Given below are two statements: Statement I: (43.25)10= (101011.01)2 Statement II: (1110.111)2= (14.625)10 In the light of the above statements, choose the correct answer from the options given below:
Statement I is true but Statement II is false.
Understanding Number Base Conversions
This question asks us to verify the correctness of two statements involving number base conversions, specifically between decimal (base 10) and binary (base 2).
Let's analyze each statement separately by performing the indicated number base conversions.
Analyzing Statement I: Converting Decimal to Binary
Statement I claims that the decimal number (43.25)10 is equal to the binary number (101011.01)2.
To convert a decimal number with a fractional part to binary, we convert the integer part and the fractional part separately.
Converting the Integer Part (43)10 to Binary
We use successive division by 2 and record the remainders:
- 43 ÷ 2 = 21 remainder 1
- 21 ÷ 2 = 10 remainder 1
- 10 ÷ 2 = 5 remainder 0
- 5 ÷ 2 = 2 remainder 1
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
Reading the remainders from bottom to top, the binary representation of 43 is 101011.
Converting the Fractional Part (0.25)10 to Binary
We use successive multiplication by 2 and record the integer parts:
- 0.25 × 2 = 0.50 (integer part is 0)
- 0.50 × 2 = 1.00 (integer part is 1)
Reading the integer parts from top to bottom, the binary representation of 0.25 is 0.01.
Combining the Parts
Combining the binary integer part and the binary fractional part, we get:
(43.25)10 = (101011.01)2
This matches the binary number given in Statement I. Therefore, Statement I is true.
Analyzing Statement II: Converting Binary to Decimal
Statement II claims that the binary number (1110.111)2 is equal to the decimal number (14.625)10.
To convert a binary number with a fractional part to decimal, we multiply each digit by the corresponding power of 2 and sum the results. The weights are powers of 2 decreasing from left to right, starting from $2^0$ at the binary point.
Converting (1110.111)2 to Decimal
Let's write out the digits and their positional weights:
| Binary Digit | Position | Weight (Power of 2) | Value (Digit × Weight) |
|---|---|---|---|
| 1 | 3 | $2^3 = 8$ | $1 \times 8 = 8$ |
| 1 | 2 | $2^2 = 4$ | $1 \times 4 = 4$ |
| 1 | 1 | $2^1 = 2$ | $1 \times 2 = 2$ |
| 0 | 0 | $2^0 = 1$ | $0 \times 1 = 0$ |
| . | Binary Point | ||
| 1 | -1 | $2^{-1} = 0.5$ | $1 \times 0.5 = 0.5$ |
| 1 | -2 | $2^{-2} = 0.25$ | $1 \times 0.25 = 0.25$ |
| 1 | -3 | $2^{-3} = 0.125$ | $1 \times 0.125 = 0.125$ |
Now, sum the values:
Decimal Value = $8 + 4 + 2 + 0 + 0.5 + 0.25 + 0.125$
Decimal Value = $14 + 0.875$
Decimal Value = $14.875$
The calculated decimal value is 14.875, which is not equal to 14.625. Therefore, Statement II is false.
Conclusion
Based on our analysis:
- Statement I: (43.25)10 = (101011.01)2 is True.
- Statement II: (1110.111)2 = (14.625)10 is False (it is 14.875)10.
We need to choose the option that states Statement I is true and Statement II is false.
Revision Table: Number Base Conversions
| Original Number | Base | Conversion Method | Result | Matches Statement? |
|---|---|---|---|---|
| 43.25 | Decimal (10) | Decimal to Binary (Integer & Fraction) | 101011.01 | Yes (Statement I) |
| 1110.111 | Binary (2) | Binary to Decimal (Positional Weight) | 14.875 | No (Statement II claimed 14.625) |
Additional Information: Number Systems and Bases
A number system is a way of representing numbers. The base (or radix) of a number system determines the number of unique digits used to represent numbers and the value of each digit based on its position.
- Decimal System (Base 10): Uses 10 digits (0-9). Positional weights are powers of 10 ($10^0, 10^1, 10^2$, etc., and $10^{-1}, 10^{-2}$, etc. for fractions).
- Binary System (Base 2): Uses 2 digits (0 and 1). Positional weights are powers of 2 ($2^0, 2^1, 2^2$, etc., and $2^{-1}, 2^{-2}$, etc. for fractions). Binary is fundamental in digital electronics and computing.
- Octal System (Base 8): Uses 8 digits (0-7).
- Hexadecimal System (Base 16): Uses 16 symbols (0-9 and A-F, where A=10, B=11, ..., F=15).
Converting between bases is a common task in computer science and digital logic. The methods demonstrated above (successive division/multiplication and positional weight summation) are standard techniques for these conversions.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
