Key Concept: Exponential Growth

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

Key Concept: Exponential Growth, Folding Paper

b) 1.024 cm
[Solution Description]
The thickness of the paper doubles with each fold. For 10 folds, the thickness increases by $2^{10} = 1024$ times.
Initial thickness: $0.001$ cm
Final thickness after 10 folds: $0.001 \times 2^{10} = 0.001 \times 1024 = 1.024$ cm

Key Concept: Exponential Growth

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: 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, 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 in Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given table, the initial thickness of the paper is $0.001 cm$. After each fold, the thickness doubles. Using the formula for exponential growth:
Thickness after $n$ folds = Initial thickness $\times 2^n$.
For $n = 7$, the thickness becomes $0.001 cm \times 2^7 = 0.128 cm$, which matches the assertion. The reason correctly explains why the thickness is $0.128 cm$ after 7 folds because it describes the doubling mechanism.

Key Concept: Exponential Growth

c) $0.008$ cm
[Solution Description]
The thickness doubles with each fold. After the first fold, it becomes $0.001 \times 2 = 0.002$ cm. After the second fold, it becomes $0.002 \times 2 = 0.004$ cm. After the third fold, it becomes $0.004 \times 2 = 0.008$ cm or $0.001 \times 2^3 = 0.008$ cm.

Key Concept: Exponential Growth

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: Exponential Notation and Operations

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

Key Concept: Exponential Notation

a) $2^8$
[Solution Description]
To solve $2^3 \times 2^5$, we use the exponent rule $n^a \times n^b = n^{a + b}$. Here, the base ($n$) is the same (which is 2), so we can directly add the exponents: $3 + 5 = 8$. Thus, $2^3 \times 2^5 = 2^{8}$.

Key Concept: 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: 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: 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

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial thickness of the paper is $0.001 \, \text{cm}$. After 1 fold, it becomes $0.001 \times 2 = 0.002 \, \text{cm}$. After 2 folds, it becomes $0.002 \times 2 = 0.004 \, \text{cm}$, and so on. After 5 folds, the thickness will be $0.001 \times 2^5 = 0.032 \, \text{cm}$. Thus, the Assertion is true. The Reason states that the thickness doubles after each fold, which is also true and correctly explains the Assertion because the exponential growth model ($2^n$) is applied here.

Key Concept: Exponential Growth, Doubling Effect

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

Key Concept: Power of a Power, Zero Exponent

b) $64$
[Solution Description]
Any non-zero number raised to the power of 0 is 1, so $(5^2)^0 = 1$. For the second part, use the rule $(n^a)^b = n^{a \times b}$, so $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$. Multiplying these results together gives $1 \times 64 = 64$.

Key Concept: Exponentiation of Negative Numbers

b) $9$
[Solution Description]
The expression $(-3)^2$ means $-3$ multiplied by itself twice. So, $(-3) \times (-3) = 9$.

Key Concept: 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: 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: 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: 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: Multiplication of exponents with the same base

b) $5^{7}$
[Solution Description]
To solve $5^3 \times 5^4$, use the rule $n^a \times n^b = n^{a + b}$. Here, the base $n = 5$, exponent $a = 3$, and exponent $b = 4$. So, $5^3 \times 5^4 = 5^{3 + 4} = 5^7$.

Key Concept: 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: 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: Numerical Evaluation

c) 392
[Solution Description]
To evaluate $7^2 \times 2^3$, first calculate the exponents:
$7^2 = 49$
$2^3 = 8$
Multiply them together: $49 \times 8 = 392$.

Key Concept: Prime factorization in exponential form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
– Step 1: Prime factorize 36. We know that 36 = 6 × 6.
– Step 2: Further factorize 6 into its prime factors: 6 = 2 × 3.
– Step 3: Substitute back into the equation: 36 = (2 × 3) × (2 × 3).
– Step 4: Regroup the terms to express them in exponential form: 36 = $2^2 \times 3^2$.
– Conclusion: Both Assertion (A) and Reason (R) are true, and R correctly explains A.

Key Concept: 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: Product of Powers with the Same Base

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

Key Concept: Power of a Power

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

Key Concept: 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: Negative Exponents

b) $\frac{1}{25}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this to $5^{-2}$:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Key Concept: Negative exponents, 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, Exponential operations

a) $\frac{1}{9}$
[Solution Description]
First, simplify the numerator using $n^a \times n^b = n^{a+b}$: $3^{2-5} = 3^{-3}$.
Now, divide by $3^{-1}$ using $n^a \div n^b = n^{a-b}$: $3^{-3-(-1)} = 3^{-2}$.
The simplified form is $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Key Concept: Negative 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 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: 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: Negative Exponents

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

Key Concept: Exponent Rules, Negative Exponents

c) $3^7$
[Solution Description]
To solve this, we first simplify the numerator and denominator separately using exponent rules. The numerator $3^5 \times 3^{-2} = 3^{5-2} = 3^3$. The denominator $3^0 \times 3^{-4} = 1 \times 3^{-4} = 3^{-4}$ because any number to the power of 0 is 1. Now, dividing numerator by denominator: $\frac{3^3}{3^{-4}} = 3^{3-(-4)} = 3^{7}$. Therefore, the simplified form is $3^7$.

Key Concept: Exponential Growth

b) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond would have been half-covered one day before it was fully covered. Hence, if full coverage occurs on the 30th day, half-coverage occurred on the 29th day.

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: 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, 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 Property Application

b) $-4$
[Solution Description]
Rewrite $\frac{1}{16}$ as a power of 2:
$\frac{1}{16} = 2^{-4} \text{ because } 16 = 2^4.$
So,
$2^x = 2^{-4} \implies x = -4.$

Key Concept: Exponent Operations

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

Key Concept: 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: 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) $10^2$
[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, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $10^{-5} = \frac{1}{10^5}$. This is correct because, by definition, a negative exponent indicates the reciprocal of the base raised to the positive exponent. The reason provides the general form of this property: for any non-zero number $n$ and integer $a$, $n^{-a} = \frac{1}{n^a}$. Since the assertion aligns with the general rule stated in the reason, and the reason correctly explains the assertion, both are true and the reason is the correct explanation of the assertion.

Key Concept: 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 – 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: 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, Large Numbers

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

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To express 7,00,00,000 in scientific notation, we first identify the coefficient and exponent. The coefficient must be a number between 1 and 10, which for this number is 7. Now, count the number of digits after the first digit (7): there are 6 zeros, so the exponent is 6. Therefore, the number can be written as $7 \times 10^6$. The reason correctly explains that the coefficient must be between 1 and 10 and how the exponent is determined by counting the digits after the first digit. Thus, both the assertion and reason are true, and the reason is the correct explanation of 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

c) $3.05 \times 10^4$
[Solution Description]
To convert 30,500 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (3.05).
2. Count the number of places moved (4 places to the left).
3. Write in the form $x \times 10^y$: $3.05 \times 10^4$.

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: 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: Exponent Rules

b) $5^5$
[Solution Description]
Using exponent rules:
1. Multiply the exponents for the numerator: $5^3 \times 5^4 = 5^{3+4} = 5^7$.
2. Divide by the denominator: $5^7 \div 5^2 = 5^{7-2} = 5^5$.

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

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

b) $2.5 \times 10^6$
[Solution Description]
To find how many times more ants there are compared to humans, divide the number of ants by the number of humans:
$\frac{2 \times 10^{16}}{8 \times 10^9} = \frac{2}{8} \times 10^{16-9} = 0.25 \times 10^7 = 2.5 \times 10^6$
Therefore, there are $2.5 \times 10^6$ (2.5 million) times more ants than humans.

Key Concept: Scientific Notation, Real-World Comparisons

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the ratio of stars to ants using their given values:
$\frac{2 \times 10^{23}}{2 \times 10^{16}} = 10^{23-16} = 10^7$
This confirms that the assertion is correct. The reason states that the ratio can be found by subtracting the exponents, which is a valid rule for division in scientific notation. Thus, the reason not only supports but also correctly explains the assertion.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Scientific notation requires the coefficient $x$ to satisfy $1 \leq x < 10$ and the exponent $y$ to be an integer. Here, $5.9$ is between 1 and 10, and $3$ is an integer. Thus, both Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Standard Form

c) $1.439 \times 10^{12}$ m
[Solution Description]
The given number is $1,439,000,000,000$ m.
Move the decimal point to the left until it is just after the first non-zero digit, which gives $1.439$.
Count the number of places moved (12 places). Therefore, the scientific notation is $1.439 \times 10^{12}$ m.

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

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: 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: Coin Weight Calculation

b) 6429 coins
[Solution Description]
First, convert Roxie’s weight from kilograms to grams:
$45 \text{ kg} = 45 \times 1000 = 45000 \text{ grams}$
Next, divide the total weight in grams by the weight of one coin:
$\text{Number of coins} = \frac{45000}{7} \approx 6428.57$
Since we can’t have a fraction of a coin, we round up to the nearest whole number: 6429 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 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: 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 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: 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: 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: 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: 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: 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: 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: 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: Power Play

d) 9 zeros
[Solution Description]
In the Indian numbering system:
$1 \, \text{arab} = 100 \, \text{crores} = 100 \times 1,00,00,000 = 1,00,00,00,000$
Counting the zeros gives 9 zeros.

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 vs Exponential Growth, Scientific Notation

c) After 21 hours
[Solution Description]
First, model both populations:
1. Exponential growth: $P_e(t) = 100 \times 2^{t/3}$
2. Linear growth: $P_l(t) = 100 + 50t$
We need to solve:
$100 \times 2^{t/3} – (100 + 50t) > 10,000$
Test option values:
a) t = 24 hours:
$P_e(24) = 100 \times 2^8 = 25,600 \\ P_l(24) = 100 + 50 \times 24 = 1,300 \\ Difference = 24,300 > 10,000$
b) t = 18 hours:
$P_e(18) = 100 \times 2^6 = 6,400 \\ P_l(18) = 100 + 50 \times 18 = 1,000 \\ Difference = 5,400 < 10,000$ c) t = 15 hours: $P_e(15) = 100 \times 2^5 = 3,200 \\ P_l(15) = 100 + 50 \times 15 = 850 \\ Difference = 2,350 10,000$ The smallest t satisfying the condition is 21 hours.

Key Concept: Exponential Growth Calculation

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

Key Concept: 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: 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: 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: Large Number Facts

b) $1 \times 10^{11}$
[Solution Description]
The number of stars in the Milky Way galaxy is approximately $1 \times 10^{11}$, which is 100 billion stars.

Key Concept: Scientific Notation, Real-World Application

c) $3.17 \times 10^{15}$ years
[Solution Description]
Step 1: Calculate the total time required in seconds.
Total stars = $10^{23}$
Time per star = 1 second
Total time = $10^{23}$ seconds
Step 2: Convert seconds to years.
Seconds in a year = $3.15 \times 10^7$
Total years = $\frac{10^{23}}{3.15 \times 10^7} = \frac{1}{3.15} \times 10^{23-7} = 0.317 \times 10^{16}$
Step 3: Adjust the coefficient to be between 1 and 10.
$0.317 \times 10^{16} = 3.17 \times 10^{15}$ years
It would take approximately $3.17 \times 10^{15}$ years.

Key Concept: Exponents and Large Numbers

c) $1 \times 10^7$
[Solution Description]
One crore is written numerically as $1,00,00,000$.
Step 1: The coefficient is $1$ (since it already lies between 1 and 10).
Step 2: The exponent is the number of zeros + 1 (for the last non-zero digit): $7$.
Thus, one crore in scientific notation is $1 \times 10^7$.

Key Concept: Scientific Notation

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: 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 ($10^{12}$).