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

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial thickness of the paper is $0.001$ cm. After each fold, the thickness doubles. So after $n$ folds, the thickness is given by $0.001 \times 2^n$ cm. For 46 folds:
$0.001 \times 2^{46} = 0.001 \times 7.03687 \times 10^{13}$ cm ≈ $7.03687 \times 10^8$ km ≈ $703,687$ km.
Hence, the assertion is true. The reason correctly explains the assertion as it describes the exponential growth mechanism where each fold doubles the thickness.

Key Concept: Exponential Growth in Folding Paper

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the thickness after 10 folds using the formula for exponential growth: $0.001 \text{ cm} \times 2^{10}$.
Step 1: Calculate $2^{10}$ which is $1024$.
Step 2: Multiply by the initial thickness: $0.001 \text{ cm} \times 1024 = 1.024 \text{ cm}$.
The calculation confirms that the assertion is true. The reason provided correctly explains the assertion by stating the exponential growth formula and how it applies to the folding process.

Key Concept: Exponential Growth

b) 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

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: 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: 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: 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

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

Key Concept: Negative Exponents

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

Key Concept: Exponential Notation and Operations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $2^3 \times 2^4$, we use the exponent rule $n^a \times n^b = n^{a+b}$. Here, the base is 2 for both terms. So, $2^3 \times 2^4 = 2^{3+4} = 2^7$. Both the assertion and reason are true, and the reason correctly explains the assertion.

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

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: Doubling Effect

b) 4.096 cm
[Solution Description]
The thickness doubles after each fold. For 12 folds, the total increase in thickness can be calculated using the formula:
$\text{Final Thickness} = \text{Initial Thickness} \times 2^n$
Here, $n = 12$ and Initial Thickness = 0.001 cm.
Substituting the values:
$\text{Final Thickness} = 0.001 \, \text{cm} \times 2^{12}$
Calculate $2^{12} = 4096$:
$0.001 \, \text{cm} \times 4096 = 4.096 \, \text{cm}$
Therefore, the thickness after 12 folds is 4.096 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: Zero Exponent

c) $1$
[Solution Description]
Any non-zero number raised to the power of zero is equal to 1. Therefore, $7^0 = 1$.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that $5^3 = 125$. This is correct because $5^3 = 5 \times 5 \times 5 = 125$. The reason explains the general definition of exponential notation, where $n^a$ means $n$ multiplied by itself $a$ times. Since the assertion is a specific example of this rule and correctly applies it, the reason is the correct explanation of the assertion.

Key Concept: 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: 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, 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: 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: 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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $3^5 \times 3^{-2}$, we use the exponent rule for multiplication:
$n^a \times n^b = n^{a + b}$.
Applying this:
$3^5 \times 3^{-2} = 3^{5 + (-2)} = 3^3$.
Thus, the Assertion is true.
The Reason correctly states the rule used to solve it, so it is also true and explains the Assertion.

Key Concept: 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: 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 in exponential form

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

Key Concept: Prime Factorization and Exponential Representation

b) $2^3 \times 3^4$
[Solution Description]
To find the prime factorization of $648$, divide it by primes:
648 ÷ 2 = 324
324 ÷ 2 = 162
162 ÷ 2 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Prime factors: $2^3 \times 3^4$.

Key Concept: 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: 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: 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: Product of Powers with the Same Base

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

Key Concept: Properties of 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: 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: Negative and Zero Exponents

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

Key Concept: Exponent 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: Combining Exponents

b) 16
[Solution Description]
First, simplify the division inside the parentheses using the exponent rule $n^a \div n^b = n^{a-b}$:
$2^{-4} \div 2^{-6} = 2^{-4-(-6)} = 2^{2} = 4$
Now, square the result:
$4^2 = 16$

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 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

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

Key Concept: 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: 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: Combining exponents

c) $3^5$
[Solution Description]
Using the exponent addition rule: $n^a \times n^b = n^{a + b}$. For $3^{4} \times 3^{-1} \times 3^{2}$:
$3^{4} \times 3^{-1} \times 3^{2} = 3^{4 - 1 + 2} = 3^5$
The simplified form is $3^5$.

Key Concept: Exponent Subtraction Rule

b) $5^{5}$
[Solution Description]
Using the exponent subtraction rule, when dividing powers with the same base, subtract the exponents:
$5^{3} \div 5^{-2} = 5^{3 - (-2)} = 5^{5}$

Key Concept: Exponent subtraction rule, Negative exponents

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

Key Concept: Negative Exponents

d) $\frac{1}{7^{4}}$
[Solution Description]
A negative exponent represents the reciprocal of the base raised to the positive exponent:
$7^{-4} = \frac{1}{7^{4}}$

Key Concept: Zero Exponent Evaluation

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

Key Concept: 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: 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: Exponent Operations

c) $\frac{1}{4}$
[Solution Description]
Using the exponent rule for multiplication with the same base:
$2^4 \times 2^{-6} = 2^{4 + (-6)} = 2^{-2} = \frac{1}{2^{2}} = \frac{1}{4}$
The correct answer is $\frac{1}{4}$.

Key Concept: 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: Powers of 10

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

Key Concept: Powers of 10

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: 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: 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, Comparison

c) Distance between Sun and Earth
[Solution Description]
To compare the distances, observe the exponents:
1. $1.4335 \times 10^{12}$ has exponent 12,
2. $1.439 \times 10^{12}$ also has exponent 12,
3. $1.496 \times 10^{11}$ has exponent 11.
Since $10^{11} < 10^{12}$, the distance between the Sun and Earth is the smallest. Even if we compare coefficients, the exponent difference outweighs any coefficient comparison.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number 3,00,00,000 can be written as $3 \times 10^6$ in scientific notation. This is correct because:
3,00,00,000 = 3 × 10,00,000 = $3 \times 10^6$.
The reason explains the general rule of scientific notation, which is accurate: a number is expressed as $x \times 10^y$, where $x$ is between 1 and 10, and $y$ is an integer.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Scientific Notation

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, 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, 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 Conversion

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

Key Concept: Scientific Notation, 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: 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

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 and Comparison of Large Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare the numbers, we look at their exponents since they are both in scientific notation. The number of ants is $2 \times 10^{16}$ and the number of trees is $3 \times 10^{12}$. The exponent for ants ($16$) is larger than the exponent for trees ($12$), so $2 \times 10^{16} > 3 \times 10^{12}$, regardless of the coefficients. Hence, the assertion is true. The reason states that the exponent determines the magnitude and correctly explains why $10^{16} > 10^{12}$.

Key Concept: Exponent Comparison

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

Key Concept: 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, Applications

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

Key Concept: Powers of 10

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

Key Concept: Scientific Notation

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: 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: Age calculation, Weight estimation

b) 116,160 hours
[Solution Description]
To find Roxie's age in hours, we need to convert her age from days to hours. There are 24 hours in a day. So, we multiply the number of days by 24.
Step-by-step calculation:
1. Number of days = 4840
2. Hours in a day = 24
3. Total hours = $4840 \times 24$
Now, let's compute $4840 \times 24$:
Break it down using distributive property:
$4840 \times 20 = 96,800$
$4840 \times 4 = 19,360$
Add them together:
$96,800 + 19,360 = 116,160$ hours.

Key Concept: Age in Hours

b) 116,160 hours
[Solution Description]
To convert days to hours, multiply the number of days by 24: $4840 \times 24$. Calculating step-by-step: First, multiply 4000 × 24 = 96,000. Then, multiply 800 × 24 = 19,200. Finally, multiply 40 × 24 = 960. Add them together: 96,000 + 19,200 = 115,200; 115,200 + 960 = 116,160 hours.

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: 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, Time scales

b) 94,500 seconds
[Solution Description]

The bacterial population doubles every hour, so we use the formula:

$P = 100 \times 2^t$

where P is the final population and t is the time in hours.

We need the population to reach 8 billion:

$100 \times 2^t = 8,000,000,000$

$2^t = \frac{8,000,000,000}{100}$

$2^t = 80,000,000$

Now,

$t = \log_2(80,000,000) \approx 26.25 \text{ hours}$

Convert hours into seconds:

$26.25 \times 3600 = 94,500 \text{ seconds}$

So, it would take approximately 94,500 seconds, or about 26.25 hours, for the bacterial population to reach 8 billion cells.

Key Concept: 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: 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: Large Number Naming Systems

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 number system, which is correct. The reason explains that a crore is 100 lakhs, and since 1 lakh is $10^5$, multiplying them gives $10^7$. This correctly justifies 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: Scientific Notation

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

Key Concept: 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: 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: Standard Form

b) 4,200,000
[Solution Description]
To convert $4.2 \times 10^6$ to standard numerical form, multiply 4.2 by $10^6$ (which is 1,000,000). This gives $4.2 \times 10^6 = 4,200,000$.

Key Concept: 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: Linear Growth

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

Key Concept: Linear vs Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if a pond is fully covered with lotuses on the 30th day and the lotuses double every day, then the pond was half-covered on the 29th day. This is correct because exponential growth involves doubling (multiplying by 2), so one day before full coverage (day 30) would be half coverage (day 29). The reason explains that exponential growth is multiplicative, which correctly justifies the assertion since doubling is an example of multiplicative growth.

Key Concept: Exponential Growth, Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial population $P_0 = 100$ cells. The population doubles every hour, so after $n = 10$ hours, the population $P$ can be calculated as:
$$ P = 100 \times 2^{10} $$
We know that $2^{10} = 1024$. Therefore:
$$ P = 100 \times 1024 = 102400 $$
In scientific notation, this is $1.024 \times 10^5$ cells.
The Assertion (A) correctly states the final population, and the Reason (R) provides the correct mathematical model for exponential growth, which explains why the assertion holds true.

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: 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, 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: 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 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

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 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: 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, $34,30,000 = 3.43 \times 10^6$.