Key Concept: Power Operations

b) $3^3$
[Solution Description]
Using the rule for division of exponents with the same base, $n^a \div n^b = n^{a - b}$. Applying this rule:
$3^5 \div 3^2 = 3^{5 - 2} = 3^3$

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

c) 0.128 cm
[Solution Description]
Using the formula for exponential growth:
$0.001 \times 2^7 = 0.001 \times 128 = 0.128 \, \text{cm}$

Key Concept: Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial thickness of the paper is $0.001$ cm. After each fold, the thickness doubles. So after $n$ folds, the thickness is given by $0.001 \times 2^n$ cm. For 46 folds:
$0.001 \times 2^{46} = 0.001 \times 7.03687 \times 10^{13}$ cm ≈ $7.03687 \times 10^8$ km ≈ $703,687$ km.
Hence, the assertion is true. The reason correctly explains the assertion as it describes the exponential growth mechanism where each fold doubles the thickness.

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 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

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, Folding Paper

b) 32.768 cm
[Solution Description]
The thickness doubles with each fold. So after 15 folds, thickness = $0.001 \times 2^{15}$ cm. Calculating step by step:
$2^1 = 2$
$2^2 = 4$
...
$2^{10} = 1024$
$2^{15} = 32768$
Final thickness = $0.001 \times 32768 = 32.768$ cm

Key Concept: Exponents

b) $10^5$
[Solution Description]
Using the exponent rule $m^a \times n^a = (m \times n)^a$, we can rewrite $2^5 \times 5^5$ as $(2 \times 5)^5 = 10^5$.

Key Concept: Operations with Exponents

b) $7^7$
[Solution Description]
Using the exponent rule $n^a \times n^b = n^{a+b}$, we can simplify $7^3 \times 7^4$ as $7^{3+4} = 7^7$.

Key Concept: Scientific Notation

c) $4.5 \times 10^5$
[Solution Description]
Scientific notation is expressed as $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. For $450,000$, we move the decimal point 5 places to the left to get $4.5$, so the exponent $y = 5$. Thus, $450,000 = 4.5 \times 10^5$.

Key Concept: Exponential Notation and Operations

a) $3^3 \times 7^2 \times 2^{-2}$
[Solution Description]
To simplify $\frac{3^5 \times 7^3 \times 2^4}{3^2 \times 7 \times 2^6}$, we use the rule $n^a \div n^b = n^{a-b}$ for each common base.
Step 1: Simplify the exponents of 3.
$3^5 \div 3^2 = 3^{5-2} = 3^3$.
Step 2: Simplify the exponents of 7.
$7^3 \div 7^1 = 7^{3-1} = 7^2$.
Step 3: Simplify the exponents of 2.
$2^4 \div 2^6 = 2^{4-6} = 2^{-2}$.
Final simplified form: $3^3 \times 7^2 \times 2^{-2}$.

Key Concept: Negative Exponents and Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the given assertion, first simplify $\left(\frac{1}{5}\right)^{-3}$ using the property of negative exponents:
$\left(\frac{1}{5}\right)^{-3} = 5^3 = 125.$
Now multiply it by $10^4$:
$125 \times 10^4 = 1.25 \times 10^6.$
Thus, Assertion is true.
Reason states the definition of negative exponents and scientific notation, which is also correct.
Furthermore, Reason correctly explains how the simplification in the Assertion was achieved.

Key Concept: Scientific Notation, Exponent Operations

c) $3.081 \times 10^8$
[Solution Description]
Scientific notation represents a number as $a \times 10^b$, where $1 \leq |a| < 10$ and $b$ is an integer. For $308100000$, we move the decimal point to after the first digit: $3.081 \times 10^8$

Key Concept: Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial thickness of the paper is $0.001 \, \text{cm}$. After 1 fold, it becomes $0.001 \times 2 = 0.002 \, \text{cm}$. After 2 folds, it becomes $0.002 \times 2 = 0.004 \, \text{cm}$, and so on. After 5 folds, the thickness will be $0.001 \times 2^5 = 0.032 \, \text{cm}$. Thus, the Assertion is true. The Reason states that the thickness doubles after each fold, which is also true and correctly explains the Assertion because the exponential growth model ($2^n$) is applied here.

Key Concept: Exponential Growth: Doubling Effect

b) 4.096 cm
[Solution Description]
The thickness doubles after each fold. For 12 folds, the total increase in thickness can be calculated using the formula:
$\text{Final Thickness} = \text{Initial Thickness} \times 2^n$
Here, $n = 12$ and Initial Thickness = 0.001 cm.
Substituting the values:
$\text{Final Thickness} = 0.001 \, \text{cm} \times 2^{12}$
Calculate $2^{12} = 4096$:
$0.001 \, \text{cm} \times 4096 = 4.096 \, \text{cm}$
Therefore, the thickness after 12 folds is 4.096 cm.

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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that $5^3 = 125$. This is correct because $5^3 = 5 \times 5 \times 5 = 125$. The reason explains the general definition of exponential notation, where $n^a$ means $n$ multiplied by itself $a$ times. Since the assertion is a specific example of this rule and correctly applies it, the reason is the correct explanation of the assertion.

Key Concept: Power Notation, Negative Exponents

c) $4$
[Solution Description]
We know that $81$ can be written as $3^4$. So, $\frac{1}{81} = 3^{-4}$. Therefore, $3^{-x} = 3^{-4}$. Since the bases are equal, the exponents must also be equal. Hence, $-x = -4$, which gives $x = 4$.

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: Exponentiation of Negative Numbers

b) $9$
[Solution Description]
The expression $(-3)^2$ means $-3$ multiplied by itself twice. So, $(-3) \times (-3) = 9$.

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: Multiplication of Exponents

b) $5^5$
[Solution Description]
To simplify $5^4 \times 5^{-2} \times 5^3$, we use the rule for multiplying exponents with the same base: $n^a \times n^b = n^{a+b}$.
Here, the exponents are added together: $4 + (-2) + 3 = 5$.
So, the simplified form is $5^5$.

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: Multiplying exponents, Dividing exponents

c) 1
[Solution Description]
First, simplify the numerator $5^6 \times 5^{-2}$ using the rule $n^a \times n^b = n^{a + b}$. So, $5^6 \times 5^{-2} = 5^{6 + (-2)} = 5^4$. Next, simplify the denominator $5^3 \div 5^{-1}$ using the rule $n^a \div n^b = n^{a - b}$. So, $5^3 \div 5^{-1} = 5^{3 - (-1)} = 5^4$. Now, the expression becomes $\left(\frac{5^4}{5^4}\right)^2 = \left(5^{4 - 4}\right)^2 = \left(5^0\right)^2 = 1^2 = 1$.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $3^5 \times 3^{-2}$, we use the exponent rule for multiplication:
$n^a \times n^b = n^{a + b}$.
Applying this:
$3^5 \times 3^{-2} = 3^{5 + (-2)} = 3^3$.
Thus, the Assertion is true.
The Reason correctly states the rule used to solve it, so it is also true and explains the Assertion.

Key Concept: Prime factorization in exponential form

a) $2^3 \times 3^4$
[Solution Description]
To find the prime factorization of 648, start dividing by the smallest prime number, which is 2:
$648 \div 2 = 324 \\ \\ 324 \div 2 = 162 \\ \\ 162 \div 2 = 81$
Now, 81 is not divisible by 2, so move to the next prime number, 3:
$81 \div 3 = 27 \\ \\ 27 \div 3 = 9 \\ \\ 9 \div 3 = 3 \\ \\ 3 \div 3 = 1$
The prime factors are $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$. In exponential form: $2^3 \times 3^4$.

Key Concept: Exponentiation, Verification

b) -6912
[Solution Description]
Step 1: Calculate $(-3)^2 = 9$
Step 2: Calculate $4^3 = 64$
Step 3: Multiply results from Step 1 and Step 2:
9 × 64 = 576
Step 4: Simplify denominator:
$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$
$(-3)^{-1} = -\frac{1}{3}$
So denominator is $\frac{1}{4} \times -\frac{1}{3} = -\frac{1}{12}$
Step 5: Divide numerator (576) by denominator (-1/12):
576 ÷ (-1/12) = 576 × (-12) = -6912
Final simplified expression evaluates to $-6912$.

Key Concept: Exponential form representation

d) $5^2 \times 7^3$
[Solution Description]
The expression $5 \times 5 \times 7 \times 7 \times 7$ represents two 5's and three 7's multiplied together.
In exponential form, this becomes $5^2 \times 7^3$.

Key Concept: Prime factorization, Exponential form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's perform the prime factorization of 3600:
1. Divide 3600 by 2 (smallest prime): 3600 ÷ 2 = 1800
2. Continue dividing by 2: 1800 ÷ 2 = 900 → 900 ÷ 2 = 450 → 450 ÷ 2 = 225 (total four divisions by 2)
3. Next divide by 3 (next prime): 225 ÷ 3 = 75 → 75 ÷ 3 = 25 (total two divisions by 3)
4. Finally divide by 5: 25 ÷ 5 = 5 → 5 ÷ 5 = 1 (total two divisions by 5)
Thus, 3600 = $2^4 \times 3^2 \times 5^2$, which matches the assertion. The reason correctly describes the process of prime factorization. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Multiplication of Powers with the Same Base

d) $3^7$
[Solution Description]
To simplify $3^5 \times 3^2$, we use the property of exponents: $n^a \times n^b = n^{a + b}$. Here, the base is the same (3), so we add the exponents:
$3^5 \times 3^2 = 3^{5+2} = 3^7$
Therefore, the simplified form of the expression is $3^7$.

Key Concept: Properties of Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $5^3 \times 5^{-1} = 5^2$. Let's verify this using the product of powers property mentioned in the reason.
Step 1: Apply the product of powers property, which states $n^a \times n^b = n^{a+b}$. Here, $n = 5$, $a = 3$, and $b = -1$.
Step 2: Calculate the exponents: $3 + (-1) = 2$.
Step 3: Thus, $5^3 \times 5^{-1} = 5^{3 + (-1)} = 5^2$.
Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Power of a Power

a) $2^{12}$
[Solution Description]
Using the power of a power property $(n^a)^b = n^{a \times b}$, we can simplify $(2^3)^4$ as follows:
$(2^3)^4 = 2^{3 \times 4} = 2^{12}$

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: 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

b) 8.5
[Solution Description]
First, evaluate $7^0$ and $4^{-2}$ separately:
$7^0 = 1 \quad \text{(Any non-zero number to the power of 0 is 1)}$
$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
Now add these results and multiply by 8:
$(1 + \frac{1}{16}) \times 8 = \left(\frac{17}{16}\right) \times 8 = \frac{17 \times 8}{16} = \frac{136}{16} = 8.5$

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: 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: Negative exponents, Zero exponent

c) 10
[Solution Description]
Using the property $n^a \times n^b = n^{a+b}$, we can rewrite the equation as:
$5^{a-3} = 5^7$.
Since the bases are equal, the exponents must also be equal: $a - 3 = 7$.
Solving for $a$, we get $a = 7 + 3 = 10$.

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: 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: Exponents and Division

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $2^{10} \div 2^4$, we use the exponent division rule: $\frac{a^m}{a^n} = a^{m-n}$ where $a \neq 0$. Here, the base is 2, so:
$2^{10} \div 2^4 = 2^{10-4} = 2^6$.
The assertion correctly applies the exponent division rule, and the reason accurately states the rule used. Therefore, both are true, and the reason explains the assertion.

Key Concept: Exponent Division

b) $2^{75}$
[Solution Description]
Using the exponent division rule, $n^a \div n^b = n^{a-b}$. Here, $n = 2$, $a = 100$, and $b = 25$. So, $2^{100} \div 2^{25} = 2^{100-25} = 2^{75}$.

Key Concept: Combining exponents

c) $3^5$
[Solution Description]
Using the exponent addition rule: $n^a \times n^b = n^{a + b}$. For $3^{4} \times 3^{-1} \times 3^{2}$:
$3^{4} \times 3^{-1} \times 3^{2} = 3^{4 - 1 + 2} = 3^5$
The simplified form is $3^5$.

Key Concept: Exponent Subtraction Rule

b) $5^{5}$
[Solution Description]
Using the exponent subtraction rule, when dividing powers with the same base, subtract the exponents:
$5^{3} \div 5^{-2} = 5^{3 - (-2)} = 5^{5}$

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: 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: Negative Exponents, Zero Exponents

b) $\frac{1}{9}$
[Solution Description]
To solve this problem, we will use the concepts of negative exponents and zero exponents.
Step 1: Evaluate $(-3)^{-2}$ using the negative exponent rule:
$(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$
Step 2: Evaluate $4^0$ using the zero exponent rule:
$4^0 = 1$
Step 3: Multiply the results from Step 1 and Step 2:
$\frac{1}{9} \times 1 = \frac{1}{9}$
The final answer is $\frac{1}{9}$.

Key Concept: Zero Exponent Evaluation

c) $\frac{10}{9}$
[Solution Description]
Break it into two parts:
$(7)^0 = 1 \text{ (any non-zero number to the power 0 is 1)}$
$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
Now add them together:
$1 + \frac{1}{9} = \frac{10}{9}.$

Key Concept: Exponent Property Application

b) $-4$
[Solution Description]
Rewrite $\frac{1}{16}$ as a power of 2:
$\frac{1}{16} = 2^{-4} \text{ because } 16 = 2^4.$
So,
$2^x = 2^{-4} \implies x = -4.$

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: Powers of 10 - Expanded Form

a) $(3 \times 10^3) + (4 \times 10^2) + (7 \times 10^1) + (5 \times 10^0)$
[Solution Description]
To express $3475$ in expanded form using powers of 10, we break it into each digit multiplied by its place value:
$3475 = (3 \times 10^3) + (4 \times 10^2) + (7 \times 10^1) + (5 \times 10^0)$

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: Large Numbers - Naming Conventions

a) Arab
[Solution Description]
According to the Indian numbering system:
$10^9 = 100 \times 10^7 = \text{arab}$

Key Concept: Scientific Notation

c) $4.5 \times 10^3$
[Solution Description]
To convert 4,500 to scientific notation:
1. Move the decimal point after the first non-zero digit: 4.5
2. Count how many places the decimal moved from its original position (from after the last zero to after 4): 3 places.
3. Thus, $4,500 = 4.5 \times 10^3$.

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number 3,00,00,000 can be written as $3 \times 10^6$ in scientific notation. This is correct because:
3,00,00,000 = 3 × 10,00,000 = $3 \times 10^6$.
The reason explains the general rule of scientific notation, which is accurate: a number is expressed as $x \times 10^y$, where $x$ is between 1 and 10, and $y$ is an integer.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Magnitude Comparison

b) $3.2 \times 10^8$
[Solution Description]
To compare magnitudes, focus on the exponent $y$ in $x \times 10^y$. A higher exponent indicates a larger magnitude regardless of the coefficient. Here, the exponents are:
- $3.2 \times 10^8$ has exponent 8,
- $5.6 \times 10^6$ has exponent 6.
Since $8 > 6$, $3.2 \times 10^8$ has a larger magnitude.

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, 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, Large Numbers

b) $3.0 \times 10^{22}$ m
[Solution Description]
To convert the given large number to scientific notation, we need to write it in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. Given number: 30,00,00,00,00,00,00,00,00,000 m Step 1: Count the number of digits after the first digit (3). The number can be written as 3 followed by 22 zeros, so total digits = 23. Step 2: Express the number as $3.0 \times 10^{22}$ m, since moving the decimal point 22 places to the right gives the original number. Therefore, the correct scientific notation is $3.0 \times 10^{22}$ m.

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: 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: Historical Context

b) 100
[Solution Description]
To find how many ayutas make up one niyuta, divide the value of a niyuta by the value of an ayuta:
$\frac{10^{11}}{10^9} = 10^{11-9} = 10^2 = 100$
Therefore, 100 ayutas make up one niyuta.

Key Concept: Scientific Notation, Real-World Comparisons

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the ratio of stars to ants using their given values:
$\frac{2 \times 10^{23}}{2 \times 10^{16}} = 10^{23-16} = 10^7$
This confirms that the assertion is correct. The reason states that the ratio can be found by subtracting the exponents, which is a valid rule for division in scientific notation. Thus, the reason not only supports but also 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 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: Powers of 10

b) $(1 \times 10^2) + (7 \times 10^1) + (2 \times 10^0)$
[Solution Description]
Break down the number $172$ into its expanded form using powers of 10 as follows:
$172 = (1 \times 100) + (7 \times 10) + (2 \times 1) = (1 \times 10^2) + (7 \times 10^1) + (2 \times 10^0)$.

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, Magnitude Comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare $3.5 \times 10^7$ and $4.8 \times 10^6$, we examine their exponents. The exponent of the first number is 7, while the exponent of the second number is 6. Since 7 > 6, $3.5 \times 10^7$ is indeed greater than $4.8 \times 10^6$, regardless of the coefficients. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Scientific Notation, Expanded Form

a) $7.2381 \times 10^6$
[Solution Description]
Step 1: Add all the terms in the expanded form to get the standard number.
$7 \times 10^6 = 7,000,000$
$2 \times 10^5 = 200,000$
$3 \times 10^4 = 30,000$
$8 \times 10^3 = 8,000$
$1 \times 10^2 = 100$
Step 2: Sum all the values: $7,000,000 + 200,000 = 7,200,000$; $7,200,000 + 30,000 = 7,230,000$; $7,230,000 + 8,000 = 7,238,000$; $7,238,000 + 100 = 7,238,100$.
Step 3: Convert $7,238,100$ to scientific notation by writing it as $7.2381 \times 10^6$.

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: 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: 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: Weight estimation, Unit conversion

c) 15,000 coins
[Solution Description]
To find the number of 1-rupee coins that equal Roxie's weight, we first convert her weight to grams and then divide by the weight of one coin.
Step-by-step calculation:
1. Roxie's weight = 45 kg = $45 \times 1000 = 45,000$ grams
2. Weight of one coin = 3 grams
3. Number of coins = $\frac{45,000}{3} = 15,000$ coins

Key Concept: Exponential Growth, Time scales

b) 94,500 seconds
[Solution Description]

The bacterial population doubles every hour, so we use the formula:

$P = 100 \times 2^t$

where P is the final population and t is the time in hours.

We need the population to reach 8 billion:

$100 \times 2^t = 8,000,000,000$

$2^t = \frac{8,000,000,000}{100}$

$2^t = 80,000,000$

Now,

$t = \log_2(80,000,000) \approx 26.25 \text{ hours}$

Convert hours into seconds:

$26.25 \times 3600 = 94,500 \text{ seconds}$

So, it would take approximately 94,500 seconds, or about 26.25 hours, for the bacterial population to reach 8 billion cells.

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: Linear Growth vs. Exponential Growth

c) Covering distance between Earth and Moon with steps of 20 cm each
[Solution Description]
Linear growth involves adding a fixed amount repeatedly, such as walking steps of equal length. Among the options, covering distance with equal-sized steps $(20 + 20 + \ldots)$ is linear growth.

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: Scientific Notation, Earth's Water Volume

b) $6.93 \times 10^{21}$
[Solution Description]
First, convert 200 ml to liters:
$200 \text{ ml} = 0.2 \text{ liters}$
Now divide the total volume by the volume of one glass:
$\frac{1.386 \times 10^{21}}{0.2} = 6.93 \times 10^{21}$

Key Concept: Exponential Notation, Universe Age

c) $4.35 \times 10^{17}$ seconds
[Solution Description]
First, convert 13.8 billion years to seconds:
$13.8 \times 10^9 \text{ years} \times 3.154 \times 10^7 \text{ seconds/year} = 13.8 \times 3.154 \times 10^{16}$
$= 43.5252 \times 10^{16} \approx 4.35 \times 10^{17} \text{ seconds}$

Key Concept: Comparing large numbers

c) 1000
[Solution Description]
1 billion is $10^9$ and 1 million is $10^6$. To find how many millions make a billion, divide 1 billion by 1 million:
$\frac{10^9}{10^6} = 10^{9-6} = 10^3 = 1000$
So, 1000 million make 1 billion.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 1 crore is equal to $10^7$ in the Indian numbering system. According to the syllabus, a crore is indeed defined as $10^7$, which is equivalent to 100 lakhs since 1 lakh equals $10^5$. Therefore, both the Assertion (A) and Reason (R) are true, and the Reason correctly explains the Assertion.

Key Concept: Scientific Notation and Large Number Naming Systems

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this, let us analyze both the Assertion (A) and Reason (R).
- **Assertion (A):** The number $5,00,00,000$ can be written in scientific notation as $5 \times 10^6$. This is correct because:
$5,00,00,000 = 5 \times 10,00,000 = 5 \times 10^6$ (since $10,00,000 = 10^6$).
- **Reason (R):** In the Indian numbering system, a lakh is $1,00,000 = 10^5$ and a crore is $1,00,00,000 = 10^7$. This is also correct.
However, the reason does not explain why $5,00,00,000$ becomes $5 \times 10^6$ in scientific notation. It only states facts about the Indian numbering system. Therefore, while both (A) and (R) are true, (R) is not the correct explanation of (A).
Hence, the correct option is (b).

Key Concept: Standard Form

b) 4,200,000
[Solution Description]
To convert $4.2 \times 10^6$ to standard numerical form, multiply 4.2 by $10^6$ (which is 1,000,000). This gives $4.2 \times 10^6 = 4,200,000$.

Key Concept: Scientific Notation

b) $3 \times 10^{20}$
[Solution Description]
To convert the given number to scientific notation:
$30,00,00,00,00,00,00,00,00,000 = 3 \times 10^{20} \, \text{meters}$
We count the number of zeroes after the digit '3', which is 20, and write it in the form $x \times 10^y$.

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: Linear vs Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if a pond is fully covered with lotuses on the 30th day and the lotuses double every day, then the pond was half-covered on the 29th day. This is correct because exponential growth involves doubling (multiplying by 2), so one day before full coverage (day 30) would be half coverage (day 29). The reason explains that exponential growth is multiplicative, which correctly justifies the assertion since doubling is an example of multiplicative growth.

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: 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

c) 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, if the pond is fully covered on the 30th day, it was half-covered on the 29th day.

Key Concept: Currency and Large Numbers

c) Hungary
[Solution Description]
In 1946, Hungary issued a currency note with the denomination of 1 sextillion pengő ($10^{21}$ pengő).

Key Concept: Scientific Notation

c) $8 \times 10^6$
[Solution Description]
To express 80,00,000 in scientific notation, we write it as $8 \times 10^6$. Here, the coefficient $x$ is 8, and the exponent $y$ is 6 because we move the decimal point 6 places to the left.

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: Large Number Facts

b) $1 \times 10^{11}$
[Solution Description]
The number of stars in the Milky Way galaxy is approximately $1 \times 10^{11}$, which is 100 billion stars.

Key Concept: Scientific Notation

b) $7.004 \times 10^{10}$
[Solution Description]
To convert 70,04,00,00,000 to scientific notation:
Step 1: Count the number of digits after the first digit (7). Here, there are 10 digits after 7.
Step 2: Write the coefficient as a number between 1 and 10. Move the decimal point after the first digit: $7.004$.
Step 3: Multiply by $10$ raised to the power equal to the number of places the decimal moved (10).
Thus, $70,04,00,00,000 = 7.004 \times 10^{10}$.

Key Concept: Scientific Notation, Real-World Application

c) $3.17 \times 10^{15}$ years
[Solution Description]
Step 1: Calculate the total time required in seconds.
Total stars = $10^{23}$
Time per star = 1 second
Total time = $10^{23}$ seconds
Step 2: Convert seconds to years.
Seconds in a year = $3.15 \times 10^7$
Total years = $\frac{10^{23}}{3.15 \times 10^7} = \frac{1}{3.15} \times 10^{23-7} = 0.317 \times 10^{16}$
Step 3: Adjust the coefficient to be between 1 and 10.
$0.317 \times 10^{16} = 3.17 \times 10^{15}$ years
It would take approximately $3.17 \times 10^{15}$ years.

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.

Key Concept: Scientific Notation, Exponent Importance

c) $8.02 \times 10^4$
[Solution Description]
To find how many times farther the exoplanet is compared to the Sun:
Step 1: Write down both distances in scientific notation.
- Exoplanet distance: $1.2 \times 10^{16}$ m
- Sun distance: $1.496 \times 10^{11}$ m
Step 2: Divide the exoplanet distance by the Sun distance.
$\frac{1.2 \times 10^{16}}{1.496 \times 10^{11}} = \frac{1.2}{1.496} \times 10^{16-11} = 0.80214 \times 10^5$
Step 3: Adjust the coefficient to be between 1 and 10.
$0.80214 \times 10^5 = 8.0214 \times 10^4$
The exoplanet is approximately $8.02 \times 10^4$ times farther than the Sun.