Explanation

To find the third proportion:

First term: a²b²

Second term: abc

The proportion is in the form:

a²b² : abc :: abc : x

To find x, set up a proportion equation:

a²b² / abc = abc / x

Simplify:

ab / c = abc / x

x = c²

Explanation

= 18 × 1/6

= 3

Explanation

The trigonometric identity is:

1 + cot²θ = csc²θ

Explanation

The volume of a parallelepiped is given by:

Volume = length × breadth × height

Substitute the given values:

Volume = 5x × 3x² × 7x⁴

Multiply the terms:

Volume = 105x⁷

Explanation

A number is divisible by 8 if its last three digits form a number that is exactly divisible by 8.

For example, 4,320 is divisible by 8 because 320 ÷ 8 = 40 (a whole number).

Explanation

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data points.

It is calculated as the inverse of the average of the inverses of the values.

Explanation

To solve this problem, find the rate at which the men weave cloth:

15 men weave 120 meters in a day

Rate = 120 meters / 15 men = 8 meters per man per day

Now, find the total cloth woven by 35 men:

35 men × 8 meters per man per day = 280 meters

Explanation

Explanation

30% of 50 students = (30/100) × 50 = 15

So, 15 students received an A grade.

Explanation

If b² - 4ac < 0, the discriminant is negative, meaning the quadratic equation has complex (imaginary) roots.

These roots are conjugate pairs and unequal.