Explanation

A function f: A → B is called onto (surjective) if Range(f) = B, meaning every element in B has at least one preimage in A.

In other words, for every b ∈ B, there exists a ∈ A such that f(a) = b.

Explanation

Formula: Total amount = Average × Number of students

Calculation: 20 × 30 = 600

So, the total pocket money of the class is Rs. 600.

Explanation

Given:

x = 1 / (√2 - 1)

To simplify, multiply the numerator and denominator by (√2 + 1):

x = 1 / (√2 - 1) × (√2 + 1) / (√2 + 1)

= (√2 + 1) / (√2² - 1²)

= (√2 + 1) / (2 - 1)

= (√2 + 1)

Now, find x - 1/x:

x - 1/x = (√2 + 1) - 1 / (√2 + 1)

= (√2 + 1) - (√2 - 1) / (√2 + 1)(√2 - 1)

= (√2 + 1) - (√2 - 1) / (2 - 1)

= (√2 + 1) - (√2 - 1)

= (√2 + 1 - √2 + 1)

= 2

Explanation

In an expression like y or 1y, the coefficient of y is 1 (since it is implicitly present).

A coefficient is the numerical factor multiplied by the variable.

Explanation

Using the Pythagorean theorem:

a² + b² = c²

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

Given:

a = 5 cm

b = 12 cm

Substitute the values:

5² + 12² = c²

25 + 144 = c²

169 = c²

Take the square root:

c = √169

= 13 cm

Explanation

The angle θ can be found using tan(θ) = height / shadow length, i.e., tan(θ) = 3 / √3 = √3.

Since tan 60° = √3, the angle between the ground and the Sun is 60°.

Explanation

A surd is an expression of the form n√a, where a is a positive irrational number and cannot be simplified into a rational number.

Examples include √2, √3, and ³√5, which are irrational and cannot be expressed as exact fractions.

Explanation

Using the identity:

(1 - ω + ω²)ⁿ = (-1)ⁿ ω²ⁿ

Substituting n = 7:

(1 - ω + ω²)⁷ = (-1)⁷ ω²⁷

= -1 × ω²⁷ (since (-1)⁷ = -1)

= -128ω² (since ω²⁷ = (ω²)⁷ × ω⁰ = (ω²)⁷ = 2⁷ω² = 128ω², and multiplying by -1)

The correct answer is:

-128ω²

Explanation

To convert 18 (decimal) to octal:

18 ÷ 8 = 2 remainder 2

2 ÷ 8 = 0 remainder 2

Reading the remainders from bottom to top, 18₁₀ = 22₈.

Explanation