IAT Mathematics Syllabus

What is the nth term and the sum of n terms of an arithmetic progression (AP)?

nth term: a_n = a + (n−1)d; Sum: S_n = n/2 [2a + (n−1)d] = n/2 (a + l).

What is the nth term and the sum of n terms of a geometric progression (GP)?

nth term: a_n = a·r^(n−1); Sum (r≠1): S_n = a(r^n − 1)/(r − 1).

What is the sum to infinity of a GP and when does it exist?

S_∞ = a/(1 − r), valid only when |r| < 1.

State the relationship between the arithmetic mean (AM) and geometric mean (GM) of two positive numbers.

AM ≥ GM, i.e., (a + b)/2 ≥ √(ab), with equality when a = b.

What is the slope of a line through points (x₁, y₁) and (x₂, y₂), and how do slopes relate for perpendicular lines?

Slope m = (y₂ − y₁)/(x₂ − x₁). For perpendicular lines, m₁·m₂ = −1; for parallel lines, m₁ = m₂.

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

d = |Ax₁ + By₁ + C| / √(A² + B²).

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

(x − h)² + (y − k)² = r².

Give the standard equation and eccentricity range for an ellipse and a hyperbola.

Ellipse: x²/a² + y²/b² = 1, eccentricity e < 1. Hyperbola: x²/a² − y²/b² = 1, eccentricity e > 1. (Parabola e = 1.)

What is the standard equation of a parabola opening rightward, and its focus and directrix?

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

State the distance formula between two points in three-dimensional space.

d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].

What is the section formula for a point dividing the join of (x₁,y₁,z₁) and (x₂,y₂,z₂) in ratio m:n internally?

((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)).

State the standard limit lim(x→0) (sin x)/x.

lim(x→0) (sin x)/x = 1 (x in radians).

What are the derivatives of sin x, cos x, and tan x?

d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x.

State the product rule and quotient rule for differentiation.

Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

What three conditions must hold for a function f to be continuous at x = a?

(1) f(a) is defined; (2) lim(x→a) f(x) exists; (3) lim(x→a) f(x) = f(a).

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

dy/dx = f'(g(x)) · g'(x), i.e., dy/dx = (dy/du)(du/dx).

What is the derivative of log_e x and of e^x?

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

How do you find whether a critical point gives a local maximum or minimum using the second derivative test?

At a point where f'(x)=0: if f''(x) < 0 it is a local maximum; if f''(x) > 0 it is a local minimum; if f''(x) = 0 the test fails.

What does it mean for a function to be increasing or decreasing on an interval in terms of its derivative?

f is increasing where f'(x) > 0 and decreasing where f'(x) < 0 on the interval.

State the power rule for integration: ∫ xⁿ dx.

∫ xⁿ dx = x^(n+1)/(n+1) + C, for n ≠ −1 (for n = −1, ∫ dx/x = ln|x| + C).

State the formula for integration by parts.

∫ u v dx = u ∫v dx − ∫ (u' ∫v dx) dx; commonly ∫ u dv = uv − ∫ v du.

What is the fundamental theorem of calculus (evaluation part) for a definite integral?

∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f.

How do you find the area bounded by a curve y = f(x), the x-axis, and lines x = a and x = b?

Area = ∫_a^b |f(x)| dx (taking absolute value where the curve is below the axis).

What is the order and degree of a differential equation?

Order: the order of the highest derivative present. Degree: the power of the highest-order derivative when the equation is polynomial in derivatives (free of radicals/fractions).

State the general solution method for a linear first-order differential equation dy/dx + Py = Q.

Use integrating factor IF = e^(∫P dx); then solution is y·(IF) = ∫ Q·(IF) dx + C.

How are the dot product and cross product of two vectors a and b defined in terms of the angle θ between them?

Dot product: a·b = |a||b|cosθ (a scalar). Cross product: |a×b| = |a||b|sinθ, directed perpendicular to both (a vector).

What is the condition for two non-zero vectors to be perpendicular and to be parallel?

Perpendicular: a·b = 0. Parallel: a×b = 0 (or a = λb for some scalar λ).

What are direction cosines of a line, and what relation do they satisfy?

Direction cosines l, m, n are the cosines of angles the line makes with the x, y, z axes; they satisfy l² + m² + n² = 1.

In linear programming, what is the corner point theorem for finding the optimal solution?

If the feasible region is bounded, the objective function attains its maximum and minimum at corner (vertex) points of the feasible region.

In mathematical reasoning, what is the contrapositive of the statement 'if p then q', and how does its truth relate to the original?

The contrapositive is 'if not q then not p'. It is logically equivalent to the original statement (both have the same truth value).