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DAY-4

MOST IMPORTANT FORMULAS

The image displays the Black-Scholes equation, a fundamental mathematical model in financial derivatives, often called the "Trillion Dollar Equation" due to its impact on modern finance.

Purpose: It is a partial differential equation used to price options and other financial derivatives.
Significance: It laid the groundwork for the multi-trillion dollar options market.
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The square of an odd integer n is always odd because it can be expressed in the form 2m+1. Since any odd integer is n = 2k + 1, its square is (2k+1)²=4k²+4k+1=2(2k²+2k)+1, which fits the definition of an odd number.

Examples

3² = 9 (Odd)

5² = 25 (Odd)

7² = 49 (Odd)
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Pi is an irrational number. It cannot be expressed as a simple fraction (a ratio of two integers), and its decimal representation never ends or repeats. While approximations like 22/7 or 3.14 are used, the actual value is approximately 3.14159 and continues infinitely.

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Zero is not a positive number; it is neither positive nor negative. It acts as the neutral, unsigned dividing point between positive and negative numbers on the number line. While zero is a non-negative integer, it does not satisfy the requirement of being greater than zero.

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Yes Cube root of 1331 is 11
11³=1331
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True. The largest 4-digit number is 9999. Any number higher (10,000 and above) requires five digits, making 9999 the maximum value possible in the thousands place.

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False.

A triangle is a two-dimensional (2D) shape, not a three-dimensional (3D) shape. It is a flat, plane polygon with three sides and three angles that only possesses length and width, but no depth. Examples of 3D shapes that use triangles in their structure include pyramids and tetrahedrons.

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False

An equilateral triangle is defined by having all three sides equal in length, not just two. While it is technically a special type of isosceles triangle (which must have at least two sides equal), the defining characteristic of an equilateral triangle is that all three sides are congruent.

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First term
d/dx (3x⁴) = 4(3)x⁴¯¹
= 12x³

Second term
d/dx (5x²) = 2(5)x²¯¹
10x

Third term is a constant
Therefore, d/dx (7)=0

Final answer
12x³-10x

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DAY-5
MOST IMPORTANT FORMULAS

The image is a social media post about Euler's identity, often celebrated as one of the most beautiful equations in mathematics.
The identity connects concepts from arithmetic, geometry, algebra, and analysis.

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