Explanation

Calculating the Ratio

Passed students = 25

Total students = 45

Ratio = Passed students/Total students = 25/45 = 5/9 = 5:9

Explanation

Explanation

Calculating the Derivative

Given: f(x) = (3x^5 + 6)^5

Using the chain rule: f'(x) = 5(3x^5 + 6)^4 . d/dx(3x^5 + 6)

= 5(3x^5 + 6)^4 . 15x^4

= 75x^4(3x^5 + 6)^4

= (3x^5 + 6)^4 . 75x^4

Explanation

A function can often be written as an equation (like y = 2x + 3), but not every equation defines a function.

For example, x² + y² = 4 is an equation but not a function because one x can give two different y values.

Explanation

Explanation

Calculating the Sum

The first 25 natural numbers starting from 10 are 10, 11, ..., 34.

This is an arithmetic series with first term a = 10, last term l = 34, and number of terms $n = 25.

Using the Formula for the Sum of an Arithmetic Series

Sn = n/2 (a + l)

= 25/2 (10 + 34)

= 25/2 × 44

= 25 × 22

= 550

Explanation

Verifying the Options

Given: sqrt{49x^4 + 42x^3 + 79x^2 + 30x + 25}

Checking the Options

We've already verified that (7x^2 + 3x + 5)^2 = 49x^4 + 42x^3 + 79x^2 + 30x + 25

Explanation

Using the Given Equations

Given: a^2 + b^2 = 18 and a + b = 8

Squaring the Second Equation

(a + b)^2 = 8^2

a^2 + 2ab + b^2 = 64

Substituting a^2 + b^2 = 18

18 + 2ab = 64

Solving for ab

2ab = 64 - 18

2ab = 46

ab = 23

Explanation

Calculating the Number of Relations

Number of elements in A = 2

Number of elements in B = 3

Number of elements in A × B = 2 × 3 = 6

Calculating the Number of Relations

Number of relations = 2^6 = 64

Explanation

Understanding the Symmetric Difference

AΔB = (A ∪ B) - (A ∩ B)

Given: A ∪ B = {a, c, e, i, m, n, o} and A ∩ B = {e, i, o}

Calculating the Symmetric Difference

AΔB = {a, c, e, i, m, n, o} - {e, i, o}

= {a, c, m, n}