Key Concept: Exponential Growth, Comparison of Folds

b) 1024
[Solution Description]
The thickness after 10 folds is $0.001 \times 2^{10} = 1.024$ cm.
The thickness after 20 folds is $0.001 \times 2^{20} = 1048.576$ cm.
The ratio of thickness after 20 folds to 10 folds is:
$\frac{1048.576}{1.024} = 1024$

Key Concept: Scientific Notation

b) $2 \times 10^7$
[Solution Description]
Scientific notation expresses numbers in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. Since $2$ crores is $20,000,000$, it can be written as: $2 \times 10^7$

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, 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: 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, Magical Pond

c) Day 9
[Solution Description]
Since the lotuses triple daily, the pond would be one-third full one day before becoming full. Therefore, if full on day 10, it was one-third full on day 9.

Key Concept: Exponential Growth in Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given table, the initial thickness of the paper is $0.001 cm$. After each fold, the thickness doubles. Using the formula for exponential growth:
Thickness after $n$ folds = Initial thickness $\times 2^n$.
For $n = 7$, the thickness becomes $0.001 cm \times 2^7 = 0.128 cm$, which matches the assertion. The reason correctly explains why the thickness is $0.128 cm$ after 7 folds because it describes the doubling mechanism.

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

b) 1.024 cm
[Solution Description]
Each fold doubles the thickness of the paper. The formula for thickness after $n$ folds is:
$\text{Thickness} = \text{Initial Thickness} \times 2^n$
For $n = 10$,
$\text{Thickness} = 0.001 \, \text{cm} \times 2^{10} = 0.001 \times 1024 = 1.024 \, \text{cm}.$

Key Concept: Negative Exponents

c) $\frac{1}{25}$
[Solution Description]
Using the negative exponent rule $n^{-a} = \frac{1}{n^a}$, we have $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$.

Key Concept: Operations with Exponents, Negative Exponents

a) $\frac{2}{625}$
[Solution Description]
To evaluate $\left(\frac{2^{-3} \times 5^2}{10^{-2}}\right)^{-1}$:
Step 1: Rewrite the denominator $10^{-2}$ as $\frac{1}{10^2}$.
The expression becomes $\left(2^{-3} \times 5^2 \times 10^2\right)^{-1}$.
Step 2: Simplify $10^2$ to $(2 \times 5)^2 = 2^2 \times 5^2$.
The expression now is $\left(2^{-3} \times 5^2 \times 2^2 \times 5^2\right)^{-1}$.
Step 3: Combine like terms using $n^a \times n^b = n^{a+b}$.
For 2: $-3 + 2 = -1$. For 5: $2 + 2 = 4$.
So, $\left(2^{-1} \times 5^4\right)^{-1}$.
Step 4: Apply the exponent $-1$ to each term inside the parentheses.
$(2^{-1})^{-1} = 2^1$, $(5^4)^{-1} = 5^{-4}$.
Final result: $2^1 \times 5^{-4} = \frac{2}{5^4}$.
Simplified numerical value: $\frac{2}{625}$.

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

b) $3^5$
[Solution Description]
The expression $3 \times 3 \times 3 \times 3 \times 3$ has the base 3 multiplied by itself 5 times. Therefore, the exponential form is $3^5$.

Key Concept: Exponential Growth, Scientific Notation

c) 32.768 cm
[Solution Description]
The thickness doubles with each fold, so after $n$ folds, the thickness is given by $0.001 \times 2^n$ cm. For $n = 15$, the calculation is:
$0.001 \times 2^{15} = 0.001 \times 32768 = 32.768 \text{ cm}$

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: Exponent Properties, Exponential Growth

c) 320
[Solution Description]
The number of bacteria doubles every hour, so after $t$ hours, the count becomes $10 \times 2^t$. For $t = 5$, the calculation is:
$10 \times 2^5 = 10 \times 32 = 320$

Key Concept: Exponential Growth

c) 0.032 cm
[Solution Description]
The thickness doubles with each fold. For 5 folds, the thickness is calculated as:
$0.001 \text{ cm} \times 2^5 = 0.001 \text{ cm} \times 32 = 0.032 \text{ cm}$

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

b) $5^4$
[Solution Description]
The expression $5 \times 5 \times 5 \times 5$ means multiplying 5 by itself 4 times. In exponential notation, this is written as $5^4$.

Key Concept: Power of a Power, Zero Exponent

b) $64$
[Solution Description]
Any non-zero number raised to the power of 0 is 1, so $(5^2)^0 = 1$. For the second part, use the rule $(n^a)^b = n^{a \times b}$, so $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$. Multiplying these results together gives $1 \times 64 = 64$.

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: Power play, Operations with exponents

c) 128
[Solution Description]
First, simplify the numerator $(2^3 \times 2^5) \div (2^2)^2$. Using $n^a \times n^b = n^{a + b}$, $2^3 \times 2^5 = 2^{3 + 5} = 2^8$. Using $(n^a)^b = n^{a \times b}$, $(2^2)^2 = 2^{2 \times 2} = 2^4$. Now, $2^8 \div 2^4 = 2^{8 - 4} = 2^4$. Next, simplify the denominator $(2^{-1})^3 = 2^{-1 \times 3} = 2^{-3}$. Finally, divide the numerator and denominator: $\frac{2^4}{2^{-3}} = 2^{4 - (-3)} = 2^7 = 128$.

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: 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: Operations with exponents, Zero exponent

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, evaluate the expression inside the parentheses: $3^5 \div 3^{-2}$. Using the division rule of exponents, $n^a \div n^b = n^{a-b}$, we get:
$3^5 \div 3^{-2} = 3^{5-(-2)} = 3^{7}.$
Now, raise this result to the power of zero: $(3^7)^0$. According to the zero exponent rule, any non-zero number raised to the power of zero is equal to $1$. Hence, $(3^7)^0 = 1$.
Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: Prime Factorization, Exponentiation

b) $2^3 \times 3^5 \times 5$
[Solution Description]
First, perform prime factorization of 9720:
Divide by 2:
9720 ÷ 2 = 4860
4860 ÷ 2 = 2430
2430 ÷ 2 = 1215
1215 is divisible by 3:
1215 ÷ 3 = 405
405 ÷ 3 = 135
135 ÷ 3 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
Finally, 5 ÷ 5 = 1
So, the prime factors are:
2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5
In exponential form:
2^3 × 3^5 × 5^1
Therefore, the correct answer is $2^3 \times 3^5 \times 5$.

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]
First, perform the prime factorization of 540:
$540 = 2 \times 270 = 2 \times 2 \times 135 = 2 \times 2 \times 3 \times 45 = 2 \times 2 \times 3 \times 3 \times 15 = 2 \times 2 \times 3 \times 3 \times 3 \times 5$
Expressing this in exponential form gives $2^2 \times 3^3 \times 5^1$. Thus, the Assertion is true.
The Reason states that prime factorization breaks down a number into the product of prime numbers raised to their respective powers, which is correct and indeed explains why the Assertion holds true. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Properties of Powers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The property of exponents states that when two powers with the same base are multiplied, their exponents must be added. For example, $3^2 \times 3^4 = 3^{2+4} = 3^6$. The assertion correctly states this property, and the reason provides the correct explanation for the assertion.

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: Exponents, Negative exponents, Power of a power

b) $3^{-6}$
[Solution Description]
Step 1: Simplify the numerator inside the bracket using the property $n^a \times n^b = n^{a+b}$:
$3^4 \times 3^{-2} = 3^{4 + (-2)} = 3^2$
Step 2: Now divide the simplified numerator by the denominator using the property $n^a \div n^b = n^{a-b}$:
$\frac{3^2}{3^5} = 3^{2-5} = 3^{-3}$
Step 3: Apply the power of a power property $(n^a)^b = n^{ab}$ to the entire expression:
$\left(3^{-3}\right)^2 = 3^{-6}$
The simplified form is $3^{-6}$.

Key Concept: Product of Powers with the Same Base

a) $3^7$
[Solution Description]
Using the property of exponents that states $n^a \times n^b = n^{a+b}$, we can simplify $3^5 \times 3^2$ as follows:
$3^5 \times 3^2 = 3^{5+2} = 3^7$

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: Negative Exponents

c) 5
[Solution Description]
To simplify $5^{-3} \times 5^4$, we use the exponent rule $n^a \times n^b = n^{a+b}$. Here, $a = -3$ and $b = 4$.
$5^{-3} \times 5^4 = 5^{-3+4} = 5^1 = 5$

Key Concept: Zero exponent, Negative exponents

d) Any non-zero real number
[Solution Description]
Substitute $y = 4$ into the equation: $(x^0 + 4^{-1})^{-1} = 2$.
Simplify $x^0 = 1$ and $4^{-1} = \frac{1}{4}$: $(1 + \frac{1}{4})^{-1} = (\frac{5}{4})^{-1} = \frac{4}{5}$.
However, $\frac{4}{5} \neq 2$, so there is no solution. But since $x^0 = 1$ for any $x \neq 0$, $x$ can be any non-zero real number. The question may have an error or missing condition.

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: Exponent Rules

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $5^0 = 1$. According to the exponent rules, any non-zero number raised to the power of zero is indeed equal to one. This is derived from the property $x^a ÷ x^a = x^{a-a} = x^0 = 1$, where $x ≠ 0$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

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 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: 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: Exponent Rules

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $2^5 \div 2^3 = 4$. Let us verify this:
Step 1: Calculate $2^5$ which is $2 \times 2 \times 2 \times 2 \times 2 = 32$.
Step 2: Calculate $2^3$ which is $2 \times 2 \times 2 = 8$.
Step 3: Divide $32 \div 8 = 4$. So, the Assertion is true.
The Reason states the exponent rule $n^a \div n^b = n^{a-b}$. This is a valid rule as per the syllabus.
Applying this rule to the Assertion:
$2^5 \div 2^3 = 2^{5-3} = 2^2 = 4$, which matches the Assertion. Thus, the Reason correctly explains the Assertion.

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, Negative exponents

a) $7^3$
[Solution Description]
First, multiply the terms in the numerator using the exponent addition rule: $7^8 \times 7^{-3} = 7^{8 + (-3)} = 7^5$. Next, divide the result by the denominator using the exponent subtraction rule: $\frac{7^5}{7^2} = 7^{5 - 2} = 7^3$. Therefore, the simplified form is $7^3$.

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, 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: 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: 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: Negative exponents, Zero exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $x^0 + x^{-1} = \frac{x + 1}{x}$. According to the rules of exponents, $x^0 = 1$ for any non-zero $x$ (given in Reason), and $x^{-1} = \frac{1}{x}$ (also given in Reason). Substituting these into the Assertion: $x^0 + x^{-1} = 1 + \frac{1}{x} = \frac{x}{x} + \frac{1}{x} = \frac{x + 1}{x}$. Thus, both Assertion and Reason are true, and the Reason correctly explains the simplification.

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: Powers of 10, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $10^{-5} = \frac{1}{10^5}$. This is correct because, by definition, a negative exponent indicates the reciprocal of the base raised to the positive exponent. The reason provides the general form of this property: for any non-zero number $n$ and integer $a$, $n^{-a} = \frac{1}{n^a}$. Since the assertion aligns with the general rule stated in the reason, and the reason correctly explains the assertion, both are true and the reason is the correct explanation of the assertion.

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: Powers of 10 - Basic Conversion

b) 5074
[Solution Description]
To solve this, multiply each term by its respective power of 10 and add them together:
$5 \times 10^3 = 5000$
$7 \times 10^1 = 70$
$4 \times 10^0 = 4$
Adding them:
$5000 + 70 + 4 = 5074$
The standard form is 5074.

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: 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: Scientific Notation, Conversion

c) $3.43 \times 10^6$
[Solution Description]
To convert 34,30,000 to scientific notation, identify the coefficient between 1 and 10. Moving the decimal point 6 places to the left gives 3.43. Thus, the number becomes $3.43 \times 10^6$.

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

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: 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

d) $7.2 \times 10^6$
[Solution Description]
To convert 7,200,000 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (7.2).
2. Count the number of places moved (6 places to the left).
3. Write in the form $x \times 10^y$: $7.2 \times 10^6$.

Key Concept: Scientific Notation Conversion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion is true because $1,000,000 = 1 \times 10^6$, where the coefficient is 1 (which lies between 1 and 10) and the exponent 6 shows the decimal moved 6 places to the left. Reason is also true because it correctly describes the definition of scientific notation: the coefficient must satisfy $1 \leq x < 10$, and the exponent $y$ represents the number of decimal shifts. Further, the Reason accurately explains why the Assertion is correct.

Key Concept: Scientific Notation

c) 1,439,000,000,000 m
[Solution Description]
To convert $1.439 \times 10^{12}$ m to standard form:
1. The exponent $10^{12}$ means moving the decimal point 12 places to the right.
2. Starting from $1.439$, moving the decimal 12 places gives $1,439,000,000,000$ m.

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: Comparing Numbers

c) Ant population is larger
[Solution Description]
Comparing exponents:
1. Ants have exponent 16, humans have exponent 9
2. Larger exponent means bigger number regardless of coefficient
3. Therefore $2 \times 10^{16} > 8 \times 10^9$

Key Concept: Exponential vs. Linear Growth, Scientific Notation

a) $6.4 \times 10^9$ m
[Solution Description]
Each fold doubles the thickness, so the thickness after $n$ folds is given by $T = T_0 \times 2^n$, where $T_0$ is the initial thickness.
Initial thickness $T_0 = 0.1$ mm = $0.1 \times 10^{-3}$ m = $1 \times 10^{-4}$ m
Thickness after 46 folds:
$T = 1 \times 10^{-4} \times 2^{46}$
Calculate $2^{10} = 1024 \approx 10^3$
So, $2^{40} = (2^{10})^4 \approx (10^3)^4 = 10^{12}$
And $2^{46} = 2^{40} \times 2^6 = 10^{12} \times 64 = 6.4 \times 10^{13}$
Now, $T = 1 \times 10^{-4} \times 6.4 \times 10^{13} = 6.4 \times 10^9$ m
The distance to the Moon is $3.84 \times 10^8$ m, which is less than $6.4 \times 10^9$ m, confirming the claim.

Key Concept: Historical Context, Powers of 10

b) $10^{11}$
[Solution Description]
According to historical context from the syllabus, the term "niyuta" was used in ancient Indian texts like the *Lalitavistara* to refer to $10^{11}$.

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, Applications

c) $1 \times 10^{10}$
[Solution Description]
Step 1: Multiply the rate of calculations by the time to get the total calculations.
$1.25 \times 10^{15} \times 8 \times 10^{-6} = (1.25 \times 8) \times 10^{15-6} = 10 \times 10^9 = 1 \times 10^{10}$
Step 2: Answer is $1 \times 10^{10}$ calculations.

Key Concept: Powers of 10

b) $8 \times 10^6$
[Solution Description]
To express 80,00,000 using powers of 10:
Step 1: The coefficient is 8 (a number between 1 and 10).
Step 2: Count the number of zeroes, which is 6.
Hence, the standard form is $8 \times 10^6$.

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, Large numbers

c) $3.17 \times 10^{15}$ years
[Solution Description]
The time taken to count all the stars can be found by dividing the total number of stars by the counting rate (1 star per second).
Step-by-step calculation:
1. Total stars = $10^{23}$
2. Counting rate = 1 star/second
3. Time in seconds = $10^{23}$ seconds
Convert seconds to years for better understanding:
1 minute = 60 seconds
1 hour = 3600 seconds
1 day = 86,400 seconds
1 year ≈ 31,536,000 seconds (assuming 365.25 days/year)
Total years = $\frac{10^{23}}{3.1536 \times 10^7} \approx 3.17 \times 10^{15}$ years.

Key Concept: Age in Hours

b) 116160 hours
[Solution Description]
To convert days into hours, multiply the number of days by 24 (since there are 24 hours in a day).
Calculation:
$\text{Hours} = 4840 \times 24 = 116160 \text{ hours}$

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: Weight-Donation Cost Calculation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the cost of jaggery donated by Nanjundappa depends on Roxie’s weight and the price per kg of jaggery, which is mathematically represented as $\text{Worth of jaggery} = \text{Roxie’s weight} \times \text{Cost of 1 kg jaggery}$. This is correct because the total cost is calculated by multiplying the weight by the unit price. The reason explains the general principle behind Tulābhāra, where the worth of donated goods depends on the individual’s weight and the commodity's unit price. Since the reason correctly explains the assertion, both are true and related.

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

b) 0.128 cm
[Solution Description]
The thickness after $n$ folds follows exponential growth as $0.001 \times 2^n$ cm. For 7 folds:
Thickness = $0.001 \times 2^7$ cm = $0.001 \times 128$ cm = 0.128 cm.

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 building a ladder to the Moon with 20 cm steps would require 1,92,20,00,000 steps. This is true because linear growth involves a fixed increase (additive process) with each step. The reason correctly explains that linear growth is additive, as it involves a constant addition (20 cm per step). Therefore, both the assertion and reason are true, and the reason properly 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: Population Calculation

b) $2 \times 10^9$
[Solution Description]
The total population of sheep and goats is the sum of their individual populations:
$10^9 + 10^9 = 2 \times 10^9$
So, the total population is $2 \times 10^9$.

Key Concept: Comparison of Large Numbers

b) $2^{100}$
[Solution Description]
We compare $100^2$ and $2^{100}$:
- $100^2 = 10,000$
- $2^{100}$ can be simplified using logarithms or exponentiation:
$2^{10} = 1024 \approx 10^3$
$2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}$
Since $10^{30} > 10,000$, $2^{100}$ is larger than $100^2$.

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: 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: 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: Scientific Notation

b) $5.976 \times 10^{24}$
[Solution Description]
To convert the given number into scientific notation, we need to express it in the form $x \times 10^y$ where $1 \leq x < 10$. Starting with the number 59,76,00,00,00,00,00,00,00,00,00,000, we move the decimal point after the first digit (5). Counting the number of places moved gives us $5.976 \times 10^{24}$ because there are 24 digits after the 5.

Key Concept: Powers of 10, Scientific notation

c) 10,000 times larger
[Solution Description]
To find how many times larger $10^{82}$ is compared to $10^{78}$, we divide them:
$\frac{10^{82}}{10^{78}} = 10^{82-78} = 10^4$.
$10^4$ means 10,000 times larger.

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: Linear Growth

c) 1,922,000,000 steps
[Solution Description]
To find the number of 20 cm steps required to cover 3,84,400 km:
First, convert 3,84,400 km to cm:
$3,84,400 \text{ km} = 3,84,400 \times 1,00,000 \text{ cm} = 3,84,40,00,00,000 \text{ cm}$
Now, divide by step size (20 cm):
$\frac{3,84,40,00,00,000}{20} = 1,92,20,00,000 \text{ steps}$
Hence, 1,92,20,00,000 steps are needed.

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

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: Scientific Notation, Large Numbers

b) $5 \times 10^{12}$
[Solution Description]
First, identify the given number of stars: $1 \times 10^{11}$. Multiply this by 50 to find the number of stars in the new galaxy.
Calculation:
$1 \times 10^{11} \times 50 = 50 \times 10^{11}$
To express this in proper scientific notation, adjust the coefficient to be between 1 and 10:
$5 \times 10^{12}$
So, the correct answer is $5 \times 10^{12}$.

Key Concept: Currency and Large Numbers

c) $1 \times 10^{14}$
[Solution Description]
100 trillion can be expressed numerically as 100,000,000,000,000.
To write this in scientific notation, move the decimal point to just after the first digit (1), giving 1.00.
Count the number of places the decimal moved: 14 places to the left. Hence, the exponent is 14.
Therefore, 100 trillion is written as $1 \times 10^{14}$ in scientific notation.

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: Understanding Large Numbers

b) $3.6 \times 10^6$
[Solution Description]
To convert 3,600,000 into scientific notation, we need to express it as a product of a number between 1 and 10 and a power of 10.
First, move the decimal point from its original position after the last zero to just after the first non-zero digit (3). This gives us 3.6.
Then, count how many places the decimal moved: it moved 6 places to the left. Thus, the exponent is 6.
So, the scientific notation for 3,600,000 is $3.6 \times 10^6$.

Key Concept: Scientific Notation Application

c) $2 \times 10^7$
[Solution Description]
2 crores = 2,00,00,000.
To write in scientific notation, express it as $x \times 10^y$ where $1 \leq x < 10$. Here, $2,00,00,000 = 2 \times 10^7$.

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 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: Exponents and Large Numbers

c) $1 \times 10^7$
[Solution Description]
One crore is written numerically as $1,00,00,000$.
Step 1: The coefficient is $1$ (since it already lies between 1 and 10).
Step 2: The exponent is the number of zeros + 1 (for the last non-zero digit): $7$.
Thus, one crore in scientific notation is $1 \times 10^7$.