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Trigonometric Ratios & Identities — JEE Main & Advanced Mathematics PYQs

189 previous year questions from Trigonometric Ratios & Identities with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

189 questionsMathematicsSolutions on every page
1

Let = (1 - 2 ( 11 ) ) (1 - 2 ( 3 11 ) ) (1 - 2 ( 9 11 ) ) (1 - 2 ( 27 11 ) ) (1 - 2 ( 81 11 ) ) . Then the value of 5 - ^2 is _____________.

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2

If A = 3^ 9^ + 9^ 27^ + 27^ 81^ and B = 81^ - 3^ , then B A is equal to _____.

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3

Let P = [0, 4 ] : ^2 1 and S = a Z : 2( ^8 - ^8 ) 2 = a^2, P . Then n(S) is:

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4

If ( 18 ) ( 5 18 ) ( 7 18 ) = K , then the value of ( 10K 3 ) is :

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5

If ( A - B ) A + ² C ² ~A =1, ~A , ~B , C (0, 2 ) , then

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6

If x= 5 12 for some x ( , 3 2 ) , then 7 x ( 13 x 2 + 13 x 2 )+ 7 x ( 13 x 2 - 13 x 2 ) is equal to

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7

The value of 3 cosec 20^ - 20^ 20^ 40^ 60^ 80^ is equal to

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8

The least value of ( ² -6 +3 ² +2 ) is

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9

Let 2 ( 15 2 )( 8 + 8 )+ ( 15 2 )( 8 - 8 ) is equal to

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10

Let and respectively be the maximum and the minimum values of the function f( )=4 ( ⁴ ( 7 2 - )+ ⁴(11 + ) )-2 ( ⁶ ( 3 2 - )+ ⁶(9 - ) ), R . Then +2 is equal to :

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11

Let ( + )=- 1 10 and ( - )= 3 8 , where 0< < 3 and 0< < 4 . If 2 = 3(1-r 5 ) 11 (s+ 5 ) , r, s N , then r+s is equal to _ _ _ _ .

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12

If ² 48^ - ² 12^ ² 24^ - ² 6^ = + 5 2 , where , N , then + is equal to _ _ _ _ _

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13

The value of cosec 10^ - 3 10^ is equal to :

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14

Let = 1 60^ 61^ + 1 62^ 63^ + + 1 118^ 119^ Then the value of ( cosec 1^ )^2 is __________

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15

If x+ ^2 x=1, x (0, 2 ) , then ( ¹² x+ ¹² x )+3 ( ¹⁰ x+ ¹⁰ x+ ^8 x+ ^8 x )+ ( ^6 x+ ^6 x ) is equal to :

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16

If _ r=1 ¹³ 1 ( 4 +(r-1) 6 ) ( 4 + r 6 ) =a 3 +b, a, b Z , then a^2+b^2 is equal to :

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17

The value of ( 70^ ) ( 10^ 70^ -1 ) is

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18

If 2 x 3 4 , then ⁻¹ ( 12 13 x+ 5 13 x ) is equal to

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19

Let | (60- ) (60+ )| 1 8 , [0,2 ] . Then, the sum of all [0,2 ] , where 3 attains its maximum value, is :

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20

If the value of 3 36^ +5 18^ 5 36^ -3 18^ is a 5 -b c , where a, b, c are natural numbers and gcd (a, c)=1 , then a+b+c is equal to :

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21

If x=- 3 5 , where < x < 3 2 , then 80 ( ^2 x- x ) is equal to

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22

If tan A = 1 x x 2 + x + 1 , tan B = x x 2 + x + 1 and tan C = x − 3 + x − 2 + x − 1 1 2 , 0 < A , B , C < π 2 , then A + B is equal to:

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23

For α , β ∈ 0 , π 2 let 3 sin ( α + β ) = 2 sin ( α - β ) and a real number k be such that tan α = tan β . Then the value of k is equal to

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24

96 cos &#960; 33 cos 2 &#960; 33 cos 4 &#960; 33 cos 8 &#960; 33 cos 16 &#960; 33 is equal to

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25

The value of 36 4 cos 2 9 &#8728; - 1 4 cos 2 27 &#8728; - 1 4 cos 2 81 &#8728; - 1 4 cos 2 243 &#8728; - 1 is

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26

The value of tan 9 o - tan 27 o - tan 63 o + tan 81 o is _____.

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27

If tan 15 &#176; + 1 tan 75 &#176; + 1 tan 105 &#176; + tan 195 &#176; = 2 a , then the value of a + 1 a is :

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28

Let &#945; and &#946; be real numbers such that - &#960; 4 &#60; &#946; &#60; 0 &#60; &#945; &#60; &#960; 4 . If sin &#945; + &#946; = 1 3 and cos &#945; - &#946; = 2 3 , then the

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29

Let &#945; = &#8721; k = 1 &#8734; sin 2 k &#960; 6 . Let g : 0 , 1 &#8594; &#8477; be the function defined by g x = 2 a x + 2 a 1 - x . Then, which of the following statements is/

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30

Let S = &#952; &#8712; 0 , &#960; 2 : &#8721; m = 1 9 sec &#952; + m - 1 &#960; 6 sec &#952; + m &#960; 6 = - 8 3 . Then

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31

The value of 2 sin &#960; 22 sin 3 &#960; 22 sin 5 &#960; 22 sin 7 &#960; 22 sin 9 &#960; 22 is equal to:

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32

If cot &#945; = 1 and sec &#946; = - 5 3 , where &#960; &#60; &#945; &#60; 3 &#960; 2 and &#960; 2 &#60; &#946; &#60; &#960; , then the value of tan &#945; + &#946; and the quadran

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33

&#945; = sin 36 &#176; is a root of which of the following equation

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34

The value of cos 2 &#960; 7 + cos 4 &#960; 7 + cos 6 &#960; 7 is equal to

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35

16 sin 20 &#176; sin 40 &#176; sin 80 &#176; is equal to

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36

If sin 2 10 &#176; sin 20 &#176; sin 40 &#176; sin 50 &#176; sin 70 &#176; = &#945; - 1 16 sin 10 &#176; , then 16 + &#945; - 1 is equal to _____.

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37

The value of 2 sin 12 &#176; - sin 72 &#176; is

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38

cosec 18 &#176; is a root of the equation:

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39

A 10 inches long pencil A B with mid point C and a small eraser P are placed on the horizontal top of a table such that P C = 5 inches and &#8736; P C B = tan - 1 ( 2 ) . The acute

JEE Main 2021 Solution
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40

The value of 2 sin &#960; 8 sin 2 &#960; 8 sin 3 &#960; 8 sin 5 &#960; 8 sin 6 &#960; 8 sin 7 &#960; 8 is :

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41

If sin &#952; + cos &#952; = 1 2 , then 16 sin 2 &#952; + cos 4 &#952; + sin 6 &#952; is equal to:

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42

The value of cot &#960; 24 is:

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43

If 15 sin 4 &#945; + 10 cos 4 &#945; = 6 , for some &#945; &#8712; R , then the value of 27 sec 6 &#945; + 8 cosec 6 &#945; is equal to :

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44

The number of integral values of k for which the equation 3 sin x + 4 cos x = k + 1 has a solution, k &#8712; R is _______.

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45

If L = sin 2 &#960; 16 - sin 2 &#960; 8 and M = cos 2 &#960; 16 - sin 2 &#960; 8

JEE Main 2020 Solution
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46

If the equation cos 4 &#952; + sin 4 &#952; + &#955; = 0 has real solutions for &#952; then &#955; lies in interval

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47

The value of cos 3 &#960; 8 . cos 3 &#960; 8 + sin 3 &#960; 8 . sin 3 &#960; 8 is:

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48

If 2 s i n &#945; 1 + c o s 2 &#945; = 1 7 and 1 - c o s 2 &#946; 2 = 1 10 , &#945; , &#8201; &#946; &#8712; 0 , &#960; 2 , then t a n &#945; + 2 &#946; , is equal to

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49

If A , B , C are in arithmetic progression and B = π 4 , then tan ⁡ A tan ⁡ B tan ⁡ C =

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50

Let y = sin - 1 sin 8 - tan - 1 tan 1 0 + cos - 1 cos 1 2 - sec - 1 sec 9 + cot - 1 cot 6 - cosec - 1 cosec 7 . If y simplifies to a π + b then a - b is

NTA Abhyas JEE Main 2020 Solution
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51

If the value of 1 + tan 1 o 1 + tan 2 o 1 + tan 3 o … … … . ( 1 + tan 44 o ) ( 1 + tan 45 o ) is 2 λ , then the sum of the digits of the number λ is

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52

If a sin 2 ⁡ x + b cos 2 ⁡ x = c , b sin 2 ⁡ y + a cos 2 ⁡ y = d and a tan ⁡ x = b tan ⁡ y , then a 2 b 2 is equal to (Here a , b , c and d are distinct)

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53

If p = sin 2 ⁡ x + cos 4 ⁡ x , then

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54

The value of 2 sin 2 ⁡ θ + 4 cos ⁡ θ + α sin ⁡ α sin ⁡ θ + cos ⁡ 2 ( α + θ ) is

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55

The value of the expression 2 ( s i n 1 ° + s i n 2 ° + s i n 3 ° + . . . . . . . . + s i n 89 ° ) 2 ( c o s 1 ° + c o s 2 ° + c o s 3 ° + . . . . . . . . + c o s 44 ° ) + 1 is equ

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56

If cos α + cos β = a , sin α + sin β = b and α - β = 2 θ , then cos 3 θ cos θ is equal to

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57

The value of 1 + c o s π 8 1 + c o s 3 π 8 1 + c o s 5 π 8 1 + c o s 7 π 8 is equal to

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58

Let the maximum and minimum value of the expression 2 cos 2 ⁡ θ + cos ⁡ θ + 1 is M and m respectively, then the value of M m is, (where [.] is the greatest integer function)

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59

t a n 6 2 0 o - 33 t a n 4 2 0 o + 27 t a n 2 2 0 o + 4 =

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60

If π < θ < 3 π 2 and cos ⁡ θ = - 3 5 , then t a n θ 4 is equal to

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61

Let f n = sin ⁡ 1 × sin ⁡ 2 × . . . . . s i n n , ∀ n ∈ N , then the number of elements in the set A = f 1 , f 2 , . . . . . . . . . . , f 6 that are positive are

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62

The difference between the greatest and the least possible value of the expression 3 - cos ⁡ x + sin 2 ⁡ x is

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63

If E = cos 2 ⁡ 71 o + cos 2 ⁡ 49 o + cos ⁡ 71 o cos ⁡ 49 o , then the value of 10 E is equal to

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64

Let f x = m a x tan ⁡ x , cot ⁡ x . Then the number of roots of the equation f x = 1 2 in 0,2 π is

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65

The value of the expression 1 + c o s e c π 4 + c o s e c π 8 + c o s e c π 16 is equal to

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66

If 2 sin ⁡ α 1 + cos ⁡ α + sin ⁡ α = 3 4 , then the value of 1 - cos ⁡ α + sin ⁡ α 1 + sin ⁡ α is equal to

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67

If sin &#8289; A and cos &#8289; A are the roots of the equation 4 x 2 - 3 x + a = 0 , sin &#8289; A + cos &#8289; A + tan &#8289; A + cot &#8289; A + sec &#8289; A + cosec A = 7 a

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68

If 4 sin ⁡ 27 o = α - β , then the value of α + β is

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69

If cos &#8289; 5 &#952; = 5 c o s &#952; &#8289; - 20 cos 3 &#8289; &#952; + a cos 5 &#8289; &#952; + b , then the value of a + b is equal to

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70

The maximum value of the expression sin ⁡ θ c o s 2 θ ∀ θ ∈ 0 , π is

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71

The value of tan ⁡ 63 o - cot ⁡ 63 o is equal to

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72

If s i n 23 π 24 = 2 p - q - 1 4 r , then the value of p 2 + q 2 - r 2 is equal to

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73

If 4 cos &#8289; 36 o + c o t 7 1 2 &#8728; = n 1 + n 2 + n 3 + n 4 + n 5 + n 6 , then the value of &#8721; i = 1 6 n i 2 10 is equal to

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74

If &#960; 2 &#60; &#945; &#60; 3 &#960; 4 , then 2 tan &#8289; &#945; + 1 c o s 2 &#945; is equal to

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75

If 3 + cot &#8289; 80 o cot &#8289; 20 o cot &#8289; 80 o + cot &#8289; 20 o = t a n &#960; k , then the value of k is (where, &#960; k is an acute angle)

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76

If 0 < α < π 16 and ( 1 + tan ⁡ α ) 1 + tan ⁡ 4 α = 2 , then the value of α is equal to

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77

The value of cot ⁡ 70 o + 4 cos ⁡ 70 o is

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78

If sin 6 ⁡ θ + cos 6 ⁡ θ + k cos 2 ⁡ 2 θ = 1, then k is equal to

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79

The angle between the minute hand and the hour hand of a clock at “half past 3 ” is equal to

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80

In a triangle A B C , if sin &#8289; A cos &#8289; B = 1 4 and 3 tan &#8289; A = tan &#8289; B , then cot 2 &#8289; A is equal to

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81

The value of cos ⁡ π 11 cos ⁡ 2 π 11 cos ⁡ 3 π 11 … … cos ⁡ 11 π 11 is equal to

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82

Three positive acute angles &#945; , &#160; &#946; and &#947; satisfy the relation tan &#8289; &#946; 2 = 1 3 cot &#8289; &#945; 2 and cot &#8289; &#947; 2 = 1 2 3 tan &#8289; &#94

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83

If sin ⁡ 2 β is the geometric mean between sin ⁡ α and cos ⁡ α , then cos ⁡ 4 β is equal to

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84

If tan ⁡ θ = 3 tan ⁡ ϕ , then the maximum value of tan 2 ⁡ θ - ϕ is (where, tan ⁡ ϕ > 0 )

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85

The value of sin ⁡ 9 π 14 sin ⁡ 11 π 14 sin ⁡ 13 π 14 is equal to

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86

If 0 &#60; &#945; &#60; &#960; 3 , then &#945; sec &#8289; &#945; is

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87

If tan 25 o = a , then the value of tan ⁡ 205 o - tan ⁡ 115 o tan ⁡ 245 o + tan ⁡ 335 o in terms of a is

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88

The expression sin ⁡ 27 ° cos ⁡ 57 ° sin ⁡ 87 ° simplifies to

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89

If 1 + cos ⁡ 2 x sin ⁡ 2 x + 3 1 + tan ⁡ x tan ⁡ x 2 sin ⁡ x = 4 , then the value of tan ⁡ x can be equal to

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90

Let f &#952; = 1 1 + tan &#8289; &#952; 2021 , then the value of &#8721; &#952; = 1 &#176; 89 &#176; f &#952; is equal to

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91

If x + y = 3 - cos ⁡ 4 θ and x - y = 4 sin ⁡ 2 θ , then the value of x + y is equal to

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92

Let A = x 4 + 4 x 3 + 2 x 2 - 4 x + 7 where x = cot &#8289; 11 &#960; 8 and B = 1 - cos &#8289; 8 &#952; tan 2 &#8289; 4 &#952; + 1 + cos &#8289; 8 &#952; cot 2 &#8289; 4 &#952; wh

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93

The arithmetic mean of the numbers 2 sin ⁡ 2 o , 4 sin ⁡ 4 o , 6 sin ⁡ 6 o , … … . 178 sin ⁡ 178 o , 180 sin ⁡ 180 o is

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94

If sin ⁡ θ = cos 2 ⁡ θ , then the value of cos 12 ⁡ θ + 3 cos 10 ⁡ θ + 3 cos 8 ⁡ θ + cos 6 ⁡ θ - 1 is equal to

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95

If cos ⁡ 2 x + 2 cos ⁡ x = 1 , then sin 2 ⁡ x 2 - cos 2 ⁡ x is equal to

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96

If F k = 1 + s i n &#960; 2 k 1 + sin &#8289; k - 1 &#960; 2 k 1 + sin &#8289; 2 k + 1 &#960; 2 k 1 + sin &#8289; 3 k - 1 &#960; 2 k , then the value of F 1 + F 2 + F 3 is equal to

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97

Let f x = sin ⁡ x + 3 sin ⁡ 3 x + 5 sin ⁡ 5 x + 3 sin ⁡ 7 x sin ⁡ 2 x + 2 sin ⁡ 4 x + 3 sin ⁡ 6 x , wherever defined. If x 1 + x 2 = π 2 , where f x is defined at x 1 and x 2 , the

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98

For non-negative integers n , let f n = &#8721; k = 0 n sin &#8289; k + 1 n + 2 &#960; sin &#8289; k + 2 n + 2 &#960; &#8721; k = 0 n sin 2 &#8289; k + 1 n + 2 &#960; Assuming cos

JEE Advanced 2019 Solution
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99

The value of sec - 1 &#8289; 1 4 &#8721; k = 0 10 sec &#8289; 7 &#960; 12 + k &#960; 2 sec &#8289; 7 &#960; 12 + k + 1 &#960; 2 in the interval - &#960; 4 , 3 &#960; 4 equals

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100

The equation y = s i n x sin &#8289; x + 2 - sin 2 &#8289; ( x + 1 ) represents a straight line lying in:

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101

The value of s i n 10 &#176; s i n 30 &#176; s i n 50 &#176; s i n 70 &#176; is:

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102

The value of cos 2 10 &#176; &#8211; cos 10 &#176; cos 50 &#176; + c o s 2 50 &#176; is

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103

If c o s &#945; + &#946; = 3 5 , sin &#8289; ( &#945; - &#946; ) = 5 13 and 0 &#60; &#945; , &#946; &#60; &#960; 4 , then tan &#8289; 2 &#945; is equal to:

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104

The maximum value of 3 cos &#8289; &#952; + 5 sin &#8289; &#952; - &#960; 6 for any real value of &#952; is :

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105

Let f_ k (x)= 1 k ( ^ k x+ ^ k x ) for k =1,2,3, Then for all x R , the value of f₄(x)-f₆(x) is equal to :

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106

The value of cos &#8289; &#960; 2 2 &#8901; cos &#8289; &#960; 2 3 &#8901; &#8230; &#8901; cos &#8289; &#960; 2 10 &#8901; sin &#8289; &#960; 2 10 is:

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107

For any &#952; &#8712; &#960; 4 , &#960; 2 , the expression 3 sin &#952; - cos &#952; 4 + 6 sin &#952; + cos &#952; 2 + 4 s i n 6 &#952; equals:

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108

Let a , b , c be three non-zero real numbers such that the equation 3 a cos &#8289; x + 2 b sin &#8289; x = c , x &#8712; - &#960; 2 , &#960; 2 has two distinct real roots &#945; a

JEE Advanced 2018 Solution
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109

The lengths of two adjacent sides of a cyclic quadrilateral are 2 units and 5 units and the angle between them is 60 o . If the area of the quadrilateral is 4 3 sq. units, then the

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110

If 5 tan 2 &#8289; x - cos 2 &#8289; x = 2 cos &#8289; 2 x + 9 , then the value of cos &#8289; 4 x is

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111

If A &#62; 0 , B &#62; 0 and A + B = &#960; 6 , then the minimum positive value of tan &#8289; A + tan &#8289; B is :

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112

If m and M are the minimum and the maximum values of 4 + 1 2 sin 2 &#8289; 2 x - 2 cos 4 &#8289; x , x &#8712; R , then M - m is equal to:

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113

The number of x &#8712; 0 , 2 &#960; for which 2 sin 4 x + 18 cos 2 x - 2 cos 4 x + 18 sin 2 x = 1 is:

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114

In a &#916; A B C , a b = 2 + 3 , and &#8736; C = 60 &#176; . Then the ordered pair ( &#8736; A , &#8736; B ) is equal to:

JEE Main 2015 Solution
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115

If cosec &#952; = p + q p - q p &#8800; q, p &#8800; 0 , then cot &#960; 4 + &#952; 2 is equals to:

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116

Let f k x = 1 k sin k x + cos k x where x &#8712; R and k &#8805;1 . Then f 4 x - f 6 x equals

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117

The expression tanA 1 - cotA + cotA 1 - tanA can be written as :

JEE Main 2013 Solution
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118

Let f:(-1,1) I R be such that f( 4 )= 2 2- ² for (0, 4 ) ( 4 , 2 ) . Then the value (s) of f ( 1 3 ) is (are)

JEE Advanced 2012 Solution
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119

The value of 255^ + 195^ is

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120

Suppose and ( 0) are such that ( + ) , sec and ( - ) are in A.P. If =k ( 2 ) for some k , then k is equal to

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