Optional Mathematics
ऐच्छिक गणित · Class 8
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chapters done
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The syllabus asks for
- Ordered pairs and equal ordered pairs
- Cartesian product of two sets
- Relation: meaning, representation (ordered pairs, table, arrow diagram, graph), domain and range, inverse relation
- Function: meaning and notation, domain, co-domain and range, image and pre-image
- Representation of a function and the vertical line test
Key points
- An ordered pair (a,b) has a first and a second component; (a,b)=(c,d) exactly when a=c and b=d.
- The Cartesian product A× B=(a,b): a∈ A, b∈ B; n(A× B)=n(A)× n(B); A× B≠ B× A unless A=B.
- A relation from A to B is a subset of A× B; it can be shown by ordered pairs, set-builder form, a table, an arrow diagram or a graph.
- Domain = set of first components; range = set of second components.
- The inverse R^-1=(y,x): (x,y)∈ R swaps the domain and the range.
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The syllabus asks for
- Polynomial expressions and polynomial functions; classification by degree and by number of terms
- Degree of a polynomial, standard form and equal polynomials
- Addition and subtraction of polynomials
- Multiplication of polynomials
Key points
- A polynomial has only whole-number powers of its variables; negative or fractional powers, roots of the variable and variables in a denominator are not allowed.
- Degree = highest power of the variable (for several variables, the highest sum of powers in a term); a non-zero constant has degree 0.
- By degree: constant (0), linear (1), quadratic (2), cubic (3), biquadratic (4). By terms: monomial, binomial, trinomial, polynomial.
- Standard form: descending powers; the first coefficient is the leading coefficient.
- Equal polynomials have equal coefficients of like powers: compare x^2 with x^2, x with x, constant with constant.
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The syllabus asks for
- Sequence: meaning, terms and the general term
- Finding terms and the rule of a sequence
- Series and the sigma (Σ) notation
Key points
- A sequence is an ordered list made by a rule; its terms are t1, t2, ldots, tn. It is finite or infinite.
- The general term tn gives the term in position n; a sequence is a function on the natural numbers.
- Equal first differences d: arithmetic, tn=a+(n-1)d. Equal ratios r: geometric, tn=ar^n-1. Equal second differences: the rule contains n^2.
- To test whether a number is a term, solve tn= number; it is a term only if n is a natural number.
- A series is the sum of the terms; Sn=t1+t2+cdots+tn and tn=Sn-Sn-1.
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The syllabus asks for
- Concept of limit from sequences of numbers
- Concept of limit from sequences of diagrams
- Value of a function and its limiting value
- Notation of limit
Key points
- A sequence has limit l if its terms come as close to l as we wish and stay that close; the limit need not be a term.
- (n)/(n+1)→1, (1)/(2^n)→0, 0.9, 0.99, ldots→1; sequences such as 1, 2, 3, ldots and 1, -1, 1, ldots have no limit.
- Sequences of diagrams have limits too: inscribed polygons tend to the circle (perimeter →2π r, area →π r^2); shading half of what is left tends to the whole square.
- The value f(a) is found by putting x=a; the limit comes from values of x close to a on both sides, never equal to a.
- The limit can exist without the value (a hole), the two can differ, or there can be no limit (a jump).
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The syllabus asks for
- Matrix: elements, rows, columns and order
- Types of matrices: row, column, zero, rectangular, square, diagonal, scalar, unit, equal, symmetric and triangular
- Addition and subtraction of matrices and the properties of matrix addition
- Multiplication of a matrix by a scalar
Key points
- A matrix is a rectangular array of numbers in rows and columns; a matrix with m rows and n columns has order m× n and mn elements; aij is the element in row i, column j.
- Row 1× n, column m×1, zero (all 0), rectangular (m≠ n), square (m=n).
- In a square matrix: diagonal (zero off the leading diagonal), scalar (diagonal with equal diagonal elements), unit I (scalar with 1s), symmetric (aij=aji), upper/lower triangular (zeros below/above the leading diagonal).
- Equal matrices have the same order and equal corresponding elements.
- A± B=[aij± bij], possible only for the same order.
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The syllabus asks for
- Distance formula and its applications
- Section formula: dividing a line segment in a given ratio; mid-point
- Slope of a line; lines parallel to the axes; slope-intercept form y = mx + c
- Area of a triangle and a quadrilateral from the coordinates of the vertices
Key points
- Distance: PQ=√(x2-x1)^2+(y2-y1)^2; from the origin √x^2+y^2. Compare squared lengths to classify triangles and quadrilaterals; AB+BC=AC means collinear.
- Section formula (internal, m1:m2): left((m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2)right).
- Mid-point: left((x1+x2)/(2), (y1+y2)/(2)right); centroid left((x1+x2+x3)/(3), (y1+y2+y3)/(3)right).
- Slope: m=(y2-y1)/(x2-x1)=tanθ; horizontal lines have slope 0, vertical lines have no defined slope; parallel lines have equal slopes.
- Lines parallel to the axes: x=a and y=b; the axes are y=0 and x=0.
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The syllabus asks for
- Measurement of angles: sexagesimal, centesimal and circular (radian) systems
- Trigonometric ratios of an acute angle
- Trigonometric identities and their conversions
- Values of trigonometric ratios of 0°, 30°, 45°, 60° and 90°
- Trigonometric ratios of (90° ± θ), (180° ± θ) and (−θ)
- Height and distance with one angle of elevation or depression
Key points
- Right angle =90^°=100^g=left((π)/(2)right)^c; 1^°=60', 1'=60''; 1^g=100^c, 1^c=100^cc; π^c=180^°=200^g; (D)/(90)=(G)/(100)=(2R)/(π); arc l=rθ.
- sinθ=(p)/(h), cosθ=(b)/(h), tanθ=(p)/(b); cosec, sec and cot are their reciprocals; tanθ=(sinθ)/(cosθ).
- sin^2θ+cos^2θ=1, 1+tan^2θ=sec^2θ, 1+cot^2θ=cosec^2θ; prove identities from one side, one change per line.
- sin0^°, sin30^°, sin45^°, sin60^°, sin90^°=0, (1)/(2), frac1√2, frac√32, 1; cosines in reverse order; tan45^°=1, tan60^°=√3, tan90^° not defined.
- Signs by quadrant: All, Sin, Tan, Cos.
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The syllabus asks for
- Vector and scalar quantities; magnitude and direction of a vector
- Types of vectors: row, column, position, unit, null, equal and negative vectors; like and unlike vectors
- Multiplication of a vector by a scalar; condition for parallel vectors
- Addition and subtraction of vectors
Key points
- A scalar has magnitude only (mass, distance, speed); a vector has magnitude and direction (displacement, velocity, force).
- overrightarrowAB=binomx2-x1y2-y1 (head minus tail); |a|=√x^2+y^2; direction tanθ=(y)/(x), with the quadrant fixed by the signs.
- Position vector overrightarrowOP=binomxy; unit vector a=(a)/(|a|); null vector binom00; i=binom10, j=binom01.
- Equal vectors: same magnitude and direction (equal components); negative -a: same magnitude, opposite direction; overrightarrowBA=-overrightarrowAB.
- ka=binomkxky, |ka|=|k||a|; parallel: a=kb, i.e. x1y2-x2y1=0; like if k>0, unlike if k<0.
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The syllabus asks for
- Transformation: meaning; isometric and non-isometric transformations
- Reflection in the x-axis and the y-axis
- Rotation about the origin through 90°, 180°, 270° and 360°
- Translation by a vector
Key points
- A transformation maps every point P of an object to a point P' of the image.
- Isometric transformations keep all lengths: reflection, rotation and translation give congruent images. Non-isometric ones, such as enlargement, change lengths and give similar images.
- Reflection reverses orientation; rotation and translation keep it. Points on the mirror line, and the centre of a rotation, are invariant.
- Reflection in the x-axis: (x,y)→(x,-y); in the y-axis: (x,y)→(-x,y). Two reflections, one in each axis, make a rotation of 180^°.
- Rotation about O: +90^°→(-y,x), 180^°→(-x,-y), +270^° (or -90^°) →(y,-x), 360^°→(x,y); distances from O are unchanged.
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The syllabus asks for
- Arithmetic mean of ungrouped and grouped data
- Median and quartiles of ungrouped data
- Mode of ungrouped and grouped data
- Range and coefficient of range
Key points
- Mean: X=(Σ x)/(n) (individual), (Σ fx)/(N) (discrete), (Σ fm)/(N) (continuous, m = mid-value); N=Σ f.
- Median: arrange in order; Md = value of the left((n+1)/(2)right)th item; for a frequency table use cumulative frequencies.
- Quartiles: Q1 = left((n+1)/(4)right)th item, Q3 = left((3(n+1))/(4)right)th item; interpolate between neighbours in individual data.
- Mode: the most frequent value; for grouped data Mo=L+(f1-f0)/(2f1-f0-f2)× i in the modal class.
- Choose the mean for ordinary measured data, the median when there are extreme values, the mode for the most common category.
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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 Basic Level Examination: Pattern, Grid and Strategy
The paper
School annual examination, 2 hr, FM 50 written
| Section | Questions | Marks each | Total |
|---|---|---|---|
| Group A objective | 10 | 1 | 10 |
| Group B short | 10 | 2 | 20 |
| Group C long | 5 | 4 | 20 |