Explanation

A complex cube root of unity is a complex number $omega$ such that:

$$omega^3 = 1,quad omega neq 1$$

So:

$$$$ω3=1omega^3 = 1

Explanation

Step 1: Rule for divisibility by 5

A number is divisible by 5 if and only if it ends in 0 or 5.
But 0 is not among the given digits, so the last digit must be 5.

Step 2: Fix the last digit as 5

So, we fix 5 as the last digit.

Now we must choose the remaining 4 digits (from the remaining digits: 2, 3, 7, 9)
These will go in the first 4 places.

• Number of ways to arrange 4 digits out of 4 = 4! = 24

Explanation

Deduction –  Reasoning from general principles to a specific conclusion

 This is a classic example of deductive reasoning, specifically a syllogism.

It demonstrates how a conclusion can be logically derived from two premises

Explanation

Step 1: Identify the first term and common difference

• First term $a = 7$

• Common difference $d=11-7=4$

Step 2: Use the formula for the n
th term of an A.P.:

$$a_n = a + (n - 1)d$$

For the 21st term:

$$a_{21} = 7 + (21 - 1) times 4 = 7 + 20 times 4 = 7 + 80 = 87$$

Explanation

Let apple = x, orange = y

5x + 4y = 340

Try options:

B) 5×40 + 4×20 = 200 + 80 = 340 

Explanation

Inductive reasoning involves observing patterns or examples and forming a generalized conclusion.

It moves from specific cases to a general rule.

Explanation

LIFO stands for Last In First Out

It is a principle used in stack data structures, where the last item added is the first one removed.

This "last-in, first-out" order is the defining characteristic of a stack

Explanation

To find the size of the class interval:

Size = (Greatest value - Lowest value) / Number of intervals

= (21 - 3) / 3

= 18 / 3

= 6

Explanation

5 ! = 5 × 4 × 3 × 2 × 1 = 120 ways.

Explanation

6P4 = 6 × 5 × 4 × 3 = 360.