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

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: 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: Exponential Growth, Real-World Application

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

Key Concept: Exponents, Combined Growth

b) $6^4$
[Solution Description]
Using the exponent rule $m^a \times n^a = (mn)^a$, we can combine the exponents:
$3^4 \times 2^4 = (3 \times 2)^4 = 6^4$

Key Concept: Exponential Growth

c) 14th 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.
$\text{Half-covered day} = \text{Full day} - 1 = 15 - 1 = 14.$

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, Folding Paper Experiment

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

Key Concept: Exponentiation

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

Key Concept: 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: 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: 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 Growth

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

Key Concept: Exponential 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: 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 Operations: Power Rules

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

Key Concept: 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: Exponential Notation and Power of a Power

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(3^4)^5$ simplifies to $3^{20}$. This is true because applying the power of a power rule $(n^a)^b = n^{a \times b}$, we get $(3^4)^5 = 3^{4 \times 5} = 3^{20}$. The reason correctly explains this simplification using the power of a power rule. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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, 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

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

Key Concept: Multiplication of Exponents with Same Base

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

Key Concept: Division of exponents with the same base

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

Key Concept: 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: Verification of exponential expressions

d) It equals $16$.
[Solution Description]
To evaluate $(-2)^4$, multiply $-2$ four times:
$(-2)^4 = (-2) \times (-2) \times (-2) \times (-2).$
Since a negative number multiplied by itself an even number of times results in a positive number, the calculation proceeds as follows:
$(-2) \times (-2) = 4 \\ \\ 4 \times (-2) = -8 \\ \\ -8 \times (-2) = 16$
Thus, $(-2)^4 = 16$.

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: 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: 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: Power of a Power

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

Key Concept: Properties of Powers

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: 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: 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: 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: 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: 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: 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: Exponent Operations

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

Key Concept: 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: 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: 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: Negative Exponents, Multiplicative Inverse

d) $5^4$
[Solution Description]
First, convert all terms to base 5: $25 = 5^2$ and $125 = 5^3$. Rewrite the expression: $5^{-3} \times (5^2)^2 \div (5^3)^{-1} = 5^{-3} \times 5^4 \div 5^{-3}$. When dividing exponents with the same base, subtract them: $5^{-3+4-(-3)} = 5^{1+3} = 5^4$.

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

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

Key Concept: Exponent subtraction rule, 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: Negative exponents

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

Key Concept: Negative 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

b) 1
[Solution Description]
For any non-zero number or variable, raising it to the power of zero gives 1. Thus:
$(5x)^0 = 1$
Therefore, the correct answer is 1.

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: 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: Powers of 10, Expanded Form

b) $(8 \times 10^3) + (4 \times 10^2) + (9 \times 10^1) + (3 \times 10^0)$
[Solution Description]
To express 8493 using powers of 10, we break it down into its place values:
$8493 = (8 \times 1000) + (4 \times 100) + (9 \times 10) + 3$
Converting each term to powers of 10:
$8493 = (8 \times 10^3) + (4 \times 10^2) + (9 \times 10^1) + (3 \times 10^0)$

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: Negative Exponents

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

Key Concept: 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, 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

c) $6.03 \times 10^6$
[Solution Description]
To convert 6,030,000 to scientific notation:
1. Move the decimal point after the first non-zero digit: 6.03
2. Count how many places the decimal moved from its original position (from after the last zero to after 6): 6 places.
3. Thus, $6,030,000 = 6.03 \times 10^6$.

Key Concept: Scientific Notation, Population Representation

b) $8.5 \times 10^7$
[Solution Description]
The given population is 8,50,00,000.
Step 1: Rewrite the number without commas: 85000000.
Step 2: Count the number of digits after the first digit (8). Total digits = 8.
Step 3: Move the decimal point to after the first non-zero digit (8.5).
Step 4: The exponent is the number of places moved: 7.
So, the scientific notation is $8.5 \times 10^7$.

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, 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, 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

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: 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: Comparing Large Quantities

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

Key Concept: Scientific Notation

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: Powers of 10

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

Key Concept: Scientific Notation, Exponent Importance

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

Key Concept: 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: 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: 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: 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: 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 linear growth (adding 100 individuals per year) takes longer to double the population size than exponential growth (fixed growth rate). This is true because linear growth increases by a fixed amount each time, whereas exponential growth multiplies the current population by a factor, accelerating the growth rate. The reason correctly explains the difference between additive (linear) and multiplicative (exponential) growth patterns, justifying why exponential growth leads to faster doubling. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: 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: Comparison of Linear and Exponential Growth

c) A bacteria population doubles every day.
[Solution Description]
Exponential growth involves multiplicative increases, unlike linear growth which is additive. Only option c) represents repeated multiplication (doubling every day).

Key Concept: Exponential Growth, Historical Comparisons

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

Key Concept: 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: 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: Exponential Notation, Universe Age

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

Key Concept: 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 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, 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: 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

b) $2^6$
[Solution Description]
Using the exponent rule $\frac{n^a}{m^a} = \left( \frac{n}{m} \right)^a$:
$\frac{10^6}{5^6} = \left( \frac{10}{5} \right)^6 = 2^6 = 64.$

Key Concept: Exponential Growth, Pond Example

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

Key Concept: 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: Exponents, Thought Experiments

c) $2.4 \times 10^{11}$
[Solution Description]
Total population = $8 \times 10^9$. Each person has 30 pieces of clothing.
Calculation:
$8 \times 10^9 \times 30 = 240 \times 10^9 = 2.4 \times 10^{11}$
The total number of clothing pieces is $2.4 \times 10^{11}$.

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: Population Counting

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

Key Concept: 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, Exponent Significance

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

Key Concept: Indian and International Naming Conventions

b) Indian: neel, International: trillion
[Solution Description]
According to the syllabus:
- In the Indian system, $10^{13}$ is called "neel".
- In the International system, $10^{13}$ is not a standard named power; the closest is trillion ($10^{12}$) or ten trillion ($10^{13}$). However, the exact match for $10^{13}$ is not explicitly mentioned, but in the Indian system, it is "neel".
Thus, $10^{13}$ corresponds to "neel" in the Indian system, while it is not directly named in the International system beyond trillion ($10^{12}$).

Key Concept: Scientific Notation Conversion

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

Key Concept: Scientific Notation, Exponent Importance

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