Key Concept: Exponential Growth

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

Key Concept: Exponential Growth, Real-World Application

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

Key Concept: Exponential Growth, Power of 2

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The thickness of the paper after each fold doubles the previous thickness. So, the formula for thickness after $n$ folds is $T = 0.001 \times 2^n$ cm. For $n = 30$, we get:
$T = 0.001 \times 2^{30} \text{ cm} = 0.001 \times 1,\!073,\!741,\!824 \text{ cm} = 1,\!073,\!741.824 \text{ cm} = 10.737 \text{ km}.$
Thus, the assertion is true. The reason provides the correct formula for calculating the thickness, making it true and the correct explanation of the assertion.

Key Concept: 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

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

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

Key Concept: 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: 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 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: Scientific Notation

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

Key Concept: 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: Negative Exponents and Scientific Notation

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

Key Concept: 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: 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 vs. Linear Growth

c) Exponential growth is faster than linear growth
[Solution Description]
Exponential growth gives:
$0.001 \text{ cm} \times 2^{10} = 1.024 \text{ cm}.$
In linear growth, for 10 steps adding $0.001 \text{ cm}$ each time, the total would be:
$0.001 \text{ cm} \times 10 = 0.01 \text{ cm}.$
Exponential growth results in much larger values compared to linear growth.

Key Concept: 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: 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: 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: 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: 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: 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: 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: 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: Power of a Power

b) $2^{12}$
[Solution Description]
Using the power of a power rule, $(n^a)^b = n^{a \times b}$.
Here, $a = 4$ and $b = 3$, so $(2^4)^3 = 2^{4 \times 3} = 2^{12}$.
Therefore, the answer is $2^{12}$.

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: 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: 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: 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: 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: Power of a Power

b) $5^6$
[Solution Description]
To simplify $(5^2)^3$, we use the power of a power property: $(n^a)^b = n^{a \times b}$. Here, the base is 5, and the exponents are 2 and 3:
$(5^2)^3 = 5^{2 \times 3} = 5^6$
Therefore, the simplified form of the expression is $5^6$.

Key Concept: Properties of Powers, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's evaluate the assertion step-by-step using the properties of exponents.
1. Given expression: $5^{-3} \times 5^2$
2. Using the product of powers property:
$n^a \times n^b = n^{a+b}$
So, $5^{-3} \times 5^2 = 5^{-3+2} = 5^{-1}$
3. Using the negative exponent property:
$n^{-a} = \frac{1}{n^a}$
Therefore, $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$
The assertion is true as it correctly simplifies to $\frac{1}{5}$.
The reason states the correct properties of exponents ($n^{-a} = \frac{1}{n^a}$ and $n^a \times n^b = n^{a+b}$), which are used to derive the assertion. Thus, the reason is the correct explanation for the assertion.

Key Concept: 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: 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: Zero Exponent

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

Key Concept: Negative 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 Exponent

d) $\frac{1}{9}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this rule:
$3^{-2} = \frac{1}{3^2} = \frac{1}{9}.$

Key Concept: 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 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: Combination Calculation

b) $26^{6}$
[Solution Description]
Each slot has 26 choices (A-Z). For 6 slots, total combinations are $26 \times 26 \times 26 \times 26 \times 26 \times 26 = 26^6$.

Key Concept: Exponent Rules, 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: Fractional Exponents, Power Division

b) 5
[Solution Description]
First, express 4 as a power of 2: $4 = 2^2$. So, $4^{10} = (2^2)^{10} = 2^{20}$. Now the equation becomes $\frac{2^{20}}{2^{15}} = 2^x$. Using the division rule for exponents: $2^{20-15} = 2^x$ which simplifies to $2^5 = 2^x$. Therefore, x must be 5.

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: 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 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: 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: 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, Zero Exponents

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

Key Concept: Zero Exponent Evaluation

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

Key Concept: Operations with Negative Exponents, Multiplication

b) $125$
[Solution Description]
To simplify the expression, we will use the rules for multiplying and dividing exponents with the same base.
Step 1: Rewrite the division as multiplication by the reciprocal:
$5^{-3} \times 5^{2} \times 5^{4}$
Step 2: Apply the product of powers rule $a^{m} \times a^{n} = a^{m+n}$:
$5^{-3 + 2 + 4} = 5^{3}$
Step 3: Calculate $5^{3}$:
$5^{3} = 125$
The simplified form of the expression is $125$.

Key Concept: Large Numbers Notation

c) 7
[Solution Description]
In the Indian numbering system:
$1 \text{ crore} = 1,00,00,000$
Counting the zeros in 1,00,00,000 gives us 7 zeros.

Key Concept: Powers of 10 - Basic Conversion

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

Key Concept: 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: Powers of 10

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's verify the assertion using the reason:
Step 1: According to the negative exponent rule, $n^a = \frac{1}{n^{-a}}$.
Step 2: Substitute $n = 10$ and $a = 3$ in the above formula.
Step 3: We get $10^3 = \frac{1}{10^{-3}}$, which matches the assertion.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Scientific Notation

b) $4.75 \times 10^6$
[Solution Description]
To express 4,750,000 in scientific notation, we need to write it as $x \times 10^y$ where $1 \leq x < 10$. First, identify the coefficient by moving the decimal point to the left until there is only one non-zero digit before the decimal: $4.75$. The exponent $y$ is the number of places the decimal was moved, which is 6. Thus, $4,750,000 = 4.75 \times 10^6$.

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

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 Conversion

b) 9,040
[Solution Description]
To convert $9.04 \times 10^3$ to standard form, multiply the coefficient $9.04$ by $10^3$ (which is 1000). This gives $9.04 \times 1000 = 9040$.

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, 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) $7.004 \times 10^{10}$
[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

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

Key Concept: Exponential vs. Linear Growth, Scientific Notation

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

Key Concept: Historical Context

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

Key Concept: Scientific Notation, Real-World Comparisons

a) $2 \times 10^{12}$
[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: 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: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To express the number $50,000$ in scientific notation:
Step 1: Identify the coefficient by moving the decimal point to the left until it is between the first and second digit. For $50,000$, the decimal point is after the last zero, so moving it left 4 places gives $5.0$ (which simplifies to $5$).
Step 2: Count the number of places the decimal was moved, which is 4, hence the exponent is $4$.
Therefore, $50,000 = 5 \times 10^4$.
The reason correctly explains that the coefficient must be between $1$ and $10$, and the exponent indicates the number of decimal places moved.

Key Concept: Scientific Notation

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: 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 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: Age in Hours

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

Key Concept: Exponential Growth

c) 1.024 cm
[Solution Description]

When a paper is folded once, its thickness becomes double. So, after every fold, the thickness keeps multiplying by 2. The formula is:

$\text{Final Thickness} = \text{Initial Thickness} \times 2^n$

Here, initial thickness $= 0.001\ \text{cm}$ and number of folds $n = 10.$

$0.001 \times 2^{10} = 0.001 \times 1024 = 1.024\ \text{cm}$

So, after folding the paper 10 times, its thickness will be $1.024\ \text{cm}.$

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: 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: 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: 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

c) $3.081 \times 10^8$
[Solution Description]
Scientific notation requires expressing the number in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. For 308,100,000: $308,100,000 = 3.081 \times 10^8$ The decimal moves 8 places to the left.

Key Concept: Large Number Comparisons, Historical Context

b) niyuta
[Solution Description]
The Lalitavistara mentions that 100 ayutas make a niyuta, and since an ayuta is $10^9$, multiplying by 100 gives:
$100 \times 10^9 = 10^2 \times 10^9 = 10^{11} \text{ (niyuta)}$

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 5000 can be written as $5 \times 10^3$ in scientific notation. This is correct because moving the decimal point three places to the left gives the coefficient 5 and the exponent 3. The reason states that in scientific notation, the coefficient must be between 1 and 10, which is true and correctly explains why the assertion is valid.

Key Concept: 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: 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: 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, 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 Comparison

d) Number of bacteria doubling every hour in a lab culture
[Solution Description]
Exponential growth involves repeated multiplication, such as doubling or tripling. Among the options: folding paper (thickness doubles with each fold), steps to cover distance (linear addition), and monthly salary increment (fixed addition) are not exponential. The number of bacteria doubling every hour is an example of exponential growth because it involves multiplication.

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: 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: Currency and Large Numbers

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

Key Concept: 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

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

Key Concept: 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: 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

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

Key Concept: Scientific Notation, 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.