Let `X={a_1, a_2,.. ,a_6}a n dY={b_1, b_2,b_3}dot` The number of functions `f` from `xtoy` such that it is onto and there are exactly three elements `x in X` such that `f(x)=b_1` is 75 (b) 90 (c) 100 (d) 120

Let A = {x_1, x_2, x_3, ,x_7},B={y_1, y_2, y_3} The total number of functions f: A->B that are onto and there are exactly three element x in A such that f(x)=y_2 is equal to a. 490 b. 510 c. 630 d. none of these

A function is defined as f:{a_1,a_2,a_3,a_4,a_5,a_6}rarr{b_1,b_2,b_3} then number of functions in which f(a_i)neb_i is divisible by

Let f(x)=x+2|x+1|+2|x-1|dot If f(x)=k has exactly one real solution, then the value of k is (a) 3 (b) 0 (c) 1 (d) 2

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot

Let f={(1,1),(2,3),(0,-1),(-1,-3)} be a function from Z to Z defined by f(x)= ax + b, for some integers a, b. Determine a, b,

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in Rdot If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to (a)6 (b) 0 (c) 3f(3) (d) 2f(0)

Let f(x)={1+(2x)/a ,0lt=x 1)f(x) exists ,then a is (a) 1 (b) -1 (c) 2 (d) -2

Let f={(1,1),(2,3),(0,−1),(−1,−3)} be a function from Z to Z defined by f(x)=ax+b for some integers a,b. Determine a,b.

Let f:{1,2,3} to {a,b,c} be one-one and onto function given by f (1) =a, f (2) =b and f (3) =c. Show that there exists a function g : {a,b,c} to {1,2,3} such that gof =I _(x) and fog =I _(y), where, X = {1,2,3} and Y={a,b,c}.

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat k and vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^2)