NIMCET 2017 Previous Year Questions (PYQs) – Page 4 of 12

✅ Given Information:

An 8-bit number in 2's complement form can represent values from \(-2^{n-1}\) to \(2^{n-1} - 1\).

✅ Step 1: Calculate the smallest integer:

Smallest value = \(-2^{8-1} = -2^7 = -128\)

✅ Final Answer:

The smallest integer that can be represented by an 8-bit number in 2's complement form is: -128

✅ Given Gates:

• I. NAND
• II. NOR

✅ NAND Gate:

The NAND gate is functionally complete. It can be used to construct any logic gate (AND, OR, NOT, etc.) by combining multiple NAND gates.

✅ NOR Gate:

The NOR gate is also functionally complete. It can be used to construct any logic gate (AND, OR, NOT, etc.) by combining NOR gates.

✅ Final Answer:

Both NAND and NOR gates are functionally complete.

✅ Given Information:

We are asked to find the total number of binary functions that can be defined using n Boolean variables.

✅ Step 1: Number of input combinations:

For n Boolean variables, the number of possible input combinations is 2ⁿ.

✅ Step 2: Number of possible outputs:

Each input combination can map to either 0 or 1, so there are 2 possible outputs for each of the 2ⁿ input combinations.

✅ Final Formula:

Total Binary Functions = 2⁽²ⁿ⁾

✅ Final Answer:

The total number of binary functions that can be defined using n Boolean variables is: 2⁽²ⁿ⁾

P(B speaks truth) = y

Since, both A and B agree on certain  statement.

Hence, Total Probability =P(A)P(B)+P(A')P(B')

=xy  + (1-x)(1-y)

If statement is true then it means both A and B speaks truth.

∴ Required Probability = $\dfrac{xy}{xy+(1-x)(1-y)}$

Short Solution:

Let the two numbers be \( a \) and \( b \).

Given: Harmonic mean is 4: $$\frac{2ab}{a + b} = 4 \quad \text{(1)}$$ Arithmetic mean \( A = \frac{a + b}{2} \),
Geometric mean \( G = \sqrt{ab} \)

Given: $$2A + G^2 = 27$$ $$2 \cdot \frac{a + b}{2} + ab = 27 \Rightarrow a + b + ab = 27 \quad \text{(2)}$$ From (1): Multiply both sides by \( a + b \): $$2ab = 4(a + b) \Rightarrow ab = 2(a + b) \quad \text{(3)}$$ Substitute (3) into (2): $$a + b + 2(a + b) = 27 \Rightarrow 3(a + b) = 27 \Rightarrow a + b = 9$$ Then from (3): $$ab = 2 \cdot 9 = 18$$ Now solve: $$x^2 - (a + b)x + ab = 0 \Rightarrow x^2 - 9x + 18 = 0$$ $$\Rightarrow x = 3, 6$$
Final Answer:
$$\boxed{3 \text{ and } 6}$$