Title Function - A & R.pdf ✅

Q.31 Assertion : The total number of function ' f ' from the set {1, 2, 3} into the set {1, 2, 3, 4, 5} such that f(i)  f(j)  i < j, is equal to 35. Reason : The number of dearrangement of 'n' objects is n!         − − + − + + n! ( 1) ... 3! 1 2! 1 1! 1 1 n . Sol. [B] Let '' is associated with 'r', r  {1, 2, 3, 4, 5}, then ' 2 ' can be associated with r, r + 1, ... 5 Let '2' is associated with j ˆ then 3 can be associated with j ˆ , j ˆ + 1, ...5 Then required number of functions = = 5 r 1          − = 5 j 1 (6 j) = = − − 5 r 1 2 (6 r)(7 r) =          − + = 5 r 1 2 (42 13r r ) 2 1 =       − + 6 5.6.11 2 6.5 42.(5) 13. 2 1 = 35  (A) is true and (R) is true but not the correct explanation of (A). Q.32 Assertion : f(x) = n 1 tan x 1 tan x − + is an odd function. Reason : Let f be a function with domain D such that (i) x  D  – x  D (ii) f(–x) = – f(x) and (iii) f(0) = 0 or f(0) is not defined Then the function f(x) is an odd function Sol. [A] Reason is true (definition) Assertion is true and correct explained by reason. Q.33 Assertion : y = f(x) = x 2x 5 x 2x 4 2 2 − + − + , x  R Range       , 1 4 3 Reason : (x – 1)2 = 1 y 4y 3 − − [A] Q.34 Assertion: The domain of the function f(x) = sin–1 x + cos–1 x + tan–1 x is [–1, 1]. Reason: sin–1 x and cos–1 x is defined | x |  1 and tan–1 x is defined for all x. Sol. [A]  sin–1 x is defined in [–1, 1] cos–1 x is defined in [–1, 1] and tan–1 x is defined in R Hence f(x) is defined in [–1, 1] Q.35 Assertion : Function f(x) = sin x + {x} is periodic with period 2. Reason : sin x and {x} are both periodic functions with period 2 and 1 respectively. [D] Q.36 Assertion: If f(x) & g(x) both are one-one, then f (g (x)) is also one-one. Reason: – 2 If, f(x1) = f (x2)  x1 = x2 then f(x) is one-one [A] Q.37 Assertion: Let f: [0, 3] → [1, 13] is defined by f(x) = x2 + x + 1 then inverse is f –1 (x) = 2 −1+ 4x − 3 Reason: Many-one function is not invertible [B] Questions Add (24–6-09) Q.38 Assertion : f : N → R ; f(x) = sin x is a one- one function. Reason : The period of sin x is 2 and 2 is an irrational number. Sol. [A] Q.39 Assertion: If f(x) is an even function defined in 'R' then it's range will be identical in both cases if it is defined R→ R or it is defined [0, ] → R. Reason: Graph of an even function is symmetrical about y axis.