Title 26.DEFINITE INTEGRALS.pdf ✅

26. DEFINITE INTEGRALS-MCQS TYPE (1.) 0 3x 2x x 6 dx e 6e 11e 6   + + + [13 APRIL 2023 (MORNING)] (a.) e 32 log 27       (b.) e 256 log 81       (c.) e 512 log 81       (d.) e 64 log 27       (2.) If f R R : → be a continuous function satisfying ( ) ( ) 2 4 0 0 f x xdx f x xdx sin2 sin cos2 cos 0    +  =  , then the value of  is [11 APRIL 2023 (AFTERNOON)] (a.) − 3 (b.) 3 (c.) − 2 (d.) 2 (3.) Let the function f : 0, 2 R   → be defined as [11 APRIL 2023 (AFTERNOON)] ( )     )    ) 2 min , log , 0,1 , 1,2 e x x x x x e x f x e x − −    =    where t denotes the greatest integer less than or equal to t . Then the value of the integral ( ) 2 0  xf x dx is [11 APRIL 2023 (AFTERNOON)] (a.) ( ) 2 1 e 1 e 2   − +     (b.) 3e 1 2 + (c.) 1 2e 2 − (d.) 2e 1− (4.) The value of the integral ( ( )) log 2 x x 2x log 2 e log e 1 e dx e e e −  + + is equal to : [11 APRIL 2023 (MORNING)] (a.) 2 e (2 5) 5 log 2 1 5   +   +     + (b.) 2 e 2(2 5) 5 log 2 1 5   +   −     + (c.) 2 2(3 5) 5 log 2 1 5 e   −   +     + (d.) 2 e 2(2 5) 5 log 2 1 5   +   −     + (5.) Let f be a differentiable function such that ( ) ( ) ( ) 2 0 2 4 , 1 3 x x f x x tf t dt f − =  = . Then 18 3 f ( ) is equal to : [10 APRIL 2023 (MORNING)] (a.) 180 (b.) 150 (c.) 210 (d.) 160 (6.) Let ( ) 1 1 5f x 4f 3, x 0 x x   + = +      . Then ( ) 2 1 18 f x dx  is equal to : [06 APRIL 2023 (MORNING)]
(a.) 10log 2 6 e − (b.) 10log 2 6 e + (c.) 5log 2 3 e − (d.) e 5log 2 3 + (7.) Let a differentiable function f satisfy ( ) ( ) 3 1, 3 x f t f x dt x x t +  = +  . Then 12 8 f ( ) is equal to: [31 Jan 2023(Morning)] (a.) 34 (b.) 19 (c.) 17 (d.) 1 (8.) The value of ( ) ( ) 2 3 2 3sin sin 1 cos x dx x x   +  + is equal to [31 Jan 2023(Morning)] (a.) e 7 3 log 3 2 − − (b.) e − + + 2 3 3 log 3 (c.) e 10 3 log 3 3 − + (d.) e 10 3 log 3 3 − − (9.) 2 2 2 3 1 2 1 lim 4 2 2 3 n n n n n →             + + + + ++ −                 is equal to [30 Jan 2023(Evening)] (a.) 12 (b.) 19 3 (c.) 0 (d.) 19 Official Ans. by NTA (4) (10.) For  , R , suppose the system of linear equations x −+= y z 5 2x 2y z 8 + + =  3x y 4z − + =  has infinitely many solutions. Then  and  are the roots of [30 Jan 2023(Evening)] (a.) 2 x 10x 16 0 − + = (b.) 2 x x + + = 18 56 0 (c.) 2 x 18x 56 0 − + = (d.) 2 x x + + = 14 24 0 Official Ans. by NTA (3) (11.) The value of the integral 4 4 4 2 cos2 x dx x    − +  − is :[01 Feb 2023(Evening)] (a.) 2 6  (b.) 2 12 3  (c.) 2 3 3  (d.) 2 6 3  (12.) 0 cotx dx    , where [.] denotes the greatest integer function, is equal to [AIEEE-2009]
(a.) 1 (b.) -1 (c.) 2  − (d.) 2  (13.) Let p x( ) be a function defined on R such that ( ) ( ) ( ) ( ) 3 lim 1 1 x f x p x p x f x → =  = −  , for all x 0,1, p (0 1 ) = and p(1 41 ) = . Then ( ) 1 0  p x dx equals [AIEEE-2010] (a.) 41 (b.) 21 (c.) 41 (d.) 42 (14.) Let [.] denote the greatest integer function, then the value of 1,5 2 0  x x dx     is [AIEEE- 2011] (a.) 3 4 (b.) 5 4 (c.) 0 (d.) 3 2 (15.) If ( ) 0 cos4 x g x tdt =  , then g x( + ) equals [AIEEE-2012] (a.) g x g ( ) + ( ) (b.) g x g ( ) − ( ) (c.) g x g ( ) ( ) (d.) ( ) ( ) g x g  (16.) At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by 100 12 dP x dx = − . If the firm employs 25 more workers, then the new level of production of items is [JEE (Main)-2013] (a.) 2500 (b.) 3000 (c.) 3500 (d.) 45000 (17.) Statement - I: The value of the integral 3 6 1 tan dx x    + is equal to 6  Statement - II : ( ) ( ) b b a a  =  + − f x dx f a b x dx . [JEE (Main)-2013] (a.) Statement - I is true; Statement - II is true; Statement - II is a correct explanation for Statement -1 . (b.) Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for Statement - 1 . (c.) Statement - I is true; Statement - II is false. (d.) Statement - I is false; Statement - II is true. (18.) The intercepts on x -axis made by tangents to the curve, 0 , x y t dt x R =   , which are parallel to the line y x = 2 , are equal to [JEE (Main)-2013] (a.) \pm 1 (b.) \pm 2 (c.) \pm 3 (d.) \pm 4 (19.) The integral 2 0 1 4sin 4sin 2 2 x xdx   + − equals [JEE (Main)-2014]
(a.) 4 3 4 − (b.) 4 3 4 3  − − (c.)  −4 (d.) 2 4 4 3 3  − − (20.) The integral ( ) 2 4 2 2 2 log log log 36 12 x dx x x x  + − + is equal to [JEE (Main)-2015] (a.) 2 (b.) 4 (c.) 1 (d.) 6 (21.) The integral 3 4 4 1 cos dx x    + is equal to [JEE (Main)-2017] (a.) 2 (b.) 4 (c.) -1 (d.) -2 (22.) The value of 2 2 2 sin 1 2x x dx   −  + is [JEE (Main)-2018] (a.) 8  (b.) 2  (c.) 4 (d.) 4  (23.) The value of 3 0 |cos | x dx   is [JEE (Main)-2019] (a.) 0 (b.) 2 3 (c.) 4 3 − (d.) 4 3 (24.) If / 3 0 tan 1 1 ,( 0) 2 sec 2 d k k      = −  , then the value of k is [JEE (Main)-2019] (a.) 4 (b.) 2 (c.) 1 (d.) 1 2 (25.) Let ( ) 4 2 2 b a I x x dx =  − . If I is minimum then the ordered pair (a b, ) is [JEE (Main)-2019] (a.) (− 2,0) (b.) (0, 2 ) (c.) ( 2, 2 − ) (d.) (− 2, 2 ) (26.) The value of     2 2 sin 4 dx x x  −  + + , where t denotes the greatest integer less than or equal to t , is [JEE (Main)-2019] (a.) ( ) 1 7 5 12  − (b.) ( ) 3 4 3 10  − (c.) ( ) 3 4 3 20  − (d.) ( ) 1 7 5 12  + (27.) If ( ) ( ) 2 1 2 0 x x  = +  f t dt x t f t dt , then f (1/ 2) is [JEE (Main)-2019] (a.) 6 25 (b.) 24 25 (c.) 4 5 (d.) 18 25