Mathematics
गणित · Class 11 · Code 0071 (CDC Mat.401)
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chapters done
- Algebra
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The syllabus asks for
- Statements; logical connectives
- Truth tables
- Theorems based on set operations
Key points
- A statement has exactly one truth value, T or F. Open sentences are not statements.
- p wedge q is true only when both are true; p vee q is false only when both are false; p → q is false only when p is T and q is F.
- p → q equiv sim p vee q equiv sim q → sim p. The converse q → p is equivalent to the inverse, not to the original.
- A truth table on n letters has 2^n rows. All T = tautology, all F = contradiction, identical final columns = logical equivalence.
- De Morgan, in logic and in sets: sim(p vee q) equiv sim p wedge sim q and (A ∪ B)^c = A^c ∩ B^c; (A ∩ B)^c = A^c ∪ B^c.
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The syllabus asks for
- Geometric representation of real numbers
- Interval
- Absolute value
Key points
- mathbbN ⊂ mathbbW ⊂ mathbbZ ⊂ mathbbQ ⊂ mathbbR; a real number is rational iff its decimal terminates or recurs.
- Each real number corresponds to exactly one point of the number line and conversely; √n is located exactly by a Pythagorean construction.
- Between any two distinct reals there are infinitely many rationals and infinitely many irrationals.
- (a,b) excludes endpoints, [a,b] includes them; ∞ always takes a round bracket; the length of either is b-a.
- |x| = x for x ≥ 0 and -x for x < 0; |x| = √x^2; |x-y| is the distance from x to y.
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The syllabus asks for
- Domain and range
- Inverse function; composite function
- Algebraic functions: linear, quadratic, cubic
- Transcendental functions: trigonometric, exponential, logarithmic
Key points
- f : A → B assigns exactly one image to each element of A; the range f(A) may be a proper subset of the co-domain B.
- One-one means distinct inputs have distinct images; onto means range = co-domain; a bijection is both, and only a bijection is invertible.
- Domain rules: no division by zero, no even root of a negative, no log of a non-positive number; take the intersection when several apply.
- (g ° f)(x) = g(f(x)) — apply f first. In general f ° g ≠ g ° f, and (f ° g)^-1 = g^-1 ° f^-1.
- To invert: write y = f(x), make x the subject, swap letters. Domain and range swap, and the graph of f^-1 is the reflection of that of f in y = x.
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The syllabus asks for
- Odd and even functions; periodicity
- Symmetry about origin, X-axis and Y-axis; monotonicity
- Graphs of quadratic, cubic, rational function of form 1/(ax+b)
- Graphs of a sin bx, a cos bx, e^x, ln x
Key points
- f is even if f(-x) = f(x) (graph symmetric about the Y-axis) and odd if f(-x) = -f(x) (symmetric about the origin). odd × odd = even.
- f is periodic with period T if f(x+T) = f(x); asin bx and acos bx have period (2π)/(|b|), and atan bx has period (π)/(|b|).
- Symmetry tests: x → -x unchanged gives Y-axis symmetry; y → -y gives X-axis symmetry; both together give origin symmetry.
- f is increasing on I if x1 < x2 ⇒ f(x1) < f(x2); prove it from the sign of f(x2)-f(x1). Monotonicity is claimed on an interval, never pointwise.
- Quadratic: vertex at x = -(b)/(2a), opens up if a>0; number of x-intercepts from the discriminant b^2-4ac.
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The syllabus asks for
- Arithmetic, geometric and harmonic sequences and series
- A.M., G.M., H.M. and their relations
- Sum of infinite geometric series
Key points
- AP: tn = a+(n-1)d and Sn = (n)/(2)[2a+(n-1)d] = (n)/(2)(a+l). Three terms are in AP if 2b = a+c.
- GP: tn = ar^ n-1 and Sn = dfraca(r^n-1)r-1 = dfraca(1-r^n)1-r for r ≠ 1. Three terms are in GP if b^2 = ac.
- HP: reciprocals form an AP, so tn = 1/[a+(n-1)d]. Invert, solve as an AP, invert back. There is no elementary sum formula.
- Means of a and b: A = dfraca+b2, G = √ab, H = dfrac2aba+b.
- G^2 = AH (so A, G, H are in GP) and A ≥ G ≥ H, with equality only when a = b.
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The syllabus asks for
- Transpose of a matrix and its properties
- Minors and cofactors
- Adjoint and inverse of a matrix
- Determinant and its properties (without proof)
Key points
- (AB)ij is row i of A against column j of B; AB needs (columns of A) = (rows of B), and AB ≠ BA in general.
- Transpose swaps rows and columns: (A^T)^T = A, (A+B)^T = A^T+B^T, (kA)^T = kA^T, and the reversal law (AB)^T = B^TA^T.
- Symmetric: A^T = A. Skew-symmetric: A^T = -A (zero diagonal). Every square A = frac12(A+A^T) + frac12(A-A^T).
- Minor Mij = determinant after deleting row i and column j; cofactor Cij = (-1)^i+jMij. Expand |A| along the row or column with most zeros.
- adj A = transpose of the cofactor matrix, and A(adj A) = (adj A)A = |A|I.
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The syllabus asks for
- Nature of roots
- Relation between roots and coefficients
- Formation of quadratic equation
- Symmetric roots; one or both roots common
Key points
- ax^2+bx+c = 0 has roots x = dfrac-b±√b^2-4ac2a.
- Discriminant D = b^2-4ac: D > 0 real distinct (D a perfect square makes them rational), D = 0 real and equal, D < 0 a conjugate complex pair.
- α+β = -b/a and αβ = c/a; and (α-β)^2 = D/a^2.
- Useful identities: α^2+β^2 = s^2-2p, α^3+β^3 = s^3-3ps, dfrac1α+dfrac1β = dfracsp.
- To form an equation: x^2 - (sum)x + (product) = 0.
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The syllabus asks for
- Imaginary unit; algebra of complex numbers
- Geometric representation
- Modulus and conjugate with properties
- Square root of a complex number
Key points
- i = √-1, i^2 = -1; powers of i repeat with period 4.
- z = a+ib; a+ib = c+id exactly when a = c and b = d.
- Divide by multiplying top and bottom by the conjugate of the denominator.
- Argand diagram: z = a+ib is the point (a,b); |z| = √a^2+b^2 and arg z = θ with tanθ = b/a, fixed by the quadrant.
- Polar form z = r(cosθ + isinθ).
- Trigonometry
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The syllabus asks for
- Definition, domain and range of inverse circular functions
- Formulae and identities of inverse circular functions
Key points
- Inverse circular functions exist only after the domain is restricted to a one-to-one principal branch; the graph of the inverse is the reflection of that branch in y=x.
- Ranges: sin^-1, csc^-1 ∈left[-(π)/(2),(π)/(2)right]; tan^-1∈left(-(π)/(2),(π)/(2)right); cos^-1, sec^-1∈[0,π]; cot^-1∈(0,π).
- sin^-1x+cos^-1x = tan^-1x+cot^-1x = sec^-1x+csc^-1x = (π)/(2).
- sin^-1(-x)=-sin^-1x and tan^-1(-x)=-tan^-1x, but cos^-1(-x)=π-cos^-1x and cot^-1(-x)=π-cot^-1x.
- tan^-1x+tan^-1y = tan^-1(x+y)/(1-xy) if xy<1; add π if x,y>0 with xy>1 and subtract π if x,y<0 with xy>1.
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The syllabus asks for
- Solution of trigonometric equations
- General values
Key points
- A trigonometric equation has infinitely many roots; the answer required is the general value, with n∈mathbbZ stated.
- sinθ=sinα ⇒ θ = nπ+(-1)^nα; cosθ=cosα ⇒ θ = 2nπ±α; tanθ=tanα ⇒ θ = nπ+α.
- Squared equations sin^2θ=sin^2α (and the cos^2, tan^2 versions) all give θ = nπ±α.
- Zero cases: sinθ=0 ⇒ θ=nπ; cosθ=0 ⇒ θ=(2n+1)(π)/(2); tanθ=0 ⇒ θ=nπ.
- Quadratics in one ratio: factorise. Mixed sin^2/cos: substitute sin^2θ = 1-cos^2θ. Sums of sines or cosines: use sum-to-product, then factorise.
- Analytic Geometry
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The syllabus asks for
- Length of perpendicular from a given point to a given line
- Bisectors of the angles between two straight lines
Key points
- General form ax+by+c=0; slope m = -a/b = tanθ where θ is the inclination. Forms: y=mx+c, y-y1=m(x-x1), x/a+y/b=1, xcosα+ysinα=p.
- Angle between two lines: tanθ = |dfracm1-m21+m1m2|; parallel if m1=m2, perpendicular if m1m2=-1 or a1a2+b1b2=0.
- Perpendicular distance from (x1,y1) to ax+by+c=0: d = dfrac|ax1+by1+c|√a^2+b^2; from the origin, d = dfrac|c|√a^2+b^2 = p.
- Distance between the parallel lines ax+by+c1=0 and ax+by+c2=0 is dfrac|c1-c2|√a^2+b^2, after making a and b identical.
- The bisectors of the angles between two lines are dfraca1x+b1y+c1√a1^2+b1^2 = ±dfraca2x+b2y+c2√a2^2+b2^2, and they are perpendicular to each other.
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The syllabus asks for
- General equation of second degree; condition for representing a pair of lines
- Homogeneous second degree equation in x and y
- Angle between the pair of lines
- Bisectors of the angles between the pair of lines
Key points
- ax^2+2hxy+by^2+2gx+2fy+c = 0 represents a pair of straight lines exactly when Δ = abc+2fgh-af^2-bg^2-ch^2 = 0. Remember h, g, f are half the coefficients of xy, x, y.
- The lines are real and distinct if h^2>ab, coincident if h^2=ab, parallel if h^2=ab with the pair not degenerate, and not real if h^2<ab.
- They meet at left(dfracbg-fhh^2-ab, dfracaf-ghh^2-abright).
- A homogeneous equation ax^2+2hxy+by^2 = 0 always represents two lines through the origin, with m1+m2 = -dfrac2hb and m1m2 = dfracab.
- Angle between the pair: tanθ = dfrac2√h^2-aba+b; they are perpendicular when a+b = 0 and coincident when h^2 = ab.
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The syllabus asks for
- Points in space
- Distance between two points
- Direction cosines and direction ratios
Key points
- Three mutually perpendicular axes give three coordinate planes (x=0, y=0, z=0) and eight octants; a point is an ordered triple (x,y,z).
- Distance of P(x,y,z) from the xy-plane is |z|; from the x-axis is √y^2+z^2; from the origin is √x^2+y^2+z^2.
- AB = √(x2-x1)^2+(y2-y1)^2+(z2-z1)^2.
- Section formula (internal, ratio m:n): left(dfracmx2+nx1m+n, dfracmy2+ny1m+n, dfracmz2+nz1m+nright); for external division replace n by -n. Midpoint is the case m=n; centroid is the average of the three vertices.
- Direction cosines l = cosα, m = cosβ, n = cosγ satisfy l^2+m^2+n^2 = 1, and l = dfracxr, m = dfracyr, n = dfraczr with r = OP.
- Vectors
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The syllabus asks for
- Collinear and non-collinear vectors
- Coplanar and non-coplanar vectors
- Linear combination of vectors
- Linearly dependent and independent vectors
Key points
- A vector needs magnitude and direction; in components r = xi+yj+zk with |r| = √x^2+y^2+z^2, unit vector r = r/|r|, and direction cosines satisfying l^2+m^2+n^2 = 1.
- Triangle law: AB+BC = AC. For two vectors at angle θ, R = √a^2+b^2+2abcosθ and tanα = dfracbsinθa+bcosθ.
- AB = b-a (head minus tail) — the key to every proof.
- Collinear: a = kb; points A,B,C collinear if BC = k AB.
- Section formula: internal r = dfracmb+nam+n, external r = dfracmb-nam-n, midpoint dfraca+b2.
- Statistics and Probability
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The syllabus asks for
- Standard deviation and variance
- Coefficient of variation
- Skewness; Karl Pearson's coefficient of skewness
Key points
- Dispersion measures spread about the average. Range = L-S; Q.D. = (Q3-Q1)/(2); M.D. = (Σ| x-x|)/(n).
- Variance σ^2 = dfracΣ(x-x)^2n = dfracΣ x^2n - x^2; standard deviation σ = √σ^2, in the original unit.
- Frequency data: x = dfracΣ fxN and σ = √dfracΣ fx^2N - x^2 with N = Σ f.
- Grouped data (step deviation, d' = (x-A)/(h)): x = A + dfracΣ fd'Nh and σ = h√dfracΣ fd'^2N - left(dfracΣ fd'Nright)^2.
- σ is unchanged by adding a constant and multiplied by | k| when every value is multiplied by k.
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The syllabus asks for
- Independent cases
- Mathematical and empirical definition of probability
- Two basic laws of probability (without proof)
Key points
- A random experiment has a sample space S; an event is a subset of S. Cases may be exhaustive, equally likely, mutually exclusive (A∩ B = varnothing) or independent (one does not affect the other).
- Mathematical definition: P(A) = dfracn(A)n(S) = dfracmn, valid only for equally likely cases. 0 ≤ P(A) ≤ 1 and P(A) = 1 - P(A).
- Empirical definition: P(A) = limn→∞ dfracmn, the limiting relative frequency in repeated trials.
- Odds in favour a:b Longleftrightarrow P(A) = dfracaa+b.
- Addition law: P(A∪ B) = P(A)+P(B)-P(A∩ B); for mutually exclusive events P(A∪ B) = P(A)+P(B).
- Calculus
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The syllabus asks for
- Concept of limit; limits of algebraic and transcendental functions
- Indeterminate forms
- Continuity and discontinuity of a function
Key points
- limx→ af(x) = L describes the behaviour of f near a, never at a; the value f(a) is irrelevant to the limit.
- The limit exists iff LHL = RHL = L, finite. Use x = a mp h, h>0, to compute one-sided limits.
- Formally: for every varepsilon>0 there is δ>0 with 0<|x-a|<δ ⇒ |f(x)-L|<varepsilon.
- Standard limits: (x^n-a^n)/(x-a)→ na^n-1; (sin x)/(x)→1; (tan x)/(x)→1; frac1-cos xx^2→frac12; (1+x)^1/x→ e; fraca^x-1x→ln a; (ln(1+x))/(x)→1; frac1x^n→0 as x→∞.
- The six indeterminate forms are frac00, (∞)/(∞), ∞-∞, 0×∞, 1^∞, 0^0 (with ∞^0). Factorise, rationalise, divide by the highest power, or force the e form.
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The syllabus asks for
- Derivative by first principle (definition)
- Rules of differentiation
- Derivatives of parametric and implicit functions
- Higher order derivatives
- Geometric interpretation of derivative
- Monotonicity of a function; intervals of increase and decrease
- Extreme values (maxima and minima)
- Concavity and points of inflection
Key points
- First principle: f'(x) = limh→0dfracf(x+h)-f(x)h. Four steps — form f(x+h), subtract, divide by h, take the limit. The trick in step 2 is expansion (polynomials), the conjugate (surds), a common denominator (fractions), sum-to-product (trigonometric) or a standard limit (exponential and logarithmic).
- Differentiable ⇒ continuous, but not the reverse: |x| has a corner at 0 where the left and right derivatives are -1 and +1.
- Rules: (u± v)' = u'± v'; (uv)' = uv'+vu'; left(frac uvright)' = fracvu'-uv'v^2; chain rule (dy)/(dx) = (dy)/(du)·(du)/(dx).
- Standard derivatives: x^n → nx^n-1; e^x → e^x; a^x → a^xln a; ln x → 1/x; sin x → cos x; cos x → -sin x; tan x → sec^2x; sin^-1x → 1/√1-x^2; tan^-1x → 1/(1+x^2). Every "co-" function carries a minus sign.
- Logarithmic differentiation for [f(x)]^g(x) and for long products/quotients: take ln of both sides, then differentiate, remembering (d)/(dx)ln y = frac1y(dy)/(dx).
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The syllabus asks for
- Basic (standard) integrals
- Integration by substitution
- Integration by parts
- Definite integral
- Area under a curve; area between two curves
Key points
- ∫ f(x)dx = F(x)+C means F'(x)=f(x). The constant C is compulsory in an indefinite integral and cancels in a definite one.
- Integration is linear: constants come out, sums split. There is no product or quotient rule.
- Standard forms to know cold: ∫ x^n dx = fracx^n+1n+1 (n≠-1), ∫(dx)/(x)=ln|x|, ∫(dx)/(a^2+x^2)=frac1atan^-1(x)/(a), ∫fracdx√a^2-x^2=sin^-1(x)/(a), ∫(dx)/(x^2-a^2)=(1)/(2a)ln(|x-a|)/(|x+a|).
- Substitution: u=g(x), du=g'(x)dx. Two workhorses: ∫(f')/(f)dx=ln|f|+C and ∫ f^n f'dx = fracf^n+1n+1+C. Surds √a^2-x^2, √a^2+x^2, √x^2-a^2 call for x=asinθ, atanθ, asecθ.
- Quadratic denominators: complete the square; for (px+q)/(ax^2+bx+c) write px+q=λ(2ax+b)+μ. Rational functions: divide if improper, then split into partial fractions by the four denominator cases.
- Computational Methods OR Mechanics
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The syllabus asks for
- Roots of algebraic and transcendental equations: bisection method
- Roots by Newton-Raphson method
- Numerical integration: trapezoidal rule
- Numerical integration: Simpson's rule
Key points
- A root of f(x)=0 is bracketed whenever f(a) f(b) < 0 and f is continuous on [a,b].
- Bisection: repeatedly take c = (a+b)/(2) and keep the half in which the sign change lies. It always converges, and |xn-α| ≤ fracb-a2^n, so n > (ln((b-a)/varepsilon))/(ln 2) iterations guarantee accuracy varepsilon.
- Newton-Raphson: xn+1 = xn - fracf(xn)f'(xn), obtained by replacing the curve with its tangent. Convergence is quadratic (correct digits double each step) but is not guaranteed; it fails when f'(xn) is zero or the start is poor.
- Special cases: xn+1 = frac12left(xn+fracNxnright) for √N, and xn+1 = (1)/(3)left(2xn+fracNxn^2right) for √[3]N.
- Trapezoidal rule with h = (b-a)/(n): ∫a^bf dx ≈ (h)/(2)left[(y0+yn)+2(y1+cdots+yn-1)right], with |ET| ≤ frac(b-a)h^212M2. Any n is allowed.
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The syllabus asks for
- Statics: forces and resultant forces; parallelogram law of forces
- Composition and resolution of forces; resultant of coplanar forces acting on a point
- Dynamics: motion of a particle in a straight line; motion with uniform acceleration
- Motion under gravity; motion down a smooth inclined plane
Key points
- A force is a vector: magnitude, direction, point of application. Forces at a point are concurrent; forces in one plane are coplanar.
- Parallelogram law: two forces P, Q at angle θ have resultant R = √P^2+Q^2+2PQcosθ making angle α = tan^-1dfracQsinθP+Qcosθ with P.
- The resultant always lies between |P-Q| and P+Q; it equals P+Q at θ = 0^°, √P^2+Q^2 at 90^°, |P-Q| at 180^°.
- Resolution: Fx = Fcosθ, Fy = Fsinθ. Any number of coplanar concurrent forces combine as X = Σ Fcosθ, Y = Σ Fsinθ, R = √X^2+Y^2, tanθ = Y/X; read the quadrant from the signs.
- Equilibrium Leftrightarrow X = 0 and Y = 0. For three forces use Lami's theorem: each force divided by the sine of the angle between the other two is the same. The equilibrant is minus the resultant.
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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