The product of n consecutive natural numbers is always

Question

1 )

Let f(x) = x - |x| then f¹ (x) = 1

(Note : [x] denotes the greatest integer less than or equal to x)

A ∀x∈ R
B ∀x∈R - {0}
C ∀x∈R-Z
D ∀x∈R - {1}

2)

lim_n→∞ ⁿC_r (m/n)^r (1 - m/n )^n-r equals

A e^-m m^r
B m^r/r!
C m^re^-m/r!
D e^-rr^m/m!

3)
Integration of cotx/1 + sin square x

4)
Consider f(x) = x²/|x|, x ≠ 0,

                  = 0, x = 0
A f(x) is discontinuous every where
B f(x) is continuous every where
C f' (x) exists for all x in (-1, 1)
D f' (x) exists for all x in (-2, 2)

5) (√(42 + (√(42 + (√42+...)) =

A 7
B -6
C 5
D 4

6) If the position vector of a point P is r^→: = x_i^{^} + y_j^{^} + zk^{^}, where x, y, z∈N and α^- is a vector given by α^→ =  i^{^} + j^{^} + k^{^}, then the total number of possible positions of point P for which r^→.α^→ = 10.
A 36
B 72
C 66
D 100

7) Question :

A person is to count 4500 currency notes. Let a_n denote the number of notes he counts in the nth minute. If a₁ = a₂ = ...... = a₁₀ = 150 and a₁₀, a₁₁,...... are in A.P with common difference (-2), then the time taken by him to count all notes is
A 34 minutes
B 125 minutes C 135 minutes
D 24 minutes

8) Question:
The product of n consecutive natural numbers is always divisible by
A 4n!
B 3n!
C 2n!
D n!

9)
The number of ways in which the candidates A₁, A₂ A₁₀ can be ranked if Al is always above A2 is
A 9! 2!
B 9!
C 10!/2
D 10!