Explanation

x = A cos(mt - a) is a wave/oscillation.

If you differentiate it twice, you get the same x back but with a minus sign and m² in front.

So the equation becomes: d²x/dt² = -m²x

Clear rule: Cos or Sin → 2nd derivative = -ω² × function

Last verified on 04-06-2026

Explanation

The Fourier series expansion of an even, periodic function contains only cosine terms and a constant term.

In other words, the sine terms are absent.

Even functions:

An even function is symmetric about the y-axis, meaning f(x) = f(-x). 

Fourier series:

A Fourier series represents a periodic function as a sum of sine and cosine waves. 

Explanation

Explanation

• Bigotry means strong prejudice or hatred.
• Fanaticism is closely related as it leads to intense intolerance or extreme views.

Last verified on 04-06-2026

Explanation

• “Everyone” is a singular indefinite pronoun.
• Singular subjects take singular auxiliary verbs like has.
• So the correct form is “has learned.”

Last verified on 04-06-2026

Explanation

y = (1/8)x^2 + (1/2)x + 1

At (0,1)

1) Slope of tangent at (0,1) 

y = (1/8)x² + (1/2)x + 1  

y' = (1/4)x + 1/2  

At x=0, m = 1/2

2) Tangent equation

y - 1 = (1/2)x → y = 0.5x + 1

3) Intercepts

x-intercept: 0 = 0.5x + 1 → x = -2  

y-intercept: y = 1  

4) Area 

Triangle with base 2, height 1  

Area = 1/2 × 2 × 1 = 1

Last verified on 04-06-2026

Explanation

Explanation

Given: 3x⁵ + 96 = ∫_c^x f(y) dy for all x

1) Differentiate both sides w.r.t x

Use Fundamental Theorem of Calculus: d/dx ∫_c^x f(y) dy = f(x)

So: d/dx [3x⁵ + 96] = f(x)  

→ f(x) = 15x⁴

2) Put x = c

LHS: 3c⁵ + 96  

RHS: ∫_c^c f(y) dy = 0

So 3c⁵ + 96 = 0  

→ c⁵ = -32  

→ c = -2

Last verified on 04-06-2026

Explanation

Explanation