Subject: Mathematics

Aims:

1. To acquire knowledge and understanding of the terms, symbols, concepts, principles, processes, proofs, etc. of mathematics.
2. To develop an understanding of mathematical concepts and their application to further studies in mathematics and science.
3. To develop skills to apply mathematical knowledge to solve real life problems.
4. To develop the necessary skills to work with modern technological devices such as calculators and computers.
5. To develop drawing skills, skills of reading tables, charts and graphs.
6. To develop an interest in mathematics.

CLASS IX

There will be one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks.

The paper will be divided into two sections, Section I (40 marks), Section II (40 marks).

Section I: will consist of compulsory short answer questions.

Section II: Candidates will be required to answer four out of seven questions.

The solution of a question may require the knowledge of more than one branch of the syllabus.

1. Pure Arithmetic

Irrational Numbers

(a) Rational, irrational numbers as real numbers, their place in the number system. Surds and rationalization of surds.

(b) Irrational numbers as non-repeating, nonterminating decimals.

(c) Classical definition of a rational number p, q Î Z, q 0. Hence, define irrational numbers as what cannot be expressed as above.

(d) Simplifying an expression by rationalising the denominator.

2. Commercial Mathematics

(i) Profit and Loss
The meaning of Marked price, selling price and discount, thus giving an idea of profit and loss on day to day dealings. Simple problems related to Profit and Loss and Discount, including inverse working.

(ii) Compound Interest Compound Interest as a repeated Simple Interest computation with a growing Principal. Use of formula -

(iii) Changing the subject of a formula.

• Concept that each formula is a perfect equation with variables.
• Concept of expressing one variable in terms, of another various operators on terms transposing the terms squaring or taking square root etc.

(iv) Linear Equations and Simultaneous (linear) Equations

• Solving algebraically (by elimination as well as substitution) and graphically.
• Solving simple problems based on these by framing appropriate formulae.

(v) Indices/ Exponents

Handling positive, fractional, negative and “zero” indices.

Simplification of expressions involving various exponents

(vi) Logarithms

(a) Logarithmic form vis-à-vis exponential form: interchanging.
(b) Laws of Logarithms and its use Expansion of expression with the help of laws of logarithm

4. Geometry

(i) Triangles, Relation between sides and angles of triangles. Types of triangles, Congruent triangles.

(a) Congruency: four cases: SSS, SAS, AAS, RHS. Illustration through cutouts. Simple applications.
(b) Problems based on: Angles opposite equal sides are equal and converse. If two sides of a triangle are unequal, then the greater angle is opposite the greater side and converse.

Sum of any two sides of a triangle is greater than the third side.

Of all straight lines that can be drawn to a given line from a point outside it, the perpendicular is the shortest. Proofs not required.

(ii) Constructions (using ruler and compasses)  Constructions of triangles involving 30°, 45°, 60°, 75°, 90°, 120°, 135° angles.

(iii) Mid Point Theorem and its converse, equal intercept theorem

(a) Proof and simple applications of mid point theorem and its converse.
(b) Equal intercept theorem: proof and simple application.

(iv) Similarity, conditions of similar triangles.
(a) As a size transformation.
(b) Comparison with congruency, keyword being proportionality.
(c) Three conditions: SSS, SAS, AA. Simple applications (proof not included).
(d) Applications of Basic Proportionality Theorem.

(v) Pythagoras Theorem Proof and Simple applications of Pythagoras Theorem and its converse.

(vi) Rectilinear Figures Rectilinear figures or polygons, Different kinds of polygons and its names interior and exterior angles and their relations. Types of regular polygons parallelograms, conditions of parallelograms, Rhombus, Rectangles. Proof and use of theorems on parallelogram.
(a) Sum of interior angles of a polygon.
(b) Sum of exterior angles of a polygon.
(c) Regular polygons.
(d) Parallelogram:

• Both pairs of opposite sides equal (without proof).
• Both pairs of opposite angles equal.
• One pair of opposite sides equal and parallel (without proof).
• Diagonals bisect each other and bisect the parallelogram.
• Rhombus as a special parallelogram whose diagonals meet at right angles.
• In a rectangle, diagonals are equal, in a square they are equal and meet at right angles.

(e) Quadrilaterals Construction of quadrilaterals (including parallelograms and rhombus) and regular hexagon using ruler and a pair of compasses only.

(f) Proof and use of area theorems on parallelograms: Parallelograms on the same base and between the same parallels are equal in area.

The area of a triangle is half that of a parallelogram on the same base and between the same parallels. Triangles between the same base and between the same parallels are equal in area (without proof).

Triangles with equal areas on the same bases have equal corresponding altitudes. Note: Proofs of the theorems given above are to be taught unless specified otherwise.

5. Statistics

Introduction, collection of data, presentation of data, Graphical representation of data, Mean, Median of ungrouped data.
(i) Understanding and recognition of raw, arrayed and grouped data.
(ii) Tabulation of raw data using tally-marks.
(iii) Understanding and recognition of discrete and continuous variables.
(iv) Mean, median of ungrouped data
(v) Class intervals, class boundaries and limits, frequency, frequency table, class size for grouped data.
(vi) Grouped frequency distributions: the need to and how to convert discontinuous intervals to continuous intervals.
(vii)Drawing a histogram and frequency polygon.
(viii) Understanding of how a histogram differs from a bar chart.

6. Mensuration

Area and perimeter of a triangle and a quadrilateral. Area and circumference of a circle. Surface area and volume of Cube, Cuboids and Cylinder.

(a) Area and perimeter of triangle (including Heron’s formula), square, rhombus, rectangle, parallelogram and trapezium.
(b) (i) Circle: Area and circumference
(ii) Simple direct problems involving inner and outer dimensions and cost.

(c) Surface area and volume of 3-D solids: cube, cuboid and cylinder including problems of type involving: Different internal and external dimensions of the solid. Cost.

Concept of volume being equal to area of cross-section x height.

Open/closed cubes/cuboids/cylinders.

7. Trigonometry
(a) Trigonometric Ratios: sine, cosine, tangent of an angle and their reciprocals.
(b) Trigonometric ratios of standard angles- 0, 30, 45, 60, 90 degrees. Evaluation of an expression involving these ratios.
(c) Simple 2-D problems involving one right-angled triangle.
(d) Concept of sine and cosine being complementary with simple, direct application.

8. Co-ordinate Geometry
Cartesian System, Plotting a point in the plane for given coordinates.
(a) Dependent and independent variables.
(b) Ordered pairs, co-ordinates of points and plotting them in the Cartesian Plane.
(c) Graphs of x=0, y=0, x=a, y=a, x=y, y= mx+c including identification and conceptual understanding of slope and y-intercept.
(d) Recognition of graphs based on the above.

INTERNAL ASSESSMENT

A minimum of three assignments are to be done during the year as prescribed by the teacher.

Suggested Assignments

• Surveys of a class of students - height, weight, number of family members, pocket money, etc.
• Correlation of body weight to body height.
• Planning delivery routes for a postman/milkman.
• Running a tuck shop/canteen.
• Visit one or two stores where sales are being offered to investigate - cost price, marked price, selling price, discount, profit/loss.
• Study ways of raising a loan to buy a car or house, e.g. bank loan or purchase a refrigerator or a television set through hire purchase

CLASS X

There will be one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks.

The paper will be divided into two sections, Section I (40 marks), Section II (40 marks).

Section I: Will consist of compulsory short answer questions.
Section II: Candidates will be required to answer four out of seven questions.

1. Commercial Mathematics

(i) Compound Interest
(a) Compound interest as a repeated Simple Interest computation with a growing Principal. Use of this in computing Amount over a period of 2 or 3-years.
(b) Use of formula A = P (1+ r /100)ⁿ. Finding CI from the relation CI = A – P.

• Interest compounded half-yearly included.

• Using the formula to find one quantity given different combinations of A, P, r, n, CI and SI; difference between CI and SI type included.

• Rate of growth and depreciation.

Note: Paying back in equal installments, being given rate of interest and installment amount, not included.

(ii) Sales Tax and Value Added Tax Computation of tax including problems involving discounts, list-price, profit, loss, basic/cost price including inverse cases.
(iii) Banking
(a) Savings Bank Accounts. Types of accounts. Idea of savings Bank Account, computation of interest for a series of months.
(b) Recurring Deposit Accounts: computation of interest using the formula:

(iv) Shares and Dividends

(a) Face/Nominal Value, Market Value, Dividend, Rate of Dividend, Premium.
(b) Formulae Income = number of shares × rate of dividend × FV. Return = (Income / Investment) ×100.

Note: Brokerage and fractional shares not included

2. Algebra

(i) Linear Inequations

Linear Inequations in one unknown for x Î N, W, Z, R. Solving

• Algebraically and writing the solution in set notation form.
• Representation of solution on the number line.

(ii) Quadratic Equations
(a) Quadratic equations in one unknown. Solving by:

• Factorisation.
• Formula.

(b) Nature of roots,
Two distinct real roots if b² – 4ac > 0
Two equal real roots if b² – 4ac = 0
No real roots if b² – 4ac < 0
(c) Solving problems.

(iii) Reflection
(a) Reflection of a point in a line: x=0, y =0, x= a, y=a, the origin.
(b) Reflection of a point in the origin.
(c) Invariant points.

(iv) Ratio and Proportion
(a) Duplicate, triplicate, sub-duplicate, sub-triplicate, compounded ratios.
(b) Continued proportion, mean proportion
(c) Componendo and dividendo, alternendo and invertendo properties.
(d) Direct applications.

(v) Factorization
(a) Factor Theorem.
(b) Remainder Theorem.
(c) Factorising a polynomial completely after obtaining one factor by factor theorem.

Note: f (x) not to exceed degree 3.
(vi) Matrices
(a) Order of a matrix. Row and column matrices.
(b) Compatibility for addition and multiplication.
(c) Null and Identity matrices.
(d) Addition and subtraction of 2×2 matrices.
(e) Multiplication of a 2×2 matrix by

• a non-zero rational number
• a matrix.

(vii) Co-ordinate Geometry

Co-ordinates expressed as (x,y) Distance between two points, section, and Midpoint formula, Concept of slope, equation of a line, Various forms of straight lines.
(a) Distance formula.
(b) Section and Mid-point formula (Internal section only, co-ordinates of the centroid of a triangle included).
(c) Equation of a line:

• Slope –intercept form y = mx + c
• Two- point form (y-y1) = m(x-x1)

Geometric understanding of ‘m’ as slope/ gradient/ tanq where q is the angle the line makes with the positive  Geometric understanding of c as the y-intercept/the ordinate of the point where the line intercepts the y axis/ the point on the line where x=0. Conditions for two lines to be parallel or perpendicular. Simple applications of all of the above.

3. Geometry

(i) Symmetry

(a) Lines of symmetry of an isosceles triangle, equilateral triangle, rhombus, square, rectangle, pentagon, hexagon, octagon (all regular) and diamondshaped figure.

(b) Being given a figure, to draw its lines of symmetry. Being given part of one of the figures listed above to draw the rest of the figure based on the given lines of symmetry (neat recognizable free hand sketches  acceptable).

(ii) Similarity

Axioms of similarity of triangles. Basic theorem of proportionality.

(a) Areas of similar triangles are proportional to the squares on corresponding sides.
(b) Direct applications based on the above including applications to maps and models.

(iii) Loci

Loci: Definition, meaning, Theorems based on Loci.

(a) The locus of a point equidistant from a fixed point is a circle with the fixed point as centre.
(b) The locus of a point equidistant from two interacting lines is the bisector of the angles between the lines.
(c) The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the points.

(iv) Circles
(a) Chord Properties:direction of the x axis.

• A straight line drawn from the center of a circle to bisect a chord which is not a diameter is at right angles to the chord.
• The perpendicular to a chord from the center bisects the chord (without proof).
• Equal chords are equidistant from the center.
• Chords equidistant from the center are equal (without proof).
• There is one and only one circle that passes through three given points not in a straight line.

(b) Arc and chord properties:

• The angle that an arc of a circle subtends at the center is double that which it subtends at any point on the remaining part of the circle.
• Angles in the same segment of a circle are equal (without proof).
• Angle in a semi-circle is a right angle.
• If two arcs subtend equal angles at the center, they are equal, and its converse.
• If two chords are equal, they cut off equal arcs, and its converse (without proof).
• If two chords intersect internally or externally then the product of the lengths of the segments are equal.

(c) Cyclic Properties:

• Opposite angles of a cyclic quadrilateral are supplementary.
• The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle (without proof).

(d) Tangent Properties:

• The tangent at any point of a circle and the radius through the point are perpendicular to each other.
• If two circles touch, the point of contact lies on the straight line joining their centers.
• From any point outside a circle two tangents can be drawn and they are equal in length.
• If a chord and a tangent intersect externally, then the product of the lengths of segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
• If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.

Note: Proofs of the theorems given above are to be taught unless specified otherwise.

(v) Constructions

(a) Construction of tangents to a circle from an external point.
(b) Circumscribing and inscribing a circle on a triangle and a regular hexagon.

4. Mensuration

Area and circumference of circle, Area and volume of solids – cone, sphere.

(a) Circle: Area and Circumference. Direct application problems including Inner and Outer area..
(b) Three-dimensional solids - right circular cone and sphere: Area (total surface and curved surface) and Volume. Direct application problems including cost, Inner and Outer volume and melting and recasting method to find the volume or surface area of a new solid. Combination of two solids included.

Note: Frustum is not included. Areas of sectors of circles other than quartercircle and semicircle are not included.

5. Trigonometry

(a) Using Identities to solve/prove simple algebraic trigonometric expressions

Courtesy: cisce.org

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