Key Concept: Exponential Growth, Power of 2

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The thickness of the paper after each fold doubles the previous thickness. So, the formula for thickness after $n$ folds is $T = 0.001 \times 2^n$ cm. For $n = 30$, we get:
$T = 0.001 \times 2^{30} \text{ cm} = 0.001 \times 1,\!073,\!741,\!824 \text{ cm} = 1,\!073,\!741.824 \text{ cm} = 10.737 \text{ km}.$
Thus, the assertion is true. The reason provides the correct formula for calculating the thickness, making it true and the correct explanation of the assertion.

Key Concept: Exponential Growth, Real-World Application

b) 10.737 km
[Solution Description]
The thickness after 30 folds is calculated as:
$0.001 \times 2^{30}$ cm
$2^{10} = 1024$, so $2^{30} = (2^{10})^3 = 1024^3 \approx 1.0737 \times 10^9$
Thickness in cm: $0.001 \times 1.0737 \times 10^9 = 1.0737 \times 10^6$ cm
Convert cm to km: $\frac{1.0737 \times 10^6}{100000} = 10.737$ km

Key Concept: Exponential Growth, Folding Paper

b) 1.024 cm
[Solution Description]
The thickness of the paper doubles with each fold. For 10 folds, the thickness increases by $2^{10} = 1024$ times.
Initial thickness: $0.001$ cm
Final thickness after 10 folds: $0.001 \times 2^{10} = 0.001 \times 1024 = 1.024$ cm

Key Concept: Exponential Growth

d) 1024 times
[Solution Description]
The thickness increases by a factor of $2^n$, where $n$ is the number of folds. For 10 folds:
$2^{10} = 1024$
So, the thickness increases 1024 times.

Key Concept: Exponential Growth

c) $0.008$ cm
[Solution Description]
The thickness doubles with each fold. After the first fold, it becomes $0.001 \times 2 = 0.002$ cm. After the second fold, it becomes $0.002 \times 2 = 0.004$ cm. After the third fold, it becomes $0.004 \times 2 = 0.008$ cm or $0.001 \times 2^3 = 0.008$ cm.

Key Concept: Exponential Growth

b) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond must have been half-covered one day before it was fully covered. Therefore, on the 29th day, the pond was half-covered.

Key Concept: Exponential Growth in Folding Paper

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the thickness after 10 folds using the formula for exponential growth: $0.001 \text{ cm} \times 2^{10}$.
Step 1: Calculate $2^{10}$ which is $1024$.
Step 2: Multiply by the initial thickness: $0.001 \text{ cm} \times 1024 = 1.024 \text{ cm}$.
The calculation confirms that the assertion is true. The reason provided correctly explains the assertion by stating the exponential growth formula and how it applies to the folding process.

Key Concept: Exponential Growth, Folding Paper Experiment

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we calculate the thickness after 46 folds using the formula for exponential growth: $T = T_0 \times 2^n$, where $T_0$ is the initial thickness (0.001 cm) and $n$ is the number of folds (46).
Step 1: Calculate $2^{46}$. Using logarithmic approximations or computational tools, $2^{46} \approx 7.04 \times 10^{13}$.
Step 2: Multiply the result by the initial thickness. $T = 0.001 \, \text{cm} \times 7.04 \times 10^{13} = 7.04 \times 10^{10} \, \text{cm}$.
Step 3: Convert centimeters to kilometers. Since $1 \, \text{km} = 10^5 \, \text{cm}$, $T = \frac{7.04 \times 10^{10}}{10^5} \, \text{km} = 7.04 \times 10^5 \, \text{km} \approx 7,00,000 \, \text{km}$.
Both the assertion and reason are true, and the reason correctly explains the assertion as it provides the mathematical basis for the exponential growth calculation.

Key Concept: Exponentiation

b) $2^4$
[Solution Description]
Using the property $\frac{m^a}{n^a} = \left(\frac{m}{n}\right)^a$,
$\frac{10^4}{5^4} = \left( \frac{10}{5} \right)^4 = 2^4.$

Key Concept: Exponential Notation and Operations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $2^3 \times 2^4$, we use the exponent rule $n^a \times n^b = n^{a+b}$. Here, the base is 2 for both terms. So, $2^3 \times 2^4 = 2^{3+4} = 2^7$. Both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Prime Factorization, Exponential Form

a) $2^5 \times 3^3 \times 5^2$
[Solution Description]
To express 21600 in prime factorization exponential form:
Step 1: Break down 21600 into its prime factors.
21600 ÷ 2 = 10800 ÷ 2 = 5400 ÷ 2 = 2700 ÷ 2 = 1350 ÷ 2 = 675 ÷ 3 = 225 ÷ 3 = 75 ÷ 3 = 25 ÷ 5 = 5 ÷ 5 = 1.
So, 21600 = $2^5 \times 3^3 \times 5^2$.

Key Concept: Negative Exponents

c) $\frac{1}{9}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this to $3^{-2}$, we get $\frac{1}{3^2} = \frac{1}{9}$.

Key Concept: Exponential Notation

a) $2^8$
[Solution Description]
To solve $2^3 \times 2^5$, we use the exponent rule $n^a \times n^b = n^{a + b}$. Here, the base ($n$) is the same (which is 2), so we can directly add the exponents: $3 + 5 = 8$. Thus, $2^3 \times 2^5 = 2^{8}$.

Key Concept: Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given information, the thickness of the paper doubles after every fold, indicating exponential growth. Specifically, after $n$ folds, the thickness is given by $0.001 \text{ cm} \times 2^n$. For $n = 10$, the thickness increases by $2^{10} = 1024$ times the initial thickness. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion since exponential growth (doubling) leads to the specified increase.

Key Concept: Exponential vs. Linear Growth

c) Exponential growth is faster than linear growth
[Solution Description]
Exponential growth gives:
$0.001 \text{ cm} \times 2^{10} = 1.024 \text{ cm}.$
In linear growth, for 10 steps adding $0.001 \text{ cm}$ each time, the total would be:
$0.001 \text{ cm} \times 10 = 0.01 \text{ cm}.$
Exponential growth results in much larger values compared to linear growth.

Key Concept: Exponential Operations: Power Rules

c) $3^3$
[Solution Description]
Using the exponent rule $n^a \times n^b = n^{a+b}$, first multiply the terms in the numerator:
$3^4 \times 3^2 = 3^{4+2} = 3^6$
Now, divide by the denominator using the rule $n^a \div n^b = n^{a-b}$:
$3^6 \div 3^3 = 3^{6-3} = 3^3$
Therefore, the simplified form is $3^3$.

Key Concept: Scientific Notation Conversion

c) $7.5 \times 10^6$
[Solution Description]
Scientific notation is written as $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. To convert 7,500,000 to scientific notation: Move the decimal point 6 places to the left to get $7.5$. Since the decimal was moved 6 places, $y = 6$. Thus, the scientific notation is: $7.5 \times 10^6$

Key Concept: Negative Exponents

b) $\frac{1}{5^4}$
[Solution Description]
A negative exponent represents the reciprocal of the base raised to the positive exponent. Thus:
$5^{-4} = \frac{1}{5^4}$
The correct equivalent form is $\frac{1}{5^4}$.

Key Concept: Exponent Rules

b) $2^{12}$
[Solution Description]
Using the power of a power rule, $(n^a)^b = n^{a \times b}$. Applying this rule to $(2^3)^4$:
$(2^3)^4 = 2^{3 \times 4} = 2^{12}$
The simplified form is $2^{12}$.

Key Concept: Exponential Notation and Operations: Power Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's analyze both the Assertion (A) and Reason (R):
1. Assertion (A): Evaluate $(-3)^4$.
- $(-3)^4 = (-3) \times (-3) \times (-3) \times (-3)$.
- Multiplying two negatives gives a positive: $(-3) \times (-3) = 9$.
- Again multiplying two positives: $9 \times 9 = 81$.
- So, $(-3)^4 = 81$, which is indeed positive. Thus, Assertion (A) is true.
2. Reason (R): The reason states that any negative base raised to an even exponent results in a positive number.
- This is a fundamental property of exponents:
For any non-zero real number $n$ and even integer $a$, $(-n)^a = n^a > 0$.
- Example: $(-2)^6 = 64$ (positive).
- Hence, Reason (R) is true and correctly explains Assertion (A).
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Zero Exponent

c) $1$
[Solution Description]
Any non-zero number raised to the power of zero is equal to 1. Therefore, $7^0 = 1$.

Key Concept: Multiplication of Exponents with Same Base

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the rule for multiplication of exponents with the same base:
$n^a \times n^b = n^{a + b}$.
Given the assertion: $3^5 \times 3^{-2}$.
Applying the rule: $3^{5 + (-2)} = 3^{3}$.
The reason correctly explains that exponents are added when multiplying with the same base.

Key Concept: Division of Exponents

b) $3^1$
[Solution Description]
First, multiply the exponents in the numerator: $3^7 \times 3^{-4} = 3^{7 + (-4)} = 3^3$.
Now, divide by the denominator using the rule $n^a \div n^b = n^{a-b}$: $3^3 \div 3^2 = 3^{3-2} = 3^1$.
The simplified form is $3^1$.

Key Concept: Negative exponents

b) $2^{2}$
[Solution Description]
To solve $2^{-3} \times 2^5$, use the rule $n^a \times n^b = n^{a + b}$. Here, the base $n = 2$, exponent $a = -3$, and exponent $b = 5$. So, $2^{-3} \times 2^5 = 2^{-3 + 5} = 2^2$.

Key Concept: Division of exponents with the same base

b) $7^{4}$
[Solution Description]
To solve $\frac{7^6}{7^2}$, use the rule $n^a \div n^b = n^{a - b}$. Here, the base $n = 7$, exponent $a = 6$, and exponent $b = 2$. So, $\frac{7^6}{7^2} = 7^{6 - 2} = 7^4$.

Key Concept: Multiplication of exponents with the same base

b) $5^{7}$
[Solution Description]
To solve $5^3 \times 5^4$, use the rule $n^a \times n^b = n^{a + b}$. Here, the base $n = 5$, exponent $a = 3$, and exponent $b = 4$. So, $5^3 \times 5^4 = 5^{3 + 4} = 5^7$.

Key Concept: Prime Factorization and Exponential Representation

b) $2^3 \times 3^4$
[Solution Description]
To find the prime factorization of $648$, divide it by primes:
648 ÷ 2 = 324
324 ÷ 2 = 162
162 ÷ 2 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Prime factors: $2^3 \times 3^4$.

Key Concept: Prime factorization in exponential form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- Step 1: Prime factorize 36. We know that 36 = 6 × 6.
- Step 2: Further factorize 6 into its prime factors: 6 = 2 × 3.
- Step 3: Substitute back into the equation: 36 = (2 × 3) × (2 × 3).
- Step 4: Regroup the terms to express them in exponential form: 36 = $2^2 \times 3^2$.
- Conclusion: Both Assertion (A) and Reason (R) are true, and R correctly explains A.

Key Concept: Numerical Evaluation

c) 392
[Solution Description]
To evaluate $7^2 \times 2^3$, first calculate the exponents:
$7^2 = 49$
$2^3 = 8$
Multiply them together: $49 \times 8 = 392$.

Key Concept: Exponential Form, Prime Factorization

b) 1
[Solution Description]
First, find prime factors of 504:
504 ÷ 2 = 252
252 ÷ 2 = 126
126 ÷ 2 = 63
63 ÷ 3 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
So, 504 = 2^3 × 3^2 × 7^1
Given equation: a + b - c = 4
From prime factors, we have:
a = 3, b = 2, c = 1
Verify equation:
3 + 2 - 1 = 4 which satisfies given condition.
Thus, value of c is 1.

Key Concept: Properties of Powers, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's evaluate the assertion step-by-step using the properties of exponents.
1. Given expression: $5^{-3} \times 5^2$
2. Using the product of powers property:
$n^a \times n^b = n^{a+b}$
So, $5^{-3} \times 5^2 = 5^{-3+2} = 5^{-1}$
3. Using the negative exponent property:
$n^{-a} = \frac{1}{n^a}$
Therefore, $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$
The assertion is true as it correctly simplifies to $\frac{1}{5}$.
The reason states the correct properties of exponents ($n^{-a} = \frac{1}{n^a}$ and $n^a \times n^b = n^{a+b}$), which are used to derive the assertion. Thus, the reason is the correct explanation for the assertion.

Key Concept: Power of a Power

b) $5^6$
[Solution Description]
To simplify $(5^2)^3$, we use the power of a power property: $(n^a)^b = n^{a \times b}$. Here, the base is 5, and the exponents are 2 and 3:
$(5^2)^3 = 5^{2 \times 3} = 5^6$
Therefore, the simplified form of the expression is $5^6$.

Key Concept: Negative Exponents

b) $\frac{1}{25}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this to $5^{-2}$:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Key Concept: Zero exponent, Negative exponents

b) $\frac{8}{9}$
[Solution Description]
Step 1: Simplify the exponents inside the brackets. Any number to the power 0 is 1 ($5^0 = 1$, $4^0 = 1$):
$\frac{1 \times 2^{-3}}{3^{-2} \times 1} = \frac{2^{-3}}{3^{-2}}$
Step 2: Rewrite negative exponents using $n^{-a} = \frac{1}{n^a}$:
$\frac{\frac{1}{2^3}}{\frac{1}{3^2}} = \frac{3^2}{2^3} = \frac{9}{8}$
Step 3: Take the reciprocal of the entire expression because of the outer $-1$ exponent:
$\left( \frac{9}{8} \right)^{-1} = \frac{8}{9}$
The final value is $\frac{8}{9}$.

Key Concept: Prime factorization, Exponents

a) $2 \times 3^2 \times 5^{-1}$
[Solution Description]
Step 1: Simplify the expression $\frac{x^2 \times y}{x \times y^2}$ to $\frac{x}{y}$:
$\frac{x^2 \times y}{x \times y^2} = \frac{x^{2-1}}{y^{2-1}} = \frac{x}{y}$
Step 2: Substitute the given values for $x$ and $y$:
$\frac{x}{y} = \frac{2^3 \times 3^4 \times 5^2}{2^2 \times 3^2 \times 5^3}$
Step 3: Simplify using the property $n^a \div n^b = n^{a-b}$ for each prime factor:
$2^{3-2} \times 3^{4-2} \times 5^{2-3} = 2^1 \times 3^2 \times 5^{-1}$
The simplified form is $2 \times 3^2 \times 5^{-1}$.

Key Concept: Negative exponents, Exponential operations

a) $\frac{1}{9}$
[Solution Description]
First, simplify the numerator using $n^a \times n^b = n^{a+b}$: $3^{2-5} = 3^{-3}$.
Now, divide by $3^{-1}$ using $n^a \div n^b = n^{a-b}$: $3^{-3-(-1)} = 3^{-2}$.
The simplified form is $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Key Concept: Negative Exponent

d) $\frac{1}{9}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this rule:
$3^{-2} = \frac{1}{3^2} = \frac{1}{9}.$

Key Concept: Negative exponents and zero exponents

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that for any non-zero number $n$, $n^0 = 1$, which is true because it follows from the exponent division rule: $n^a ÷ n^a = n^{a–a} = n^0$. The reason explains why $0^0$ is undefined, as applying the exponent division rule would involve division by zero, which is not allowed. Both statements are true, but the reason does not explain the assertion; it provides a related but separate explanation about $0^0$.

Key Concept: Exponent Operations

c) $\frac{1}{49}$
[Solution Description]
Using the exponent division rule: $n^a ÷ n^b = n^{a-b}$. Here, $7^4 ÷ 7^6 = 7^{4-6} = 7^{-2}$.
By the negative exponent rule, $7^{-2} = \frac{1}{7^2} = \frac{1}{49}$.

Key Concept: Exponents, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $\frac{1}{10^{-5}} = 10^5$. Using the property of negative exponents, we know that $n^{-a} = \frac{1}{n^a}$. Therefore, $\frac{1}{10^{-5}}$ can be rewritten as $10^5$ because $\frac{1}{\frac{1}{10^5}} = 10^5$. The reason provides the general form of the negative exponent rule, which correctly explains why the assertion is true. Hence, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Exponent Rules, Negative Exponents

c) $3^7$
[Solution Description]
To solve this, we first simplify the numerator and denominator separately using exponent rules. The numerator $3^5 \times 3^{-2} = 3^{5-2} = 3^3$. The denominator $3^0 \times 3^{-4} = 1 \times 3^{-4} = 3^{-4}$ because any number to the power of 0 is 1. Now, dividing numerator by denominator: $\frac{3^3}{3^{-4}} = 3^{3-(-4)} = 3^{7}$. Therefore, the simplified form is $3^7$.

Key Concept: Combination Calculation

b) $26^{6}$
[Solution Description]
Each slot has 26 choices (A-Z). For 6 slots, total combinations are $26 \times 26 \times 26 \times 26 \times 26 \times 26 = 26^6$.

Key Concept: Exponential Growth

b) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond would have been half-covered one day before it was fully covered. Hence, if full coverage occurs on the 30th day, half-coverage occurred on the 29th day.

Key Concept: Negative exponents, Division of powers

a) $1$
[Solution Description]
First, simplify the expression inside the parentheses: Multiply $5^3 \times 5^{-5} = 5^{3 + (-5)} = 5^{-2}$. Then, divide by $5^{-2}$: $\frac{5^{-2}}{5^{-2}} = 5^{-2 - (-2)} = 5^0 = 1$. Finally, take the reciprocal as per the negative exponent outside the parentheses: $1^{-1} = 1$.

Key Concept: Exponent subtraction rule and negative exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $5^{-3} \div 5^2$, we apply the exponent subtraction rule: $5^{-3} \div 5^2 = 5^{-3 - 2} = 5^{-5}$. The Reason correctly states the rule for dividing exponents with the same base. Therefore, both Assertion and Reason are true, and Reason explains Assertion.

Key Concept: Exponent subtraction rule

b) $2^2$
[Solution Description]
To simplify $2^{5} \div 2^{3}$, we use the exponent subtraction rule: $n^a \div n^b = n^{a - b}$. Here, $n = 2$, $a = 5$, and $b = 3$. Applying the rule:
$2^{5} \div 2^{3} = 2^{5 - 3} = 2^2$
Calculating $2^2$ gives $4$.

Key Concept: Negative exponents

c) $\frac{1}{25}$
[Solution Description]
We know that $n^{-k} = \frac{1}{n^k}$. Thus, for $5^{-2}$:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
The simplified form is $\frac{1}{25}$.

Key Concept: Negative and Zero Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $5^0 = 1$, which is correct according to the exponent rule for zero exponents. The reason provides the general rule that any non-zero number raised to the power of zero equals one ($n^0 = 1$), which correctly explains why $5^0 = 1$. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Negative Exponent Simplification

c) $\frac{1}{125}$
[Solution Description]
Using the rule for negative exponents, $n^{-a} = \frac{1}{n^a}$. Here, $n = 5$ and $a = 3$. Thus,
$5^{-3} = \frac{1}{5^3} = \frac{1}{125}.$

Key Concept: Zero Exponents, Real-world Context

d) $96$ grams
[Solution Description]
To find the original size of the sample, we will use the concept of negative exponents in exponential decay.
Step 1: Given the equation $S \times 2^{-5} = 3$, rewrite $2^{-5}$ as $\frac{1}{2^5}$:
$S \times \frac{1}{32} = 3$
Step 2: Solve for $S$ by multiplying both sides by 32:
$S = 3 \times 32$
Step 3: Perform the multiplication:
$S = 96$
The original size of the sample was $96$ grams.

Key Concept: Exponent Operations

c) $\frac{1}{4}$
[Solution Description]
Using the exponent rule for multiplication with the same base:
$2^4 \times 2^{-6} = 2^{4 + (-6)} = 2^{-2} = \frac{1}{2^{2}} = \frac{1}{4}$
The correct answer is $\frac{1}{4}$.

Key Concept: Negative Exponents

c) $10^2$
[Solution Description]
Using the rule for negative exponents:
$\frac{1}{10^{-a}} = 10^{a}$
Here, $a = 2$, so:
$\frac{1}{10^{-2}} = 10^{2}$
Hence, the answer is $10^2$.

Key Concept: Powers of 10

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $10^3$ can be written as $\frac{1}{10^{-3}}$. From the given syllabus, we know that for any non-zero number $n$, the relation $n^a = \frac{1}{n^{-a}}$ holds true. Substituting $n = 10$ and $a = 3$ in the general form gives us $10^3 = \frac{1}{10^{-3}}$. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Negative Exponents, Powers of 10

b) 5
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Here, $n = 10$ and $-a = -5$, so $a = 5$.

Key Concept: Powers of 10

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's verify the assertion using the reason:
Step 1: According to the negative exponent rule, $n^a = \frac{1}{n^{-a}}$.
Step 2: Substitute $n = 10$ and $a = 3$ in the above formula.
Step 3: We get $10^3 = \frac{1}{10^{-3}}$, which matches the assertion.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To express 7,00,00,000 in scientific notation, we first identify the coefficient and exponent. The coefficient must be a number between 1 and 10, which for this number is 7. Now, count the number of digits after the first digit (7): there are 6 zeros, so the exponent is 6. Therefore, the number can be written as $7 \times 10^6$. The reason correctly explains that the coefficient must be between 1 and 10 and how the exponent is determined by counting the digits after the first digit. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Scientific Notation, Comparison

c) Distance between Sun and Earth
[Solution Description]
To compare the distances, observe the exponents:
1. $1.4335 \times 10^{12}$ has exponent 12,
2. $1.439 \times 10^{12}$ also has exponent 12,
3. $1.496 \times 10^{11}$ has exponent 11.
Since $10^{11} < 10^{12}$, the distance between the Sun and Earth is the smallest. Even if we compare coefficients, the exponent difference outweighs any coefficient comparison.

Key Concept: Scientific Notation Conversion

b) 9,040
[Solution Description]
To convert $9.04 \times 10^3$ to standard form, multiply the coefficient $9.04$ by $10^3$ (which is 1000). This gives $9.04 \times 1000 = 9040$.

Key Concept: Scientific Notation

b) $4.75 \times 10^6$
[Solution Description]
To express 4,750,000 in scientific notation, we need to write it as $x \times 10^y$ where $1 \leq x < 10$. First, identify the coefficient by moving the decimal point to the left until there is only one non-zero digit before the decimal: $4.75$. The exponent $y$ is the number of places the decimal was moved, which is 6. Thus, $4,750,000 = 4.75 \times 10^6$.

Key Concept: Scientific Notation, Magnitude and Precision

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert $7,200,000,000,000$ to scientific notation:
Step 1: Move the decimal to the left until it is after the first non-zero digit (here, 7). This requires moving the decimal 12 places to the left.
Step 2: The coefficient $x$ must satisfy $1 \leq x < 10$. Here, $x = 7.2$. Step 3: The exponent $y$ is the number of places the decimal was moved, so $y = 12$. Thus, $7,200,000,000,000 = 7.2 \times 10^{12}$. The Reason states that the exponent must match the number of decimal places moved, which is correct and explains why the Assertion is true. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Scientific Notation

c) $3.05 \times 10^4$
[Solution Description]
To convert 30,500 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (3.05).
2. Count the number of places moved (4 places to the left).
3. Write in the form $x \times 10^y$: $3.05 \times 10^4$.

Key Concept: Exponent Rules

b) $5^5$
[Solution Description]
Using exponent rules:
1. Multiply the exponents for the numerator: $5^3 \times 5^4 = 5^{3+4} = 5^7$.
2. Divide by the denominator: $5^7 \div 5^2 = 5^{7-2} = 5^5$.

Key Concept: Scientific Notation

c) $4.5 \times 10^5$
[Solution Description]
To convert 450,000 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (4.5).
2. Count the number of places moved (5 places to the left).
3. Write in the form $x \times 10^y$: $4.5 \times 10^5$.

Key Concept: Comparing Large Quantities

b) $2.39 \times 10^{30}$
[Solution Description]
To find the number of mosquitoes equivalent to the Earth's mass, divide the Earth's mass by the mass of one mosquito:
$\frac{5.976 \times 10^{24}}{2.5 \times 10^{-6}} = \frac{5.976}{2.5} \times 10^{24-(-6)} = 2.3904 \times 10^{30}$
Therefore, approximately $2.39 \times 10^{30}$ mosquitoes would weigh as much as the Earth.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To write a number in scientific notation, we express it as $x \times 10^y$, where $1 \leq x < 10$. Here, $5,000$ is indeed equal to $5 \times 10^3$ because moving the decimal point three places to the left gives $5.000 = 5.0$ and the exponent becomes $3$. The Reason correctly explains that the exponent ($3$ here) represents the number of places the decimal moves, which corresponds to the number of zeros ($3$) after the digit $5$ in $5,000$.

Key Concept: Scientific Notation

b) $2.5 \times 10^6$
[Solution Description]
To find how many times more ants there are compared to humans, divide the number of ants by the number of humans:
$\frac{2 \times 10^{16}}{8 \times 10^9} = \frac{2}{8} \times 10^{16-9} = 0.25 \times 10^7 = 2.5 \times 10^6$
Therefore, there are $2.5 \times 10^6$ (2.5 million) times more ants than humans.

Key Concept: Scientific Notation, Real-World Comparisons

a) $2 \times 10^{12}$
[Solution Description]
To find how many times more stars are in the observable universe compared to the Milky Way, divide the number of stars in the universe by the number of stars in the Milky Way.
Number of stars in the universe = $2 \times 10^{23}$
Number of stars in the Milky Way = $1 \times 10^{11}$
Ratio = $\frac{2 \times 10^{23}}{1 \times 10^{11}} = 2 \times 10^{(23-11)} = 2 \times 10^{12}$
So, the observable universe has $2 \times 10^{12}$ times more stars than the Milky Way.

Key Concept: Scientific Notation

d) $4.25 \times 10^4$
[Solution Description]
To convert 42,500 to scientific notation:
Step 1: Rewrite the number as 4.25 (between 1 and 10).
Step 2: Count how many places the decimal moves from its original position to after 4.25, which is 4.
Thus, the standard form is $4.25 \times 10^4$.

Key Concept: Scientific Notation

c) $5.9853 \times 10^4$
[Solution Description]
To express $59,853$ in scientific notation:
First, move the decimal point to the left until it is just after the first non-zero digit. Here, we get $5.9853$.
Count the number of places moved (4 places), which gives the exponent for 10. Thus, $59,853 = 5.9853 \times 10^4$.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To express the number $50,000$ in scientific notation:
Step 1: Identify the coefficient by moving the decimal point to the left until it is between the first and second digit. For $50,000$, the decimal point is after the last zero, so moving it left 4 places gives $5.0$ (which simplifies to $5$).
Step 2: Count the number of places the decimal was moved, which is 4, hence the exponent is $4$.
Therefore, $50,000 = 5 \times 10^4$.
The reason correctly explains that the coefficient must be between $1$ and $10$, and the exponent indicates the number of decimal places moved.

Key Concept: Scientific Notation, Exponent Importance

b) 11.52 times
[Solution Description]
Step 1: Divide the distance to Jupiter by the distance to Mars to find the ratio.
$\frac{6.29 \times 10^8}{5.46 \times 10^7} = \frac{6.29}{5.46} \times 10^{8-7} = 1.152 \times 10^1 = 11.52$
Step 2: Answer is approximately $11.52$ times.

Key Concept: Coin Weight Calculation

b) 6429 coins
[Solution Description]
First, convert Roxie's weight from kilograms to grams:
$45 \text{ kg} = 45 \times 1000 = 45000 \text{ grams}$
Next, divide the total weight in grams by the weight of one coin:
$\text{Number of coins} = \frac{45000}{7} \approx 6428.57$
Since we can't have a fraction of a coin, we round up to the nearest whole number: 6429 coins.

Key Concept: Weight-Based Donation Calculations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The worth of donated jaggery is calculated using the formula:
$\text{Worth of jaggery} = \text{Roxie’s weight} \times \text{cost of 1 kg jaggery}$.
Given Roxie's weight is 45 kg and cost of 1 kg jaggery is Rs.70, the calculation is $45 \times 70 = Rs.3150$.
The reason correctly describes the general method for calculating the worth of any donated item based on weight and unit cost. Therefore, both Assertion and Reason are true, and Reason explains the Assertion correctly.

Key Concept: Age in Hours

b) 116,160 hours
[Solution Description]
To convert days to hours, multiply the number of days by 24: $4840 \times 24$. Calculating step-by-step: First, multiply 4000 × 24 = 96,000. Then, multiply 800 × 24 = 19,200. Finally, multiply 40 × 24 = 960. Add them together: 96,000 + 19,200 = 115,200; 115,200 + 960 = 116,160 hours.

Key Concept: Age calculation, Weight estimation

b) 116,160 hours
[Solution Description]
To find Roxie's age in hours, we need to convert her age from days to hours. There are 24 hours in a day. So, we multiply the number of days by 24.
Step-by-step calculation:
1. Number of days = 4840
2. Hours in a day = 24
3. Total hours = $4840 \times 24$
Now, let's compute $4840 \times 24$:
Break it down using distributive property:
$4840 \times 20 = 96,800$
$4840 \times 4 = 19,360$
Add them together:
$96,800 + 19,360 = 116,160$ hours.

Key Concept: Exponential Growth, Historical Comparisons

a) n is 5 less than dinosaur exponent
[Solution Description]
Chola duration: 900 years $= 3 × 10^{10}$ seconds (given)
Dinosaurs appeared: 200 million years ago $= 2 × 10^8 $years.
Convert to seconds: $2 × 10^8 × 3.15 × 10^7 ≈ 6.3 × 10^{15}$ seconds.
Difference in exponents: 15 - 10 = 5 orders of magnitude.
Comparison: The dinosaur era was about $10^5 (100,000)$ times longer than the Chola reign.

Key Concept: Powers of 10

c) 7
[Solution Description]
The number $10^7$ has 1 followed by 7 zeros: 1,00,00,000. Thus, it has exactly 7 zeros.

Key Concept: Exponential Growth

c) 1.024 cm
[Solution Description]

When a paper is folded once, its thickness becomes double. So, after every fold, the thickness keeps multiplying by 2. The formula is:

$\text{Final Thickness} = \text{Initial Thickness} \times 2^n$

Here, initial thickness $= 0.001\ \text{cm}$ and number of folds $n = 10.$

$0.001 \times 2^{10} = 0.001 \times 1024 = 1.024\ \text{cm}$

So, after folding the paper 10 times, its thickness will be $1.024\ \text{cm}.$

Key Concept: Linear Growth vs. Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that linear growth (adding 100 individuals per year) takes longer to double the population size than exponential growth (fixed growth rate). This is true because linear growth increases by a fixed amount each time, whereas exponential growth multiplies the current population by a factor, accelerating the growth rate. The reason correctly explains the difference between additive (linear) and multiplicative (exponential) growth patterns, justifying why exponential growth leads to faster doubling. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Understanding large numbers

c) $10^7$
[Solution Description]
1 crore is defined as $10^7$, which is 1 followed by 7 zeros (1,00,00,000). Let's verify each option:
a) $10^5$ = 1 lakh (1,00,000) – Incorrect.
b) $10^6$ = 1 million (1,000,000) – Incorrect.
c) $10^7$ = 1 crore (1,00,00,000) – Correct.
d) $10^8$ = 10 crore (10,00,00,000) – Incorrect.

Key Concept: Scientific Notation Conversion

b) $10^{23}$ seconds
[Solution Description]
To find the time taken to count all the stars, we divide the total number of stars by the counting rate (1 star per second). Given the number of stars as $10^{23}$, the time taken is $10^{23}$ seconds.
Hence, the answer is $10^{23}$ seconds.

Key Concept: Large Number Comparisons, Historical Context

b) niyuta
[Solution Description]
The Lalitavistara mentions that 100 ayutas make a niyuta, and since an ayuta is $10^9$, multiplying by 100 gives:
$100 \times 10^9 = 10^2 \times 10^9 = 10^{11} \text{ (niyuta)}$

Key Concept: Scientific notation

c) $3.081 \times 10^8$
[Solution Description]
Scientific notation requires expressing the number in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. For 308,100,000: $308,100,000 = 3.081 \times 10^8$ The decimal moves 8 places to the left.

Key Concept: Powers of 10

c) $10^9$
[Solution Description]
In the Indian numbering system, 1 arab is equal to 100 crores. Since 1 crore is $10^7$, then 1 arab would be $100 \times 10^7 = 10^9$.

Key Concept: Power Play

d) 9 zeros
[Solution Description]
In the Indian numbering system:
$1 \, \text{arab} = 100 \, \text{crores} = 100 \times 1,00,00,000 = 1,00,00,00,000$
Counting the zeros gives 9 zeros.

Key Concept: Indian and International number systems, Conversion

b) 500 billion
[Solution Description]
1 kharab in Indian system = $10^{11}$ (100 billion in International).
Therefore, 5 kharab = $5 \times 10^{11} = 500,000,000,000$.
To convert to billions: Divide by $10^9$ (since 1 billion = $10^9$),
$\frac{5 \times 10^{11}}{10^9} = 5 \times 10^2 = 500$ billion.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 5000 can be written as $5 \times 10^3$ in scientific notation. This is correct because moving the decimal point three places to the left gives the coefficient 5 and the exponent 3. The reason states that in scientific notation, the coefficient must be between 1 and 10, which is true and correctly explains why the assertion is valid.

Key Concept: Exponential Growth, Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial population $P_0 = 100$ cells. The population doubles every hour, so after $n = 10$ hours, the population $P$ can be calculated as:
$$ P = 100 \times 2^{10} $$
We know that $2^{10} = 1024$. Therefore:
$$ P = 100 \times 1024 = 102400 $$
In scientific notation, this is $1.024 \times 10^5$ cells.
The Assertion (A) correctly states the final population, and the Reason (R) provides the correct mathematical model for exponential growth, which explains why the assertion holds true.

Key Concept: Linear vs Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion (A) correctly defines linear and exponential growth. Linear growth is additive (e.g., $20 + 20 + 20 \ldots$), while exponential growth is multiplicative (e.g., $0.001 \times 2 \times 2 \ldots$). Reason (R) provides a valid example of linear growth, where each step adds a fixed distance (20 cm). Thus, both are true, and (R) explains (A).

Key Concept: Exponent Rules

d) $2^4$
[Solution Description]
Using the exponent rule $\frac{m^a}{n^a} = \left(\frac{m}{n}\right)^a$:
$\frac{10^4}{5^4} = \left(\frac{10}{5}\right)^4 = 2^4$
The simplified exponential form is $2^4$.

Key Concept: Exponential Growth, Pond Example

c) Day 27
[Solution Description]
Since coverage doubles daily:
Day 30 = 100% → Day 29 = 50% → Day 28 = 25% → Day 27 = 12.5%
Using exponential formula:
$\text{Coverage} = 100\% \times 2^{-(30-n)} = 12.5\%$
Solving: $2^{-(30-n)} = 0.125 = 2^{-3}$ ⇒ n = 27

Key Concept: Population and Demographics, Scientific Notation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the world population is approximately $8 \times 10^9$ and highlights its importance in resource planning, which is true based on the syllabus. The Reason explains that scientific notation simplifies the representation and comparison of large numbers, which is also correct. However, the Reason does not directly explain why the population figure is essential for resource planning—it only describes the utility of scientific notation. Thus, both statements are true, but the Reason does not correctly explain the Assertion.

Key Concept: Scientific Notation and Large Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of stars in the Milky Way is $1 \times 10^{11}$, which is correct. The reason explains that scientific notation simplifies large numbers by using a coefficient and a power of 10, which is also true. Furthermore, the reason correctly explains why the assertion uses scientific notation to represent the large number of stars in the Milky Way.

Key Concept: Population Counting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the global population is approximately $8 \times 10^9$, which is true as per the given syllabus context. The reason explains that large numbers like $8 \times 10^9$ are used for resource allocation and urban planning, which is also true as mentioned in the syllabus. Moreover, the reason correctly explains why the assertion is significant, as it highlights the practical application of such large numbers in real-world scenarios like governance and planning. Thus, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Real-World Applications

c) Jupiter is about 318 times heavier than Earth.
[Solution Description]
To compare the masses of Jupiter and Earth, divide Jupiter's mass by Earth's mass:
$\frac{1.898 \times 10^{27}}{5.976 \times 10^{24}} = \frac{1.898}{5.976} \times 10^{27-24} = 0.3176 \times 10^3 \approx 317.6$
Therefore, Jupiter's mass is approximately 318 times that of Earth's mass.

Key Concept: Scientific Notation and Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine which number is larger, compare their exponents first. Here, $2.5 \times 10^6$ has an exponent of 6, whereas $3.4 \times 10^5$ has an exponent of 5. Since 6 > 5, $2.5 \times 10^6$ is indeed greater than $3.4 \times 10^5$, regardless of the coefficients. Therefore, the Assertion is true. The Reason states that the magnitude of a number in scientific notation is primarily determined by its exponent, which is also true and correctly explains why the Assertion holds. Thus, both are true, and the Reason is the correct explanation for the Assertion.

Key Concept: Scientific Notation, Exponent Significance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of stars in the observable universe ($10^{23}$) is much larger than the number of grains of sand on Earth's beaches ($7.5 \times 10^{18}$). This is true as $10^{23} > 7.5 \times 10^{18}$. The reason states that the exponent in scientific notation indicates the order of magnitude, which is correct and explains why the comparison is straightforward. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Scientific Notation Conversion

b) 1,43,35,00,00,000 m
[Solution Description]
To convert $1.4335 \times 10^{12}$ to the Indian number system:
First, calculate $10^{12} = 1$ lakh crore.
Then, multiply by the coefficient: $1.4335 \times 1$ lakh crore $= 1.4335$ lakh crore.
This equals approximately 1,43,35,00,00,000 m.

Key Concept: Scientific Notation Basics

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $6.02 \times 10^{23}$ is written in scientific notation. This is correct because the coefficient (6.02) lies between 1 and 10, and the exponent (23) is an integer, satisfying the definition of scientific notation.
The Reason correctly defines the conditions for a number to be in scientific notation: the coefficient must be between 1 and 10, and the exponent must be an integer.
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.