Title 19.CONTINUTY AND DIFFERENTIABILITY.pdf ✅

19. CONTINUTY AND DIFFERENTIABILITY (1.) Let A be a 2 2  matrix Statement-1 : adj adjA A ( ) = Statement-2 : adjA = A [AIEEE-2009] (a.) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 (b.) Statement-1 is true, Statement-2 is false (c.) Statement- 1 is false, Statement- 2 is true (d.) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 (2.) Let abc , , be such that b a c ( + ) 0 . If 2 1 1 1 1 1 1 1 1 1 1 1 0, 1 1 ( 1) ( 1) ( 1) + + + − + + − − + − + − − + = − + − − − n n n a a a a b c b b b a b c c c c a b c then the value of n is [AIEEE-2009] (a.) Any even integer (b.) Any odd integer (c.) Any integer (d.) Zero (3.) Consider the system of linear equations: 1 2 3 1 2 3 1 2 3 2 3 2 3 3 3 5 2 1 + + = + + = + + = x x x x x x x x x The system has [AIEEE-2010] (a.) Infinite number of solutions (b.) Exactly 3 solutions (c.) A unique solutions (d.) Nosolution (4.) If the trivial solution is the only solution of the system of equations 0 3 0 3 0 − + = + − = + − = x ky z kx y kz x y z then the set of all values of k is [AIEEE-2011] (a.) R − − 3 (b.) 2, 3−  (c.) R − − 2, 3 (d.) R −2 (5.) Let P and Q be 3 3 matrices with P Q . If 3 3 P Q= and 2 2 P Q Q P = , then determinant of ( ) 2 2 P Q+ is equal to [AIEEE-2012] (a.) 1 (b.) 0 (c.) -1 (d.) -2 (6.) The number of values of k , for which the system of equations
( ) ( ) 1 8 4 3 3 1 + + = + + = − k x y k kx k y k has no solution, is [JEE (Main)-2013] (a.) Infinite (b.) 1 (c.) 2 (d.) 3 (7.) If 1 3 1 3 3 2 4 4      =       P is the adjoint of a 3 3 matrix A and A = 4 , then  is equal to [JEE (Main)-2013] (a.) 4 (b.) 11 (c.) 5 (d.) 0 (8.) If  , 0  , and ( ) = +   n n f n and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 3 1 1 1 2 1 1 1 2 1 3 (1 ) (1 ) ( ) 1 2 1 3 1 4     + + + + + = − − − + + + f f f f f K f f f , then K is equal to [JEE (Main)-2014] (a.) 1 (b.) -1 (c.)  (d.) 1  (9.) The set of all values of  for which the system of linear equations 1 2 3 1 1 2 3 2 1 2 3 2 2 2 3 2 2    − + = − + = − + = x x x x x x x x x x x has a non-trivial solution [JEE (Main)-2015] (a.) Is an empty set (b.) Is a singleton (c.) Contains two elements (d.) Contains more than two elements (10.) The system of linear equations 0 0 0    + − = − − = + − = x y z x y z x y z has a non-trivial solution for [JEE (Main)-2016] (a.) Exactly one value of  (b.) Exactly two values of  (c.) Exactly three values of  (d.) Infinitely many values of  (11.) Let  be a complex number such that 2 1 + = z where z = −3 . If 2 2 2 7 1 1 1 1 1 3 1     − − = k , then k is equal to (a.) z (b.) -1 (c.) 1 (d.) −z

(17.) Let d R  , and ( ) ( ) ( ) ( ) 2 4 sin 2 1 sin 2 , 5 2sin sin 2 2       − + −   = +     − − + +   d A d d d   0, 2  . If the minimum value of det( A) is 8 , then a value of d is [JEE (Main)-2019] (a.) -5 (b.) 2 2 1 ( + ) (c.) -7 (d.) 2 2 2 ( + ) (18.) If the system of equations 5 2 3 9 3   + + = + + = + + = x y z x y z x y z has infinitely many solutions, then  − equals [JEE (Main)-2019] (a.) 18 (b.) 21 (c.) 8 (d.) 5 (19.) The number of values of   (0, ) for which the system of linear equations ( ) ( ) 3 7 0 4 7 0 sin3 cos2 2 0   + + = − + + = + + = x y z x y z x y z has a non-trivial solution, is [JEE (Main)-2019] (a.) Four (b.) One (c.) Three (d.) Two (20.) If the system of linear equations 2 2 3 3 5 3 2 + + = − + = − + = x y z a x y z b x y z c where abc , , are non-zero real numbers, has more than one solution, then [JEE (Main)-2019] (a.) b c a − + = 0 (b.) b c a + − = 0 (c.) abc + + = 0 (d.) b c a − − = 0 (21.) If ( ) 2 2 2 2 ( 2 2 − − − − = + + − − a b c a a b b c a b a b c x c c c a b 2 + + +  a b c x ) , 0 and abc + +  0 , then x is equal to [JEE (Main)-2019] (a.) 2(abc + + ) (b.) − + + (abc) (c.) abc (d.) − + + 2(abc) (22.) An ordered pair ( , ) for which the system of linear equations