Elementary Mathematics (Up to Class X Level)

What the official syllabus says

Preliminary Examination, Paper-II, General Studies-II:

• Elementary Mathematics upto Class X level: Arithmetic, Algebra, Geometry and Statistics.

Elementary Mathematics (Upto Class X Level):

Arithmetic:

• Number systems: Natural Numbers, Integers, Rational and Irrational numbers, Real numbers, Divisors of an Integer, prime Integers, L.C.M. and H.C.F. of integers and their Interrelationship.
• Average.
• Ratio and proportion.
• Percentage.
• Profit and Loss.
• Simple and Compound Interests.
• Work and Time.
• Speed, Time and Distance.

Algebra:

• Factors of polynomials, L.C.M. and H.C.F. of polynomials and their Interrelationship, Remainder theorem, simultaneous linear equations, quadratic equations.
• Set Theory: Set, null set, subsets and proper subsets of a set, operations (Union, Intersections, difference, symmetric difference) between sets, venn diagram.

Geometry:

• Constructions and theorems regarding triangle, rectangle, square, trapezium and circles, their perimeter and area.
• Volume and surface area of sphere, right circular cylinder, right circular Cone and Cube.

Statistics:

• Collection of data, Classification of data, frequency, frequency distribution, tabulation, cumulative frequency. Representation of data: Bar diagram, Pie chart, histogram, frequency polygon, cumulative frequency curves (ogives), Measures of Central tendency: Arithmetic Mean, Median and Mode.

Topics in detail

Arithmetic

• Number system: natural numbers, whole numbers, integers, rational, irrational and real numbers, and how each set contains the one before
• Terminating and recurring decimals; converting a recurring decimal into a fraction
• Irrational numbers such as the square root of 2 and pi; surds
• Divisibility rules for 2, 3, 4, 5, 6, 8, 9 and 11
• Prime and composite numbers; co-primes; prime factorisation
• Number of divisors of an integer from its prime factorisation
• H.C.F. and L.C.M. by prime factorisation and by division; H.C.F. and L.C.M. of fractions
• Interrelationship: for two numbers, H.C.F. × L.C.M. = product of the numbers
• Remainders and the unit digit of powers
• Average: simple and weighted averages, average of consecutive numbers, effect of adding, removing or replacing a member
• Ratio and proportion: compound ratio, mean, third and fourth proportional, direct and inverse proportion
• Partnership; mixtures and alligation
• Percentage: fraction-percentage conversion, percentage increase and decrease, successive changes
• Population growth and depreciation; expenditure and consumption problems; percentages in examinations and elections
• Profit and loss: cost price, selling price, marked price, discount, successive discounts, dishonest dealer (false weights)
• Simple interest: SI = P × R × T / 100
• Compound interest: amount = P(1 + R/100) to the power n (n years), compounding half-yearly and quarterly; difference between CI and SI for two years
• Work and time: work done per day, efficiency, people working together or in turns; pipes and cisterns; work and wages
• Speed, time and distance: converting km/h and m/s (multiply by 5/18), average speed, relative speed
• Trains crossing poles, platforms and each other; boats and streams; races

Algebra

• Polynomials: degree, coefficients, value of a polynomial, zeroes
• Algebraic identities: (a + b)², (a − b)², a² − b², (a + b)³, a³ + b³, a³ − b³
• Factorising polynomials: common factors, grouping, splitting the middle term, using identities
• Remainder theorem: the remainder when p(x) is divided by (x − a) is p(a); factor theorem
• H.C.F. and L.C.M. of polynomials, and their relationship with the product of two polynomials
• Simultaneous linear equations in two variables: substitution, elimination and cross-multiplication
• Conditions for a unique solution, no solution or infinitely many solutions
• Word problems leading to linear equations: ages, numbers, money, speed
• Quadratic equations: standard form, solving by factorisation and by the quadratic formula
• Discriminant (b² − 4ac) and the nature of the roots
• Sum of the roots (−b/a) and product of the roots (c/a)
• Set theory: roster and set-builder forms; finite, infinite, null (empty) and universal sets
• Subsets and proper subsets; number of subsets of a set with n elements (2 to the power n)
• Union, intersection, difference and symmetric difference of sets; complement
• Venn diagrams; n(A ∪ B) = n(A) + n(B) − n(A ∩ B); problems with two and three sets

Geometry

• Triangles: angle-sum property, exterior angle, kinds of triangles
• Congruence of triangles (SSS, SAS, ASA, RHS) and similarity (AA, SSS, SAS)
• Basic proportionality theorem; areas of similar triangles
• Pythagoras theorem and its converse
• Area of a triangle: half base × height, Heron's formula, equilateral triangle
• Rectangle and square: properties of diagonals, perimeter and area
• Trapezium: area as half the sum of parallel sides × height
• Circles: chords, equal chords and their distance from the centre
• Angle at the centre is twice the angle at the circumference; angle in a semicircle is a right angle
• Cyclic quadrilateral: opposite angles add up to 180°
• Tangents: perpendicular to the radius; two tangents from an outside point are equal
• Circumference and area of a circle; length of an arc and area of a sector
• Standard constructions: dividing a line segment in a given ratio, tangents to a circle, triangles similar to a given triangle
• Cube: surface area 6a², volume a³
• Right circular cylinder: curved surface 2πrh, total surface 2πr(r + h), volume πr²h
• Right circular cone: slant height, curved surface πrl, volume one-third πr²h
• Sphere: surface area 4πr², volume four-thirds πr³
• Problems combining or recasting solids (melting one solid into another)

Statistics

• Collection of data: primary and secondary data
• Classification: raw data, class intervals (inclusive and exclusive), class limits and class marks
• Frequency, frequency distribution tables and tally marks
• Cumulative frequency: less-than and more-than tables
• Bar diagrams: simple, multiple and sub-divided
• Pie chart: the angle of each sector as its share of 360°
• Histogram, frequency polygon and ogives (less-than and more-than); reading the median from ogives
• Arithmetic mean of ungrouped and grouped data: direct, assumed-mean and step-deviation methods
• Median of ungrouped and grouped data
• Mode of ungrouped and grouped data
• Empirical relationship: mode = 3 median − 2 mean

Overlaps

• Number patterns used in General Mental Ability rest on the number system above.
• Venn diagrams link set theory with Logical Reasoning.