Mathematics
गणित · Class 12 · Code 0081 (CDC Mat.402)
0 / 17
chapters done
- Algebra
-
The syllabus asks for
- Basic principle of counting
- Permutation of objects all different
- Permutation of objects not all different
- Circular permutation; permutation with repeated use of objects
- Combination of things all different
- Properties of combination
Key points
- Multiplication principle (AND): multiply the counts. Addition principle (OR): add them. A tree diagram makes which one applies visible.
- n! = n(n-1)cdots 1 and 0! = 1.
- Permutation (order matters): ^nPr = n(n-1)cdots(n-r+1) = dfracn!(n-r)!, and ^nPn = n!.
- n objects with p, q, s alike arrange in dfracn!p! q! s! ways.
- Round a circle: (n-1)! arrangements; a garland or necklace, which can be flipped: dfrac(n-1)!2.
-
The syllabus asks for
- Binomial theorem for a positive integral index
- General term; binomial coefficients
- Binomial theorem for any index (without proof)
- Approximation using binomial expansion
- Euler's number e
- Expansion of e^x, a^x and log(1+x) (without proof)
Key points
- (a+b)^n = Σr=0^n^nCra^n-rb^r for a positive integer n; it has n+1 terms and is proved by induction using Pascal's rule.
- General term: Tr+1 = ^nCra^n-rb^r. To find a particular term, simplify the power of x in Tr+1 and equate it to the power wanted.
- Middle term: T(n)/(2)+1 if n is even; T(n+1)/(2) and T(n+3)/(2) if n is odd.
- Binomial coefficients: ^nCr = ^nCn-r, they build Pascal's triangle, Σ^nCr = 2^n, and the alternating sum is 0.
- For any index and |x| < 1: (1+x)^n = 1 + nx + dfracn(n-1)2!x^2 + cdots — always state |x| < 1, and factor out constants first so the bracket starts with 1.
-
The syllabus asks for
- Polar form of a complex number
- De Moivre's theorem and its application to roots
- Properties of cube roots of unity
- Euler's formula
Key points
- Polar form: z = r(cosθ + isinθ) with r = |z| = √x^2+y^2 and tanθ = y/x; fix θ from the quadrant of (x, y), principal value in -π < θ ≤ π.
- Multiply: moduli multiply, arguments add. Divide: moduli divide, arguments subtract.
- De Moivre: left[r(cosθ+isinθ)right]^n = r^n(cos nθ + isin nθ), proved by induction for positive integers and extended to negative and rational indices.
- Equating real and imaginary parts in (cosθ+isinθ)^n = cos nθ + isin nθ gives the multiple-angle formulas, e.g. cos 3θ = 4cos^3θ - 3cosθ.
- n^th roots: z^1/n = r^1/nleft[cosdfracθ+2kπn + isindfracθ+2kπnright], k = 0, 1, ldots, n-1 — exactly n roots, equally spaced on a circle of radius r^1/n, forming a regular n-gon.
-
The syllabus asks for
- Sum of first n natural numbers
- Sum of squares and cubes of first n natural numbers
- Principle of mathematical induction
Key points
- Σk=1^n k = dfracn(n+1)2, Σk=1^n k^2 = dfracn(n+1)(2n+1)6, Σk=1^n k^3 = dfracn^2(n+1)^24 = left(Σ kright)^2.
- Sigma is linear: Σ(ak ± bk) = Σ ak ± Σ bk and Σ c ak = cΣ ak, with Σk=1^n c = nc.
- To sum a series, write tk as a polynomial in k, split with the sigma rules, substitute the three standard sums, factorise, then test on n = 1.
- For a part-sum, Σk=m^n = Σk=1^n - Σk=1^m-1.
- Induction: state P(n); prove P(1); assume P(k); use the assumption to prove P(k+1); conclude for all n ∈ mathbbN.
-
The syllabus asks for
- Cramer's rule (up to three variables)
- Matrix method: row-equivalent (Gauss) method
- Matrix method: inverse matrix method
Key points
- Every linear system is AX = B with A the coefficient matrix, X the column of unknowns and B the column of constants.
- Cramer: x = Dx/D, y = Dy/D, z = Dz/D, where D = |A| and each Dx replaces the matching column of D by B. Requires D ≠ 0.
- Row-equivalent (Gauss): apply elementary row operations (Ri leftrightarrow Rj, Ri → cRi, Ri → Ri + cRj) to [A | B] until the coefficient block is upper triangular, then back substitute. Row operations never change the solution set.
- Gauss-Jordan: carry on until [I | X]; the solution is the last column.
- Inverse method: X = A^-1B = dfrac1|A|(adj A)B, valid only when |A| ≠ 0. Multiply on the left.
- Trigonometry
-
The syllabus asks for
- Sine law
- Cosine law
- Tangent law
- Projection laws
- Half angle laws
Key points
- Standard notation: a faces A, s = (a+b+c)/(2), area Δ.
- Sine law: (a)/(sin A) = (b)/(sin B) = (c)/(sin C) = 2R, so a = 2Rsin A — the switch from sides to sines.
- Cosine law: a^2 = b^2+c^2-2bccos A, i.e. cos A = fracb^2+c^2-a^22bc; negative cosine means an obtuse angle.
- Tangent law: tan(A-B)/(2) = (a-b)/(a+b)cot(C)/(2), with (A+B)/(2) = 90^°-(C)/(2).
- Projection laws: a = bcos C + ccos B, and its two cyclic partners.
-
The syllabus asks for
- Solution of triangles (simple cases)
- Heights and distances
Key points
- Three parts solve a triangle if at least one is a side; AAA gives shape only.
- SSS: cosine law cos A = fracb^2+c^2-a^22bc, or half angles tan(A)/(2) = √((s-b)(s-c))/(s(s-a)).
- SAS: a^2 = b^2+c^2-2bccos A; or Napier, tan(A-B)/(2) = (a-b)/(a+b)cot(C)/(2) with (A+B)/(2) = 90^°-(C)/(2).
- ASA/AAS: third angle first, then (a)/(sin A) = (b)/(sin B).
- SSA is ambiguous: with h = bsin A and A acute — none if a<h, one (right-angled) if a = h, two if h<a<b, one if a ≥ b.
- Analytic Geometry
-
The syllabus asks for
- Condition of tangency of a line to a circle
- Tangent and normal to a circle
- Standard equation of parabola
- Tangent and normal to a parabola
- Standard equation of ellipse
- Standard equation of hyperbola
Key points
- A conic is the locus of P with PS/PM = e, where S is the focus and the directrix gives PM. e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola. The latus rectum is the focal chord perpendicular to the axis.
- Circle: (x-h)^2+(y-k)^2=r^2; general form x^2+y^2+2gx+2fy+c=0 has centre (-g,-f) and r=√g^2+f^2-c; diameter form (x-x1)(x-x2)+(y-y1)(y-y2)=0.
- A line touches a circle when the perpendicular distance from the centre equals the radius; for x^2+y^2=a^2 and y=mx+c this is c^2=a^2(1+m^2), with contact point (-a^2m/c, a^2/c).
- Tangent to x^2+y^2=a^2 at (x1,y1) is xx1+yy1=a^2; the normal always passes through the centre; the length of the tangent from (x1,y1) is √S1; the director circle is x^2+y^2=2a^2.
- Parabola: y^2=4ax has vertex (0,0), focus (a,0), directrix x=-a, latus rectum 4a, parametric point (at^2,2at). Three sign/axis variants give the other three orientations.
- Vectors
-
The syllabus asks for
- Scalar (dot) product; angle between two vectors
- Geometric interpretation and properties of scalar product
- Applications of scalar product in geometry and trigonometry
- Vector (cross) product; geometrical interpretation and properties
- Applications of vector product in geometry and trigonometry
Key points
- Scalar product: acdotb = |a||b|cosθ = a1b1+a2b2+a3b3, a number. The two forms agree by the cosine law applied to the triangle on a, b and b-a.
- Angle and perpendicularity: cosθ = dfracacdotb|a||b|; for non-zero vectors aperpb Leftrightarrow acdotb = 0. Also acdota = |a|^2, icdoti = 1, icdotj = 0.
- Properties of ·: commutative, distributive over addition, (ma)cdotb = m(acdotb), and |acdotb| ≤ |a||b|. You may not cancel a vector factor.
- Projection: the scalar projection of b on a is dfracacdotb|a| (signed), and the vector projection is dfracacdotb|a|^2a. Work done is W = Fcdotd.
- Dot-product geometry: cosine law a^2 = b^2+c^2-2bccos A; projection law a = bcos C + ccos B; cos(A± B) = cos Acos B mp sin Asin B; diagonals of a rhombus are perpendicular because (a+b)·(a-b) = |a|^2-|b|^2; the angle in a semicircle is 90^°.
- Statistics and Probability
-
The syllabus asks for
- Correlation and its nature
- Karl Pearson's coefficient of correlation and interpretation
- Properties of correlation coefficient (without proof)
- Rank correlation (elementary concept)
- Regression equations: line of y on x and x on y
Key points
- Correlation measures how two variables move together: positive, negative or zero; linear or curvilinear; simple, multiple or partial. The scatter diagram shows it at a glance.
- Karl Pearson: r = dfracΣ dx dy√Σ dx^2Σ dy^2 (actual means), or r = dfracnΣ xy - Σ xΣ y√[nΣ x^2-(Σ x)^2][nΣ y^2-(Σ y)^2] (raw scores, or deviations from assumed means).
- 1 ≤ r ≤ 1; r is symmetric, unit-free and unchanged by a change of origin and scale; r = ± 1 means all points lie on a line; r = 0 means no linear relation; r^2 is the fraction of variation explained.
- Spearman: rs = 1 - dfrac6Σ D^2n^3-n, with Σ D^2 replaced by Σ D^2 + Σdfracm^3-m12 when ranks are tied. Rank both variables from the same end; check Σ D = 0.
- Least squares gives the normal equations Σ y = na + bΣ x and Σ xy = aΣ x + bΣ x^2, whence byx = dfracΣ dx dyΣ dx^2 and bxy = dfracΣ dx dyΣ dy^2.
-
The syllabus asks for
- Dependent cases
- Conditional probability (without proof)
Key points
- Events are dependent when one changes the probability of the other; drawing without replacement is the standard example.
- Multiplication theorem: P(A ∩ B) = P(A)P(B mid A) = P(B)P(Amid B), and P(A∩ B∩ C) = P(A)P(Bmid A)P(Cmid A∩ B). Independent means P(A ∩ B) = P(A)P(B).
- Conditional probability: P(A mid B) = dfracP(A∩ B)P(B), i.e. B becomes the new sample space.
- On a tree diagram: multiply along a branch, add across branches; all branch probabilities from one node add to 1.
- "At least one" = 1 - "none".
- Calculus
-
The syllabus asks for
- Derivatives of hyperbolic and inverse hyperbolic functions
- L'Hospital's rule (0/0 and infinity/infinity forms)
- Differentials
- Tangent and normal to a curve
- Derivative as a rate measure
Key points
- The standard derivatives, the product, quotient and chain rules, and the techniques for implicit (y gives a factor (dy)/(dx)), parametric ((dy)/(dx) = (dy/dt)/(dx/dt)) and logarithmic differentiation (take ln of both sides, for u^v forms) are the working toolkit.
- Hyperbolic: sinh x = frace^x-e^-x2, cosh x = frace^x+e^-x2, cosh^2x-sinh^2x=1, and (sinh x)' = cosh x, (cosh x)' = sinh x, (tanh x)' = sech^2x. Inverses: (sinh^-1x)' = frac1√x^2+1, (cosh^-1x)' = frac1√x^2-1, (tanh^-1x)' = frac11-x^2.
- L'Hospital: for frac00 or (∞)/(∞) only, lim(f)/(g) = lim(f')/(g'), repeated as needed. Convert 0·∞, ∞-∞, 0^0, 1^∞ to a quotient first.
- Differentials: dy = f'(x)dx and f(x+Δ x)≈ f(x) + f'(x)Δ x; relative error (dy)/(y).
- Tangent at (x1,y1): y-y1 = m(x-x1) with m = f'(x1); normal: y-y1 = -(1)/(m)(x-x1).
-
The syllabus asks for
- Standard integrals
- Integrals reducible to standard forms
- Integrals of rational functions
Key points
- Anti-differentiation reverses differentiation: ∫ f(x) dx = F(x)+C where F' = f. The +C is compulsory, and it makes the answer a whole family of parallel curves.
- Learn the table of standard integrals to the point of recall, including ∫(dx)/(x) = ln|x|, the nine quadratic forms, and ∫tan x dx = ln|sec x|, ∫sec x dx = ln|sec x+tan x|.
- Substitution works when the integrand contains g(x) together with g'(x). The master pattern is ∫(g'(x))/(g(x))dx = ln|g(x)|+C.
- By parts: ∫ u dv = uv - ∫ v du, with u chosen by ILATE. Polynomial times sine or exponential needs repeated parts; e^axsin bx is cyclic, so apply parts twice and solve for I; and ∫ e^xleft[f+f'right]dx = e^xf(x)+C in one step.
- A quadratic denominator is handled by completing the square; an x in the numerator by splitting px+q = λ(denominator)' + μ.
-
The syllabus asks for
- Order and degree of a differential equation
- Differential equations of first order and first degree
- Separable variables
- Homogeneous differential equations
- Linear differential equations
- Exact differential equations
Key points
- A differential equation relates y to its derivatives. Its order is the highest derivative present; its degree is the power of that derivative once all radicals in the derivatives have been cleared. The degree may be undefined.
- The general solution of an n-th order equation contains n arbitrary constants. Extra conditions turn it into a particular solution. To form the equation of a family with n constants, differentiate n times and eliminate the constants.
- A first-order first-degree equation has a one-parameter family of solutions, and the slope field is the picture of that family.
- Separable: write f(y) dy = g(x) dx and integrate both sides. Test this first, because it is the cheapest method.
- Homogeneous: if (dy)/(dx) = Fleft(frac yxright), put y = vx so that (dy)/(dx) = v + x(dv)/(dx). The result always separates.
- Computational Methods OR Mechanics
-
The syllabus asks for
- System of linear equations: Gauss Elimination Method
- System of linear equations: Gauss-Seidel Method
- Linear programming problems (LPP): simplex method (two variables)
Key points
- Gauss elimination. Form [A | b]; use Ri → Ri - mikRk with mik = aik/akk to make it upper triangular; back substitute. Swap rows if a pivot is 0 (partial pivoting).
- A row 0;0;0 | k with k ≠ 0 means no solution; 0;0;0 | 0 means infinitely many.
- Gauss–Seidel. x^(k+1) = frac1a11(b1 - a12y^(k) - a13z^(k)), and so on, always using the newest value available. Converges for certain when |aii| > Σj≠ i|aij| in every row — reorder the equations if it is not so.
- Direct methods (elimination) finish in a set number of steps; iterative methods (Gauss–Seidel) are stopped when successive iterates agree to the required accuracy.
- LPP. Maximise Z = c1x + c2y over a convex feasible region; the optimum lies at a corner.
-
The syllabus asks for
- Statics: triangle law of forces; Lami's theorem
- Dynamics: Newton's laws of motion
- Dynamics: projectile motion
Key points
- Resultant of P and Q at angle θ: R = √P^2+Q^2+2PQcosθ, at tanα = dfracQsinθP+Qcosθ to P. Resolution: components Fcosθ, Fsinθ.
- Triangle law. Three concurrent forces are in equilibrium exactly when they form a closed head-to-tail triangle; with more forces, a closed polygon.
- Lami's theorem. For three concurrent forces in equilibrium, dfracPsinα = dfracQsinβ = dfracRsinγ, each angle being between the other two forces.
- Moment M = F× d; a rigid body is in equilibrium when Σ Fx = 0, Σ Fy = 0 and Σ M = 0.
- Newton's laws. F = ma; impulse FΔ t = mv - mu; momentum m1u1 + m2u2 = m1v1 + m2v2 is conserved in every collision.
-
The syllabus asks for
- Elementary Group Theory: binary operation, group, abelian group, finite and infinite groups
- Rolle's theorem and Mean Value Theorem
- Numerical integration: trapezoidal and Simpson's rule (revision)
Key points
- A binary operation on G assigns to each ordered pair (a, b) a unique element a * b of G; the closure requirement is part of the definition.
- (G, ) is a group if it is closed, associative, has an identity e and every element has an inverse; it is abelian if ab = b*a for all a, b, and the order |G| is the number of elements.
- The Cayley table shows closure (all entries in G), the identity (a row repeating the heading), inverses (one e in each row and column) and the abelian property (symmetry about the leading diagonal).
- Standard examples: (mathbbZ, +) infinite abelian; (Z4, +4) and 1, -1, i, -i under × finite abelian of order 4; (1,2,3,4 ×5) abelian of order 4; S3 non-abelian of order 6.
- In any group the identity and each inverse are unique.
No chapter matches. Try another word.
The paper
3hr, FM 75 written (+25 internal)
| Section | Questions | Marks each | Total |
|---|---|---|---|
| Group A MCQ | 11 | 1 | 11 |
| Group B short | 8 | 5 | 40 |
| Group C long | 3 | 8 | 24 |
Where the hours go
Teaching hours are the board's guide to how much of the paper each part carries.
- Calculus48 h
- Algebra44 h
- Computational Methods OR Mechanics24 h
- Analytic Geometry20 h
- Trigonometry12 h
- Vectors12 h
- Statistics and Probability12 h