ESAT ESAT Mathematics 3

Question 1

What is the area enclosed by y = x² and y = x between x = 0 and x = 1?

Accepted Answer

1/6

Answer

∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6.

Question 2

What is the value of sin(π/6)?

Accepted Answer

1/2

Answer

sin(30°) = sin(π/6) = 1/2 is a standard trigonometric value.

Question 3

If a vector a = (3, 4), what is |a|?

Accepted Answer

5

Answer

|a| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Question 4

What is the general solution to dy/dx = 2y?

Accepted Answer

y = Ce^(2x)

Answer

Separating variables: dy/y = 2dx → ln|y| = 2x + const → y = Ce^(2x).

Question 5

How many complex roots does x⁴ + 1 = 0 have?

Accepted Answer

4

Answer

A degree-4 polynomial always has exactly 4 roots (counting multiplicity) in the complex numbers.

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