) 4 (b) 2 sin 2
(c) –4sin2 (d) 4sin2
[2005]
23. If in a ABC, the altitudes from the vertices A, B,C on opposite sides are in H.P., then sin A, sin B,sin C are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) Arithmetic-Geometric Progression
[2005]
AIEEE ANALYSIS [2006]
24. If 0 < x < , and cos x + sin x = 2
1, then tan x is
(a) (1 + 7)/4 (b) (1 – 7)/4
(c) (4 – 7)/3 (d) –(4 + 7)/3
AIEEE ANALYSIS [2007]
25 If 24
5eccos
5
xsin 11
then a value of x is
(a) 4 (b) 5
(c) 1 (d) 3
26. A tower stands at the centre of a circular park. Aand B are two points on the boundary of the parksuch that AB (=a) subtends an angle of 600 at thefoot of the tower, and the angle of elevation of thetop of the tower from A or B is 300. The height ofthe tower is
(a) a/3 (b) a3
(c) 2a/3 (d) 2a3
ANSWERS AIEEE ANALYSIS
1. c 2. b 3. b 4. c 5. b 6. b 7. d
8. d 9. c 10. a 11. a 12. b 13. d 14. a
15. d 16. d 17. a 18. c 19. a 20. c 21. a
22. d 23. a 24. d 25. d 26. a
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MT – 15
TEST YOURSELF
1. If A and B be acute positive angles satisfying
3sin2A + 2sin2B = 1, 3 sin 2A – 2 sin 2B = 0
then
(a) B = /4 – A/2 (b) A = /4 – 2B
(c) B = /2 – A/4 (d) A = /4 – B/2
2. If 5
ztan
3
ytan
2
xtan and x + y + z = , then the
value of tan2x + tan2y + tan2z is
(a) 38/3 (b) 38
(c) 114 (d) 19
3. If tan A – tan B = x and cot B – cot A = y, then thevalue of cot (A – B) is
(a)xy
yx (b) 22 y
1
x
1
(c)xy
yx (d) xy
4. The value of cos–1 (cos (–17/5), is equal to
(a) –17/5 (b) 3/5
(c) 2/5 (d) none of these
5. The value of sin (cot–1(cos(tan–1x))) is
(a)1x
2x2
2
(b)
2x
1x2
2
(c)2x
x
2 (d)
2x
1
2
6.ca
)cba(btan
bc
)cba(atan 11
ab
)cba(ctan 1
is equal to
(a) /4 (b) /2
(c) (d) 0
ANSWERS
1. a
2. a
3. c
4. b
5. b
6. d
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