AMUEEE Mathematics Syllabus

For ax²+bx+c=0, give the sum and product of the roots.

Sum of roots = −b/a; product of roots = c/a.

What is the conjugate of z = a + bi and a key property of it?

Conjugate z̄ = a − bi; z·z̄ = a² + b² = |z|², and the conjugate of a real-coefficient polynomial's complex root is also a root.

What is the slope of a line making angle θ with the positive x-axis?

slope m = tan θ.

State the slope-intercept form of a straight line.

y = mx + c, where m is the slope and c is the y-intercept.

Give the point-slope form of a line through (x₁, y₁) with slope m.

y − y₁ = m(x − x₁).

What are the conditions for two lines to be parallel and perpendicular in terms of slopes?

Parallel: m₁ = m₂; Perpendicular: m₁·m₂ = −1.

State the formula for the distance from point (x₁, y₁) to line Ax + By + C = 0.

d = |Ax₁ + By₁ + C| / √(A² + B²).

What is the angle θ between two lines with slopes m₁ and m₂?

tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.

Name the four conic sections obtained by slicing a double cone.

Circle, ellipse, parabola, and hyperbola (degenerate cases: point, line, pair of lines).

Give the standard equation of a parabola opening rightward and its focus.

y² = 4ax; focus at (a, 0), directrix x = −a, vertex at origin.

State the standard equation of an ellipse with a > b and its eccentricity.

x²/a² + y²/b² = 1; eccentricity e = √(1 − b²/a²), with 0 < e < 1.

State the standard equation of a hyperbola centered at origin and its eccentricity range.

x²/a² − y²/b² = 1; eccentricity e = √(1 + b²/a²), with e > 1.

What is the eccentricity of a circle and a parabola?

Circle: e = 0; Parabola: e = 1.

Give the length of the latus rectum of the parabola y² = 4ax.

Length of latus rectum = 4a.

What is the general equation of a circle with centre (h, k) and radius r?

(x − h)² + (y − k)² = r².

State the limit definition of the derivative of f at x.

f'(x) = lim_{h→0} [f(x+h) − f(x)] / h, when the limit exists.

What is the value of lim_{x→0} (sin x)/x?

1 (x in radians).

State the product rule for differentiation.

d/dx[u·v] = u'v + uv' (where u, v are functions of x).

State the quotient rule for differentiation.

d/dx[u/v] = (u'v − uv')/v².

What is lim_{x→0} (e^x − 1)/x?

1.

Give the derivative of xⁿ with respect to x.

d/dx(xⁿ) = n·x^(n−1) (power rule).

State the three conditions for a function f to be continuous at x = a.

f(a) is defined; lim_{x→a} f(x) exists; and lim_{x→a} f(x) = f(a).

State the chain rule for differentiating a composite function f(g(x)).

d/dx f(g(x)) = f'(g(x))·g'(x).

What is the relationship between differentiability and continuity?

If a function is differentiable at a point, it is continuous there; but continuity does not imply differentiability.

Give the derivative of ln x and of e^x.

d/dx(ln x) = 1/x (x > 0); d/dx(e^x) = e^x.

What is logarithmic differentiation used for?

To differentiate functions of the form [f(x)]^{g(x)} or products/quotients of many factors, by taking ln of both sides first.

In mathematical reasoning, what is a 'statement' (mathematically acceptable statement)?

A sentence that is either true or false but not both; ambiguous or opinion-based sentences are not statements.

What is the negation of the statement 'p: 2 is an even number'?

~p: '2 is not an even number' (or 'It is false that 2 is an even number').

Define a compound statement and name the common connecting words.

A statement formed by combining two or more statements using connectives: 'and', 'or', 'if-then', 'if and only if'.

Distinguish between the quantifiers 'There exists' and 'For all'.

'For all' (∀) is universal — true for every element; 'There exists' (∃) is existential — true for at least one element.

What does it mean to validate a statement of the form 'if p then q' by the direct method?

Assume p is true and show by a logical chain of steps that q must then be true.

Describe validating 'if p then q' by contrapositive.

Prove the contrapositive '~q ⇒ ~p', which is logically equivalent to 'p ⇒ q'.

How do you validate a statement using proof by contradiction (reductio ad absurdum)?

Assume the statement is false (assume ~p), then derive a logical contradiction, forcing p to be true.

How can you show a statement of the form 'for all x, p(x)' is false?

By giving a counterexample — a single case where p(x) fails — which disproves the universal claim.