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COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) Syllabus
The Mathematics (NAT-IE / NAT-ICS) syllabus for COMSATS NTS-based Test in full: 5 chapters and 16 topics, every one listed below, plus 50 Mathematics (NAT-IE / NAT-ICS) flashcards with answers.
Mathematics (NAT-IE / NAT-ICS) syllabus: all 5 chapters
Expand any chapter to see its topics and sub-topics.
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Algebra and Functions
4 topics- Sets and Functions
- Sequences and Series
- Permutations and Combinations
- Binomial Theorem
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Trigonometry
3 topics- Trigonometric Ratios and Identities
- Trigonometric Equations
- Inverse Trigonometric Functions
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Calculus
3 topics- Limits and Continuity
- Differentiation
- Integration
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Analytic Geometry
3 topics- Straight Lines
- Conic Sections
- Vectors
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Matrices and Determinants
3 topics- Matrix Operations
- Determinants
- Solving Systems of Equations
COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) flashcards — 50 questions with answers
Define a function from set A to set B.
A relation that assigns to each element of A exactly one element of B. A is the domain; B is the codomain.
State De Morgan's laws for sets A and B.
(A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
What is the number of subsets and the number of proper subsets of a set with n elements?
Subsets = 2^n; proper subsets = 2^n − 1.
Distinguish a one-to-one (injective) function from an onto (surjective) function.
Injective: distinct inputs give distinct outputs (no two x map to same y). Surjective: every element of the codomain is an image of some input (range = codomain).
State the formula for the nth term of an arithmetic progression (AP).
a_n = a + (n − 1)d, where a is the first term and d the common difference.
State the sum of the first n terms of an arithmetic progression.
S_n = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
State the nth term and the sum of n terms of a geometric progression (GP).
a_n = a r^{n−1}; S_n = a(r^n − 1)/(r − 1) for r ≠ 1.
State the sum to infinity of a geometric series and its condition.
S_∞ = a/(1 − r), valid only when |r| < 1.
What is the relationship between the arithmetic mean (A) and geometric mean (G) of two positive numbers?
A ≥ G, with equality if and only if the two numbers are equal (AM ≥ GM).
Insert the formula for a single geometric mean between two positive numbers a and b.
G = √(ab).
State the formula for the number of permutations of n distinct objects taken r at a time.
nPr = n! / (n − r)!.
State the formula for the number of combinations of n distinct objects taken r at a time.
nCr = n! / [r!(n − r)!].
How many distinct arrangements are there of n objects where some are alike (p alike, q alike, etc.)?
n! / (p! q! r! ...), dividing by the factorials of the counts of each group of identical objects.
How many circular permutations of n distinct objects are there?
(n − 1)! (or (n − 1)!/2 if clockwise and anticlockwise are considered identical, e.g. a necklace).
State the symmetry property of combinations.
nCr = nC(n−r).
State the Binomial Theorem for (a + b)^n where n is a positive integer.
(a + b)^n = Σ_{r=0}^{n} nCr · a^{n−r} · b^r.
Write the general (r+1)th term in the expansion of (a + b)^n.
T_{r+1} = nCr · a^{n−r} · b^r.
How do you find the term independent of x (constant term) in a binomial expansion?
Write the general term, simplify the power of x, set the exponent of x equal to zero, solve for r, then substitute back.
State the binomial series for (1 + x)^n when n is not a positive integer, and its validity condition.
(1 + x)^n = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ..., valid for |x| < 1.
Give the exact values of sin, cos, and tan of 30°.
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.
Show the other 30 Mathematics (NAT-IE / NAT-ICS) flashcards
State the three fundamental Pythagorean trigonometric identities.
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.
State the sine and cosine of a sum: sin(A+B) and cos(A+B).
sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB.
State the double-angle formulas for sin 2θ and cos 2θ.
sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
State the formula for tan(A + B).
tan(A+B) = (tanA + tanB) / (1 − tanA tanB).
Express the product 2 sinA cosB as a sum.
2 sinA cosB = sin(A+B) + sin(A−B).
State the law of cosines for a triangle with sides a, b, c.
c² = a² + b² − 2ab cosC (and cyclic forms).
Give the general solution of sinθ = sinα.
θ = nπ + (−1)^n α, where n is any integer.
Give the general solution of cosθ = cosα.
θ = 2nπ ± α, where n is any integer.
Give the general solution of tanθ = tanα.
θ = nπ + α, where n is any integer.
State the domain and range of the principal-value function y = sin⁻¹x.
Domain: [−1, 1]; Range: [−π/2, π/2].
State the principal-value range of y = cos⁻¹x and y = tan⁻¹x.
cos⁻¹x: [0, π]; tan⁻¹x: (−π/2, π/2).
State the addition formula for inverse tangents: tan⁻¹x + tan⁻¹y.
tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)], when xy < 1.
State the formal definition that the limit of f(x) as x→a equals L.
For every ε > 0 there exists δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ.
State the standard limit of sin x / x as x → 0.
lim_{x→0} sin x / x = 1 (x in radians).
State the limit defining e: lim (1 + 1/n)^n as n → ∞.
lim_{n→∞} (1 + 1/n)^n = e ≈ 2.71828.
State the three conditions for f(x) 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 limit definition of the derivative of f(x).
f'(x) = lim_{h→0} [f(x + h) − f(x)] / h.
State the power rule and the derivative of sin x and cos x.
d/dx(x^n) = n x^{n−1}; d/dx(sin x) = cos x; d/dx(cos x) = −sin x.
State the product rule and quotient rule of differentiation.
Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².
State the chain rule for y = f(g(x)).
dy/dx = f'(g(x)) · g'(x), i.e. dy/dx = (dy/du)(du/dx).
How do you classify a stationary point using the second derivative test?
At f'(x)=0: if f''(x) > 0 it is a local minimum; if f''(x) < 0 a local maximum; if f''(x) = 0 the test is inconclusive.
Give derivatives of e^x, ln x, and tan x.
d/dx(e^x) = e^x; d/dx(ln x) = 1/x; d/dx(tan x) = sec²x.
State the power rule for integration of x^n.
∫ x^n dx = x^{n+1}/(n+1) + C, for n ≠ −1; for n = −1, ∫ (1/x) dx = ln|x| + C.
State the Fundamental Theorem of Calculus for evaluating a definite integral.
∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f.
State the integration by parts formula.
∫ u dv = uv − ∫ v du.
Give ∫ sin x dx and ∫ cos x dx.
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C.
State the distance formula between two points (x₁,y₁) and (x₂,y₂) and the slope formula.
Distance = √[(x₂−x₁)² + (y₂−y₁)²]; slope m = (y₂−y₁)/(x₂−x₁).
State the condition for two lines with slopes m₁ and m₂ to be parallel and to be perpendicular.
Parallel: m₁ = m₂. Perpendicular: m₁·m₂ = −1.
State the standard equation of a circle and of an ellipse centered at the origin.
Circle: x² + y² = r². Ellipse: x²/a² + y²/b² = 1.
State the standard equations of a parabola (vertex at origin, opening right) and a hyperbola centered at the origin.
Parabola: y² = 4ax. Hyperbola: x²/a² − y²/b² = 1.
COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) FAQ
What is in the COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) syllabus?
COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) is split into 5 chapters: Algebra and Functions, Trigonometry, Calculus, Analytic Geometry and Matrices and Determinants. Together they contain 16 topics.
How many chapters are in COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS)?
5 chapters, containing 16 topics. The largest chapter is Algebra and Functions, with 4 topics.
Are there flashcards for COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS)?
Yes — a 50-card Mathematics (NAT-IE / NAT-ICS) deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.