TRIGONOMETRIC FUNCTIONS AND …einsteinclasses.com/Trigonometry.pdfMT – 6 Einstein Classes, Unit...

MT – 1

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road

New Delhi – 110 018, Ph. : 9312629035, 8527112111

TRIGONOMETRIC FUNCTIONS AND TRIGONOMETRIC EQUATIONS

C1 Trigonometric Functions :

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}

Circular Definition of Trigonometric Functions :

OP

OMcos,

OP

PMsin , 0cos,

cos

sintan

, 0sin,

sin

coscot

, 0cos,

cos

1sec

0sin,sin

1eccos

C2 Graphs of Trigonometric functions :

(a) y = sin x, x R; y [–1, 1]

(b) y = cos x, x R; y [–1, 1]

(c) y = tan x, x R – (2n + 1) /2, n I; y R

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(d) y = cot x, x R – n, n I; y R

(e) y = cosec x, x R – n, n I; y (–, –1] [1, )

(f) y = sec x, x R – (2n + 1)/2, n I, y (–, –1] [1, )

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

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

3. If cosec A + cot A = 2

11 then tan A equals

(a)22

21(b)

16

15(c)

117

44(d)

43

117

[Answers : (1) a (2) a (3) c]

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C3 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

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]

C4 Trigonometric Functions of Sum and 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(

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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

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]

C5 Factorisation of the Sum and Difference of Two Sines and Cosines :

(a)2

DCcos

2

DCsin2DsinCsin

(b)2

DCsin

2

DCcos2DsinCsin

(c)2

DCcos

2

DCcos2DcosCcos

(d)2

DCsin

2

DCsin2DcosCcos

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C6 Transformation of Products into Sum and 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)

C7 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

(g)Atan31

AtanAtan3A3tan

2

3

C8 Important Trigonometric Ratios :

(a) sin n = 0; cos n = (–1)n; tan n = 0, where n I

(b)12

5cosor75cos

22

13

12sinor15sin 00

12

5sinor75sin

22

13

12cosor15cos 00

0000 15cot3213

1375tan;75cot32

13

1315tan

(c)4

15

5cosor36cos&

4

1518sinor

10sin 00

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

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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

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

12. Prove the following statements :

(a) cos4A – sin4A + 1 = 2 cos2A (b) (sinA + cosA) (1 – sinA . cosA) = sin3A + cos3A

(c) ecAcos2Asin

Acos1

Acos1

Asin

(d) cos6A + sin6A = 1 – 3sin2A . cos2A

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(e) AtanAsecAsin1

Asin1

(f) Acos

AtanAcot

ecAcos

(g) (1 + cotA – cosecA) (1 + tanA + secA) = 2

(h)Asin

1

AcotecAcos

1

AcotecAcos

1

Asin

1

(i) Btan.AcotAtanBcot

BtanAcot

(j)

2222 sineccos

1

cossec

1

22

2222

sin.cos2

sin.cos1sin.cos

[Answers : (1) a (2) c (3) a (4) d (5) b (6) d (7) c (8) c (9) c (10) d (11) c]

C9 Range of Trigonometric Expression :

E = a sin + b cos

22 baE sin ( + ), where a

btan

b

atanwhere),cos(ba 22

Hence for any real value of 2222 baEba,

Practice Problems :

1. Find the value of x for which following expression will have maximum value :

(a) 3 cos x + sin x (b) cos x + sin x

2. Find the range of following trigonometric expression

(a) 3 cos x + sin x (b) cos x + sin x

(c) 3sin x + 4 cos x

C10 Sine and Cosine Series :

2

1nsin

2sin

2

nsin

1nsin....)2sin()sin(sin

2

1ncos

2sin

2

nsin

1ncos....)2cos()cos(cos

C11 Trigonometric Equations : Equation involving trigonometric functions of a variable are calledtrigonometric equations. The trigonometric equation may have infinite number of a solutions and can beclassified as :

(i) Principal solution : The solution of a trigonometric equations for which 0 x < 2 are called principalsolution.

(ii) General solution

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Important Points :

1. sin = 0 = n

2. cos = 0 = (2n + 1) 2

3. tan = 0 = n

4. sin = sin = n + (–1)n , where

2,

2

5. cos = cos = 2n ± , where [0, ]

6. tan = tan = n + , where

2,

2

7. sin2 = sin2, cos2 = cos2, tan2 = tan2 = n ±

8. sin = 1 = (4n + 1) 2

9. cos = 1 = 2n

10. cos = –1 = (2n + 1)

11. sin = sin and cos = cos = 2n + 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)

6n

(c)

3n2

(d)

3n

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

7. The complete solution of the equation 7cos2x + sinxcosx – 3 = 0 is given by

(a) )In(2

n (b) )In(

4n

(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)

4n2 (b)

4n2 (c)

4n (d)

4n

9. The general solution of the equation sin100x – cos100x = 1 is

(a) In,3

n2 (b) In,

2n2

(c) In,4

n (d) In,

3n2

[Answers : (1) c (2) c (3) b (4) b (5) a (6) d (7) d (8) b (9) b]

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ADDITIONAL PRACTICE PROBLEMS

1. Prove that

03

4sin

3

2sinsin

2. If A + B + C = , then prove that

(i) sin2A + sin2B + sin2C = 4 sinA sinB sinC

(ii) sinA + sinB + sinC = 4 cos 2

A cos

2

B cos

2

C

(iii) cos 2A + cos 2B + cos 2C = – 1 – 4 cos Acos B cos C

(iv) cos A + cos B + cos C = 1 + 4 sin 2

A sin

2

B sin

2

C

(v) tan A + tan B + tan C = tanA tanB tanC

(vi)

12

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

(vii)

2

Ccot.

2

Bcot.

2

Acot

2

Ccot

2

Bcot

2

Acot

(viii) cot A cot B + cot B cot C + cot C cot A = 1

3. If A + B + C = 2

then prove that tan A tan B + tan

B tan C + tan C tan A = 1

4. If A + B = 450 show that (1 + tanA) (1 + tanB) = 2

5. Prove that :

(a)

tan

5cos7cos

5sin7sin

(b) A5sec.A4cosA2sinA8sin

AsinA7sin

(c)A2sinA4sin

A4cosA2cos

AsinA3sin

AcosA3cos

A3cos.A2cos

Asin

(d))A2B4cos()B2A4cos(

)A2B4sin()B2A4sin(

)BAtan(

(e)A7cosA5cosA3cosAcos

A7sinA5sinA3sinAsin

A4tan

(f)A5sin

A3sin

A7sinA5sin2A3sin

A5sinA3sin2Asin

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SINGLE CORRECT CHOICE TYPE

1. The value of sin 200 . sin 400 . sin 600 . sin 800 isequal to

(a)8

3(b)

16

3

(c)32

3(d)

64

3

2. The value of tan 200 . tan 400 . tan 600 . tan 800 isequal to

(a) 3 (b)16

3

(c)32

3(d)

64

3

3. If BcotAcosCcos

CsinAsin

, then A, B, C are in

(a) A.P. (b) G.P.

(c) H.P. (d) none of these

4. If

3

4cos2

3

2cosycosx then the

value of z

1

y

1

x

1 is equal to

(a) 1 (b) 2

(c) 0 (d) 3cos

5. If sin x + sin2x = 1 then the value of expression

cos12x + 3cos10x + 3cos8x + cos6x – 1 is equal to

(a) 0 (b) 1

(c) –1 (d) 2

6. For 2

0 , if

0n

n2cosx ,

0n

n2siny ,

0n

n2n2 sin.cosz then

(a) xyz = xz + y

(b) xyz = xy + z

(c) xyz = x + y + z

(d) both (b) and (c) are correct

7. If A + B + C = (A, B, C > 0) and the angle C isobtuse then

(a) tan A . tan B > 1

(b) tan A . tan B < 1

(c) tan A . tan B = 1

(d) none of these

8. If A, B, C are acute positive angles such thatA + B + C = and cot A . cot B . cot C = k then

(a)33

1k (b)

33

1k

(c)9

1k (d)

3

1k

9. 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

10. If sin 5x + sin 3x + sin x = 0 then the value of x other

than 0 lying between 0 x 2

is

(a)6

(b)

12

(c)3

(d)

4

11. If sin + cos = 2 cos then the general value of is

(a)

4n2 (b)

4n2

(c)

4n (d)

4n

12. cos 2x + a sin x = 2a – 7 posses a solution for

(a) all a (b) a > 6

(c) a < 2 (d) a [2, 6]

13. The equation a sin x + b cos x = c where

|c| > 22 ba has

(a) one solution

(b) two solutions

(c) no solution

(d) infinite solutions

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14. The general solution of the equationsin100x – cos100x = 1 is

(a) In,3

n2 (b) In,

2n2

(c) In,4

n (d) In,

3n2

15. The solution of the equation cos103x – sin103x = 1 are

(a)2

(b) 0

(c)2

(d) both (a) and (b) are correct

16. The number of solutions of the equation2cosx = |sin x| in [–2, 2] is

(a) 1 (b) 2

(c) 3 (d) 4

17. The general solution of the equation

xsinxcos 22

2.312 is

(a) n (b) n +

(c) n – (d) none of these

18. The least positive nonintegral solution ofsin (x2 + x) – sinx2 = 0

(a) rational

(b) irrational of the form p

(c) irrational of the form 4

1p , where p is

an odd integer

(d) irrational of the form 4

1p , where p is

an even integer

19. If 4sin2x – 8sin x + 3 0, 0 x 2, then thesolution set for x is

(a)

6,0 (b)

6

5,0

(c)

2,

6

5(d)

6

5,

6

20. The number of values of x [0, 4] satisfying

|3cos x – sin x| 2 is

(a) 2 (b) 0

(c) 4 (d) 8

ANSWERS

(SINGLE CORRECT CHOICE TYPE)

1. b

2. a

3. a

4. c

5. a

6. d

7. b

8. a

9. d

10. c

11. b

12. d

13. c

14. b

15. d

16. d

17. a

18. c

19. d

20. c

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EXERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

Comprehension-1

Consider the cubical equation qx3 – px2 – x – p = 0.This equation has roots tan , tan and tan .

1. + + equals to

(a) /3 (b) /2

(c) 2/3 (d)

2. The value of sin . sin . sin is

(a) independent of p

(b) independent of q

(c) indepedent of both p and q

(d) none

3. The value of cos . cos . cos is

(a) independent of p

(b) independent of q

(c) indepedent of both p and q

(d) none

MATRIX-MATCH TYPE

Matching-1

Column - A Column - B

(A)

9

1r

2

18

rsin (p) 3/2

(B)

10

1r

3

3

rcos (q) 5

(C)

4

1r

4

8

)1r2(cos (r) –9/8

(D)

4

1r

4

8

)1r2(sin (s) –6/5

Matching-2


(A) The number of solutions (p) 0

of the equation

8tan2 + 9 = 6 sec

in the interval (–/2, /2)

is

(B) The number of solutions (q) 2

of the equation

tan x + sec x = 2 cos x

lying in the interval [0, 2] is

(C) The number of values (r) 3

x [0, 4] satysfying

|3cos x – sin x| 2

(D) The number of real (s) 4

solutions of (x, y) where

|y| = sin x, y = cos–1(cos x),

–2 x 2, is

Matching-3


(A) cos4x – (a + 2) cos2x – (p) any irrational

(a + 3) = 0 has real number

solution if a is

(B) ax2

cosxcos

(q) –3

has real solution if a is

(C) 1 + sin2ax = cos x has (r) –1

exactly one solution if

a is

(D) (sin x+3 cos x) sin 4x = a (s) 2

has no solution if a is

Matching-4

Consider a triangle ABC


(A) The maximum value of (p) 3/2

cosA + cosB + cosC

(B) The maximum value of (q) 33/8

sinA sinB sinC

(C) The minimum value of (r) 3/4

cos2A + cos2B + cos2C

(D) The maximum value of (s) 1/8

(1 – cos A )(1 – cos B)(1 – cos C)

Matching-5

Consider an acute angle triangle ABC


(A) The minimum value of (p) 1

2

Ctan

2

Btan

2

Atan 222

(B) The minimum value of (q) 33

tan A . tan B. tan C

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(C) The minimum value of (r) 9

tan2A + tan2B + tan2C

(D) The maximum value of (s) 3/2

cos2A + cos2B + cos2C

MULTIPLE CORRECT CHOICE TYPE

1. If )sincos()cossin( , then

(a)22

1

4cos

(b)4

32sin

(c)4

12sin

(d)2

1

4cos

2. Let A, B, C be three angles of a triangles such that

4A and tan B . tan C = p. The possible value of

p is

(a) –1 (b) –2

(c) 8 (d) 10

3. If f(, ) =

2

4

2

4

sin

sin

cos

cos then

(a) f(, ) = 1 (b) f(, ) = 1

(c) f(, ) 1 (b) f(, ) 1

4. Let 2

tan)(fn

. (1 + sec ) (1 + sec 2)

(1 + sec 4) .... (1 + sec2n). Then

(a) 116

f2

(b)

nn

02

)(flim

(c) 164

f4

(d)

nn

2f = 0

5.

n

1m

m3 ax2sin.xcos sin mx is an identity in x.

Then

(a) 0a,8

3a 23 (b)

2

1a,6n 1

(c)4

1a,5n 1 (d)

4

3am

6. If [–2, 2] and 2

sin2

cos

=

2(cos 360 – sin 180) then the value of is

(a)6

7(b)

6

(c)6

5 (d)

6

7. Let [x] = the greatest integer less than or equal to x.The equation sin x = [1 + sin x] + [1 – cos x] has

(a) no solution in

2,

2

(b) no solution in

,

2

(c) solution in

2

3,

(d) no solution for x R

8. If 2

1cossin and

3

2sincos then

(a) sin ( + ) = 12

5

(b) sin ( + ) = 12

5

(c) 7342

tan

(d) 7342

tan

9. If + and – are solutions of for theequation tan2 – 4tan = –1 where

20and

20

. Then

(a)4

(b)

3

(c)6

(d)

4

10. The possible real values of x which satisfy

xtan

6xtan

(a) 2 (b) 1

(c) 4 (d) 6

MT – 14


New Delhi – 110 018, Ph. : 9312629035, 8527112111

(Answers) EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE1. d 2. b 3. aMATRIX-MATCH TYPE1. [A-q; B-r; C-p; D-p] 2. [A-p; B-q; C-s; D-r] 3. [A-q; B-r, s; C-p; D-q]4. [A-p; B-q; C-r; D-s] 5. [A-p; B-q; C-r; D-s]MULTIPLE CORRECT CHOICE TYPE1. a, b 2. a, b, c, d 3. a, b 4. a, b, d 5. a, c, d 6. a, d7. a, b, d 8. a, c, d 9. a, c 10. c, d 11. a, d 12. a, bASSERTION-REASON TYPE1. C 2. A 3. A 4. A 5. B 6. A7. B 8. B 9. B 10. B

11. Consider the equation sec + cosec = c. Then

(a) equation has two roots between 0 and 2if c2 < 8

(b) equation has four roots between 0 and 2if c2 < 8

(c) equation has two roots between 0 and 2if c2 > 8

(d) equation has four roots between 0 and 2if c2 > 8

12. If A = {/2cos2 + sin 2} and

2

3

2/B . Then A B is

(a)2

3 (b)

6

5

2

(c)6

5

4

(d)

2

Assertion-Reason Type

Each question contains STATEMENT-1 (Assertion)and STATEMENT-2 (Reason). Each question has4 choices (A), (B), (C) and (D) out of which ONLYONE is correct.

(A) Statement-1 is True, Statement-2 is True;Statement-2 is a correct explanationfor Statement-1

(B) Statement-1 is True, Statement-2 is True;Statement-2 is NOT a correctexplanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

1. STATEMENT-1 : The solution of this equationsin x + cos x – 22 sin x . cos x = 0 is

4

3m2

3

1and

4

1m2 where m

STATEMENT-2 : If both sides of an equation willbe squared then we will always get the correctsolution.

2. STATEMENT-1 : If 22

22

yx

yxeccos

(x y) then

has no real value if x and y are non-zero real.

STATEMENT-2 : x2 + y2 > x2 – y2

3. STATEMENT-1 : The equation sec + cosec = chas two roots between 0 and 2 if 2 < c2 < 8.

STATEMENT-2 : –1 sin 2 1

4. STATEMENT-1 : If x sin2 + y cos2 = 2

1 sin 2

and x sin – y cos = 0, n – /4, then thelocus of point (x, y) is a straight line.

STATEMENT-2 : Locus represents the relationbetween x and y.

5. STATEMENT-1 : There is no real x for

esin x – e–sin x = 4.

STATEMENT-2 : The minimum value of

esin x + e–sin x is 2.

6. STATEMENT-1 : sin(cos x) = cos(sin x) does notpossess any real solution for x.

STATEMENT-2 : The maximum value ofsin x + cos x is 2.

7. STATEMENT-1 : The equation

2cos x + cos 2kx = 3 has only one solution if k isirrational.

STATEMENT-2 : The equation

2cos x + cos 2kx = 3 has many solutions if k isrational.

8. STATEMENT-1 : The values of that satisfycos 3 + cos 2 are in AP.

STATEMENT-2 : sin2n + cos2n 1 for all andn 1.

9. STATEMENT-1 : In a triangle ABC,

a2 + b2 + c2 43

STATEMENT-2 : AM GM

10. STATEMENT-1 : The harmonic mean of theexradii of a triangle is three times the inradius.

STATEMENT-2 : AM HM

MT – 15


New Delhi – 110 018, Ph. : 9312629035, 8527112111

INITIAL STEP EXERCISE

(SUBJECTIVE)

1(a). If 2 sin . cos. sin = sin. sin( + ), then showthat tan, tan and tan are in harmonicprogression.

(b) If ba

1

b

cos

a

sin 44

then, show that

33

8

3

8

)ba(

1

b

cos

a

sin

2. If A + B + C = then prove that

(a) cos A + cos B – cos C =

– 1 + 4 cos 2

A . cos

2

B. sin

2

C

(b) sin2A + sin2B + sin2C =

2 + 2cosA . cosB . cosC

(c) sin (B + C – A) + sin (C + A – B) +sin (A + B – C) = 4 sin A . sin B . sin C.

3. If A + B + C = 2s then prove that :

(a) sin (s – A) sin (s – B) + sin(s) . sin(s – C) =

sin A . sin B

(b) 4 sin s . sin (s – A) . sin (s – B) . sin(s – C)

= 1 – cos2A – cos2B – cos2C + 2 cos A . cos B . cos C.

(c) cos2s + cos2(s – A) + cos2(s – B) + cos2(s – C) =

2 + 2 cos A . cos B . cos C

4(a). Solve for x : sec x – 1 = (2 – 1) tan x

(b) Find x (–, ) if

648 to.....|xcos|xcos|xcos|1 32

.

(c) If sin ( cot ) = cos ( tan ). Prove that either

cosec 2 or cot 2 is equal to 4

1n , where ‘n’ is a

positive or negative integer.

5. (a) If x + y + z = xyz, then show that

z1

z2

y1

y2

x1

x2222

)z1)(y1)(x1(

xyz8222

(b) If xy + yz + zx = 1, then prove that

)z1)(y1)(x1(

2

z1

z

y1

y

x1

x

222222

FINAL STEP EXERCISE

(SUBJECTIVE)

1. Prove that tan + 2 tan 2 + 22 tan 22 + .....+2n – 1

tan 2n–1 + 2n cot 2n = cot.

2. If m sin ( + ) = cos ( – ). Prove that

2m1

2

2sinm1

1

2sinm1

1

.

3. Show that 1 + sin2 + sin2 > sin + sin +sin. sin

4. If + + + = 2, prove that

(i) .2

cos4coscoscoscos

02

cos.2

cos

(ii) sin – sin + sin – sin

+ 4 cos2

. sin

2

. cos

2

)( = 0

(iii) tan + tan + tan + tan = tan . tan .tan. tan (cot + cot + cot + cot).

5. If cos (x – y), cos x, cos(x + y) are in harmonic

progression then evaluate 2

ysec.xcos .

4 (a)4

n2,n2 where n (b)

3

2,

3

ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)

5. ± 2

ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)

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