The output s as a Boolean expression in the inputs x₁, x₂ and x₃ for the logic circuit in the following figure is

• x₁ x^'₂ + x^'₂ + x₃

• x₁ + x'₂x₃ + x₃

• (x₁x₂)' + x₁x'₂x₃

• x₁x'₂ + x'₂x₃

Let D₇₀ = {1, 2, 57, 10, 14, 35, 70} Define '+', '·' and '" by a + b = lcm (a, b), a . b = gcd (a, b) and a' = $\frac{70}{2}$ for all a, b $\in$ D₇₀. The value of (2 + 7)(14 . 10)' is

• 7

• 14

• 35

• 5

Let a be any element in a Boolean Algebra B. If a + x = 1 and ax = 0, then :

• x = 1

• x = 0

• x = a

• x = a'

then s is equal to :

• x . (y' + z)

• x . (y' + z')

• x . (y + z)

• (x + y) . z

If N_a = {an : n $\in$ N}, then N₅ n N₇ is equal to :

• N₇

• N

• N₃₅

• N₅

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{\alpha x}}{\mathrm{x} + 1}, \mathrm{x} \ne - 1$, for what value of $\mathrm{\alpha }$ is f$\left[\mathrm{f}\left(\mathrm{x}\right)\right]$ = x ?

• $\sqrt{2}$

• $- \sqrt{2}$

• 1

• - 1

Let D = {1, 2, 35, 6, 10, 15, 30}. Define the operattons '+', ' . ' and ' ' ' on D as follows a + b = LCM(a, b), a . b = GCD(a, b) and a' = $\frac{30}{\mathrm{a}}$ Then (15' + 6) · 10 1s equal to :

• 1

• 2

• 3

• 5

Let f(x) = a^x(a > 0) be written as f(x) = f₁(x) + f₂(x), where f₁(x) is an even function and f₂(x) is an odd function. Then f₁(x + y) + f₁(x – y) equals :

• 2f₁(x)f₁(y)

• 2f₁(x + y)f₁(x - y)

• 2f₁(x + y)f₂(x - y)

• 2f₁(x)f₂(y)

If the function f : R $- \left\{1, - 1\right\} \to \mathrm{A}$ defined by $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^2}{1 - {\mathrm{x}}^2}$, is surjective, then A is equal to :

• R - (- 1, 0)

• R - [- 1, 0)

• - {- 1}

• $[0, \infty )$

Let $\sum _{\mathrm{k} = 1}^0\mathrm{f}\left(\mathrm{a} + \mathrm{k}\right) = 16\left(2^{10} - 1\right)$, where function satisfies f(x + y) = f(x)f(y) for all natural numbers x, y and f(1) = 2 . Then the natural number ‘a’ is

• 16

• 22

• 20

• 25