Key Concept: Modern Timekeeping

c) 60 ghatis.
[Solution Description]
In the ancient Indian timekeeping system, a 24-hour day was divided into 60 ghatis. Each ghati lasted approximately 24 minutes.

Key Concept: Evolution of timekeeping

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Ghatika-yantra was widely used due to its precision and reliability. The syllabus mentions that this water clock was indeed used in monasteries, palaces, and public spaces for time measurement. The Reason provides an explanation about its measurement interval (24-minute ghatis), which is correct as per the syllabus. However, while the Reason is true and explains one feature of the Ghatika-yantra, it does not directly justify why it was "precise and reliable" as claimed in the Assertion. Therefore, both statements are true, but the Reason does not fully explain the Assertion.

Key Concept: Pendulum clock, Ghatika-yantra

c) 720
[Solution Description]
First, convert 24 minutes into seconds since the pendulum's swing is in seconds. We know that 1 minute = 60 seconds, so 24 minutes = $24 \times 60 = 1440$ seconds. The pendulum takes 2 seconds per swing. To find the number of swings, divide the total time by the period: $\frac{1440}{2} = 720$. Therefore, the pendulum completes 720 swings before the bowl sinks.

Key Concept: Water Clock, Unit Conversion

b) 2.1 h
[Solution Description]
Convert ghatikas to hours: $5\,\text{ghatikas} \times \frac{24\,\text{min}}{1\,\text{ghatika}} \times \frac{1\,\text{h}}{60\,\text{min}} = 2\,\text{h}$. Convert palas to hours: $15\,\text{palas} \times \frac{24\,\text{s}}{1\,\text{pala}} \times \frac{1\,\text{h}}{3600\,\text{s}} = 0.1\,\text{h}$. Total duration = $2\,\text{h} + 0.1\,\text{h} = 2.1\,\text{h}$.

Key Concept: Atomic Clocks, Error Analysis

b) 76.67 min
[Solution Description]
Calculate error rate per year: $\frac{1\,\text{s}}{3 \times 10^6\,\text{years}}$. Total error in 13.8 billion years: $\frac{1\,\text{s}}{3 \times 10^6\,\text{years}} \times 13.8 \times 10^9\,\text{years} = 4600\,\text{s}$. Convert to minutes: $\frac{4600\,\text{s}}{60\,\text{s/min}} \approx 76.67\,\text{min}$.

Key Concept: Simple Pendulum

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the time period of a simple pendulum depends on the length of the string and the acceleration due to gravity, which is true. The formula for the time period $T$ of a simple pendulum is given by $T = 2\pi\sqrt{\dfrac{l}{g}}$, where $l$ is the length of the string and $g$ is the acceleration due to gravity. The reason states that the time taken by the pendulum to complete one oscillation is called its time period, which is also true and matches the definition provided in the syllabus. However, the reason does not explain why the time period depends on $l$ and $g$. It only defines what a time period is. Thus, both statements are true, but the reason does not correctly explain the assertion.

Key Concept: Speed Comparison, Real-Life Application

b) Men's 400 m runner
[Solution Description]
Calculate the average speed for each runner. For the women's 200 m: $\text{Speed} = \frac{200 \text{ m}}{21.78 \text{ s}} \approx 9.18 \text{ m/s}$. For the men's 400 m: $\text{Speed} = \frac{400 \text{ m}}{43.03 \text{ s}} \approx 9.30 \text{ m/s}$. The men's 400 m runner had a higher average speed.

Key Concept: Unit Conversion, Time Measurement

b) 1440 s
[Solution Description]
First, convert 24 hours to seconds: $24 \text{ h} = 24 \times 60 \text{ min} = 1440 \text{ min}$. Then, $1440 \text{ min} = 1440 \times 60 \text{ s} = 86400 \text{ s}$. Since 60 ghatis equal 86400 s, one ghati is $\frac{86400 \text{ s}}{60} = 1440 \text{ s}$.

Key Concept: Speed, Time Measurement

a) 100 m
[Solution Description]
First, calculate the speed of both runners. Speed is given by $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$. For Runner A: $\text{Speed}_A = \frac{100 \text{ m}}{10.2 \text{ s}} \approx 9.8039 \text{ m/s}$. For Runner B: $\text{Speed}_B = \frac{100 \text{ m}}{10.5 \text{ s}} \approx 9.5238 \text{ m/s}$. The difference in speeds is $9.8039 - 9.5238 \approx 0.2801 \text{ m/s}$, but the question states it as 0.5 m/s, so there must be an error in interpretation. Instead, use the time taken by Runner B to find how far Runner A has run. Since Runner A finishes in 10.2 s, in 10.5 s, Runner A would have run $9.8039 \times 10.5 \approx 102.941 \text{ m}$, which exceeds the race distance. Thus, reconsider: the distance covered by Runner A when Runner B finishes is simply 100 m since Runner A has already finished.

Key Concept: Ancient Methods of Measuring Time

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that sundials were used in ancient times to measure time during the day by observing the position of shadows. This is true as per the syllabus, which mentions sundials as one of the devices used for timekeeping. The reason given is that the position of the shadow changes as the Sun moves across the sky, allowing for measurement of time intervals. This is also true and directly explains why sundials could be used to measure time. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Water clocks, Hourglasses

d) 4 times
[Solution Description]
Time taken for the water clock to sink twice: $2 \times 24 = 48$ minutes. Number of times the hourglass empties in 48 minutes: $\frac{48}{12} = 4$ times.

Key Concept: Ghatika-yantra

c) Ghatika
[Solution Description]
The question refers to the Ghatika-yantra, a type of water clock used in ancient India. The time unit measured by this clock was called "ghatika" or "ghati," which was the standard unit of time measurement until the late 19th century.

Key Concept: Ancient Time Measurement

c) Arthasastra
[Solution Description]
The Arthasastra by Kautilya, composed between the 2nd century BCE and 3rd century CE, contains the earliest reference to shadow-based time measurement in ancient India.

Key Concept: Historical Context

b) Jaipur, Rajasthan
[Solution Description]
The Samrat Yantra, the world's largest stone sundial, is located at the Jantar Mantar in Jaipur, Rajasthan, India. It is a UNESCO World Heritage site.

Key Concept: Sundials

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that a sundial measures time by tracking the movement of the Sun, which is true because sundials rely on sunlight and the position of the Sun to indicate time. The Reason explains that the shadow cast by the gnomon changes position as the Sun moves, which is also true and directly explains how the sundial functions. Hence, both are true, and the Reason correctly explains the Assertion.

Key Concept: Water Clocks - Ghatika-yantra

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Ghatika-yantra measured time intervals of 24 minutes, which is correct as per historical references from Aryabhata. The reason explains why this interval occurred: because the hole in the bowl was specifically sized to allow the bowl to fill and sink in 24 minutes. Since both statements are factually accurate and the reason directly explains the assertion, the correct answer is option a).

Key Concept: Inflow Type, Historical Context

c) Their flow rate changed as water levels dropped
[Solution Description]
Early water clocks (outflow type) were inaccurate because their flow rate decreased over time. The \textit{Ghatika-yantra} (inflow type) solved this by using a floating bowl that sank at a fixed interval, ensuring consistent time measurement regardless of water flow variations.

Key Concept: Water clocks, Ghatika-yantra

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the *Ghatika-yantra* was not very accurate due to the decreasing flow rate of water as the water level dropped. This is true because the pressure at the hole decreases with the water level, reducing the flow rate over time. The Reason claims that the accuracy of a water clock relies on a constant flow rate, which is also correct. Furthermore, the Reason correctly explains why the Assertion is true, as the varying flow rate directly impacts the accuracy of time measurement in water clocks.

Key Concept: Effect of flow rate change, Proportional reasoning

b) 30 minutes
[Solution Description]
The time taken is inversely proportional to the flow rate. Doubling the flow rate will halve the time taken.
$\text{New Time} = \frac{\text{Original Time}}{2} = \frac{60\,\text{minutes}}{2} = 30\,\text{minutes}$
The correct answer is 30 minutes.

Key Concept: Hourglass Usage

c) Timing speeches
[Solution Description]
Hourglasses were commonly used for measuring durations like speeches, cooking, and navigation at sea due to their simplicity and reliability.

Key Concept: Hourglass Design

b) The hourglass with finer sand and narrower passage
[Solution Description]
According to the syllabus, the accuracy of an hourglass depends on the size of the sand grains and the width of the connecting passage. Finer sand and a narrower passage result in more precise time measurement. Therefore, the hourglass with finer sand grains and a narrower connecting passage will measure time more accurately.

Key Concept: Candle Clock Functionality

c) 12 minutes
[Solution Description]
The candle burns 5 cm in 60 minutes. Therefore, 1 cm is burned in $60/5 = 12$ minutes. Each marking represents 12 minutes.

Key Concept: Factors Affecting Accuracy

b) Air currents and wax composition
[Solution Description]
The accuracy of candle clocks was affected by variations in burning rates caused by air currents and wax composition. Protective containers helped but did not eliminate these issues.

Key Concept: Historical Use of Candle Clocks

c) In monasteries and households
[Solution Description]
Candle clocks were commonly used in monasteries and households during ancient times. They provided a way to measure time at night or in places where sundials couldn't be used due to lack of sunlight.

Key Concept: Oscillation count, Time calculation

d) 120 s
[Solution Description]
First, find the original time period ($T_1$) for length $L$: Given 20 oscillations take 40 seconds, so $T_1 = \frac{40}{20} = 2$ s. The time period formula is $T = 2\pi \sqrt{\frac{L}{g}}$. When length becomes $4L$, the new time period ($T_2$) is: $T_2 = 2\pi \sqrt{\frac{4L}{g}} = 2 \times 2\pi \sqrt{\frac{L}{g}} = 2T_1 = 4 \text{ s}$ For 30 oscillations, total time = $30 \times T_2 = 30 \times 4 = 120$ s.

Key Concept: Effect of Bob Mass

c) Both have the same time period
[Solution Description]
The time period of a simple pendulum does not depend on the mass of the bob. It only depends on the length of the pendulum and the acceleration due to gravity. Therefore, if two pendulums have the same length, they will have the same time period regardless of their bob masses.

Key Concept: Factors affecting time period

b) Mass of the bob
[Solution Description]
The time period of a simple pendulum depends only on the length of the pendulum and acceleration due to gravity ($T = 2\pi\sqrt{\frac{L}{g}}$). It is independent of the mass of the bob or the material of the string.

Key Concept: Observation from Experiment

b) 2 seconds
[Solution Description]
To find the time period, divide the total time by the number of oscillations: $Time\ Period = \frac{Total\ Time}{Number\ of\ Oscillations} = \frac{40\ s}{20} = 2\ s$.

Key Concept: Simple Pendulum, Time Period

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, the time period ($T$) of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$, where $l$ is the length of the string and $g$ is the acceleration due to gravity. This formula shows that the time period does not depend on the mass of the bob. Therefore, changing the mass of the bob while keeping the length constant will not affect the time period. Hence, the Assertion is true. The Reason correctly explains why the time period remains unchanged because it states that the time period depends only on the length and $g$. Both statements are true, and the Reason is the correct explanation for the Assertion.

Key Concept: Time Period of Simple Pendulum

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the time period of a simple pendulum does not change with the mass of the bob, which is correct according to the syllabus observations. The reason correctly explains this by stating that the time period depends only on the pendulum's length and gravitational acceleration ($T = 2\pi\sqrt{\frac{L}{g}}$). Thus, both are true, and the reason justifies the assertion.

Key Concept: Time period, Length dependence

d) 4 seconds
[Solution Description]
The time period $T$ of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length and $g$ is acceleration due to gravity. If the original length is $L$, then the new length is $4L$. Substituting the new length:
$T_{new} = 2\pi\sqrt{\frac{4L}{g}} = 2\pi \times 2\sqrt{\frac{L}{g}} = 2T = 4 \text{ seconds}$.

Key Concept: Oscillations, Time measurement

b) 1.8 seconds
[Solution Description]
Time measured for 20 oscillations = 40 seconds.
Stopwatch error: +1 second per 10 seconds ⇒ Total error = $\frac{40}{10} \times 1 = 4$ seconds.
Corrected time = 40 - 4 = 36 seconds.
Time period $T = \frac{36}{20} = 1.8$ seconds.

Key Concept: Effect of length on time period

b) It doubles
[Solution Description]
The time period $T$ of a simple pendulum is related to its length $l$ by:
$T = 2\pi \sqrt{\frac{l}{g}}$
When length becomes $4l$, new time period:
$T' = 2\pi \sqrt{\frac{4l}{g}} = 2 \times 2\pi \sqrt{\frac{l}{g}} = 2T$
So, the time period doubles.

Key Concept: Oscillatory Motion, Time Measurement

b) Decreased, because shorter pendulums have smaller time periods
[Solution Description]
If the clock loses time, it means the pendulum is oscillating slower than required. The time period ($T$) is directly proportional to the square root of the length ($L$):
$T \propto \sqrt{L}$
To make the clock faster (reduce $T$), we must decrease the length ($L$) of the pendulum.

Key Concept: Pendulum Properties, SI Units

b) The time period depends on the mass of the bob
[Solution Description]
1. True: The time period depends on the length of the pendulum.
2. False: The time period does **not** depend on the mass of the bob.
3. True: The SI unit of time is the second ($s$).
4. True: The pendulum's time period is constant for a given length.
Thus, option 2 is false.

Key Concept: Simple Pendulum Time Period

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the time period of a simple pendulum does not change with the mass of the bob, which is correct as per the syllabus. The reason provided is also correct because the time period ($T$) of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$, where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity. This formula clearly shows that the time period is independent of the mass of the bob. Therefore, the reason correctly explains the assertion.

Key Concept: Effect of Mass on Time Period

c) The time period remains unchanged regardless of the mass.
[Solution Description]
According to the given syllabus, the time period of a simple pendulum depends only on its length ($L$) and acceleration due to gravity ($g$). It is independent of the mass of the bob, as stated by the formula $T = 2\pi \sqrt{\frac{L}{g}}$.

Key Concept: Time period formula

c) $T = \frac{\text{Time taken for 10 oscillations}}{10}$
[Solution Description]
The time period ($T$) of a simple pendulum is calculated using the formula:
$T = \frac{\text{Time taken for 10 oscillations}}{10}$
This formula is derived by measuring the total time for 10 oscillations and then dividing it by 10 to get the time period for one oscillation.

Key Concept: Factors affecting time period

a) Length of the pendulum[Solution Description]According to the syllabus, the time period of a simple pendulum depends only on its length and not on the mass of the bob. Therefore, changing the length will affect the time period, but changing the mass will not.

Key Concept: Experimental error analysis, Multiple concepts

c) $\pm 0.020$ s
[Solution Description]
First calculate actual time period: $T = \frac{20.2}{10} = 2.02$ s. Error in time measurement: $\pm 0.1$ s for total time, so error in single period = $\frac{0.1}{10} = \pm 0.01$ s. Error from length: 1\% of 1.00 m = $\pm 0.01$ m. Since $T \propto \sqrt{L}$, error contribution from length = $\frac{1}{2}\times 1\% = 0.5\%$ of T = $0.005\times 2.02 = \pm 0.0101$ s. Total maximum error = $0.01 + 0.0101 = \pm 0.0201$ s.

Key Concept: Mass and time period independence

c) The time period remains unchanged when the mass of the bob changes.
[Solution Description]
According to the syllabus, the time period of a simple pendulum depends only on its length and not on the mass of the bob. Therefore, changing the mass of the bob does not affect the time period of the pendulum.

Key Concept: Time period calculation, Length relationship

c) 2T
[Solution Description]
The time period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$. For pendulum A: $T = 2\pi\sqrt{\frac{L}{g}}$. For pendulum B with length 4L: $T_B = 2\pi\sqrt{\frac{4L}{g}} = 2\pi\times 2\sqrt{\frac{L}{g}} = 2T$. Thus, when length becomes 4 times, time period becomes 2 times.

Key Concept: Effect of mass on time period

c) The time period remains unchanged
[Solution Description]
The time period of a simple pendulum does not depend on the mass of the bob. It only depends on the length of the pendulum and the acceleration due to gravity ($g$). Therefore, changing the mass from 50 g to 200 g will not affect the time period. Hence, the time period remains unchanged.

Key Concept: Dependence on length

c) $2T$
[Solution Description]
The time period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$, where $l$ is the length and $g$ is the acceleration due to gravity. Since $T$ is proportional to $\sqrt{l}$, if the length increases by a factor of 4, the time period increases by $\sqrt{4} = 2$. Thus, if the time period of pendulum A is $T$, then the time period of pendulum B with length 400 cm will be $2T$.

Key Concept: Simple Pendulum Time Period

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The time period of a simple pendulum is calculated using the formula $T = 2\pi \sqrt{\frac{L}{g}}$. This formula clearly shows that $T$ depends only on the length $L$ of the pendulum and the acceleration due to gravity $g$ at the location. There is no mention of mass in this formula, which means the mass of the bob does not affect the time period. Hence, the Assertion is true. The Reason correctly explains the Assertion because it provides the exact formula that proves the independence of $T$ from the mass of the bob.

Key Concept: Pendulum Clocks, Quartz and Atomic Clocks

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the accuracy of a pendulum clock decreases significantly with changes in temperature. This is true because the time period $T$ of a pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity. If the temperature changes, the length $L$ of the pendulum will expand or contract due to thermal expansion or contraction, changing the time period $T$. Since the clock measures time based on the pendulum's oscillations, any change in $T$ directly affects the clock's accuracy.
The reason states that the length of the pendulum expands or contracts with temperature variations, altering its oscillation period. This is also correct for the same explanation above: thermal expansion affects $L$, which in turn alters $T$.
Moreover, the reason correctly explains why the assertion is true because it identifies the specific mechanism (thermal expansion/contraction affecting $L$) that causes the assertion's claim (decreased accuracy).

Key Concept: Time Units Conversion

b) 13500 s
[Solution Description]
To convert 3 hours and 45 minutes to seconds:
1. Convert hours to seconds: $3 \text{ h} = 3 \times 60 \text{ min} = 180 \text{ min}$. Then $180 \text{ min} = 180 \times 60 \text{ s} = 10800 \text{ s}$.
2. Convert minutes to seconds: $45 \text{ min} = 45 \times 60 \text{ s} = 2700 \text{ s}$.
3. Add both results: $10800 \text{ s} + 2700 \text{ s} = 13500 \text{ s}$.

Key Concept: Modern Clocks

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that atomic clocks use vibrations from specific atoms for high precision time measurement, which is true as per the syllabus. The reason provided explains why atomic vibrations are suitable for this purpose—because they are highly stable and consistent over long durations. This explanation directly links to the assertion, as the stability of atomic vibrations justifies their use in atomic clocks. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Pendulum clock, Gears, Oscillation time

c) 450 oscillations
[Solution Description]
The second hand moves 6 degrees per oscillation. Since the minute hand is driven by the second hand's gear system, we need to find the number of oscillations needed to move the minute hand by 90 degrees.
First, calculate the gear ratio between the second hand and minute hand. The minute hand moves 360 degrees in 60 minutes (3600 seconds), so it moves at $0.1$ degrees per second. The second hand moves 6 degrees per 2 seconds (3 degrees per second). Thus, the gear ratio is $\frac{0.1}{3} = \frac{1}{30}$.
To move the minute hand by 90 degrees, the second hand must move $90 \times 30 = 2700$ degrees. Since each oscillation moves the second hand by 6 degrees, the number of oscillations required is $\frac{2700}{6} = 450$.

Key Concept: Clock Components

d) The hanging weights
[Solution Description]
In early mechanical clocks, weights were used as the primary energy storage component. They would gradually descend due to gravity, converting potential energy into kinetic energy to power the clock's movement. This was before springs became common in portable clocks.

Key Concept: Mechanical Clocks

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that pendulum clocks improved accuracy due to relying on periodic oscillations, which is correct as per the syllabus. The reason correctly explains that Galileo's discovery about the pendulum's time period being constant for a given length is foundational to pendulum clocks. Therefore, both statements are true, and the reason correctly explains the assertion.

Key Concept: Accuracy Comparison

c) Quartz clocks are less accurate than atomic clocks but more accurate than pendulum clocks
[Solution Description]
According to the syllabus, quartz clocks are more accurate than pendulum clocks but less precise than atomic clocks. Atomic clocks lose only one second in millions of years, whereas quartz clocks are not as precise. This makes quartz clocks intermediate in accuracy between pendulum and atomic clocks.

Key Concept: Quartz Clocks, Periodic Processes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that quartz clocks are more accurate than pendulum clocks due to their use of quartz crystal vibrations. This is true because quartz crystals vibrate at a very high and stable frequency, making them highly accurate.
The reason explains why quartz clocks are more accurate: it points out the rapid and stable vibrations of the quartz crystal, which directly contribute to their precision. Thus, the reason correctly explains the assertion.

Key Concept: Quartz clocks principle

b) Vibrations from a quartz crystal
[Solution Description]
Quartz clocks measure time using the principle of periodically repeating processes, specifically the vibrations of a quartz crystal. The key point from the syllabus states: "Quartz clocks use vibrations from a quartz crystal to measure time."

Key Concept: Atomic Clocks, Precision in Time Measurement

d) 1 billion years
[Solution Description]
The original precision is 1 second lost in $10^8$ years. Improving this by a factor of 10 means the new precision is 1 second lost in $10^9$ years.
1. Original precision = 1 second / $10^8$ years
2. Improved precision = Original precision $\times$ 10 = 1 second / $10^9$ years
Therefore, the clock will now take 1 billion years to lose 1 second.

Key Concept: Atomic Clocks

b) Vibrations of specific atoms
[Solution Description]
Atomic clocks measure time using periodically repeating processes, specifically the vibrations of certain atoms. This ensures high precision in time measurement.

Key Concept: Atomic Clock Precision

d) About 3.65 billion times more accurate
[Solution Description]
First, calculate the error in seconds per day for both clocks:
- Pendulum clock error = 10 seconds/day
- Atomic clock error = 1 second / (1,000,000 years $\times$ 365 days/year) = $1 / 365,000,000$ seconds/day ≈ $2.74 \times 10^{-9}$ seconds/day
Now, compare the two errors:
Accuracy improvement factor = Pendulum error / Atomic error = $10 / (2.74 \times 10^{-9})$ ≈ $3.65 \times 10^9$ times
The atomic clock is roughly 3.65 billion times more accurate.

Key Concept: SI Unit of Time

c) Second
[Solution Description]
The SI unit of time is second, represented by the symbol $s$.

Key Concept: Time Conversion

b) 8100 s
[Solution Description]
First, convert the hours to minutes: 2 hours = $2 \times 60$ minutes = 120 minutes.
Next, add the extra 15 minutes: Total minutes = 120 + 15 = 135 minutes.
Now, convert the total minutes to seconds: 135 minutes = $135 \times 60$ seconds = 8100 seconds.

Key Concept: SI Unit of Time

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the symbol for the SI unit of time is written as 'S', which is incorrect because the correct symbol is '$s$' (lowercase). The reason states that SI unit symbols are always written in lowercase letters unless derived from a proper noun, which is true. Since the assertion is false but the reason is true, the correct answer is d).

Key Concept: Unit representation

a) h
[Solution Description]
According to the syllabus, the correct symbol for hour is 'h'.
The other options are incorrect because:
- 'hr' is not the standard symbol.
- 'H' is uppercase and incorrect.
- 'hrs' is plural and incorrect.

Key Concept: Measurement of Time

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the standard unit for measuring time in the SI system is the second (s), which is true according to the syllabus. The reason claims that the second is widely used in various fields like sports, medicine, and technology due to its precision, which is also true and directly explains why the second is the standard unit. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Units of time

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the correct symbol for the unit of time 'second' is 's', which is true according to the syllabus. The reason given is that SI units always use lowercase letters for their symbols unless derived from a proper noun, which is also true. However, the reason does not correctly explain why the symbol for second is 's' because there’s no mention of proper nouns in this context. The correctness comes from standard conventions, not the mentioned rule about proper nouns.

Key Concept: Unit Conversion

c) 180 min
[Solution Description]
We know that $1$ hour is equal to $60$ minutes. To find the number of minutes in $3$ hours, multiply $3$ by $60$.
Calculation:
$3 \text{ h} \times 60 \text{ min/h} = 180 \text{ min}$
Therefore, the correct answer is $180$ minutes.

Key Concept: Unit Conversion

b) 300 s
[Solution Description]
We know that $1$ minute is equal to $60$ seconds. To find the number of seconds in $5$ minutes, multiply $5$ by $60$.
Calculation:
$5 \text{ min} \times 60 \text{ s/min} = 300 \text{ s}$
Therefore, the correct answer is $300$ seconds.

Key Concept: Units of Time

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the symbol for minute is 'min' and is always written in lowercase letters. This is correct as per the syllabus: "Their symbols ‘s’, ‘min’, and ‘h’ are also written in lowercase letters and in singular." The reason states that all derived SI units are written in lowercase letters unless named after a person. However, this is not entirely accurate because derived SI units can be capitalized if they honor a scientist (e.g., Newton 'N'), but 'min' is not derived from a person's name. Therefore, the assertion is true, but the reason is false.

Key Concept: Correct Usage of Symbols

c) 2 h
[Solution Description]
The correct symbol for hour is 'h' (lowercase). It should be written in singular form without a full stop and with a space between the number and the unit.
Therefore, the correct representation is "2 h".

Key Concept: Unit Conversion

b) 3 min
[Solution Description]
We know that $60\ s = 1\ min$. To convert 180 seconds to minutes:
Step 1: Divide the given seconds by 60 to get minutes.
$180\ s \div 60 = 3\ min$
So, 180 seconds equals 3 minutes.

Key Concept: Correct Writing of Units

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the symbol for minute is 'min' and should not have a full stop after it. This is correct as per the syllabus, which specifies that unit symbols are written in lowercase and without a full stop unless ending a sentence. The reason correctly explains the assertion by citing the SI convention for writing unit symbols. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Speed Calculation

c) 30 km/h
[Solution Description]
To find the average speed, use the formula $Speed = \frac{Total\ distance\ covered}{Total\ time\ taken}$. Here, the total distance is 45 km and the total time taken is 1.5 hours. So, $Speed = \frac{45\ km}{1.5\ h} = 30\ km/h$.

Key Concept: Speed and Motion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a car covers 120 km in 2 hours, and its speed is $60 \text{ km/h}$. This is correct because according to the formula for speed:
$\text{Speed} = \frac{\text{Total Distance Covered}}{\text{Total Time Taken}}$
Substituting the values:
$\text{Speed} = \frac{120 \text{ km}}{2 \text{ hrs}} = 60 \text{ km/h}.$
The reason given states that speed is calculated by dividing the total distance covered by the total time taken, which is exactly the definition of speed as per the syllabus. Therefore, the reason correctly explains the assertion.

Key Concept: Speed, Distance, Time

c) 144 km
[Solution Description]
First, convert the time taken at the first speed from minutes to hours: 45 minutes = 0.75 hours.
Calculate the distance covered in the first part using $\text{Distance} = \text{Speed} \times \text{Time}$:
$72 \, \text{km/h} \times 0.75 \, \text{h} = 54 \, \text{km}$.
Next, calculate the distance covered in the second part:
$90 \, \text{km/h} \times 1 \, \text{h} = 90 \, \text{km}$.
Add both distances to get the total distance: $54 \, \text{km} + 90 \, \text{km} = 144 \, \text{km}$.

Key Concept: Speed and Distance Comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if two cars cover the same distance but take different times, the one taking less time is faster. This is true because speed is inversely proportional to time when distance is constant. The reason correctly defines speed as the ratio of distance to time, which is the basis for comparing speeds. Hence, the reason explains why the assertion is true.

Key Concept: Comparison of Speeds

a) Runner A
[Solution Description]
First, calculate the speed of both runners using:
$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$.
For Runner A:
Speed = $\frac{200 \text{m}}{25 \text{s}} = 8 \text{m/s}$.
For Runner B:
Speed = $\frac{300 \text{m}}{40 \text{s}} = 7.5 \text{m/s}$.
Since 8 m/s > 7.5 m/s, Runner A is faster.

Key Concept: Non-uniform motion, Average speed

b) 48 km/h
[Solution Description]
Total distance = $60\ km + 60\ km = 120\ km$.
Time taken for first part = $\frac{60\ km}{40\ km/h} = 1.5\ h$.
Time taken for second part = $\frac{60\ km}{60\ km/h} = 1\ h$.
Total time = $1.5\ h + 1\ h = 2.5\ h$.
Average speed = $\frac{120\ km}{2.5\ h} = 48\ km/h$.

Key Concept: Speed and Distance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the speed of the car:
1. Given, total distance = 120 km
2. Total time taken = 2 hours
3. Using the formula: $Speed = \frac{Total \ distance \ covered}{Total \ time \ taken}$
4. Substituting values: $Speed = \frac{120 \ km}{2 \ h} = 60 \ km/h$
The assertion correctly calculates the speed, and the reason provides the correct formula for calculating speed.

Key Concept: Unit Conversion, Speed Calculation

d) 180 km
[Solution Description]
First, convert the speed from m/s to km/h:
1 km = 1000 m; 1 h = 3600 s.
Speed in km/h = $25 \times \frac{3600}{1000}$ = 90 km/h.
Now, calculate the distance covered in 2 hours:
Distance = Speed × Time = 90 km/h × 2 h = 180 km.

Key Concept: Speed Calculation

c) 18 km/h
[Solution Description]
First, find the speed in m/s using the formula $Speed = \frac{Distance}{Time}$. Here, Distance = 450 m and Time = 90 s. So, $Speed = \frac{450}{90} = 5\ m/s$. To convert this to km/h, multiply by $\frac{3600}{1000}$ (since 1 km = 1000 m and 1 h = 3600 s). Thus, $Speed = 5 \times \frac{3600}{1000} = 18\ km/h$.

Key Concept: Speed, Time-Distance Relationship

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we first convert 90 minutes into hours because the speed is given in km/h.
90 minutes = $\frac{90}{60} = 1.5$ hours.
Using the distance formula: $\text{Distance} = 60 \, \text{km/h} \times 1.5 \, \text{h} = 90 \, \text{km}$.
The calculation confirms the assertion is true.
The reason states the correct formula for distance, which is indeed $\text{Distance} = \text{Speed} \times \text{Time}$.
Moreover, the reason correctly explains how the assertion was derived, as the formula was directly applied to arrive at the answer.

Key Concept: Speed: Calculation of Speed

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To calculate the speed of the car, we use the formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$. Here, Distance = 120 km and Time = 2 hours. Substituting these values into the formula gives $\text{Speed} = \frac{120 \text{ km}}{2 \text{ h}} = 60 \text{ km/h}$. This matches the assertion. The reason correctly states the formula used to calculate speed. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Unit Conversion

c) 20 m/s
[Solution Description]
To convert km/h to m/s, multiply by $(1000 \text{m})/(3600 \text{s})$ which simplifies to $5/18$.
So, 72 km/h * (5/18) = 20 m/s.

Key Concept: Speed and Unit Conversion

c) 20 m/s
[Solution Description]
First, convert the distance from kilometers to meters:
$72 \text{ km} = 72 \times 1000 = 72000 \text{ m}$.
Next, convert time from hours to seconds:
$1 \text{ h} = 3600 \text{ s}$.
Now, calculate the speed using the formula:
$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{72000}{3600} = 20 \text{ m/s}.$

Key Concept: Speed Comparison

c) Both have the same speed.
[Solution Description]
Convert Train A's speed from km/h to m/s for comparison:
$90 \text{ km/h} = \frac{90 \times 1000 \text{ m}}{3600 \text{ s}} = 25 \text{ m/s}$.
Train B's speed is already given as 25 m/s.
Hence, both trains have the same speed.

Key Concept: Units of Speed

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that speed can be measured in units of $m/s$ or $km/h$, which is true based on the syllabus. The Reason explains how speed is calculated, which is also correct ($Speed = \frac{Distance}{Time}$). Moreover, the Reason correctly justifies why speed has these units (distance divided by time). Therefore, both Assertion and Reason are true, and Reason explains Assertion correctly.

Key Concept: Uniform Motion

c) 200 m
[Solution Description]
First, find the speed of the object using the formula:
$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{80\,\text{m}}{10\,\text{s}} = 8\,\text{m/s}$
Since the speed is constant, the distance covered in 25 seconds is:
$\text{Distance} = \text{Speed} \times \text{Time} = 8\,\text{m/s} \times 25\,\text{s} = 200\,\text{m}$
Hence, the object will cover $200\,\text{m}$ in 25 seconds.

Key Concept: Speed-Distance-Time Relationship

c) 60 km/h
[Solution Description]
The average speed is calculated using the formula:
$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
Substituting the given values:
$\text{Speed} = \frac{300\,\text{km}}{5\,\text{h}} = 60\,\text{km/h}$
Thus, the average speed of the car is $60\,\text{km/h}$.

Key Concept: Unit Conversion

c) 54 km/h
[Solution Description]
To convert meters per second (m/s) to kilometers per hour (km/h), use the conversion factor:
$1\,\text{m/s} = 3.6\,\text{km/h}$
Multiply the given speed by this factor:
$15\,\text{m/s} \times 3.6 = 54\,\text{km/h}$
Therefore, the converted speed is $54\,\text{km/h}$.

Key Concept: Time calculation - train

c) 3 hours
[Solution Description]
Time = $\frac{\text{Distance}}{\text{Speed}} = \frac{540}{180} = 3 \text{ hours}$

Key Concept: Speed, Distance, Time Conversion

d) 300
[Solution Description]
First, convert the speed from km/h to m/s for consistency in units: $108 \text{ km/h} = \frac{108 \times 1000 \text{ m}}{3600 \text{ s}} = 30 \text{ m/s}$. The distance between the stations is 540 km, which is $540 \times 1000 = 540,000$ meters. Using the formula for time taken, $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$, we get $\frac{540,000 \text{ m}}{30 \text{ m/s}} = 18,000 \text{ seconds}$. To convert this into minutes, divide by 60: $\frac{18,000}{60} = 300 \text{ minutes}$.

Key Concept: Distance calculation - bus

b) 90 km
[Solution Description]
Convert time to hours: $1 \text{ hour } 30 \text{ min} = 1.5 \text{ hours}$
Distance = Speed × Time = $60 \times 1.5 = 90 \text{ km}$

Key Concept: Uniform and Non-uniform Linear Motion, Speed Calculation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's calculate the total distance covered by the car:
- First hour: 60 km/h $\times$ 1 hour = 60 km
- Second hour: 40 km/h $\times$ 1 hour = 40 km
Total distance: 60 km + 40 km = 100 km
Total time taken: 2 hours
Average speed: $\text{Average speed} = \frac{100 \text{ km}}{2 \text{ hours}} = 50 \text{ km/h}$
The reason states that average speed is calculated by dividing the total distance by total time taken, which is correct. The assertion correctly calculates the average speed based on this formula. Thus, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Uniform vs. Non-uniform Motion

b) A cyclist covering 10 km in the first 30 minutes and 15 km in the next 30 minutes.
[Solution Description]
Non-uniform motion occurs when an object covers unequal distances in equal intervals of time. We analyze each option:
- Option a) describes uniform motion as the train moves at a constant speed.
- Option b) describes non-uniform motion because the distances covered are unequal in equal time intervals.
- Options c) and d) describe uniform motion as distances remain consistent over time.

Key Concept: Uniform motion, Speed calculation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we analyze the assertion:
- Given, speed = 60 km/h.
- Distance covered in 30 minutes (0.5 hours) = Speed $\times$ Time = $60 \times 0.5 = 30$ km.
Since this repeats every 30 minutes, the assertion correctly describes uniform motion where equal distances are covered in equal intervals of time.
Next, the reason states that distance is directly proportional to time in uniform motion.
- Mathematically, for uniform motion, $Distance = Speed \times Time$. Here, speed is constant, so distance is indeed directly proportional to time.
Additionally, the reason explains why the assertion holds true because proportionality justifies the constancy of distances over identical time intervals.
Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Uniform motion, time interval comparison

c) Both cars have the same speed.
[Solution Description]
Calculate the speed of both cars. For Car A: Speed = $\frac{120}{2}$ = 60 km/h. For Car B: Speed = $\frac{180}{3}$ = 60 km/h. Both cars have the same speed since they cover equal distances in equal time intervals proportionally.

Key Concept: Distance-time analysis, Non-uniform motion identification

b) The object is in non-uniform motion.
[Solution Description]
For non-uniform motion, the distances covered in equal time intervals are unequal. Here, the distances are 15 m, 25 m, and 35 m in consecutive 1-second intervals.
Since the distances are increasing by 10 m each second, the motion is non-uniform and accelerated. However, the question focuses on whether the motion is uniform or non-uniform.
The correct description is that the object is in non-uniform motion since unequal distances are covered in equal time intervals.

Key Concept: Non-uniform motion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Non-uniform motion is characterized by unequal distances covered in equal time intervals because the speed of the object is not constant and varies with time. The assertion states that an object covers unequal distances in equal time intervals, which is true for non-uniform motion. The reason correctly explains that the speed changes with time, leading to unequal distances being covered in equal intervals. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Real-life Non-uniform Motion

b) 75 s
[Solution Description]
Calculate the time taken for each segment separately. For the first 500 meters:
$\text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{500 \, \text{m}}{20 \, \text{m/s}} = 25 \, \text{s}$
For the next 500 meters:
$\text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{500 \, \text{m}}{10 \, \text{m/s}} = 50 \, \text{s}$
Total time taken:
$\text{Total Time} = \text{Time}_1 + \text{Time}_2 = 25 \, \text{s} + 50 \, \text{s} = 75 \, \text{s}$
The total time taken is 75 seconds.

Key Concept: Uniform Linear Motion

c) 240 km
[Solution Description]
To find the total distance covered by the train, we use the formula:
$\text{Distance} = \text{Speed} \times \text{Time}$
Given: Speed = 80 km/h, Time = 3 hours.
So,
$\text{Distance} = 80 \, \text{km/h} \times 3 \, \text{h} = 240 \, \text{km}$
Therefore, the total distance covered is 240 km.

Key Concept: Time Measurement, Precision in Modern Applications

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that modern timekeeping accuracy is crucial for applications like sports and medicine, which is true. Highly precise timing ensures fair competition in sports and reliable medical data. The reason claims that ECG machines require exact time intervals, which is also true. The reason correctly explains why precision is vital in medical diagnostics. Thus, both assertion and reason are true, and the reason justifies the assertion.

Key Concept: Time Measurement Devices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that sundials were used to measure time by observing the shadow cast by the Sun, which is true as per the syllabus. The reason explains that the position of the Sun changes throughout the day, causing the shadow to move in a predictable manner, which is also true and directly explains why sundials could measure time based on the movement of shadows. Hence, both Assertion and Reason are true, and Reason correctly explains the Assertion.