Advanced Engineering Mathematics
Study Guide Latest Update 2026
Questions With Complete Solutions.A+
Confirmed ||Highly Graded Exam
Quadratic Equation - ANSWER-Form: Ax2 + Bx + C = 0
Arithmetic Progression - ANSWER-an = a1 + (n −1)d
Geometric Progression - ANSWER-an = a1 rn−1
Harmonic Progression - ANSWER-Reciprocal of arithmetic progression
Sine Law - ANSWER-sin 𝑎/sin 𝐴 = sin 𝑏/sin 𝐵 = sin 𝐶/sin 𝑐
Cosine Law for sides - ANSWER-cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝐴
Cosine Law for angles - ANSWER-cos 𝐴 = −cos 𝐵 cos 𝐶 + sin 𝐵 sin 𝐶 cos 𝑎
Sum of Roots - ANSWER-x1 + x2 = −B
Product of Roots - ANSWER-x1 ∙ x2 = + C
Double Angle Identities - ANSWER-sin 2A = 2 sin A cos A
Central Angle, β - ANSWER-β = 360°
Area of Triangle - ANSWER-A = 1/2 bh
Area of Rectangle - ANSWER-A = bh
Area of Square - ANSWER-A = s2
Area of Parallelogram - ANSWER-A = bh
Area of Trapezoid - ANSWER-A = 1/2 (a + b)h
Volume of Sphere - ANSWER-V = 4/3 πR3
Volume of Cylinder - ANSWER-V = πR2h
,Distance from a point to a line - ANSWER-d = |Ax + By + C| / √(A2 + B2)
Eccentricity of a conic - ANSWER-e = c/a for ellipse
Slope-intercept form - ANSWER-y = mx + b
Point-slope form - ANSWER-y - y1 = m(x - x1)
Distance between two points - ANSWER-d = √(y2 − y1)² + (x2 − x1)²
Circle - ANSWER-The locus of points equidistant from a fixed point
Ellipse - ANSWER-The locus of points where the sum of distances from two foci is
constant
Hyperbola - ANSWER-The locus of points where the difference of distances from two
foci is constant
Parabola - ANSWER-The locus of points equidistant from a fixed point and a fixed line
Euler Line - ANSWER-The line that passes through the orthocenter, circumcenter, and
centroid of the triangle
Inscribed Circle - ANSWER-The circle inscribed within a triangle, tangent to its sides
Circumscribing Circle - ANSWER-The circle that passes through all vertices of a triangle
General Equation - ANSWER-x2 + y2 + Dx + Ey + F = 0
Standard Equation - ANSWER-(x −h)2 + (y −k)2 = r2
Versed cosine - ANSWER-covers A = 1 −sin A
Compound Interest - ANSWER-F = P(1 + 𝑖)n
Future Worth (F) - ANSWER-Future worth of an investment
Principal (P) - ANSWER-Principal or present worth
Interest Rate (i) - ANSWER-Interest rate per interest period
Nominal Interest Rate (r) - ANSWER-Nominal interest rate
Number of Interest Periods (n) - ANSWER-No. of interest periods
,Effective Rate (ER) - ANSWER-Effective rate
Half versed sine - ANSWER-hav A = 1 −cos A
Exsecant - ANSWER-exsec A = sec A −1
Rate of Return (RR) - ANSWER-RR = annual net profit / capital
Continuous Compounding Interest - ANSWER-F = Pert
Break-even Analysis - ANSWER-cost = revenue
Annuity - ANSWER-𝑥→𝑥+ 𝑥1
Inflation - ANSWER-𝑖f = 𝑖+ f + 𝑖f
Depreciation - ANSWER-d = FC −SV
Capitalized Cost (C) - ANSWER-C = FC + OM
Sinking Fund - ANSWER-d = (FC −SV) [(1 + i)n −1 / i]
Maxima & Minima (Critical Points) - ANSWER-𝑑𝑦/𝑑𝑥= y′ = 0
Sum-of-the-Years-Digit (SYD) - ANSWER-d = (FC −SV) [n −m + 1 / ∑years]
Single-payment-compound-amount factor - ANSWER-(F/P, 𝑖, n) = (1 + 𝑖)n
Single-payment-present-worth factor - ANSWER-(P/F, 𝑖, n) = (1 + 𝑖)−n
Equal-payment-series-compound-amount factor - ANSWER-(F/A, 𝑖, n) = [(1 + 𝑖)n′ −1 / 𝑖]
Standard Deviation - ANSWER-σ = √(variance)
Z-score - ANSWER-z = (x − μ) / σ
Coefficient of Linear Correlation - ANSWER-−1 ≤ r ≤ +1
Mean/Average Deviation - ANSWER-Mean/average value
μ - ANSWER-Mean value, x̅
σ - ANSWER-Standard deviation
t - ANSWER-Expansion pressure
, pavement density - ANSWER-
Exponential Distribution - ANSWER-
Stiffness factor of pavement - ANSWER-
mv - ANSWER-1/b − a ∫f(x)dx
q - ANSWER-Rate of flow in vehicles/hour
k - ANSWER-Density in vehicles/km
uS - ANSWER-Space mean speed in kph
Minimum time headway - ANSWER-1/q
Spacing of vehicles - ANSWER-1/k
Peak hour factor (PHF) - ANSWER-q/qmax
Mean value - ANSWER-3
SF - ANSWER-√
P(x ≥ a) - ANSWER-e−λa
P(x ≤ a) - ANSWER-1 − e−λa
P(a ≤ x ≤ b) - ANSWER-e−λa − e−λb
RMS - ANSWER-√1/b − a ∫f(x)2dx
ES - ANSWER-Modulus of elasticity of subgrade
EP - ANSWER-Modulus of elasticity of pavement
Binomial Probability Distribution - ANSWER-P(x) = C(n, x) px qn−x
Geometric Probability Distribution - ANSWER-P(x) = p(qx−1)
Fibonacci Numbers - ANSWER-x2 − x − 1 = 0
Poisson Probability Distribution - ANSWER-P(x) = μxe−μ/x!
Period (T) - ANSWER-Interval over which the graph of function repeats
Study Guide Latest Update 2026
Questions With Complete Solutions.A+
Confirmed ||Highly Graded Exam
Quadratic Equation - ANSWER-Form: Ax2 + Bx + C = 0
Arithmetic Progression - ANSWER-an = a1 + (n −1)d
Geometric Progression - ANSWER-an = a1 rn−1
Harmonic Progression - ANSWER-Reciprocal of arithmetic progression
Sine Law - ANSWER-sin 𝑎/sin 𝐴 = sin 𝑏/sin 𝐵 = sin 𝐶/sin 𝑐
Cosine Law for sides - ANSWER-cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝐴
Cosine Law for angles - ANSWER-cos 𝐴 = −cos 𝐵 cos 𝐶 + sin 𝐵 sin 𝐶 cos 𝑎
Sum of Roots - ANSWER-x1 + x2 = −B
Product of Roots - ANSWER-x1 ∙ x2 = + C
Double Angle Identities - ANSWER-sin 2A = 2 sin A cos A
Central Angle, β - ANSWER-β = 360°
Area of Triangle - ANSWER-A = 1/2 bh
Area of Rectangle - ANSWER-A = bh
Area of Square - ANSWER-A = s2
Area of Parallelogram - ANSWER-A = bh
Area of Trapezoid - ANSWER-A = 1/2 (a + b)h
Volume of Sphere - ANSWER-V = 4/3 πR3
Volume of Cylinder - ANSWER-V = πR2h
,Distance from a point to a line - ANSWER-d = |Ax + By + C| / √(A2 + B2)
Eccentricity of a conic - ANSWER-e = c/a for ellipse
Slope-intercept form - ANSWER-y = mx + b
Point-slope form - ANSWER-y - y1 = m(x - x1)
Distance between two points - ANSWER-d = √(y2 − y1)² + (x2 − x1)²
Circle - ANSWER-The locus of points equidistant from a fixed point
Ellipse - ANSWER-The locus of points where the sum of distances from two foci is
constant
Hyperbola - ANSWER-The locus of points where the difference of distances from two
foci is constant
Parabola - ANSWER-The locus of points equidistant from a fixed point and a fixed line
Euler Line - ANSWER-The line that passes through the orthocenter, circumcenter, and
centroid of the triangle
Inscribed Circle - ANSWER-The circle inscribed within a triangle, tangent to its sides
Circumscribing Circle - ANSWER-The circle that passes through all vertices of a triangle
General Equation - ANSWER-x2 + y2 + Dx + Ey + F = 0
Standard Equation - ANSWER-(x −h)2 + (y −k)2 = r2
Versed cosine - ANSWER-covers A = 1 −sin A
Compound Interest - ANSWER-F = P(1 + 𝑖)n
Future Worth (F) - ANSWER-Future worth of an investment
Principal (P) - ANSWER-Principal or present worth
Interest Rate (i) - ANSWER-Interest rate per interest period
Nominal Interest Rate (r) - ANSWER-Nominal interest rate
Number of Interest Periods (n) - ANSWER-No. of interest periods
,Effective Rate (ER) - ANSWER-Effective rate
Half versed sine - ANSWER-hav A = 1 −cos A
Exsecant - ANSWER-exsec A = sec A −1
Rate of Return (RR) - ANSWER-RR = annual net profit / capital
Continuous Compounding Interest - ANSWER-F = Pert
Break-even Analysis - ANSWER-cost = revenue
Annuity - ANSWER-𝑥→𝑥+ 𝑥1
Inflation - ANSWER-𝑖f = 𝑖+ f + 𝑖f
Depreciation - ANSWER-d = FC −SV
Capitalized Cost (C) - ANSWER-C = FC + OM
Sinking Fund - ANSWER-d = (FC −SV) [(1 + i)n −1 / i]
Maxima & Minima (Critical Points) - ANSWER-𝑑𝑦/𝑑𝑥= y′ = 0
Sum-of-the-Years-Digit (SYD) - ANSWER-d = (FC −SV) [n −m + 1 / ∑years]
Single-payment-compound-amount factor - ANSWER-(F/P, 𝑖, n) = (1 + 𝑖)n
Single-payment-present-worth factor - ANSWER-(P/F, 𝑖, n) = (1 + 𝑖)−n
Equal-payment-series-compound-amount factor - ANSWER-(F/A, 𝑖, n) = [(1 + 𝑖)n′ −1 / 𝑖]
Standard Deviation - ANSWER-σ = √(variance)
Z-score - ANSWER-z = (x − μ) / σ
Coefficient of Linear Correlation - ANSWER-−1 ≤ r ≤ +1
Mean/Average Deviation - ANSWER-Mean/average value
μ - ANSWER-Mean value, x̅
σ - ANSWER-Standard deviation
t - ANSWER-Expansion pressure
, pavement density - ANSWER-
Exponential Distribution - ANSWER-
Stiffness factor of pavement - ANSWER-
mv - ANSWER-1/b − a ∫f(x)dx
q - ANSWER-Rate of flow in vehicles/hour
k - ANSWER-Density in vehicles/km
uS - ANSWER-Space mean speed in kph
Minimum time headway - ANSWER-1/q
Spacing of vehicles - ANSWER-1/k
Peak hour factor (PHF) - ANSWER-q/qmax
Mean value - ANSWER-3
SF - ANSWER-√
P(x ≥ a) - ANSWER-e−λa
P(x ≤ a) - ANSWER-1 − e−λa
P(a ≤ x ≤ b) - ANSWER-e−λa − e−λb
RMS - ANSWER-√1/b − a ∫f(x)2dx
ES - ANSWER-Modulus of elasticity of subgrade
EP - ANSWER-Modulus of elasticity of pavement
Binomial Probability Distribution - ANSWER-P(x) = C(n, x) px qn−x
Geometric Probability Distribution - ANSWER-P(x) = p(qx−1)
Fibonacci Numbers - ANSWER-x2 − x − 1 = 0
Poisson Probability Distribution - ANSWER-P(x) = μxe−μ/x!
Period (T) - ANSWER-Interval over which the graph of function repeats
