RBSE SS Mathematics cheat sheet

A relation R from A to B is a subset of A×B; its domain is {a : (a,b)∈R} and range is {b : (a,b)∈R}.

R is a function A→B iff every a∈A appears in exactly one ordered pair; equal ranges with two preimages fail the vertical-line test.

Injective (one-one): f(a₁)=f(a₂) ⟹ a₁=a₂. Surjective (onto): range f = codomain B. Bijective = both, hence invertible.

Composition: (g∘f)(x)=g(f(x)) defined only when range f ⊆ domain g; generally g∘f ≠ f∘g.

Inverse exists iff f is bijective; then f⁻¹∘f = id_A and f∘f⁻¹ = id_B, and (f⁻¹)⁻¹ = f.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m.

Number of one-one functions from an m-set into an n-set is ⁿP_m (zero if m>n); number of bijections on an n-set is n!.

Equivalence relation needs reflexivity (aRa), symmetry (aRb⟹bRa) and transitivity (aRb, bRc⟹aRc); congruence mod n is the standard example.

f:A→B and g:B→C both one-one ⟹ g∘f one-one; both onto ⟹ g∘f onto. Converse of each fails in general.

Principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:ℝ→(−π/2,π/2); cot⁻¹:ℝ→(0,π); sec⁻¹:(−∞,−1]∪[1,∞)→[0,π]∖{π/2}; cosec⁻¹ same domain →[−π/2,π/2]∖{0}.

Reciprocal identities hold only within principal ranges: sin⁻¹(1/x)=cosec⁻¹x for |x|≥1, cos⁻¹(1/x)=sec⁻¹x for |x|≥1.

Complementary pairs: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; sec⁻¹x + cosec⁻¹x = π/2, for x in the respective domains.

Negative arguments: sin⁻¹(−x)=−sin⁻¹x, tan⁻¹(−x)=−tan⁻¹x, cosec⁻¹(−x)=−cosec⁻¹x, but cos⁻¹(−x)=π−cos⁻¹x and cot⁻¹(−x)=π−cot⁻¹x.

Sum formulas valid when xy<1: tan⁻¹x+tan⁻¹y = tan⁻¹[(x+y)/(1−xy)]; if xy>1 and x,y>0 add π, if x,y<0 subtract π.

Difference: tan⁻¹x−tan⁻¹y = tan⁻¹[(x−y)/(1+xy)] when xy>−1; adjust by ±π when xy<−1.

Double angle: 2tan⁻¹x = tan⁻¹[2x/(1−x²)] for |x|<1; equals π+tan⁻¹[...] for x>1 and −π+tan⁻¹[...] for x<−1.

sin⁻¹(sin θ)=θ only when θ∈[−π/2,π/2]; otherwise reduce θ into the principal range before simplifying.

Standard substitutions: for √(a²−x²) put x=a sin θ; for √(a²+x²) put x=a tan θ; for √(x²−a²) put x=a sec θ.

(AB)ᵀ=BᵀAᵀ and (AB)⁻¹=B⁻¹A⁻¹, both requiring conformable sizes and, for inverses, A and B invertible.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix splits as (A+Aᵀ)/2 + (A−Aᵀ)/2.

For n×n matrices, |kA|=kⁿ|A|, |AB|=|A||B|, |Aᵀ|=|A|, and |adj A|=|A|ⁿ⁻¹.

A is invertible iff |A|≠0; then A⁻¹ = adj A / |A|, and A(adj A)=(adj A)A=|A|I.

A square matrix is singular iff |A|=0; a homogeneous system AX=0 has a non-trivial solution iff |A|=0.

Cramer/system rule: AX=B has a unique solution X=A⁻¹B when |A|≠0; if |A|=0 the system is inconsistent or has infinitely many solutions.

Row/column operations do not change the value of a determinant; swapping two rows or columns multiplies it by −1.

If a row (or column) is a linear combination of others, or two rows are identical, the determinant is 0.

Area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) = ½|determinant of the 3×3 matrix with a row of 1s|.

f is continuous at a iff lim(x→a⁻)f = lim(x→a⁺)f = f(a); all three must exist and agree.

Differentiable ⟹ continuous, but not conversely (|x| at 0 is continuous, not differentiable).

Left derivative must equal right derivative for f′(a) to exist; check piecewise definitions at the join point.

Rolle's theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⟹ some c has f′(c)=0.

Lagrange MVT: under continuity on [a,b] and differentiability on (a,b), f′(c)=[f(b)−f(a)]/(b−a) for some c∈(a,b).

Increasing on an interval iff f′(x)≥0 throughout; decreasing iff f′(x)≤0. Sign of f′ decides monotonicity.

Local maximum at c: f′(c)=0 and f″(c)<0; local minimum: f′(c)=0 and f″(c)>0. If f″(c)=0, use the first-derivative sign change.

Absolute extrema on [a,b]: evaluate f at all critical points in (a,b) and at endpoints a and b; largest and smallest values win.

Rate of change: if a quantity Q(t) changes, dQ/dt is its instantaneous rate; related-rates problems link variables through one equation before differentiating in t.

Tangent at (x₁,y₁) to y=f(x): y−y₁=f′(x₁)(x−x₁); normal slope is −1/f′(x₁) where f′(x₁)≠0.

Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b)−F(a) where F′=f; the definite integral is independent of the variable name.

Properties: ∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵇ f = ∫ₐᶜ f + ∫꜀ᵇ f; ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx (useful for symmetric limits).

∫₋ₐᵃ f = 0 if f is odd, 2∫₀ᵃ f if f is even.

Standard antiderivatives: ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a); ∫dx/√(a²−x²) = sin⁻¹(x/a); ∫dx/(x²−a²) = (1/2a)ln|(x−a)/(x+a)|.

Integration by parts: ∫u v′ dx = uv − ∫u′v dx; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

Area under y=f(x) from a to b (f≥0) = ∫ₐᵇ f dx; area between f and g = ∫ₐᵇ |f−g| dx, splitting at intersection points.

Area bounded by x=g(y) with the y-axis = ∫꜀ᵈ g(y)dy; take absolute values so total area stays positive.

Area of a circle of radius r is πr² and of the ellipse x²/a²+y²/b²=1 is πab, both obtainable by definite integration.

For ∫dx/(ax²+bx+c), complete the square first, then use tan⁻¹ or logarithmic standard forms.

Order = highest derivative present; degree = power of the highest-order derivative after the equation is made polynomial in derivatives.

General solution of an nth-order ODE contains n arbitrary constants; a particular solution fixes them via initial conditions.

Variable separable: write dy/dx = f(x)g(y), then ∫dy/g(y) = ∫f(x)dx.

Homogeneous equation dy/dx = F(x,y) with F(tx,ty)=F(x,y): substitute y=vx, so dy/dx = v + x dv/dx, then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^∫P dx, solution y·IF = ∫Q·IF dx.

Linear in x: dx/dy + P₁x = Q₁ with IF = e^∫P₁ dy, used when the equation is not linear in y.

Exact equation M dx + N dy = 0 needs ∂M/∂y = ∂N/∂x; solution is ∫M dx (y constant) + terms of N free of x = C.

Growth/decay model dN/dt = kN gives N = N₀e^{kt}; Newton cooling dT/dt = −k(T−Tₛ) gives T−Tₛ = (T₀−Tₛ)e^{−kt}.

Family of curves y = f(x) differentiated once and the constant eliminated yields the governing first-order ODE.

a·b = |a||b|cos θ = a₁b₁+a₂b₂+a₃b₃; a·b=0 iff a⊥b (for nonzero vectors).

a×b = |a||b|sin θ n̂, computed by the 3×3 determinant with i,j,k in the first row; a×b=0 iff a∥b.

|a×b| is the area of the parallelogram with sides a and b; half of it is the area of the triangle.

Scalar triple product [a b c]=a·(b×c) equals the volume of the parallelepiped; it is 0 iff the three vectors are coplanar.

Projection of a on b is (a·b)/|b|; the vector projection is [(a·b)/|b|²]b.

Direction cosines satisfy l²+m²+n²=1; direction ratios are proportional to direction cosines.

Line through point a with direction b: r = a + λb; Cartesian form (x−x₁)/l = (y−y₁)/m = (z−z₁)/n.

Angle between lines with directions b₁,b₂: cos θ = |b₁·b₂|/(|b₁||b₂|); lines are perpendicular iff b₁·b₂=0.

Plane through point a with normal n: (r−a)·n=0, i.e. ax+by+cz=d; distance from (x₁,y₁,z₁) is |ax₁+by₁+cz₁−d|/√(a²+b²+c²).

Shortest distance between skew lines r=a₁+λb₁ and r=a₂+μb₂ is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

A linear programming problem has a linear objective Z=cx+dy to maximise or minimise subject to linear constraints and non-negativity x,y≥0.

The feasible region is the intersection of the half-planes defined by the constraints; it is convex.

Corner point theorem: if an optimum exists, it is attained at a vertex of the feasible region (possibly along an entire edge).

Graphical method: plot constraints, shade the feasible region, evaluate Z at each corner, and pick the largest or smallest value.

Bounded feasible region ⟹ both maximum and minimum exist; unbounded region may give no maximum (or no minimum).

If two corner points give the same optimal Z, every point on the segment joining them is also optimal (multiple optimal solutions).

Slope method: compare the slope of the objective line −c/d with constraint slopes to confirm which vertex is optimal.

A constraint like x+y≤n with x,y≥0 gives a triangular region with vertices (0,0), (n,0), (0,n).

P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability P(A|B)=P(A∩B)/P(B), P(B)>0; A and B are independent iff P(A∩B)=P(A)P(B).

Multiplication rule: P(A∩B)=P(A)P(B|A)=P(B)P(A|B).

Bayes' theorem: P(Aᵢ|E)=P(Aᵢ)P(E|Aᵢ)/ΣⱼP(Aⱼ)P(E|Aⱼ), with {Aⱼ} a partition of the sample space.

Total probability: P(E)=ΣⱼP(Aⱼ)P(E|Aⱼ) over a partition {Aⱼ}.

Binomial distribution: P(X=r)=ⁿCᵣ pʳ qⁿ⁻ʳ, q=1−p; mean=np, variance=npq, SD=√(npq).

Bernoulli trials require fixed n, two outcomes per trial, constant p, and independence across trials.

Mean of a discrete random variable: E(X)=Σxᵢpᵢ; variance Var(X)=E(X²)−[E(X)]².

P(A′)=1−P(A); P(exactly one of A,B)=P(A)+P(B)−2P(A∩B).

RBSE internal assessment for senior secondary mathematics is typically 20 marks, split across project work, practical/lab activity and a portfolio/viva; confirm the current split in the official RBSE syllabus.

A mathematics project must state an aim, a mathematical formulation, data or derivation, analysis, conclusion, and a bibliography.

Practical work uses the prescribed mathematics lab manual: activities on relations, functions, matrices, derivatives, integrals and probability using graphs and models.

Portfolio entries should include dated records of activities performed, observations, graphs drawn, and errors corrected.

Viva questions usually test the mathematical reasoning behind the recorded activity, not memorised results.

Assessment criteria weight correct procedure, accurate computation, presentation/neatness, and interpretation of results.

Projects commonly model real data (growth, cooling, population, games of chance) with a function, derivative, integral or probability distribution.

Record the formula used and the units of every computed quantity; unexplained results lose marks.

A relation R from A to B is a subset of A×B; its domain is {a : (a,b)∈R} and range is {b : (a,b)∈R}.

R is a function A→B iff every a∈A appears in exactly one ordered pair; equal ranges with two preimages fail the vertical-line test.

Injective (one-one): f(a₁)=f(a₂) ⟹ a₁=a₂. Surjective (onto): range f = codomain B. Bijective = both, hence invertible.

Composition: (g∘f)(x)=g(f(x)) defined only when range f ⊆ domain g; generally g∘f ≠ f∘g.

Inverse exists iff f is bijective; then f⁻¹∘f = id_A and f∘f⁻¹ = id_B, and (f⁻¹)⁻¹ = f.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m.

Number of one-one functions from an m-set into an n-set is ⁿP_m (zero if m>n); number of bijections on an n-set is n!.

Equivalence relation needs reflexivity (aRa), symmetry (aRb⟹bRa) and transitivity (aRb, bRc⟹aRc); congruence mod n is the standard example.

f:A→B and g:B→C both one-one ⟹ g∘f one-one; both onto ⟹ g∘f onto. Converse of each fails in general.

Principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:ℝ→(−π/2,π/2); cot⁻¹:ℝ→(0,π); sec⁻¹:(−∞,−1]∪[1,∞)→[0,π]∖{π/2}; cosec⁻¹ same domain →[−π/2,π/2]∖{0}.

Reciprocal identities hold only within principal ranges: sin⁻¹(1/x)=cosec⁻¹x for |x|≥1, cos⁻¹(1/x)=sec⁻¹x for |x|≥1.

Complementary pairs: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; sec⁻¹x + cosec⁻¹x = π/2, for x in the respective domains.

Negative arguments: sin⁻¹(−x)=−sin⁻¹x, tan⁻¹(−x)=−tan⁻¹x, cosec⁻¹(−x)=−cosec⁻¹x, but cos⁻¹(−x)=π−cos⁻¹x and cot⁻¹(−x)=π−cot⁻¹x.

Sum formulas valid when xy<1: tan⁻¹x+tan⁻¹y = tan⁻¹[(x+y)/(1−xy)]; if xy>1 and x,y>0 add π, if x,y<0 subtract π.

Difference: tan⁻¹x−tan⁻¹y = tan⁻¹[(x−y)/(1+xy)] when xy>−1; adjust by ±π when xy<−1.

Double angle: 2tan⁻¹x = tan⁻¹[2x/(1−x²)] for |x|<1; equals π+tan⁻¹[...] for x>1 and −π+tan⁻¹[...] for x<−1.

sin⁻¹(sin θ)=θ only when θ∈[−π/2,π/2]; otherwise reduce θ into the principal range before simplifying.

Standard substitutions: for √(a²−x²) put x=a sin θ; for √(a²+x²) put x=a tan θ; for √(x²−a²) put x=a sec θ.

(AB)ᵀ=BᵀAᵀ and (AB)⁻¹=B⁻¹A⁻¹, both requiring conformable sizes and, for inverses, A and B invertible.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix splits as (A+Aᵀ)/2 + (A−Aᵀ)/2.

For n×n matrices, |kA|=kⁿ|A|, |AB|=|A||B|, |Aᵀ|=|A|, and |adj A|=|A|ⁿ⁻¹.

A is invertible iff |A|≠0; then A⁻¹ = adj A / |A|, and A(adj A)=(adj A)A=|A|I.

A square matrix is singular iff |A|=0; a homogeneous system AX=0 has a non-trivial solution iff |A|=0.

Cramer/system rule: AX=B has a unique solution X=A⁻¹B when |A|≠0; if |A|=0 the system is inconsistent or has infinitely many solutions.

Row/column operations do not change the value of a determinant; swapping two rows or columns multiplies it by −1.

If a row (or column) is a linear combination of others, or two rows are identical, the determinant is 0.

Area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) = ½|determinant of the 3×3 matrix with a row of 1s|.

f is continuous at a iff lim(x→a⁻)f = lim(x→a⁺)f = f(a); all three must exist and agree.

Differentiable ⟹ continuous, but not conversely (|x| at 0 is continuous, not differentiable).

Left derivative must equal right derivative for f′(a) to exist; check piecewise definitions at the join point.

Rolle's theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⟹ some c has f′(c)=0.

Lagrange MVT: under continuity on [a,b] and differentiability on (a,b), f′(c)=[f(b)−f(a)]/(b−a) for some c∈(a,b).

Increasing on an interval iff f′(x)≥0 throughout; decreasing iff f′(x)≤0. Sign of f′ decides monotonicity.

Local maximum at c: f′(c)=0 and f″(c)<0; local minimum: f′(c)=0 and f″(c)>0. If f″(c)=0, use the first-derivative sign change.

Absolute extrema on [a,b]: evaluate f at all critical points in (a,b) and at endpoints a and b; largest and smallest values win.

Rate of change: if a quantity Q(t) changes, dQ/dt is its instantaneous rate; related-rates problems link variables through one equation before differentiating in t.

Tangent at (x₁,y₁) to y=f(x): y−y₁=f′(x₁)(x−x₁); normal slope is −1/f′(x₁) where f′(x₁)≠0.

Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b)−F(a) where F′=f; the definite integral is independent of the variable name.

Properties: ∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵇ f = ∫ₐᶜ f + ∫꜀ᵇ f; ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx (useful for symmetric limits).

∫₋ₐᵃ f = 0 if f is odd, 2∫₀ᵃ f if f is even.

Standard antiderivatives: ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a); ∫dx/√(a²−x²) = sin⁻¹(x/a); ∫dx/(x²−a²) = (1/2a)ln|(x−a)/(x+a)|.

Integration by parts: ∫u v′ dx = uv − ∫u′v dx; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

Area under y=f(x) from a to b (f≥0) = ∫ₐᵇ f dx; area between f and g = ∫ₐᵇ |f−g| dx, splitting at intersection points.

Area bounded by x=g(y) with the y-axis = ∫꜀ᵈ g(y)dy; take absolute values so total area stays positive.

Area of a circle of radius r is πr² and of the ellipse x²/a²+y²/b²=1 is πab, both obtainable by definite integration.

For ∫dx/(ax²+bx+c), complete the square first, then use tan⁻¹ or logarithmic standard forms.

Order = highest derivative present; degree = power of the highest-order derivative after the equation is made polynomial in derivatives.

General solution of an nth-order ODE contains n arbitrary constants; a particular solution fixes them via initial conditions.

Variable separable: write dy/dx = f(x)g(y), then ∫dy/g(y) = ∫f(x)dx.

Homogeneous equation dy/dx = F(x,y) with F(tx,ty)=F(x,y): substitute y=vx, so dy/dx = v + x dv/dx, then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^∫P dx, solution y·IF = ∫Q·IF dx.

Linear in x: dx/dy + P₁x = Q₁ with IF = e^∫P₁ dy, used when the equation is not linear in y.

Exact equation M dx + N dy = 0 needs ∂M/∂y = ∂N/∂x; solution is ∫M dx (y constant) + terms of N free of x = C.

Growth/decay model dN/dt = kN gives N = N₀e^{kt}; Newton cooling dT/dt = −k(T−Tₛ) gives T−Tₛ = (T₀−Tₛ)e^{−kt}.

Family of curves y = f(x) differentiated once and the constant eliminated yields the governing first-order ODE.

a·b = |a||b|cos θ = a₁b₁+a₂b₂+a₃b₃; a·b=0 iff a⊥b (for nonzero vectors).

a×b = |a||b|sin θ n̂, computed by the 3×3 determinant with i,j,k in the first row; a×b=0 iff a∥b.

|a×b| is the area of the parallelogram with sides a and b; half of it is the area of the triangle.

Scalar triple product [a b c]=a·(b×c) equals the volume of the parallelepiped; it is 0 iff the three vectors are coplanar.

Projection of a on b is (a·b)/|b|; the vector projection is [(a·b)/|b|²]b.

Direction cosines satisfy l²+m²+n²=1; direction ratios are proportional to direction cosines.

Line through point a with direction b: r = a + λb; Cartesian form (x−x₁)/l = (y−y₁)/m = (z−z₁)/n.

Angle between lines with directions b₁,b₂: cos θ = |b₁·b₂|/(|b₁||b₂|); lines are perpendicular iff b₁·b₂=0.

Plane through point a with normal n: (r−a)·n=0, i.e. ax+by+cz=d; distance from (x₁,y₁,z₁) is |ax₁+by₁+cz₁−d|/√(a²+b²+c²).

Shortest distance between skew lines r=a₁+λb₁ and r=a₂+μb₂ is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

A linear programming problem has a linear objective Z=cx+dy to maximise or minimise subject to linear constraints and non-negativity x,y≥0.

The feasible region is the intersection of the half-planes defined by the constraints; it is convex.

Corner point theorem: if an optimum exists, it is attained at a vertex of the feasible region (possibly along an entire edge).

Graphical method: plot constraints, shade the feasible region, evaluate Z at each corner, and pick the largest or smallest value.

Bounded feasible region ⟹ both maximum and minimum exist; unbounded region may give no maximum (or no minimum).

If two corner points give the same optimal Z, every point on the segment joining them is also optimal (multiple optimal solutions).

Slope method: compare the slope of the objective line −c/d with constraint slopes to confirm which vertex is optimal.

A constraint like x+y≤n with x,y≥0 gives a triangular region with vertices (0,0), (n,0), (0,n).

P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability P(A|B)=P(A∩B)/P(B), P(B)>0; A and B are independent iff P(A∩B)=P(A)P(B).

Multiplication rule: P(A∩B)=P(A)P(B|A)=P(B)P(A|B).

Bayes' theorem: P(Aᵢ|E)=P(Aᵢ)P(E|Aᵢ)/ΣⱼP(Aⱼ)P(E|Aⱼ), with {Aⱼ} a partition of the sample space.

Total probability: P(E)=ΣⱼP(Aⱼ)P(E|Aⱼ) over a partition {Aⱼ}.

Binomial distribution: P(X=r)=ⁿCᵣ pʳ qⁿ⁻ʳ, q=1−p; mean=np, variance=npq, SD=√(npq).

Bernoulli trials require fixed n, two outcomes per trial, constant p, and independence across trials.

Mean of a discrete random variable: E(X)=Σxᵢpᵢ; variance Var(X)=E(X²)−[E(X)]².

P(A′)=1−P(A); P(exactly one of A,B)=P(A)+P(B)−2P(A∩B).

RBSE internal assessment for senior secondary mathematics is typically 20 marks, split across project work, practical/lab activity and a portfolio/viva; confirm the current split in the official RBSE syllabus.

A mathematics project must state an aim, a mathematical formulation, data or derivation, analysis, conclusion, and a bibliography.

Practical work uses the prescribed mathematics lab manual: activities on relations, functions, matrices, derivatives, integrals and probability using graphs and models.

Portfolio entries should include dated records of activities performed, observations, graphs drawn, and errors corrected.

Viva questions usually test the mathematical reasoning behind the recorded activity, not memorised results.

Assessment criteria weight correct procedure, accurate computation, presentation/neatness, and interpretation of results.

Projects commonly model real data (growth, cooling, population, games of chance) with a function, derivative, integral or probability distribution.

Record the formula used and the units of every computed quantity; unexplained results lose marks.