This is the NET mathematics formula sheet, grouped by chapter, with the results that come up most in MCQs: quadratic roots, series sums, counting rules, trig identities, standard derivatives and integrals, lines and conics. Learn the formulas, then test them on the chapter drills linked under each block.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences and 30% of NET-Architecture
- The syllabus follows the HSSC Part-I and Part-II mathematics books
- NET is 200 MCQs in 180 minutes, so a formula has to come to mind in seconds
Source: NUST, Subjects included in NET with weightings, checked October 2026
Algebra
i² = −1 i³ = −i i⁴ = 1 i^n depends only on n mod 4
z = a + bi conjugate z̄ = a − bi |z| = √(a² + b²)
z · z̄ = a² + b² = |z|² 1/z = z̄ / |z|²
Polar form: z = r(cos θ + i sin θ), r = |z|, tan θ = b/a (check the quadrant)
|z₁z₂| = |z₁||z₂| arg(z₁z₂) = arg z₁ + arg z₂
De Moivre: (cos θ + i sin θ)^n = cos nθ + i sin nθ
ax² + bx + c = 0 x = (−b ± √(b² − 4ac)) / 2a
Discriminant D = b² − 4ac
D > 0 real and distinct D = 0 real and equal D < 0 complex conjugates
Sum of roots α + β = −b/a Product αβ = c/a
Equation with roots α, β: x² − (α + β)x + αβ = 0
α² + β² = (α + β)² − 2αβ
α³ + β³ = (α + β)³ − 3αβ(α + β)
Roots reciprocal of each other: c = a
Cube roots of unity: 1, ω, ω² ω³ = 1 1 + ω + ω² = 0 ω = (−1 + i√3)/2
Remainder theorem: p(x) ÷ (x − a) leaves remainder p(a)
AP: aₙ = a + (n − 1)d Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l)
GP: aₙ = a·r^(n−1) Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1
S∞ = a/(1 − r), only when |r| < 1
Means between a and b: A = (a + b)/2 G = ±√(ab) H = 2ab/(a + b)
For positive a, b: A ≥ G ≥ H and G² = AH
HP: the reciprocals of the terms form an AP
Σn = n(n + 1)/2 Σn² = n(n + 1)(2n + 1)/6 Σn³ = [n(n + 1)/2]²
ⁿPᵣ = n!/(n − r)! ⁿCᵣ = n!/[r!(n − r)!] 0! = 1
ⁿCᵣ = ⁿCₙ₋ᵣ ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ
Arrangements with repeats: n!/(p! q! r!)
Round table: (n − 1)! Necklace or garland: (n − 1)!/2
Diagonals of an n-sided polygon: ⁿC₂ − n = n(n − 3)/2
Probability: P(E) = n(E)/n(S), 0 ≤ P(E) ≤ 1, P(not E) = 1 − P(E)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent events: P(A ∩ B) = P(A)·P(B)
Induction and binomial theorem
(a + b)ⁿ = Σ ⁿCᵣ a^(n−r) b^r, r = 0 to n
General term: Tᵣ₊₁ = ⁿCᵣ a^(n−r) b^r
Number of terms = n + 1 Sum of coefficients = 2ⁿ
Middle term: n even → T(n/2 + 1) n odd → T((n+1)/2) and T((n+3)/2)
Any n, |x| < 1: (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + ...
The trap in binomial questions is the sign. In (x² − 3/(2x))¹⁰ the second term carries (−1)^r, so check whether r is odd before you pick a positive option.
Matrices and vectors
A = [a b; c d] |A| = ad − bc A⁻¹ = (1/|A|)·[d −b; −c a]
A⁻¹ = adj(A)/|A| A is singular when |A| = 0 (no inverse)
(AB)⁻¹ = B⁻¹A⁻¹ (AB)ᵗ = BᵗAᵗ
|AB| = |A||B| |Aᵗ| = |A| |A⁻¹| = 1/|A| |kA| = kⁿ|A| for an n × n matrix
|A| = 0 if two rows (or columns) are equal, or one row is all zeros
Swapping two rows changes the sign of |A|
Cramer's rule: x = |Aₓ|/|A|, y = |Aᵧ|/|A|, z = |A_z|/|A|
v = xi + yj + zk |v| = √(x² + y² + z²) unit vector v̂ = v/|v|
a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
|a × b| = |a||b| sin θ
i·i = j·j = k·k = 1 i·j = j·k = k·i = 0
i × j = k j × k = i k × i = j (reverse order flips the sign)
Perpendicular: a · b = 0 Parallel: a × b = 0
Area of parallelogram = |a × b| Area of triangle = ½|a × b|
Projection of a on b = (a · b)/|b|
Scalar triple product a · (b × c) = determinant of the components
= volume of the parallelepiped; zero when a, b, c are coplanar
Direction cosines: cos²α + cos²β + cos²γ = 1
Trigonometry
π rad = 180° 1° = π/180 ≈ 0.01745 rad 1 rad ≈ 57.3°
Arc length l = rθ Sector area = ½r²θ (θ in radians)
sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
tan 2A = 2 tan A/(1 − tan²A)
sin²A = (1 − cos 2A)/2 cos²A = (1 + cos 2A)/2
sin 3A = 3 sin A − 4 sin³A cos 3A = 4 cos³A − 3 cos A
sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2)
sin C − sin D = 2 cos((C+D)/2) sin((C−D)/2)
cos C + cos D = 2 cos((C+D)/2) cos((C−D)/2)
cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2)
Periods: sin x, cos x → 2π tan x, cot x → π sin(ax) → 2π/a
Range of a sin x + b cos x: from −√(a² + b²) to +√(a² + b²)
Signs: all positive in Q1, sin in Q2, tan in Q3, cos in Q4
Triangles
Sine rule: a/sin A = b/sin B = c/sin C = 2R
Cosine rule: a² = b² + c² − 2bc cos A
Area = ½ab sin C Hero: Δ = √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2
Circumradius R = abc/(4Δ) Inradius r = Δ/s
Functions, limits and differentiation
x → 0: sin x / x → 1 (x in radians) (aˣ − 1)/x → ln a (1 + x)^(1/x) → e
n → ∞: (1 + 1/n)ⁿ → e
x → a: (xⁿ − aⁿ)/(x − a) → n·a^(n−1)
Rational function as x → ∞:
top degree lower → 0 equal → ratio of leading coefficients higher → ±∞
f'(x) = lim (h → 0) [f(x + h) − f(x)]/h
d/dx xⁿ = n·x^(n−1)
d/dx sin x = cos x d/dx cos x = −sin x
d/dx tan x = sec²x d/dx cot x = −cosec²x
d/dx sec x = sec x tan x d/dx cosec x = −cosec x cot x
d/dx eˣ = eˣ d/dx aˣ = aˣ ln a d/dx ln x = 1/x d/dx logₐx = 1/(x ln a)
d/dx sin⁻¹x = 1/√(1 − x²) d/dx cos⁻¹x = −1/√(1 − x²) d/dx tan⁻¹x = 1/(1 + x²)
Product: (uv)' = u'v + uv' Quotient: (u/v)' = (u'v − uv')/v²
Chain: dy/dx = (dy/du)(du/dx) Degree sign: d/dx sin x° = (π/180) cos x°
nth derivative: eᵃˣ → aⁿeᵃˣ sin(ax) → aⁿ sin(ax + nπ/2) cos(ax) → aⁿ cos(ax + nπ/2)
Tangent at (x₁, y₁): y − y₁ = m(x − x₁), m = f'(x₁) Normal slope = −1/m
Stationary point f'(x) = 0: f''(x) < 0 → maximum f''(x) > 0 → minimum
Integration
∫ xⁿ dx = x^(n+1)/(n + 1) + C, n ≠ −1 ∫ (1/x) dx = ln|x| + C
∫ eᵃˣ dx = eᵃˣ/a + C ∫ aˣ dx = aˣ/ln a + C
∫ sin x dx = −cos x + C ∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C ∫ cosec²x dx = −cot x + C
∫ sec x tan x dx = sec x + C ∫ cosec x cot x dx = −cosec x + C
∫ tan x dx = ln|sec x| + C ∫ cot x dx = ln|sin x| + C
∫ sec x dx = ln|sec x + tan x| + C ∫ cosec x dx = ln|cosec x − cot x| + C
∫ dx/(a² + x²) = (1/a) tan⁻¹(x/a) + C
∫ dx/√(a² − x²) = sin⁻¹(x/a) + C
∫ dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)| + C
∫ f'(x)/f(x) dx = ln|f(x)| + C
∫ [f(x)]ⁿ f'(x) dx = [f(x)]^(n+1)/(n + 1) + C
∫ eˣ[f(x) + f'(x)] dx = eˣ f(x) + C
By parts: ∫ u dv = uv − ∫ v du
Definite integrals:
Notation: ∫[a→b] means the integral from x = a to x = b
∫[a→b] f(x) dx = F(b) − F(a) ∫[a→b] f = −∫[b→a] f ∫[a→a] f = 0
∫[−a→a] f = 0 if f is odd ∫[−a→a] f = 2∫[0→a] f if f is even
∫[0→a] f(x) dx = ∫[0→a] f(a − x) dx
Area between curves = ∫ (upper − lower) dx
The eˣ[f(x) + f'(x)] rule looks obscure, but it answers a whole family of MCQs in one line. ∫eˣ(tan⁻¹x + 1/(1 + x²)) dx is eˣ tan⁻¹x + C, because the second bracket term is the derivative of the first.
Coordinate geometry and conics
Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
Point dividing in ratio m:n (internal) = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
Centroid = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
Slope m = (y₂ − y₁)/(x₂ − x₁) = tan θ
Parallel: m₁ = m₂ Perpendicular: m₁m₂ = −1
Lines: y = mx + c y − y₁ = m(x − x₁) x/a + y/b = 1 x cos α + y sin α = p
Point to line ax + by + c = 0: d = |ax₁ + by₁ + c| / √(a² + b²)
Angle between lines: tan θ = |(m₂ − m₁)/(1 + m₁m₂)|
Triangle area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
Pair of lines ax² + 2hxy + by² = 0: m₁ + m₂ = −2h/b, m₁m₂ = a/b
perpendicular when a + b = 0
Circle: (x − h)² + (y − k)² = r²
x² + y² + 2gx + 2fy + c = 0 → centre (−g, −f), r = √(g² + f² − c)
Tangent to x² + y² = a² at (x₁, y₁): xx₁ + yy₁ = a²
y = mx + c touches x² + y² = a² when c² = a²(1 + m²)
Parabola y² = 4ax: vertex (0, 0), focus (a, 0), directrix x = −a, latus rectum 4a
Tangent at (x₁, y₁): yy₁ = 2a(x + x₁) y = mx + c is a tangent when c = a/m
Ellipse x²/a² + y²/b² = 1, a > b:
c² = a² − b², e = c/a (0 < e < 1), foci (±c, 0), directrices x = ±a/e, latus rectum 2b²/a
Hyperbola x²/a² − y²/b² = 1:
c² = a² + b², e = c/a (e > 1), foci (±c, 0), asymptotes y = ±(b/a)x, latus rectum 2b²/a
Eccentricity: circle 0 ellipse between 0 and 1 parabola 1 hyperbola above 1
Rectangular hyperbola (a = b): e = √2
How to use this sheet
Reading a formula sheet feels productive and teaches very little. Use it the other way round. Cover a block, write out every formula you remember, then check. Whatever you missed goes on a card.
After that, do the chapter drill. A formula you can recite but cannot spot inside a question is not yet yours. The NET mathematics guide explains which chapters to drill first and how much time each question deserves.
Physics on NET-Engineering has its own sheet: the NET physics formula sheet.
FAQ
Does NUST give a formula sheet in NET?
Do not count on one. Learn the formulas by chapter and test them on MCQs, so you can recall each one in a few seconds under the clock.
Which maths formulas are most important for NET?
Standard derivatives and integrals, the trig identities, quadratic root relations, AP and GP sums, ⁿCᵣ and ⁿPᵣ, and the line and conic equations. Those blocks cover the calculus, trigonometry and geometry chapters, which make up more than half the chapter list.
How do I memorise maths formulas for NET?
Write each block from memory, check it, and put every miss on a card. Then use the formulas on chapter MCQs, since recognising when to use one is the skill the test measures.
