Title 20.APPLICATION OF DERIVATIVES.pdf ✅

20. APPLICATION OF DERIVATIVES-MCQS TYPE (1.) Given ( ) 4 3 2 P x x ax bx cx d = + + + + such that x = 0 is the only real root of P x ( ) = 0 . If P P (−  1 1 ) ( ) , then in the interval −1,1 [AIEEE-2009] (a.) P(−1) is not minimum but P(1) is the maximum of P (b.) P(−1) is minimum but P(1) is not the maximum of P (c.) Neither P(−1) is the minimum nor P(1) is the maximum of P (d.) P(−1) is the minimum and P(1) is the maximum of P (2.) The shortest distance between the line y x − =1 and the curve 2 x y = is [AIEEE- 2009] (a.) 2 3 8 (b.) 3 2 5 (c.) 3 4 (d.) 3 2 8 (3.) The equation of the tangent to the curve 2 4 y x x = + , that is parallel to the x -axis, is [AIEEE-2010] (a.) y = 0 (b.) y =1 (c.) y = 2 (d.) y = 3 (4.) Let f R R : → be defined by ( ) 2 , if 1 2 3, if 1 k x x f x x x  −  − =   +  − If f has a local minimum at x =−1 , then a possible value of k is [AIEEE-2010] (a.) 1 (b.) 0 (c.) 1 2 − (d.) -1 (5.) The curve that passes through the point (2,3) and has the property that the segment of any tangent to it lying between the coordinate axes bisected by the point of contact is given by [AIEEE-2011] (a.) 2 2 x y + =13 (b.) 2 2 2 2 3     x y     + =     (c.) 2 3 0 y x − = (d.) 6 y x = (6.) Let f be a function defined by ( ) tan , 0 1, 0 x x f x x x    =    = Statement-1: x = 0 is point of minima of f . Statement-2: f (0 0 ) = . [AIEEE-2011] (a.) Statement-1 is true, statement- 2 is false.
(b.) Statement-1 is false, statement-2, is true. (c.) Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1 (d.) Statement-1 is true, statement-2 is true; statement-2 is 0 a correct explanation for statement-1 (7.) A spherical balloon is filled with 4500 cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72 cubic meters per minute, then the rate (in meters per unit) at which the radius of the balloon decreases 49 minutes after the leakage began is [AIEEE-2012] (a.) 7/9 (b.) 2 / 9 (c.) 9 / 2 (d.) 9 / 7 (8.) Let a b R ,  be such that the function f given be ( ) 2 f x x bx ax x = + +  ln , 0 has extreme values at x =−1 and x = 2 . Statement-1: f has local maximum at x =−1 and at x = 2 . Statement-2: 1 2 a = and 1 4 b − = . [AIEEE-2012] (a.) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1. (b.) Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1. (c.) Statement-1 is true, statement-2 is false. (d.) Statement-1 is false, statement-2 is true. (9.) If x =−1 and x = 2 are extreme points of ( ) 2 f x x x x = + +   log then [JEE (Main)-2014] (a.) 1 2, 2   = = − (b.) 1 2, 2   = = (c.) 1 6, 2   = − = (d.) 1 6, 2   = − = − (10.) The normal to the curve, 2 2 x xy y + − = 2 3 0 at (1,1) [JEE (Main)-2015] (a.) Does not meet the curve again (b.) Meets the curve again in the second quadrant (c.) Meets the curve again in the third quadrant (d.) Meets the curve again in the fourth quadrant (11.) Let f x( ) be a polynomial of degree four having extreme values at x =1 and x = 2 . If ( ) 0 2 lim 1 3 x f x x →     + =   , then f (2) is equal to [JEE (Main)-2015] (a.) -8 (b.) -4 (c.) 0 (d.) 4
(12.) A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then [JEE (Main)-2016] (a.) (4 − =   ) x r (b.) x r = 2 (c.) 2x r = (d.) 2 4 x r = + ( ) (13.) The normal to the curve y x x x ( − − = + 2 3 6 )( ) at the point where the curve intersects the y -axis passes through the point [JEE (Main)-2017] (a.) 1 1 , 2 2       (b.) 1 1 , 2 3     −   (c.) 1 1 , 2 3       (d.) 1 1 , 2 2     − −   (14.) Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m ) of the flower-bed, is [JEE (Main)-2017] (a.) 10 (b.) 25 (c.) 30 (d.) 12.5 (15.) If the curves 2 2 2 y x x by = + = 6 ,9 16 intersect each other at right angles, then the value of b is [JEE (Main)-2018] (a.) 6 (b.) 7 2 (c.) 4 (d.) 9 2 (16.) Let ( ) 2 2 1 f x x x = + and ( )   1 g x x x R , 1,0,1 x = −  − − . If ( ) ( ) ( ) f x h x g x = , then the local minimum value of h x( ) is [JEE (Main)-2018] (a.) 3 (b.) -3 (c.) −2 2 (d.) 2 2 (17.) If  denotes the acute angle between the curves, 2 y x = − 10 and 2 y x = +2 at a point of their intersection, then tan is equal to [JEE (Main)-2019] (a.) 8 15 (b.) 7 17 (c.) 8 17 (d.) 4 9 (18.) The maximum volume (in cu. m ) of the right circular cone having slant height 3 m is [JEE (Main)-2019] (a.) 4 3  (b.) 2 3 (c.) 3 3 (d.) 6 (19.) The shortest distance between the point 3 ,0 2       and the curve y x x =  ,( 0) , is [JEE (Main)-2019] (a.) 3 2 (b.) 5 4 (c.) 3 2 (d.) 5 2
(20.) The tangent to the curve 2 x y xe = passing through the point (I e, ) also passes through the point [JEE (Main)-2019] (a.) (2,3e) (b.) 4 ,2 3        (c.) (3,6e) (d.) 5 ,2 3        (21.) A helicopter is flying along the curve given by ( ) 3/2 y x x − =  7, 0 . A soldier positioned at the point 1 ,7 2       wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is [JEE (Main)-2019] (a.) 1 7 6 3 (b.) 5 6 (c.) 1 2 (d.) 1 7 3 3 (22.) The maximum value of the function ( ) 3 f x x = − 3 2 18 27 40 x x + − on the set  2 S x R x =  +  : 30 11 }x is [JEE (Main)-2019] (a.) 122 (b.) -122 (c.) 222 (d.) -222 (23.) Let x y, be positive real numbers and mn, positive integers. The maximum value of the expression ( )( ) 2 2 1 1 m n m n x y + + x y is [JEE (Main)-2019] (a.) 1 2 (b.) 6 m n mn + (c.) 1 (d.) 1 4 (24.) Let ( ) ( ) 2 2 2 2 , ( ) x d x f x x R a x b d x − = −  + + − where a , b and d are non-zero real constants. Then [JEE (Main)-2019] (a.) f is an increasing function of x (b.) f is a decreasing function of x (c.) f is neither increasing nor decreasing function of x (d.) f  is not a continuous function of x (25.) The tangent to the curve 2 y x x = − + 5 5 , parallel to the line 2 4 1 y x = + , also passes through the point [JEE (Main)-2019] (a.) 1 7 , 4 2       (b.) 1 , 7 8     −   (c.) 7 1 , 2 4       (d.) 1 ,7 8     −   (26.) If a curve passes through the point (1, 2− ) and has slope of the tangent at any point ( x y, ) on it as 2 x y 2 x − , then the curve also passes through the point [JEE (Main)- 2019]