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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the thickness of a paper after 10 folds is 1.024 cm when the initial thickness is 0.001 cm. Using the given formula $\text{Thickness} = 0.001 \times 2^n \text{ cm}$, let's verify this:
For $n = 10$,
$0.001 \times 2^{10} = 0.001 \times 1024 = 1.024 \text{ cm}.$
So, the assertion is true. The reason provides the correct formula used in the calculation, which directly explains why the assertion holds true. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: 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 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: 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: 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: 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 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: 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 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: 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: Exponential Growth, Doubling Effect

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that the thickness of a paper folded 20 times would exceed the height of Mount Everest, while the Reason provides the formula for calculating the thickness after $n$ folds and calculates the thickness for 20 folds as 10.48576 m, which is less than the height of Mount Everest (8848 m). Here, the Assertion is false because the calculated thickness (10.48576 m) is significantly less than the height of Mount Everest (8848 m). However, the Reason correctly explains the calculation using the exponential growth formula and arrives at the correct numerical result. Therefore, the Assertion is false, but the Reason is true.

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: 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: 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 Form

c) $(-3)^3 \times 2^2$
[Solution Description]
To convert the product into exponential form, count how many times each base is multiplied by itself. Here, $-3$ is multiplied three times, and $2$ is multiplied twice. Therefore:
$(-3) \times (-3) \times (-3) \times 2 \times 2 = (-3)^3 \times 2^2$
The correct exponential form is $(-3)^3 \times 2^2$.

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: 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: 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: 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: 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: 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: Multiplying exponents, Generalized forms

a) $-1$
[Solution Description]
Using the rule $n^a \times n^b = n^{a + b}$, we can combine all the exponents: $x^{-4} \times x^5 \times x^{-2} = x^{-4 + 5 + (-2)} = x^{-1}$. Therefore, the value of $k$ is $-1$.

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 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: 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: 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: 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: Division of Powers with the Same Base

c) $7^3$
[Solution Description]
To simplify $\frac{7^8}{7^5}$, we use the division of powers property: $n^a \div n^b = n^{a - b}$. Here, the base is the same (7), so we subtract the exponents:
$\frac{7^8}{7^5} = 7^{8-5} = 7^3$
Therefore, the simplified form of the expression is $7^3$.

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: 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: 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: 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: 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 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: Negative and Zero Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, the assertion states that $5^0 = 1$. According to the properties of exponents, any non-zero number raised to the power of 0 equals 1. Therefore, the assertion is true.
The reason given is that for any non-zero number $n$, $n^0 = 1$. This is a fundamental property of exponents and correctly explains why $5^0 = 1$. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: Exponent Division

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

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: Negative Exponents

b) $\frac{1}{10^5}$
[Solution Description]
The general form for negative exponents is $n^{-a} = \frac{1}{n^a}$. Applying this to $10^{-5}$, we get $\frac{1}{10^{+5}}$.

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: Combining Exponents

c) $4$
[Solution Description]
When multiplying powers with the same base, add the exponents:
$2^{-3} \times 2^{5} = 2^{-3 + 5} = 2^{2} = 4$

Key Concept: Exponent subtraction rule, Negative exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $\frac{n^5 \times n^{-2}}{n^{-3}}$, first multiply the numerator terms by adding their exponents: $n^5 \times n^{-2} = n^{5 + (-2)} = n^3$. Then, divide by the denominator using the exponent subtraction rule: $\frac{n^3}{n^{-3}} = n^{3 - (-3)} = n^{6}$. The assertion is correct as it simplifies to $n^6$. The reason correctly states the rules applied in the simplification, explaining why the assertion holds true.

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: 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: 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: 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: 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: 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: Large Numbers, Powers of 10

c) 11
[Solution Description]
In the Indian numbering system:
$1 \text{ kharab} = 100 \text{ arab} = 100 \times 1,00,00,00,000 = 1,00,00,00,00,000$
Counting the zeros: 11 zeros.

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: 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: Scientific Notation, Exponent Importance

d) Assertion is false, but Reason is true.
[Solution Description]
To compare $3.6 \times 10^5$ and $36 \times 10^4$, we first convert both to the same form.
$3.6 \times 10^5 = 360,000$ and $36 \times 10^4 = 360,000$. Thus, both numbers are equal.
Assertion (A) is false because the two numbers are equal, not greater or smaller.
Reason (R) is also false because the magnitude depends on both the coefficient and the exponent, not just the exponent.
Therefore, the correct answer is (d).

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

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, Large Numbers

c) $7.89 \times 10^6$
[Solution Description]
To convert 7,89,00,000 to scientific notation, we first write it as 7.89 multiplied by a power of 10. Counting the digits after the first digit (7), we have 7 digits (including the 8 and 9). Therefore, the exponent is 6 (since we move the decimal point 6 places to the left from its original position). Hence, the scientific notation is $7.89 \times 10^6$.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert $70,04,00,00,000$ to scientific notation:
1. First, rewrite the number as $7,004 \times 10,00,00,000$.
2. Count the number of zeroes after 7,004, which is 9. However, moving the decimal from the end of 7,004 to after the first digit gives 7.004. So, add the count of zeroes plus the digits moved (9 + 3 = 12).
3. Therefore, the scientific notation is $7.004 \times 10^{10}$ because we moved the decimal 10 places to the left.
The assertion is correct as $7.004$ is between 1 and 10, and the exponent $10$ correctly represents the movement of the decimal point. The reason explains the rule for scientific notation accurately and justifies the 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, Mass Conversion

a) $5.976 \times 10^{24}$ kg
[Solution Description]
The given mass of the Earth is 59,76,00,00,00,00,00,00,00,00,00,000 kg.
Step 1: Rewrite the number without commas: 5976000000000000000000000 kg.
Step 2: Count the number of digits after the first digit (5). Total digits = 25.
Step 3: Move the decimal point to after the first non-zero digit (5.976).
Step 4: The exponent is the number of places moved: 24.
So, the scientific notation is $5.976 \times 10^{24}$ kg.

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, 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: 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, 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: Exponent Comparison

b) Stars in universe ($2 \times 10^{23}$)
[Solution Description]
Compare exponents of given quantities:
1. Milky Way stars ($1 \times 10^{11}$)
2. Stars in universe ($2 \times 10^{23}$)
3. Grains of sand ($1 \times 10^{21}$)
4. Highest exponent is 23 for stars in universe

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, 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: 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

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: 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: Worth of Donated Items

b) Rs.3150
[Solution Description]
The worth of donated jaggery can be calculated using the formula: $\text{Worth of jaggery} = \text{Roxie's weight} \times \text{Cost of 1 kg jaggery}$. Substituting the given values: $45 \times 70 = Rs.3150$.

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 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: Linear Growth vs. Exponential Growth

b) 1,92,20,00,000 steps
[Solution Description]
To find the number of steps, convert the total distance into centimeters and divide by the step size.
3,84,400 km = 3,84,400 $\times$ 100,000 cm = 38,44,00,00,000 cm.
Number of steps = Total distance / Step size = 38,44,00,00,000 cm / 20 cm = 1,92,20,00,000 steps.

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, 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 vs. Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that folding a paper 46 times makes its thickness exceed the Earth-Moon distance, which is true because exponential growth (represented as $0.001 \times 2^{46}$ km) results in a thickness of over 700,000 km, exceeding the Moon's distance (384,400 km). The Reason correctly explains why this happens: exponential growth multiplies the quantity by 2 at each step, leading to a rapid increase. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

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: 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: 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, 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: Scientific notation, Indian and International number systems

b) $5.976 \times 10^{24}$ kg
[Solution Description]
First, convert the given Indian number to International format by removing commas and grouping digits in sets of three from the right:
$59,76,00,00,00,00,00,00,00,00,00,000$ becomes $5,976,000,000,000,000,000,000,000$ kg.
Now express this in scientific notation: Identify the coefficient (between 1 and 10) and exponent.
Coefficient = $5.976$
Count the number of places the decimal moves from the end to after the first digit: 24 places.
Thus, scientific notation is $5.976 \times 10^{24}$ kg.

Key Concept: Standard Form

c) 7
[Solution Description]
To express 2 crores in standard form:
$2 \, \text{crores} = 2 \times 1,00,00,000 = 2 \times 10^7$
Here, the exponent is 7.

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: 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: Exponential Growth Calculation

c) 29th day[Solution Description]Exponential growth involves multiplication, where the quantity doubles each day. If the pond is fully covered on the $30^{th}$ day, it means the coverage doubled from the previous day. Hence, on the $29^{th}$ day, the pond must have been half-covered because doubling the coverage on the $29^{th}$ day results in full coverage on the $30^{th}$ day.

Key Concept: Linear vs Exponential, Paper Folding

a) Folding once exceeds all steps
[Solution Description]
1. Folded paper thickness:
$0.001 \times 2^{42} \text{ cm} = 4.398 \times 10^9 \text{ km}$
2. Earth-Moon steps:
$Each step = 20 cm → Total steps = 384,400 km / 0.0002 km = 1.922 \times 10^9 steps$
Comparison:
$4.398 \times 10^9 \text{ km} \text{ (folded paper)} > 384,400 \text{ km} \text{ (steps)}$

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: 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: 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: 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: 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: Indian System, Exponents

c) 250,000 crore
[Solution Description]
From the Indian system, 1 kharab $= 10^{11}$ rupees and 1 crore $= 10^7$ rupees. To convert kharab to crore, divide:
$\frac{25 \times 10^{11}}{10^7} = 25 \times 10^4 = 250000 \text{ crore}$
Thus, 25 kharab rupees equals 250,000 crore rupees.

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 $1 \times 10^7$.

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 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.