maths doubts:miscellaneous

[Mauli]

q1)
the no. of continuous functions f:[-1,1]--->R
such that (f(x))^2=x^2 for all x belonging to [-1,1] is
a) 4
b) 2
c) 1
d) infinitely many.

q2)
the no. of functions f from {1,2,..20} onto {1,2..20} such that f(k) is a multiple of 3 whenever k is a multiple of 4 is
a) 5!6!9!
b)5^6* 15!
c) 6^5*14!
d)15!6!

q3) given integral from 0 to 2 f(x) dx=3 and integral from 2 to 4 f(x) dx =5 , calculate integral from 0 to 2 f(2x)dx.
a) 3/2
b)3
c)4
d)6.

[Amit Goyal]

Q1 (a) 4
(f(x) = x, g(x) = -x, h(x) = |x|, t(x) = - |x|)

Q2 (d) 15!6!
(there are five multiple of 4s in the domain of f, {4, 8, 12, 16, 20}, which can be assigned any five out of following six multiples of 3 in the range of f, {3, 6, 9, 12, 15, 18}.  There are 6 ways in which 5 out of 6 number can be selected from {3, 6, 9, 12, 15, 18} for assignment and there are 5! ways in which they can be assigned to {4, 8, 12, 16, 20} and the remaining 15 numbers can be assigned in 15! ways. Thus, total number of functions equals 6 x 5! x 15! = 6! x 15!)

Q3 (c) 4
(Use substitution, put 2x = y and then write the integral in y - Make sure you substitute dx by dy/2 and adjust the limits of the integral)

[Mauli]

thankyou sir:)