Relations and Functions

Let f : R $\to$ R be a function which satisfies f(x + y) = f(x) + f(y) $\forall$ x, y $\in$ R. If f(1) = 2 and g(n) = $\sum _{\mathrm{k} = 1}^{\left(\mathrm{n} - 1\right)}\mathrm{f}\left(\mathrm{k}\right) \mathrm{n} \in \mathrm{N}$, then the value of g(n) = 20, is

• 9

• 20

• 5

• 4

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \left( - 1, \infty \right) \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{by} \mathrm{f}\left(0\right) = 1 \\ \mathrm{and} \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{\mathrm{x}}{\log }_{\mathrm{e}}\left(1 + \mathrm{x}\right), \mathrm{x} \ne 0. \mathrm{Then} \mathrm{the} \mathrm{function} \mathrm{f} : \end{gathered}$$

• increases in ( – 1, 0) and decreases in (0, $\infty$).

• $\mathrm{decreases} \mathrm{in} ( – 1, \infty )$

• decreases in ( – 1, 0) and increases in (0, $\infty$).

• increases in ( – 1, $\infty$)

If the lines x + y = a and x – y = b touch the curve y = x² – 3x + 2 at the points where the curve intersects the x–axis, then bais equal to ...

If 

• is an empty set

• contains exactly one element.

• contains exactly two elements.

• contains exactly two elements.

For x ε R, f (x)=|log2−sinx| and g(x)=f(f(x)), then :

• g is not differentiable at x=0

• g'(0)=cos(log2)

• g'(0)=-cos(log2)

• g'(0)=-cos(log2)

Let two fair six-faced dice A and B be thrown simultaneously. If E₁ is the event that die A shows up four, E₂ is the event that die B shows up two and E₃ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

• E₁ and E₂ are independent

• E₂ and E₃ are independent.

• E₁ and E₃ are independent.

• E₁ and E₃ are independent.

If g is the inverse of a function f and f'(x) =  then g'(x) is equal to 

• 1+ x⁶

• 5x⁴

• 
• 

If f_k(x) = 1/k (sin^k x + cos^k x), where x ε R and k ≥1, then f₄ (x)-f_o (x) equal to 

• 1/6

• 1/3

• 1/4

• 1/4

Let A and B be two events such that  where . Then, the events A and B are

• independent but not equally likely

• independent and equally likely

• mutually exclusive and independent

• mutually exclusive and independent

If: R →R is a function defined by  where [x] denotes the greatest integer function, then f is

• continuous for every real x

• discontinous only at x = 0

• discontinuous only at non-zero integral values of x

• discontinuous only at non-zero integral values of x