Title 3.TRIGONOMETRIC FUNCTIONS.pdf ✅

3. TRIGONOMETRIC FUNCTIONS (1.) Let A and B denote the statements : A:cos cos cos 0    + + = B : sin sin sin 0    + + = If ( ) ( ) ( ) 3 cos cos cos 2       − + − + − = − , then [AIEEE-2009] (a.) A is false and B is true (b.) Both A and B are true (c.) Both A and B are false (d.) A is true and B is false (2.) Let ( ) 4 cos 5   + = and let ( ) 5 sin 13   − = , where 0 , 4      . Then tan2 = [AIEEE-2010] (a.) 25 16 (b.) 56 33 (c.) 19 12 (d.) 20 7 (3.) For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is [AIEEE-2010] (a.) There is a regular polygon with 1 2 = r R (b.) There is a regular polygon with 1 2 = r R (c.) There is a regular polygon with 2 3 = r R (d.) There is a regular polygon with 3 2 = r R (4.) The possible values of   (0, ) such that sin sin 4 sin 7 0 ( ) + + = ( ) ( ) are [AIEEE-2011] (a.) 2 2 3 8 , , , , , 9 4 2 3 4 9       (b.) 2 4 3 8 , , , , , 9 4 9 2 4 9       (c.) 5 2 3 8 , , , , , 4 12 2 3 4 9       (d.) 2 2 3 35 , , , , , 9 4 2 3 4 36       (5.) In a PQR , if 3sin 4cos 6 P Q + = and 4sin 3cos 1 Q P + = , then the angle R is equal to [AIEEE-2012] (a.) 6  (b.) 4  (c.) 3 4  (d.) 5 6  (6.) ABCD is a trapezium such that AB and CD are parallel and BC CD ⊥ . If   ADB BC p = = , and CD q = , then AB is equal to [JEE (Main)-2013] (a.) ( ) 2 2 sin cos sin    + + p q p q (b.) 2 2cos cos sin    + + p q p q
(c.) 2 2 2 2 cos sin   + + p q p q (d.) ( ) 2 2 2 sin ( cos sin )    + + p q p q (7.) The expression tan cot 1 cot 1 tan + − − A A A A can be written as [JEE (Main)-2013] (a.) sin cos 1 A A+ (b.) sec cosec 1 A A+ (c.) tan cot A A + (d.) sec cosec A A + (8.) Let ( ) ( ) 1 = + sin cos k k k f x x x k where x R  and k 1 . Then f x f x 4 6 ( ) − ( ) equals [JEE (Main)-2014] (a.) 1 4 (b.) 1 12 (c.) 1 6 (d.) 1 3 (9.) A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45 . It flies off horizontally straight away from the point O . After one second, the elevation of the bird from O is reduced to 30 . Then the speed (in m / s ) of the bird is [JEE (Main)-2014] (a.) 20 2 (b.) 20 3 1 ( − ) (c.) 40 2 1 ( − ) (d.) 40 3 2 ( − ) (10.) If the angles of elevation of the top of a tower from three collinear points AB, and C , on a line leading to the foot of the tower, are 30 , 45 and 60 respectively, then the ratio, AB BC : , is [JEE (Main)-2015] (a.) 3 :1 (b.) 3 : 2 (c.) 1: 3 (d.) 2:3 (11.) If 0 2  x  , then the number of real values of x , which satisfy the equation cos cos2 cos3 cos4 0 x x x x + + + = , is [JEE (Main)-2016] (a.) 5 (b.) 7 (c.) 9 (d.) 3 (12.) A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30 . After walking for 10 minutes from A in the same direction, at a point B , he observes that the angle of elevation of the top of the pillar is 60 . Then the time taken (in minutes) by him, from B to reach the pillar, is [JEE (Main)-2016] (a.) 10 (b.) 20 (c.) 5 (d.) 6 (13.) If ( ) 2 2 5 tan cos 2cos2 9 x x x − = + , then the value of cos4x is [JEE (Main)-2017] (a.) 1 3 (b.) 2 9 (c.) 7 9 − (d.) 3 5 −
(14.) Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP AB = 2 . If   BPC = then tan is [JEE (Main)-2017] (a.) 1 4 (b.) 2 9 (c.) 4 9 (d.) 6 7 (15.) If sum of all the solutions of the equation 1 8cos cos cos 1 6 6 2          +  − − =             x x x in 0,  is k , then k is equal to : [JEE (Main)-2018] (a.) 2 3 (b.) 13 9 (c.) 8 9 (d.) 20 9 (16.) PQR is a triangular park with PQ PR = = 200 m . A T.V. tower stands at the mid- point of QR . If the angles of elevation of the top of the tower at PQ, and R are respectively 45 ,30 and 30 , then the height of the tower (in m ) is [JEE (Main)- 2018] (a.) 100 (b.) 50 (c.) 100 3 (d.) 50 2 (17.) For any , 4 2          , the expression 3(sin − 4 2 6 cos ) 6(sin cos ) 4sin     + + + equals [JEE (Main)-2019] (a.) 2 2 2 13 4cos 6sin cos − +    (b.) 2 4 13 4cos 6cos − +   (c.) 6 13 4cos −  (d.) 4 2 2 13 4cos 2sin cos − +    (18.) If 0 2   x , then the number of values of x for which sin sin2 sin3 0 x x x − + = , is [JEE (Main)-2019] (a.) 2 (b.) 3 (c.) 1 (d.) 4 (19.) Consider a triangular plot ABC with sides AB = 7 m , BC = 5 m and CA= 6 m . A vertical lamp-post at the midpoint D of AC subtends an angle 30 at B . The height (in m ) of the lamp-post is [JEE (Main)-2019] (a.) 2 21 (b.) 7 3 (c.) 2 21 3 (d.) 3 21 2 (20.) The sum of all values of 0, 2         satisfying 2 4 3 sin 2 cos 2 4   + = is [JEE (Main)- 2019] (a.) 5 4  (b.) 2  (c.) 3 8  (d.) 
(21.) If 2 5,5 ,5 r r are the lengths of the sides of a triangle, then r cannot be equal to [JEE (Main)-2019] (a.) 3 2 (b.) 7 4 (c.) 3 4 (d.) 5 4 (22.) With the usual notation, in ABC , if  A+  B a = = + 120 , 3 1 and b = − 3 1 , then the ratio   A B : , is [JEE (Main)-2019] (a.) 7:1 (b.) 3:1 (c.) 9:7 (d.) 5:3 (23.) The value of 2 3 10 10 cos cos cos sin 2 2 2 2        is [JEE (Main)-2019] (a.) 1 512 (b.) 1 256 (c.) 1 2 (d.) 1 1024 (24.) Let ( ) ( ) 1 = + sin cos k k k f x x x k for k =  1,2,3, . Then for all x R  , the value of f x f x 4 6 ( ) − ( ) is equal to [JEE (Main)-2019] (a.) 1 12 − (b.) 1 12 (c.) 5 12 (d.) 1 4 (25.) Given 11 12 13 + + + = = b c c a a b for ABC with usual notation. If cos cos cos    = = A B C , then the ordered triplet (   , , ) has a value [JEE (Main)-2019] (a.) (3, 4,5) (b.) (7,19,25) (c.) (19,7,25) (d.) (5,12,13) (26.) The maximum value of 3cos 5sin 6      + −     for any real value of  is [JEE (Main)-2019] (a.) 34 (b.) 19 (c.) 79 2 (d.) 31 (27.) If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30 and the angle of depression of reflection of the cloud in the lake from P be 60 , then the height of the cloud (in meters) from the surface of the lake is [JEE (Main)- 2019] (a.) 45 (b.) 50 (c.) 42 (d.) 60 (28.) If 4 4 sin 4cos 2 4 2sin cos ; , [0       + + =  , ] , then cos cos (    + − − ) ( ) is equal to [JEE (Main)-2019] (a.) 2 (b.) − 2 (c.) -1 (d.) 0