Explanation

The logarithmic equation t = log𝑏 m means that b raised to the power t equals m.

Converting to exponential form: m = bᵗ.

Explanation

The formula for the volume of a cone is:

Volume = (1/3)πr²h

Where:

r = radius of the base

h = height of the cone

Explanation

The complement of a null set (∅) includes all elements of the universal set (U).

Since ∅ contains nothing, its complement is everything in U, so ∅' = U.

Explanation

Given: √x = 10

Squaring both sides: (√x)² = 10² → x = 100

Solution set: {100}

Explanation

The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both.

Since (a - b)^4 and (a - b)^3 are both powers of (a - b), their LCM is the highest power of (a - b) between them:

LCM = (a - b)^4

Explanation

Simplify inside the parentheses: (3 - 5 + 2) = 0

Solve the expression: 8 - 0 = 8

Explanation

Given:

A ∪ B = A ∩ B

This equation implies that every element that is in A or B (or both) is also in both A and B.

This can only be true if:

A = B

or, in other words, A and B have exactly the same elements.

Explanation

Given:

x ∝ 1/y

y ∝ 1/z

We can rewrite these proportions as:

x = k₁/y (where k₁ is a constant)

y = k₂/z (where k₂ is a constant)

Substitute the expression for y into the first equation:

x = k₁/(k₂/z)

Simplify:

x = k₁z/k₂

x ∝ z

Explanation

Hamza gives 1/5 of 500 = Rs. 100 to Adil, leaving Rs. 400.

He then gives half of 400 = Rs. 200 to Bilal, so he is left with Rs. 200.

Explanation

To find the ratio of passed to failed students:

Number of students who passed = Total students - Failed students

= 112 - 14

= 98

Ratio of passed to failed students = 98 : 14

Simplify the ratio by dividing both numbers by 14:

Ratio = 98/14 : 14/14

= 7 : 1