Optional Mathematics
ऐच्छिक गणित · Class 10 · Code SEE
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chapters done
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The syllabus asks for
- Arithmetic sequence and the nth term
- Arithmetic mean
- Sum of an arithmetic series
- Geometric sequence and the nth term
- Geometric mean
- Sum of a geometric series
- Exercises of the SEE type
Key points
- AP: tn=a+(n-1)d with d=t2-t1; three numbers in AP: a-d, a, a+d.
- AM: A=(a+b)/(2); to insert n means, d=(b-a)/(n+1).
- Sum of an AP: Sn=(n)/(2)[2a+(n-1)d]=(n)/(2)(a+l); tn=Sn-Sn-1. Sums of the first n natural, even and odd numbers: (n(n+1))/(2), n(n+1), n^2.
- GP: tn=ar^n-1 with r=(t2)/(t1); three numbers in GP: (a)/(r), a, ar.
- GM: G=√ab; to insert n means, r=left((b)/(a)right)^(1)/(n+1); A≥ G.
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The syllabus asks for
- Quadratic equations: roots and their nature
- Graph of a quadratic function
- Graph of a cubic function
- Solving a quadratic and a linear equation together
- Exercises of the SEE type
Key points
- Quadratic formula: x=frac-b±√b^2-4ac2a, D=b^2-4ac. D>0: two real different roots; D=0: equal roots; D<0: no real roots.
- Sum and product: α+β=-(b)/(a), αβ=(c)/(a); equation x^2-(α+β)x+αβ=0.
- Parabola y=ax^2+bx+c: axis x=-(b)/(2a); vertex found by substituting; y-intercept c; x-intercepts are the roots; opens up if a>0 (minimum), down if a<0 (maximum).
- Cubic y=ax^3: rises through quadrants 3, 1 if a>0; falls through quadrants 2, 4 if a<0; symmetric about the origin.
- To draw a graph: table centred on the vertex, plot, smooth curve, label axes and intercepts.
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The syllabus asks for
- Determinant of a matrix
- Inverse of a matrix
- Solving linear equations by the matrix method
- Cramer's rule
- Exercises of the SEE type
Key points
- Determinant: |A|=ad-bc for A=[(a,b),(c,d)]. A is singular if |A|=0. |AB|=|A||B|, |kA|=k^2|A|.
- Inverse: A^-1=(1)/(ad-bc)[(d,-b),(-c,a)], exists only if |A|≠0. AA^-1=A^-1A=I; (AB)^-1=B^-1A^-1; |A^-1|=(1)/(|A|).
- Matrix method: AX=B⇒ X=A^-1B (inverse on the left); needs |A|≠0.
- Cramer's rule: x=(Dx)/(D), y=(Dy)/(D) with D=a1b2-a2b1, Dx=c1b2-c2b1, Dy=a1c2-a2c1.
- D≠0: one solution; D=0 with Dx≠0: none; D=Dx=Dy=0: infinitely many.
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The syllabus asks for
- Compound angles: a quick review
- Multiple angles 2A and 3A
- Submultiple angles A/2
- Transformation formulas: sums, differences and products
- Exercises of the SEE type
Key points
- Compound angles: sin(A± B)=sin Acos B±cos Asin B; cos(A± B)=cos Acos Bmpsin Asin B; tan(A± B)=(tan A±tan B)/(1mptan Atan B).
- Double angle: sin2A=2sin Acos A=(2tan A)/(1+tan^2A); cos2A=cos^2A-sin^2A=2cos^2A-1=1-2sin^2A=(1-tan^2A)/(1+tan^2A); tan2A=(2tan A)/(1-tan^2A).
- Triple angle: sin3A=3sin A-4sin^3A; cos3A=4cos^3A-3cos A; tan3A=(3tan A-tan^3A)/(1-3tan^2A).
- Half angle: sin^2(A)/(2)=(1-cos A)/(2); cos^2(A)/(2)=(1+cos A)/(2); tan(A)/(2)=(sin A)/(1+cos A)=(1-cos A)/(sin A). Sign from the quadrant of (A)/(2).
- Product to sum: 2sin Acos B=sin(A+B)+sin(A-B); 2cos Acos B=cos(A+B)+cos(A-B); 2sin Asin B=cos(A-B)-cos(A+B).
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The syllabus asks for
- Conditional trigonometric identities
- Trigonometric equations
- Height and distance
- Exercises of the SEE type
Key points
- Triangle conditions (A+B+C=180^°): sin(A+B)=sin C, cos(A+B)=-cos C, tan(A+B)=-tan C, sin(A+B)/(2)=cos(C)/(2).
- Identities: sin2A+sin2B+sin2C=4sin Asin Bsin C; tan A+tan B+tan C=tan Atan Btan C; sin A+sin B+sin C=4cos(A)/(2)cos(B)/(2)cos(C)/(2); cos A+cos B+cos C=1+4sin(A)/(2)sin(B)/(2)sin(C)/(2); cos2A+cos2B+cos2C=-1-4cos Acos Bcos C.
- If A+B+C=90^°: tan Atan B+tan Btan C+tan Ctan A=1.
- Equations: find the principal value α, then use quadrants: sinθ=k: α, 180^°-α; cosθ=k: α, 360^°-α; tanθ=k: α, 180^°+α (for positive k). Factorise, never divide by a ratio, and respect the range.
- Height and distance: tanθ=(h)/(d); angle of depression from a point equals the angle of elevation of that point from below (alternate angles). Draw the figure first.
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The syllabus asks for
- Angle between two straight lines, parallel and perpendicular lines
- Pair of straight lines through the origin: the homogeneous equation of second degree
- Angle between the pair of lines given by a homogeneous equation
- Exercises of the SEE type
Key points
- Slope m=tanθ; for ax+by+c=0, m=-dfrac ab.
- Angle between lines of slopes m1,m2: tanθ=left|dfracm1-m21+m1m2right|.
- For a1x+b1y+c1=0 and a2x+b2y+c2=0: tanθ=left|dfraca1b2-a2b1a1a2+b1b2right|.
- Parallel: m1=m2. Perpendicular: m1m2=-1, i.e. a1a2+b1b2=0.
- The combined equation of two lines through the origin is the product of their equations; it is ax^2+2hxy+by^2=0.
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The syllabus asks for
- Conic sections: circle, parabola, ellipse and hyperbola
- Equation of a circle: centre and radius form
- Diameter form and the general equation of a circle
- Circle through three points and circles under given conditions
- Exercises of the SEE type
Key points
- Conic sections: circle (cut parallel to base), ellipse (slanted cut), parabola (cut parallel to a generator), hyperbola (cut of a double cone steeper than a generator).
- A circle is the locus of a point at a fixed distance (radius) from a fixed point (centre).
- Centre–radius form: (x-h)^2+(y-k)^2=r^2; with centre at the origin, x^2+y^2=r^2.
- Diameter form: (x-x1)(x-x2)+(y-y1)(y-y2)=0.
- General form: x^2+y^2+2gx+2fy+c=0 with centre (-g,-f) and radius √g^2+f^2-c.
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The syllabus asks for
- Scalar (dot) product of two vectors
- Angle between vectors, perpendicular and parallel vectors
- Vector geometry: section formula and collinear points
- Theorems on triangles
- Theorems on quadrilaterals and the semicircle
- Exercises of the SEE type
Key points
- acdotb=|a||b|cosθ=a1b1+a2b2; the result is a number.
- icdoti=jcdotj=1, icdotj=0; acdota=|a|^2.
- |apmb|^2=|a|^2±2acdotb+|b|^2; (a+b)·(a-b)=|a|^2-|b|^2.
- cosθ=dfracacdotb|a||b|; perpendicular Leftrightarrowacdotb=0; parallel Leftrightarrow a1b2-a2b1=0.
- Projection of a on b =dfracacdotb|b|.
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The syllabus asks for
- Linear inequalities in two variables and systems of inequalities
- Linear programming: objective function, constraints and corner points
- Composition of transformations
- Inversion transformation and the inversion circle
- Transformation using matrices
- Exercises of the SEE type
Key points
- Graph of ax+by+c≥ 0: boundary line (solid for ≥ ,≤ ; dashed for >,<) and the side confirmed by a test point; a system's solution is the overlap.
- Linear programming: maximise or minimise z=px+qy under linear constraints with x,y≥ 0; the optimum occurs at a corner point of the feasible region.
- Method: variables, objective function, constraints, feasible region, corners, evaluate z, choose.
- Basic images: reflection in x-axis (x,-y); in y-axis (-x,y); in y=x (y,x); in y=-x (-y,-x); rotation 90^° (-y,x); 180^° (-x,-y); enlargement (kx,ky); translation (x+a,y+b).
- T2° T1 means T1 first, then T2; order matters.
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The syllabus asks for
- Limit of a function: left-hand and right-hand limits
- Continuity and discontinuity on the number line and in graphs
- Continuity of a function at a point
- Exercises of the SEE type
Key points
- displaystylelimx→ a^-f(x) is the left-hand limit; displaystylelimx→ a^+f(x) the right-hand limit. The limit exists only if they are equal.
- The limit depends on values near a, not on f(a) itself.
- To find a limit: substitute; if dfrac00, factorise and cancel.
- The real number line is continuous (no gaps); natural numbers, integers and rational numbers are not.
- A graph is continuous at x=c if it has no break, jump, gap or hole there.
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The syllabus asks for
- Quartile deviation and its coefficient
- Mean deviation and its coefficient
- Standard deviation, variance and coefficient of variation
- Exercises of the SEE type
Key points
- Quartiles: Q1, Q2, Q3 divide ordered data into four parts. Positions: (n+1)/(4) and (3(n+1))/(4) (individual/discrete); frac N4 and (3N)/(4) with Q=l+frac(kN)/(4)-c.f.f× i (grouped).
- Quartile deviation =dfracQ3-Q12; coefficient =dfracQ3-Q1Q3+Q1.
- Mean deviation =dfracΣ f|x-bar x|N; coefficient =dfracM.D.bar x.
- Standard deviation σ=√dfracΣ f(x-bar x)^2N=√dfracΣ fx^2N-bar x^2; variance =σ^2.
- Step deviation: d=dfracm-Ah; bar x=A+dfracΣ fdNh; σ=h√dfracΣ fd^2N-left(dfracΣ fdNright)^2.
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The syllabus asks for
- Relations: ordered pairs, domain and range
- Functions: definition, vertical line test, values of a function
- Domain and range of a function; one-one, onto, constant and identity functions
- Composite functions
- Inverse functions and their graphs
- Polynomials: degree, standard form and the division algorithm
- Remainder theorem and factor theorem
- Synthetic division, zeros of a polynomial and factorisation
- Exercises of the SEE type
Key points
- Relation: a set of ordered pairs; domain = first components, range = second components.
- Function: every element of the domain has exactly one image. Vertical line test; horizontal line test for one-one.
- Domain: exclude zero denominators; keep square-root arguments ≥ 0.
- Composite: (f° g)(x)=f(g(x)); in general f° g≠ g° f.
- Inverse: solve y=f(x) for x, then swap letters; the graph of f^-1 is the mirror image of that of f in y=x.
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Objective questions
Optional Mathematics: Unit-wise Objective Question Bank
Subjective questions
Optional Mathematics: Unit-wise Subjective Question Bank
Model sets and predicted questions
Optional Mathematics: Model Question Sets and Predicted Questions
Exam guide
The SEE Examination: Pattern, Grid and Strategy
The paper
SEE, 3 hr, FM 75 written (+25 internal)