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
= 18 × 1/6
= 3
The trigonometric identity is:
1 + cot²θ = csc²θ
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
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).
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.
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
30% of 50 students = (30/100) × 50 = 15
So, 15 students received an A grade.
If b² - 4ac < 0, the discriminant is negative, meaning the quadratic equation has complex (imaginary) roots.
These roots are conjugate pairs and unequal.