Test: Significant Figures & Dimensions - JEE MCQ

• Dimensional Equations are used to derive the Relations between the Physical quantities.
• Also, they are used to check the correctness of the Equation by the Principle of the homogeneity of the dimensions.

Hence, Option (a) and option (b) are true for the Dimensional equation. Thus, It is not correct for the given questions.

• Dimensional Equations are not valid for Pure numbers since they are non-dimensional constants. Also, they do not provide the value of the Constant of the Proportionality.
• Thus, we can also say that all equations which are correct must be dimensionally correct but all those equations that are dimensionally correct may or may not be accurately correct. Thus, Option (c) is the correct answer for the given question.
• Dimensional Equations are used for the conversions of the units thus (d) option can not be correct for the given question.

Hence, option C is the correct answer as it is an incorrect statement. 

The significant figures in any number are all certain digits plus one doubtful digit. Certain rules are followed which are as under:

1) All one zero digits in a number are significant.

2) The zeroes between two non-zero digits are always significant.

3) The zeros written to the left of the first non-zero digit in a number are non-significant. They simply indicate the position of the decimal point.

4) All zeros placed to the right of a decimal point in a number are significant.

Since, 0.0268 has three significant figures whereas 2.608 cm, 26.08 mm and 2.068 cm have four significant figures, A is the correct answer. 

Dimension of work and energy are the same i.e. [ML²T⁻²].
 Power = Energy/Time,
 Energy = Power × Time (in terms of hour instead of second),
 Energy = Watt × Hour,
= [ML²T⁻³] [T],
= [ML²T⁻²].
But, Momentum(p) = Mass × Velocity,
 p = [M] [LT⁻¹],
 p = [MLT⁻¹].
Hence the Dimension of work, energy and watt-hour is the same i.e. [ML²T⁻²] but the dimension of momentum is different ([MLT⁻¹]).

Step 1: Identify the number in scientific notation
The given number is 0.310×10³. In scientific notation, the significant figures are determined by the digits in the coefficient (the part before the multiplication by 10ⁿ).

Step 2: Analyze the coefficient
The coefficient here is 0.310. To find the significant figures, we need to look at each digit:
- The digit '3' is significant.
- The digit '1' is significant.
- The digit '0' after the decimal point is also significant because it comes after non-zero digits.

Step 3: Count the significant figures
In 0.310, we have:
- 3 significant figures (3, 1, and 0).

Step 4: Conclusion
Thus, the number of significant figures in 0.310×10³ is 3.

As the power of 10 is irrelevant in the determination of significant figures hence, the numbers of significant figures are 5 i.e. 9,4,6,0,5

As the power of 10 is irrelevant to the determination of significant figures hence, the numbers of significant figures are 3 (1, 8 and 4).

The multiplication of the numbers:
10.610 × 0.210 = 2.2281

Since, the numbers 10.610 and 0.210 have 5 and 3 significant digits, respectively. So, the final result must also have 3 significant digits.

The final answer is 2.23 after rounding off 2.2281. 

• As per rule, zero between two non zero digits are significant.
• The number of a significant figure in the length of the side is 4.
• So the calculated volume should be rounded off to 4 significant figures. Multiplying numbers do not increase significant figures.

Therefore, the correct answer is 4. 

• Scientific notation is the method of writing, or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10. 
• To convert a regular number to scientific notation, we first rewrite it as a decimal, then multiply it by a power of 10.
• There is an infinite number of ways to do this for any given number, but we always prefer the one that has only a single digit in front of the decimal point. 

As the power of 10 is irrelevant to the determination of significant figures hence, the numbers of significant figures are 2 (9 and 1).

Among all the given quantities, work, force and area can be expressed as products of some quantities which have non-zero dimensions, but the angle is defined as the ratio of arc length to the radius which is dimensionless because both arc length and radius have the same dimensions which when taken a ratio of, will get cancelled out. 

Additional Information: A dimensionless quantity is the one for which no physical dimension is assigned. It is also known as a bare, pure, or scalar quantity or also as a quantity of dimension one.

the correct answer should be C because a meter rod can only read upto two decimal points

In rounding off numbers, the last figure kept should be unchanged if the first figure dropped is less than 5.
Example: If only one decimal is to be kept, then 6.422 becomes 6.4.

Hence the correct answer is C. 

Volume of water = Volume of the tank = Length * breadth * height = 5.6 * 8.2 * 12.8 = 587.776

Volume, correct up to 4 significant figure = 587.8

Pressure = Force/Area = ML¹T^-2/ L² = [M L^-1 T^-2]
Stress = Force/Area   = ML¹T^-2/ L² = [ML^-1T^-2]
Coefficient of elasticity = Stress/Strain = [ML^-1T^-2]

Hence, the correct answer is C. 

In option a, b and c, we only have 4 significant figures in the number

In option d, the number has 5 significant figures.

To determine the number of significant figures, we only consider the digits in the base number 0.310, as the power of 10 does not affect the count of significant figures.

• In 0.310:
  • The digit 3 is significant.
  • The digit 1 is significant.
  • The digit 0 after the decimal is also significant (since it follows non-zero digits in a decimal number).

Therefore, there are 3 significant figures.

• Dimensionless quantities are the quantities that do not possess any dimensions.
• An example of such a quantity is specific gravity (ratio of absolute density of an object to absolute density of water).
• As these quantities are often a pure ratio between similar physical quantities they do not possess dimension and hence no units. 
• But when we look at the dimensionless quantity- angle, it has a unit assigned to it, the radians or degrees.
• So we cannot say that all dimensionless quantities are unitless.

Therefore, the correct answer is D. 

Calculating the Area:

• The area A of a rectangle is given by: A = length × breadth
• Here, the length is 16.2 cm (3 significant figures), and the breadth is 10.1 cm (3 significant figures).
• Multiplying these, we get: A = 16.2 × 10.1 = 163.62 cm²
• Since both measurements have 3 significant figures, the area should be reported to 3 significant figures, giving: A = 164 cm²

Calculating the Error:

• The formula for the error in the area ΔA when multiplying two quantities with errors ΔL and ΔB is: ΔA / A = ΔL / L + ΔB / B
• Assuming the smallest unit for each measurement as the error (0.1 cm), we have ΔL = 0.1 cm and ΔB = 0.1 cm
• So, ΔA / 164 = 0.1 / 16.2 + 0.1 / 10.1
• Calculating each term: 0.1 / 16.2 ≈ 0.00617 and 0.1 / 10.1 ≈ 0.0099
• Adding these: ΔA / 164 ≈ 0.00617 + 0.0099 = 0.01607
• Therefore, ΔA ≈ 164 × 0.01607 ≈ 2.6 cm²
• Rounding the error to one significant figure gives ΔA ≈ 3 cm²

Thus, the area of the sheet is 164 ± 3 cm².