Key Concept: Exponential Growth

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

Key Concept: Exponential Growth

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

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

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: 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, 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

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 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 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 and Operations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(-3)^4 \times (-3)^5 = (-3)^9$. According to the rule of exponents for multiplication with the same base, $n^a \times n^b = n^{a+b}$. Here, the base is $-3$, and the exponents are $4$ and $5$. Adding them gives $4 + 5 = 9$. Therefore, $(-3)^4 \times (-3)^5 = (-3)^{9}$ is correct. The reason correctly explains the assertion by stating the exponent addition rule for multiplication with the same base. Both the assertion and reason are true, and the reason justifies the assertion.

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: Prime Factorization, Exponential Form

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

Key Concept: 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: 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: Scientific Notation

b) $1.31 \times 10^2$ cm
[Solution Description]
The thickness is approximately $131 \text{ cm}$. In scientific notation, this is written as:
$1.31 \times 10^2 \text{ cm}.$

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 $0.001 \, \text{cm}$. 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: Negative Exponents

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

Key Concept: Exponent Rules

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

Key Concept: 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 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: 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: 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: 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: 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: 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 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: 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: Expressing Numbers in Exponential Form

b) $5^2 \times 7^3$
[Solution Description]
To express $5 \times 5 \times 7 \times 7 \times 7$ in exponential form, count the number of times each base is multiplied by itself.
5 is multiplied by itself 2 times: $5^2$.
7 is multiplied by itself 3 times: $7^3$.
The exponential form is $5^2 \times 7^3$.

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: 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: 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: 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: 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: 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: Negative exponents and zero exponents

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

Key Concept: 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: 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: Exponential Growth

b) $2^{29}$
[Solution Description]
Since the lotuses double each day, if the pond is fully covered on the 30th day, it must have been half-covered on the 29th day. Let the full coverage be $2^{30}$, then half coverage is $\frac{2^{30}}{2} = 2^{29}$.

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: 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: 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: 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: 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: Negative exponents, Exponent subtraction rule

c) 3
[Solution Description]
First, multiply the terms in the numerator: $10^{-4} \times 10^6 = 10^{-4 + 6} = 10^2$. Then, divide by the denominator using the exponent subtraction rule: $\frac{10^2}{10^{-1}} = 10^{2 - (-1)} = 10^3$. Thus, $x = 3$.

Key Concept: Exponent Subtraction Rule

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, apply the exponent subtraction rule: $n^a \div n^b = n^{a-b}$. Here, $a = 3$ and $b = -1$.
Calculation: $5^3 \div 5^{-1} = 5^{3 - (-1)} = 5^{4}$.
The Assertion is true because $5^3 \div 5^{-1} = 5^4$.
The Reason is also true as it correctly states the exponent subtraction rule. Moreover, the Reason explains why the Assertion holds true.

Key Concept: Negative and Zero Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that $5^0 = 1$, which is true as per the general rule that any non-zero number raised to the power of 0 is 1. The reason provides the correct explanation for this: when you divide $n^a$ by itself, it equals $n^{a-a} = n^0$, and any number divided by itself is 1. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Negative and Zero Exponents

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

Key Concept: 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 Exponents

c) $\frac{1}{9}$
[Solution Description]
To find the value of $3^{-2}$, we use the property of negative exponents:
$3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}$
Therefore, the correct answer is $\frac{1}{9}$.

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, 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: 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

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

c) $7.2 \times 10^4$
[Solution Description]
To convert 72,000 to scientific notation:
1. Move the decimal point after the first non-zero digit: 7.2
2. Count how many places the decimal moved from its original position (from after the last zero to after 7): 4 places.
3. Thus, $72,000 = 7.2 \times 10^4$.

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

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, 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, 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

b) $7.004 \times 10^{10}$
[Solution Description]
To express $70,04,00,00,000$ in scientific notation:
1. Identify the coefficient: Move the decimal point to the left until it is after the first non-zero digit: $7.004$.
2. Count the number of places moved: The decimal moves 10 places to the left.
3. Write in scientific notation: $7.004 \times 10^{10}$.

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, 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

b) $5.976 \times 10^{24}$ kg
[Solution Description]
To write the number in scientific notation:
1. Place the decimal after first digit: $5.976$
2. Count how many places we moved decimal from original number: 24 places
3. So the scientific notation is $5.976 \times 10^{24}$

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: 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: 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

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

Key Concept: Scientific Notation

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, 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-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: Tulābhāra Donation Cost

b) Rs.3150
[Solution Description]
To find the worth of the donated jaggery, multiply Roxie's weight by the cost per kg of jaggery.
Given:
- Roxie's weight = 45 kg
- Cost of 1 kg jaggery = Rs.70
Calculation:
$\text{Worth of jaggery} = 45 \times 70 = Rs.3150$

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the worth of donated jaggery, we multiply Roxie's weight (45 kg) by the cost per kg of jaggery (Rs.70). So, $45 \times 70 = Rs.3150$. The assertion is true as the calculation is correct. The reason is also true because it correctly explains the formula used to calculate the worth of donated goods. Additionally, the reason is the correct explanation for the assertion since it provides the underlying principle for the calculation.

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: 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: Linear vs Exponential, Real-World Context

b) Exponential becomes more than 50,000x faster by 20 folds
[Solution Description]

For exponential growth:

$T = 0.001 \times 2^n$

For linear growth:

$L = 0.001 + 0.002n$

Here, T represents exponential thickness and L represents linear thickness.

For example, after 20 folds:

$T = 0.001 \times 2^{20}$

$T = 0.001 \times 1,048,576 = 1048.576 \text{ cm}$

Linear growth after 20 folds:

$L = 0.001 + 0.002(20)$

$L = 0.001 + 0.04 = 0.041 \text{ cm}$

Now compare them:

$\frac{1048.576}{0.041} \approx 25,575$

So, after 20 folds, exponential growth gives a thickness about 25,575 times greater than linear growth.

Also, the increase during the $20^{\text{th}}$ fold is:

$0.001 \times 2^{19} = 524.288 \text{ cm}$

Linear growth adds only:

$0.002 \text{ cm}$

$\frac{524.288}{0.002} = 262,144$

So, by the $20^{\text{th}}$ fold, the rate of exponential growth is more than 50,000 times faster than 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: 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: 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: 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: 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: Indian and International Number Systems, Powers of 10

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this, we first verify the assertion:
1. According to the Indian system, $1 \text{ kharab} = 100 \times 1 \text{ arab} = 100 \times 1,00,00,00,000 = 1,00,00,00,00,000$ or $10^{11}$.
2. In the international system, $100 \text{ billion} = 100 \times 1,000,000,000 = 100,000,000,000$ or $10^{11}$.
Thus, the Assertion is true.
Next, we examine the Reason:
- In the Indian system, each term is multiplied by $100$ (e.g., $1 \text{ lakh} = 100 \times 10^3$, $1 \text{ crore} = 100 \times 10^5$).
- In the international system, each term is multiplied by $1000$ (e.g., $1 \text{ million} = 1000 \times 10^3$, $1 \text{ billion} = 1000 \times 10^6$).
Therefore, the Reason is also true. However, while the numerical value equivalence in the assertion is correct, the reason does not directly explain why $1 \text{ kharab}$ equals $100 \text{ billion}$ as it only describes the multiplicative steps in each system. Hence, the Reason does not correctly explain the Assertion.

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: 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: 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 and Magnitude

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Scientific notation is written as $x \times 10^y$, where $x$ is a number between 1 and 10, and $y$ is an integer exponent. The exponent $y$ indicates the power of 10 by which the coefficient $x$ is multiplied, thus representing the order of magnitude of the quantity. For example, $3 \times 10^{20} \, \text{m}$ (distance to Milky Way center) has a much larger magnitude than $5.976 \times 10^{24} \, \text{kg}$ (Earth's mass) despite having a smaller coefficient because $20 < 24$. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

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

Key Concept: Exponent Rules

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

Key Concept: Exponential Growth 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: Population and Demographics, Scientific Notation

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

Key Concept: 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, 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: 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: Standard Form Conversion

c) $3.43 \times 10^6$
[Solution Description]
To convert 34,30,000 to standard form:
Step 1: Identify the coefficient by placing the decimal after the first non-zero digit: $3.43$.
Step 2: Count the number of places the decimal has moved from its original position (6 places).
Thus, $34,30,000 = 3.43 \times 10^6$.

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 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 and Exponents

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