Trigonometric Ratio [Level] (1)

IIT – ian’s P A C E

Practice Question LEVEL– 1 Trigonometric Ration

Q.1 If 5 tan = 4, then       cos 2 sin cos 3 sin 5 = (A) 5/9 (B) 14/5 (C) 9/5 (D) 5/14 Q.2 tan2_{ sec}2_(cot2_{ – cos}2_{) =}

(A) 0 (B) – 1

(C) 1 (D) 2

Q.3 If sin  + cosec  = 2 then the value of sin8_{ + cosec}8_{ is equal}

to-(A) 2 (B) 28

(C) 24 _{(D) None of these}

Q.4 If sin x + sin2 _{x = 1, then}

cos8 _{x + 2cos}6 _{x + cos}4 _{x = ....}

(A) 0 (B) – 1 (C) 2 (D) 1 Q.5 x = y cos 3 2 = z cos 3 4 , then xy + yz + zx = (A) – 1 (B) 0 (C) 1 (D) 2

Q.6 If cos2_{3 + cos}4_{= 16 cos}6_{ + 9 cos}2_{is an}

identity

then-(A)  = 1,  = 18 (B)  = 1,  = 24 (C)  = 3,  = 24 (D)  = 4,  = 2

Q.7 sin2 _{5º + sin}2 _{10º + sin}2 _{15º + ...}

sin2 _{85º + sin}2 _90º= (A) 7 (B) 8 (C) 9 2 1 (D) 10

Q.8 cosA + sin(270º+A)– sin(270º–A)+cos(180º + A) =

(A) – 1 (B) 0

(C) 1 (D) None of these

Q.9 If sin  = – 2 1

and tan = 1, then  lies in which

quadrant-(A) First (B) Second (C) Third (D) Fourth

Q.10 If A lies in the second quadrant & 3 tan A + 4 = 0, the value 2 cot A – 5 cos A + sin A is equals to-(A) 23/11 (B) 22/10

(C) 23/10 (D) None of these Q.11 cos 24º + cos 5º + cos 175º + cos 204º + cos 300º =

(A) 1/2 (B) – 1/2 (C) 2 3 (D) 1 Q.12 tan20º + 2 tan50º =

(A) tan 70º (B) cot 70º (C) sin 70º (D) tan 30º

Q.13 If A – B = 4 

, then (1 + tan A) (1 – tan B) =

(A) 1 (B) 2

(C) –1 (D) –2 Q.14 tan 5x tan 3x tan 2x = ...

(A) tan 5x – tan 3x – tan2x

(B) x 2 cos x 3 cos x 5 cos x 2 sin x 3 sin x 5 sin     (C) 0 (D) None of these Q.15 2 sin        12 5 sin        12 = (A) – 2 1 (B) 2 1 (C) 4 1 (D) 6 1 Q.16 cos2 _{48° – sin}2 _{12° =} (A) 4 1 5 (B) 8 1 5 (C) 4 1 3 (D) 2 2 1 3

Q.17 2 sin2_{+ 4 cos()sin  sin + cos 2 ( +)=}

(A) sin 2 (B) cos 2 (C) cos 2 (D) sin 2 Q.18               9 cos 7 cos 5 cos 3 cos 9 sin 7 sin 5 sin 3 sin = (A) tan 6 (B) tan 3 (C) cot 2 (D) cot 6 Q.19 If cos A = m cos B,

then-(A) cot 2 B A = 1 m 1 m   tan 2 A B (B) tan 2 B A = 1 m 1 m   cot 2 A B (C) cot 2 B A = 1 m 1 m   tan 2 B A (D) None of these

Q.20 If cos  + cos  = 0 = sin + sin , then cos 2 + cos 2 is equal to

-(A) –2sin ( + ) (B) –2cos ( + ) (C) 2sin ( + ) (D) 2cos ( + ) Q.21 If cos2B = C) cos(A C) cos(A   , then-(A) tan A, tan B, tan C are in A.P. (B) tan A, tan B, tan C are in G.P. (C) tan A, tan B, tan C are in H.P. (D) None of these Q.22        2 cos cos 1 2 sin sin = (A) 2 1 tan  (B) 2 1 cot  (C) tan  (D) cot  Q.23 1 – 2 sin2       _ 4 =

(A) cos 2 (B) – cos 2 (C) sin 2 (D) – sin 2 Q.24    2 2cos4 2 ;  0 <  < /4 is (A) cos  (B) sin  (C) 2 cos  (D) 2 sin 

Q.25 cos2 _{A (3 – 4 cos}2 _A)2 _{+ sin}2 _{A(3 – 4 sin}2 _A)2 _is

equal

to-(A) cos 4A (B) sin 4A (C) 1 (D) None of these

Q.26 The maximum value of

sin(+ /6) + cos( + /6) is attained at

(A) 12  (B) 6  (C) 3  (D) 2 

Q.27 The maximum value of 12 sin – 9 sin2_

is-(A) 3 (B) 4

(C) 5 (D) None of these Q.28 Minimum value of 5 sin2 + 4 cos2 is

-(A) 1 (B) 2 (C) 3 (D) 4 Q.29 cos 5  cos 5 2 cos 5 4 cos 5 8 = (A) 16 1 (B) 0 (C) – 8 1 (D) – 16 1

Q.30 The value of tan 6º tan 42º tan 66º tan 78º is-(A) 1 (B) 1/2 (C) 1/4 (D) 1/8 Q.31 A quadratic equation whose roots are

cosec2_{and sec}2_{, can be}

-(A) x2 _{– 2x + 2 = 0 (B) x}2 _{– 3x + 3 = 0}

(C) x2 – 5x + 5 = 0 (D) x2 + 4x – 4 = 0 Q.32 Find the number of solution of the equation

30 |sin x| = x in 0  x  2

(A) 4 (B) 2 (C) 8 (D) 6 Q.33 Total number of solution of the equation

3x + 2 tan x = 2 5

in x  [0, 2] = (A) 1 (B) 2 (C) 3 (D) 4 Q.34 The number of solutions of

|cos x| = sin x, 0  x  4 is

-(A) 8 (B) 4

(C) 2 (D) None of these

LEVEL-2

Q.1 Which of the following is correct-(A) sin 1º > sin 1 (B) sin1º< sin 1

(C) sin 1º = sin 1 (D) sin 1º = 180  sin 1 Q.2 If cos A= 4 3

then the value of sin 2 A sin 2 A 5 is-(A) 32 1 (B) 8 11 (C) 32 11 (D) 16 11 Q.3 If      cos sin 1 sin 2 =  then       sin 1 cos sin 1 is equal to-(A)  1 (B)  (C) 1 –  (D) 1 +  Q.4 The least value of

cos2_{– 6 sin . cos + 3 sin}2_{+ 2}

is-(A) 4 + 10

(B) 4 – 10

(C) 0 (D) 4

Q.5 If sin  + sin = a and cos – cos = b, then

tan        2 is equal to-(A) – b a (B) – a b (C) 2 2 _b a  (D) None of these Q.6 If ABCD is a cyclic quadrilateral, then the value

of cos A – cos B + cos C – cos D-(A) 0

(B) 1

(C) 2 (cos B – cos D)

(D) 2 (cos A – cos C)

Q.7 If x = r sin .cos ; y = r sin .sin  and z = r cos  then the value of x2 _{+ y}2 _{+ z}2 _is

independent

of-(A) , (B) r,  (C) r,  (D) r

Q.8 The value of cos 7  + cos 7 2 + cos 7 3 + cos 7 4 + cos 7 5 + cos 7 6 + cos 7 7 is-(A) 1 (B) – 1 (C) 1/2 (D) 0 Q.9 If tan        2 and tan        2

are the roots of the equation 8x2 _{– 26x + 15 = 0 then cos ( + ) is}

equal to-(A) – 725 627 (B) 725 627 (C) – 627 725 (D) – 1 Q.10 If sin ₁ + sin ₂ + sin ₃ = 3, then

cos ₁ + cos ₂ + cos ₃ =

(A) 3 (B) 2

(C) 1 (D) 0

Q.11 If sin A, cos A and tan A are in G.P., then cos3_{A + cos}2_{A is equal}

to-(A) 1 (B) 2 (C) 4 (D) None of these Q.12 If x + x 1 = 2 cos , then x3 ₊ 3 x 1 = (A) cos3 (B) 2cos3

(C) 2 1 cos3 (D) 3 1 cos3

Q.13 Exact value of tan 200º (cot 10º – tan 10º) is-(A) 1 (B) 2 (C) 0 (D) None of these Q.14 º 80 sin º 10 sin º 50 sin º 70 sin 8 º 20 cos 2  is equal to -(A) 1 (B) 2 (C) 4 3 (D) None of these Q.15 The sign of the product sin 2 sin 3 sin 5 is

-(A) Negative (B) Positive (C) 0 (D) None of these Q.16 sin3θ 3sinθ cos3θ 3cosθ   is equal to-(A) 1 + cot2 (B) cot4 (C) cot3 (D) 2 cot 

Q.17 If 3 sin  + cos  = 2 then 3 cos  – sin  is equal to-(A) – 6 (B) 6 (C) 5 (D) – 5

Q.18 No. of solution in the equation x = 4 sinx when x  [0, 2

]-(A) 1 (B) 2

(C) 3 (D) 4

Q.19 The numerical value of

tan 3  + 2tan 3 2 + 4tan 3 4 +8tan 3 8 is equal to-(A) – 5 3 (B) 3 5  (C) 5 3 (D) 3 5 Q.20 The value of



Q.21 If (sec A – tan A) (sec B – tan B) (sec C – tan C) = (sec A + tan A)

(sec B + tan B) (sec C + tan C), then every side is equal to

-(A) ±1 (B) 1

LEVEL-3

Q.1 The value of sin 10° + sin 20° + sin 30° + ... + sin 360° is equal

to-(A) 0 (B) 1 (C) 3

(D) 2 Q.2 The value of the expression

( 3 sin 75º – cos 75º) is -(A) 2 sin 15° (B) 1 + 3 (C) 2 sin 105° (D) 2

Q.3 cos 52° + cos 68° + cos 172° =

(A) 0 (B) 1 (C) 2 (D) None of these Q.4 If cosec A + cot A = 2 11 , then tan A is -(A) 22 21 (B) 16 15 (C) 117 44 (D) 43 117 Q.5 If triangle ABC, C = 3 2

, then the value of cos2_{A + cos}2_{B – cos A . cos B is equal}

to-(A) 4 3 (B) 2 3 (C) 2 1 (D) 4 1 Q.6 If f() = 5 cos  + 3 cos       _ 3 + 3, then range of f() is-(A) [–5, 11] (B) [–3, 9] (C) [–2, 10] (D) [–4, 10] Q.7 5 4 cos 5 3 cos2 _ 2  is equal to -(A) 4/5 (B) 5/2 (C) 5/4 (D) 3/4

Q.8 If 1 and 2 are two values lying in [0, 2] for

which tan  = , then

2 tan 2 tan1 _. 2 is -(A) Zero (B) –1 (C) 2 (D) 1 Q.9 If A+B +C = , then  2 B tan 2 A tan is equal to -(A) 0 (B) –1 (C) 1/2 (D) 1

Q.10 If a = sin 170° + cos 170°, then -(A) a > 0 (B) a < 0 (C) a = 0 (D) a = 1

Q.11 sin2_{A + sin}2_{(A – B) + 2 sinA cosB sin(B – A) is}

equal to

-(A) sin2_{A (B) sin}2_B

(C) cos2_{A (D) cos}2_B

Q.12 If U_n = 2 cos n, then U₁U_n – U_n–1 is equal to -(A) U_n+2 (B) U_n+1

(C) U₂U_n+1 (D) None of these Q.13 The number of real solutions of the equation

sin (ex_{) = 2}x _{+ 2}–x _is

-(A) 1 (B) 0 (C) 2 (D) Infinite Q.14 If cos 5 = a cos5 _{ + b cos}3 _{ + c cos  then c is}

equal to -(A) –5 (B) 1 (C) 5 (D) None of these Q.15 If f (x) = x sin x 3 sin

, x  n, then the range of values of f(x) for real values of x is

-(A) [–1, 3] (B) (–, –1] (C) (3, +) (D) [–1, 3)

Q.16 If A = tan 6° tan 42° and B = cot 66° cot 78°, then

-(A) A = 2B (B) A = 3 1

B (C) A = B (D) 3A = 2B

Q.17 If sin  = n sin ( + 2), then tan ( + ) is equal to -(A)    _tan n 2 n 1 (B)    _tan n 1 n 1 (C) tan  (D)    _tan n 1 n 1

Statement type Questions

Each of the questions given below consist of Statement -I and Statement- II. Use the following key to choose the appropriate answer.

(A) If both Statement-I Statement-II are true, and Statement-II is the correct explanation of Statement-I.

(B) If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement-I

(C) If Statement-I is true but Statement-II is false (D) If Statement-I is false but Statement-II is true. Q.18 Statement- I : cos 36º > tan 36º

Statement- II : cos 36º > sin 36º

Q.19 Statement- I : If A, B, C are the angles of a triangle such that angle A is obtuse, then tan B tan C > 1.

Statement- II : In any triangle,

tan A = 1 C tan B tan C tan B tan   .

Q.20 Statement-I : The number of roots of the

equation sin px = x2 _{– x +} 4 5 is 2. Statement-II : In [0, 2], sin x = 2 1 has exactly two solutions.

Q.21 Statement-I : sin1 cos2 tan 3 having positive sign.

Statement-II : 1C _{= 57°17'45''}

Q.22 Statement-I : log_ tan

      _  4 3 = 0 Statement-II : tan       _ 4 = tan 4 

Q.23 Statement-I : sec2_{. cosec}2_

= sec2_{ + cosec}2_

Statement-II : 1 + tan2_{ = sec}2_{ &}

1 + cot2_{ = cosec}2_

Passage based Questions

Passage-1

Measurement of an angle by three ways. First is degree and others are grade and radian. The

relation between them 90 D = 100 G =  C 2 Q.24 45º is equal to -(A) 50g _{(B) 45}g (C) 40g _{(D) 39}g Q.25 4 23c is equal to -(A) 675º (B) 1080º (C) 745º (D) 1035º Q.26 200g _{is equal to} -(A) 180º and 2 3c (B) 180º and c (C) 200º and c _{(D) None} Passage-2

Increasing product series

=   sin 2 2 sin n n if  n = 1 if  = 2 k = –1 if  = (2k + 1) k where kI (Integer)

Q.27 The value of cos 14 2 . cos 14 4 . cos 14 8 is -(A) – 8 1 (B) 8 1 (C) 16 1 (D) – 16 1

Q.28 The value of 8 sin 48  .cos 48  .cos 24  .cos 12  is equal to -(A) 2 1 (B) – 2 1 (C) 4 1 (D) 8 1

ANSWER KEY

LEVEL- 1

LEVEL- 2