Maths – Content (Class-VI to Intermediate Present syllabus) (40 Marks)
1. Arithmetic
BODMAS rule: Order of operations (Brackets, Of, Division, Multiplication, Addition, Subtraction).
Ratios and Proportions (Direct, Inverse):
Ratio: Comparison of two quantities.
Proportion: Equality of two ratios.
Direct proportion: Two quantities increase or decrease in the same ratio.
Inverse proportion: Two quantities increase or decrease in the opposite ratio.
Comparing quantities using ratios, proportion, percentage and their applications: Solving problems involving comparisons.
Profit and Loss:
Cost Price (CP), Selling Price (SP), Profit, Loss.
Profit = SP - CP
Loss = CP - SP
Profit percentage = (Profit / CP) * 100
Loss percentage = (Loss / CP) * 100
Discount: Reduction in the marked price.
Discount = Marked Price - Selling Price
Discount percentage = (Discount / Marked Price) * 100
Sales Tax/Value Added Tax/Goods and Services Tax: Tax on the sale of goods and services.
Simple Interest: I = (P R T) / 100
P = Principal, R = Rate of interest, T = Time.
Compound Interest: A = P(1 + R/100)^n
A = Amount, P = Principal, R = Rate of interest, n = Number of periods.
Applications of Simple and Compound Interest: Real-world problems involving interest calculations.
2. Number System
Numbers: Different types of numbers.
Four fundamental operations (Addition, Subtraction, Multiplication, Division): Basic arithmetic operations.
Knowing about Numbers: Understanding the concept of numbers.
Hindu-Arabic system of numeration (Indian system of numeration): Place value system used in India (ones, tens, hundreds, thousands, lakhs, crores, etc.).
International system of numeration (British system of numeration): Place value system used internationally (ones, tens, hundreds, thousands, millions, billions, etc.).
Place value and Face values of a digit in a number:
Place value: The value of a digit based on its position in the number.
Face value: The actual value of the digit.
Comparing and Ordering of Numbers: Determining which number is greater or smaller, and arranging numbers in ascending or descending order.
Whole Numbers: Non-negative integers (0, 1, 2, 3, ...).
Factors and Multiples:
Factor: A number that divides another number evenly.
Multiple: A number that is the product of a given number and an integer.
Prime and Composite numbers:
Prime number: A number greater than 1 that has only two factors, 1 and itself.
Composite number: A number greater than 1 that is not prime.
Even and Odd numbers:
Even number: A number divisible by 2.
Odd number: A number not divisible by 2.
Tests for Divisibility of Numbers: Rules to determine if a number is divisible by another number (e.g., divisibility by 2, 3, 4, 5, 6, 8, 9, 10, 11).
Common Factors and Common Multiples:
Common factor: A factor that is common to two or more numbers.
Common multiple: A multiple that is common to two or more numbers.
Prime factorization: Expressing a number as a product of its prime factors.
Highest Common Factor (G.C.D): The largest common factor of two or more numbers.
Lowest Common Multiple: The smallest common multiple of two or more numbers.
Integers: The set of whole numbers and their negatives (... -3, -2, -1, 0, 1, 2, 3, ...).
Properties and fundamental operations: Addition, subtraction, multiplication, and division of integers.
Fractions and decimals:
Fraction: A number that represents part of a whole.
Decimal: A number expressed in the base-10 system using a decimal point.
Types of fractions: Proper, improper, mixed fractions.
Comparison: Determining which fraction or decimal is greater or smaller.
Applications of fractions in daily life: Real-world problems involving fractions.
Four fundamental operations on fractions and decimals: Addition, subtraction, multiplication, and division.
Euclid's Division Lemma and its application: a = bq + r, where 0 ≤ r < b. Used to find the HCF.
Rational Numbers: Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Properties of Rational Numbers: Closure, commutative, associative, distributive, identity, inverse.
Representation of Rational Numbers on the Number line.
Rational Numbers between two rational numbers: Finding rational numbers between two given rational numbers.
Four fundamental Operations on Rational Numbers.
Rational numbers and their decimal expansions: Terminating and non-terminating decimals.
Non-terminating, recurring decimals in rational numbers.
Product of reciprocals: The product of a number and its reciprocal is 1.
Squares: The result of multiplying a number by itself (x²).
Square roots (Numbers and Decimals): A number that, when multiplied by itself, equals a given number (√x).
Properties of Square Numbers.
Cubes: The result of multiplying a number by itself three times (x³).
Cube roots of Numbers: A number that, when multiplied by itself three times, equals a given number (∛x).
Playing with Numbers: Number puzzles and games.
Games with Numbers.
Letters for Digits.
Irrational numbers: Numbers that cannot be expressed in the form p/q (e.g., √2, π).
Real Numbers and their Decimal Expansions: The set of all rational and irrational numbers.
Operations on Real Numbers: Addition, subtraction, multiplication, and division of real numbers.
Laws of Exponents for Real Numbers:
a^m * a^n = a^(m+n)
a^m / a^n = a^(m-n)
(a^m)^n = a^(mn)
(ab)^n = a^n * b^n
(a/b)^n = a^n / b^n
a^0 = 1
Properties & Laws of logarithms:
log_b(x) = y <=> b^y = x
log_b(mn) = log_b(m) + log_b(n)
log_b(m/n) = log_b(m) - log_b(n)
log_b(m^n) = n * log_b(m)
log_b(b) = 1
log_b(1) = 0
Sets and their representation (Roster form and Set builder form):
Roster form: Listing all elements of a set.
Set-builder form: Defining a set by a property that its elements must satisfy.
Classification of sets:
Empty set (null set): A set with no elements.
Universal set: A set that contains all elements under consideration.
Subset: A set whose elements are all contained in another set.
Finite & Infinite sets: Sets with a countable or uncountable number of elements.
Disjoint sets: Sets that have no elements in common.
Difference of sets: A - B = {x : x ∈ A and x ∉ B}
Equal sets: Sets that contain the same elements.
Using diagrams to represent sets: Visual representation of sets.
Venn diagrams and cardinality of sets:
Venn diagram: A diagram that shows the relationships between sets.
Cardinality of a set: The number of elements in a set.
Basic operations on sets:
Union: A ∪ B = {x : x ∈ A or x ∈ B}
Intersection: A ∩ B = {x : x ∈ A and x ∈ B}
3. Geometry
Basic geometrical concepts:
Point, Line, Line segment, Ray, Curves, Polygons, Angles
Measuring of Lines.
Pairs of Lines:
Intersecting Lines and Non-intersecting Lines.
Lines parallel to the same line.
Elements of Angles: Vertex, arms.
Measuring of Angles: Degrees.
Types of Angles: Acute, right, obtuse, straight, reflex, complete.
Pairs of Angles: Complementary, supplementary, adjacent, linear pair, vertically opposite.
Naming of the given 2D figures of Triangles, Square and Rectangle.
The Triangle: A polygon with three sides.
Types of Triangles and its Properties:
Based on sides: Scalene, isosceles, equilateral.
Based on angles: Acute, right, obtuse.
Properties: Angle sum property, exterior angle property.
Congruence and some properties of Triangles: Triangles having the same shape and size.
Some more criteria for Congruence of Triangles: SSS, SAS, ASA, AAS, RHS congruence rules.
Criteria for similarity of triangles: AA, SAS, SSS similarity rules.
Areas of similar triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Pythagoras theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
Classification of Polygons: Triangle, quadrilateral, pentagon, hexagon, etc.
Angle sum property: The sum of the interior angles of a polygon with n sides is (n-2) * 180°.
Kinds of Quadrilaterals:
Trapezium, Kite, Parallelogram
Some special parallelograms:
Rhombus, Rectangle, Square
Constructing different types of Quadrilaterals.
Views of 3D-Shapes: Top view, side view, front view.
Identification of Edges, Vertices and Faces of 3D figures (Euler’s Rule): F + V - E = 2
Euler’s Rule.
Nets for building 3D shapes.
Introduction to Euclid’s geometry:
Euclid’s definitions, axioms and postulates.
Angle Subtended by a Chord at a Point.
Perpendicular from the Centre to a Chord.
Equal Chords and Their Distances from the Centre.
Angle Subtended by an Arc of a Circle.
Cyclic Quadrilaterals.
Tangents of a circle: A line that touches the circle at exactly one point.
Number of Tangent to a Circle from any point.
Segment of a circle formed by a Secant: A region of a circle bounded by a chord and an arc.
4. Mensuration
Measuring Length, Weight, Capacity, Time-Seasons, Calendar, Money
Units of measurement and conversions (SI system).
Time: Relationships between seconds, minutes, hours, days, weeks, months, years.
Calendar: Leap years, days in each month.
Money: Basic arithmetic operations, currency exchange.
Area
Symmetry (Line and Rotational):
Line symmetry: A figure has line symmetry if it can be folded along a line so that the two halves match exactly.
Rotational symmetry: A figure has rotational symmetry if it can be rotated around a central point by an angle less than 360 degrees and still look the same.
Perimeter: The total length of the boundary of a closed figure.
Triangle: P = a + b + c
Square: P = 4a
Rectangle: P = 2(l + w)
Rhombus: P = 4a
Trapezium: P = a + b + c + d
Parallelogram: P = 2(a + b)
Circle: C = 2πr
Polygon: Sum of all sides
Properties of a Parallelogram:
Opposite sides are parallel and equal.
Opposite angles are equal.
Diagonals bisect each other.
The Mid-point Theorem: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half its length.
Area of a Quadrilateral:
General quadrilateral: Area can be found by dividing it into two triangles.
Special quadrilaterals (see individual shapes above).
Surface Area and Volume:
Cube:
Surface Area = 6a²
Volume = a³
Cuboid:
Surface Area = 2(lb + bh + hl)
Volume = lbh
Cylinder:
Curved Surface Area = 2πrh
Total Surface Area = 2πr(r + h)
Volume = πr²h
Sphere:
Surface Area = 4πr²
Volume = (4/3)πr³
Right Circular Cone:
Curved Surface Area = πrl
Total Surface Area = πr(l + r)
Volume = (1/3)πr²h
Volume and capacity: Capacity is the amount a container can hold (usually liquid volume). 1 liter = 1000 cm³
Surface area of combination of solids: Find the sum of the areas of the individual faces/surfaces, excluding any overlapping areas.
Volume of combination of solids: Add the volumes of the individual solids.
Conversion of solid from one shape to another: The volume remains constant. Use this principle to find new dimensions.
5. Algebra
Patterns - making rules: Generalizing patterns with variables.
The idea of variables: Symbols (letters) that represent unknown quantities.
Formation of algebraic expressions: Combining variables and constants with arithmetic operations.
Terms, Factors and Coefficients:
Term: A single number or variable, or the product of several.
Factor: A number or variable that divides evenly into a term.
Coefficient: The numerical factor of a term.
Linear equations in one variable: ax + b = 0
Linear equations in two variables: ax + by + c = 0
Solutions of Pair of Linear Equations in Two Variables:
Graphical method: Intersection point.
Algebraic methods:
Substitution method
Elimination method
Cross-multiplication method
Equations reducible to a pair of linear equations in two variables: Transforming equations (e.g., reciprocal equations).
Solution of a quadratic equation: ax² + bx + c = 0
Factorization
Completing the square
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Nature of roots: Discriminant (D = b² - 4ac)
D > 0: Two distinct real roots
D = 0: Two equal real roots
D < 0: No real roots (complex roots)
Terms and types of algebraic expressions:
Monomial, binomial, trinomial, polynomial
Finding the value of an expression: Substituting values for variables.
Addition, Subtraction and Multiplication of Algebraic Expressions: Combining like terms, distributive property.
Multiplying a Monomial by a Monomial and polynomial: Distributive property.
Multiplying a Polynomial by a Polynomial: Distributive property.
Standard Identities and their applications:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(x + a)(x + b) = x² + (a + b)x + ab
Applications of simple equations to practical situations: Word problems.
Exponents and Powers: a^n
Negative exponents: a^-n = 1/a^n
Laws of exponents:
a^m * a^n = a^(m+n)
a^m / a^n = a^(m-n)
(a^m)^n = a^(mn)
(ab)^n = a^n * b^n
(a/b)^n = a^n / b^n
a^0 = 1
Expressing large numbers in the standard form: Scientific notation (a x 10^n)
Factorisation: Expressing an algebraic expression as a product of its factors.
Division of Algebraic Expressions Continued (Polynomial ÷ Polynomial): Long division method.
Linear Graphs: Graphing linear equations.
Polynomials in one variable: p(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_0
Degree, Value, zeroes of a polynomial:
Degree: Highest power of the variable.
Value: The result when a specific value is substituted for the variable.
Zeroes: Values of the variable for which the polynomial equals zero.
Geometrical meaning of the Zeroes of a Polynomial: x-intercepts of the graph.
Graphical representation of linear, Quadratic and Cubic Polynomials: Shapes of the graphs.
Factorisation of Polynomials: Splitting the middle term, factor theorem, etc.
Algebraic Identities: Equations that are true for all values of the variables.
Working with Polynomials: Addition, subtraction, multiplication, and division.
Division algorithm for polynomials: Dividend = Divisor * Quotient + Remainder
Arithmetic progressions (AP): A sequence with a constant difference between consecutive terms.
Parameters of Arithmetic progressions:
First term (a)
Common difference (d)
nth term of an Arithmetic progression: a_n = a + (n - 1)d
Sum of first n terms in Arithmetic progression: S_n = n/2 [2a + (n - 1)d] or S_n = n/2(a + l)
Geometric progressions (GP): A sequence with a constant ratio between consecutive terms.
nth term of a GP: a_n = a * r^(n-1)
Functions:
Ordered pair: (x, y)
Cartesian product of sets: A x B = {(a, b) : a ∈ A, b ∈ B}
Relation: A set of ordered pairs.
Function: A relation where each element of the domain is associated with a unique element of the range.
Types of function:
One-to-one (injective)
Onto (surjective)
Bijective (one-to-one and onto)
Image and pre-image: If f(x) = y, then y is the image of x, and x is the pre-image of y.
Definitions of functions.
Inverse functions and Theorems: A function g is the inverse of f if f(g(y)) = y and g(f(x)) = x.
Domain, Range, Inverse of real-valued functions: The set of input values, the set of output values, and the reverse mapping.
Mathematical Induction:
Principle of Mathematical Induction: A method to prove statements for all natural numbers.
Base case: Prove the statement for n = 1.
Inductive step: Assume the statement is true for n = k, and prove it for n = k + 1.
Theorems.
Applications of Mathematical Induction: Proving formulas, properties of sequences, etc.
Problems on divisibility: Using induction to prove that an expression is divisible by a certain number.
Matrices:
Types of matrices:
Row matrix, column matrix, square matrix, rectangular matrix, diagonal matrix, scalar matrix, identity matrix, zero matrix.
Scalar multiple of a matrix and multiplication of matrices: Multiplying a matrix by a constant, multiplying two matrices.
Transpose of a matrix: Interchanging rows and columns.
Determinants: A scalar value that can be computed from the elements of a square matrix.
Adjoint and Inverse of a matrix:
Adjoint: The transpose of the matrix of cofactors.
Inverse: A^(-1) = adj(A) / det(A)
Consistency and inconsistency of Equations- Rank of a matrix:
Consistent: System has a solution.
Inconsistent: System has no solution.
Rank: The number of linearly independent rows or columns in a matrix.
Solution of simultaneous linear equations: Matrix method (A X = B).
Complex Numbers:
Complex number as an ordered pair of real numbers: z = (a, b)
Fundamental operations: Addition, subtraction, multiplication, and division.
Representation of complex numbers in the form a+ib: where i = √(-1)
Modulus and amplitude of complex numbers:
Modulus: |z| = √(a² + b²)
Amplitude (argument): θ = arctan(b/a)
Geometrical and Polar Representation of complex numbers in Argand plane- Argand diagram: Representing complex numbers as points in a plane.
De Moivre’s Theorem:
De Moivre’s theorem: (cos θ + i sin θ)^n = cos nθ + i sin nθ
Integral and Rational indices.
nth roots of unity: Solutions to the equation z^n = 1.
Geometrical Interpretations – Illustrations.
Quadratic Expressions:
Quadratic expressions, equations in one variable: ax² + bx + c
Sign of quadratic expressions – Change in signs.
Maximum and minimum values: Occur at the vertex of the parabola.
Quadratic in-equations: Solving inequalities involving quadratic expressions.
Theory of Equations:
The relation between the roots and coefficients in an equation.
Solving the equations when two or more roots of it are connected by certain relation.
Equation with real coefficients, occurrence of complex roots in conjugate pairs and its consequences.
Transformation of equations – Reciprocal Equations.
Permutations and Combinations:
Fundamental Principle of counting: Multiplication and addition rules.
Linear and circular permutations:
Linear permutations: Arranging objects in a line.
Circular permutations: Arranging objects in a circle.
Permutations of ‘n’ dissimilar things taken ‘r’ at a time: nPr = n! / (n-r)!
Permutations when repetitions allowed: n^r
Circular permutations.
Permutations with constraint repetitions.
Binomial Theorem:
Binomial theorem for positive integral index: (a + b)^n = Σ (nCr a^(n-r) b^r)
Binomial theorem for rational Index (without proof).
Approximations using Binomial theorem.
Partial fractions:
Partial fractions of f(x)/g(x) when g(x) contains non –repeated linear factors.
Partial fractions of f(x)/g(x) when g(x) contains repeated and/or non-repeated linear factors.
Partial fractions of f(x)/g(x) when g(x) contains irreducible factors.
6. Statistics
DATA HANDLING -Frequency Distribution Tables and Graphs- Grouped data- ungrouped data: Organizing data into tables and charts (bar graphs, histograms, frequency polygons, etc.).
Measures of Central Tendency:
Mean: The average of a set of numbers.
Median: The middle value in a sorted set of numbers.
Mode: The most frequent value in a set of numbers.
Grouped and ungrouped data.
Ogive curves: Cumulative frequency graphs.
MEASURES OF DISPERSION:
Range: The difference between the highest and lowest values.
Mean deviation: The average of the absolute deviations from the mean.
Variance and standard deviation of ungrouped/grouped data:
Variance: The average of the squared deviations from the mean.
Standard deviation: The square root of the variance.
Coefficient of variation and analysis of frequency distribution with equal means but different variances: Comparing the relative variability of different datasets.
7. Probability
Probability: The measure of the likelihood that an event will occur.
Linking chances to probability - Chance and probability related to real life.
Probability - a theoretical approach.
Mutually exclusive events: Events that cannot occur at the same time. P(A or B) = P(A) + P(B)
Finding probability.
Complementary events and probability: P(A') = 1 - P(A)
Impossible and certain events:
Impossible event: P(∅) = 0
Certain event: P(S) = 1
Deck of Cards and Probability.
Use and Applications of probability.
Random experiments and events.
Classical definition of probability, Axiomatic approach and addition theorem of probability.
Independent and dependent events conditional probability- multiplication theorem and Bayee’s theorem.
Random Variables: A variable whose value is a numerical outcome of a random phenomenon.
Theoretical discrete distributions:
Binomial Distribution.
Poisson Distributions.
8. Coordinate Geometry
Cartesian System: A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates.
Distance between two points:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Distance between two points on a line parallel to the co-ordinate axis: |x2-x1| or |y2-y1|
Section formula:
To find the coordinates of a point that divides a line segment in a given ratio.
Centroid of a triangle:
The point of intersection of the medians of a triangle.
Tri-sectional points of a line: The points that divide the line segment into three equal parts.
Area of the triangle:
Using coordinates: Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Heron’s formula: Area = √(s(s - a)(s - b)(s - c)), where s is the semi-perimeter.
Collinearity: Points that lie on the same straight line.
Straight lines:
Slope of the straight line: m = (y₂ - y₁) / (x₂ - x₁)
Slope of a line joining two points.
Locus:
Definition of locus: The set of all points that satisfy a given condition.
Illustrations.
To find equations of locus - Problems connected to it.
Transformation of Axes:
Transformation of axes - Rules, Derivations and Illustrations.
Rotation of axes - Derivations – Illustrations.
The Straight Line:
Revision of fundamental results.
Straight line - Normal form – Illustrations.
Straight line - Symmetric form.
Straight line - Reduction into various forms.
Intersection of two Straight Lines.
Family of straight lines - Concurrent lines.
Condition for Concurrent lines.
Angle between two lines.
Length of perpendicular from a point to a Line.
Distance between two parallel lines.
Concurrent lines - properties related to a triangle.
Pair of Straight lines:
Equations of pair of lines passing through origin, angle between a pair of lines.
Condition for perpendicular and coincident lines, bisectors of angles.
Pair of bisectors of angles.
Pair of lines - second degree general equation.
Conditions for parallel lines - distance between them, Point of intersection of pair of lines.
Homogenizing a second degree equation with a first degree equation in X and Y.
Circle:
Equation of circle -standard form-centre and radius of a circle with a given line segment as diameter & equation of circle through three non collinear points - parametric equations of a circle.
Position of a point in the plane of a circle – power of a point-definition of tangent-length of tangent
Position of a straight line in the plane of circle-conditions for a line to be tangent – chord joining two points on a circle – equation of the tangent at a point on the circle-point of contact-equation of normal.
Chord of contact - pole and polar-conjugate points and conjugate lines - equation of chord with given middle point.
Relative position of two circles- circles touching each other externally, internally common tangents-centres of similitude- equation of pair of tangents from an external point.
System of circles:
Angle between two intersecting circles.
Radical axis of two circles- properties- Common chord and common tangent of two circles – radical centre.
Intersection of a line and a Circle.
Parabola:
Conic sections –Parabola- equation of parabola in standard form-different forms of parabola- parametric equations.
Equations of tangent and normal at a point on the parabola (Cartesian and parametric) - conditions for straight line to be a tangent.
Ellipse:
Equation of ellipse in standard form- Parametric equations.
Equation of tangent and normal at a point on the ellipse (Cartesian and parametric) - condition for a straight line to be a tangent.
Hyperbola:
Equation of hyperbola in standard form- Parametric equations.
Equations of tangent and normal at a point on the hyperbola (Cartesian and parametric) - conditions for a straight line to be a tangent- Asymptotes.
Three Dimensional Coordinates:
Coordinates.
Section formulas - Centroid of a triangle and tetrahedron.
Direction Cosines and Direction Ratios:
Direction Cosines.
Direction Ratios.
Plane:
Cartesian equation of Plane - Simple Illustrations.
9. Trigonometry
Trigonometry – Naming the sides in a Right triangle – Trigonometric Ratios – Defining Trigonometric Ratios – Trigonometric ratios of some specific and complementary angles – Trigonometric identities – Applications of Trigonometry – Drawing figures to solve problems – solutions for two triangles.
Trigonometric Ratios up to Transformations:
Graphs and Periodicity of Trigonometric functions.
Trigonometric ratios and Compound angles.
Trigonometric ratios of multiple and sub- multiple angles.
Transformations - Sum and Product rules.
Trigonometric Equations:
General Solution of Trigonometric Equations.
Simple Trigonometric Equations – Solutions.
Inverse Trigonometric Functions:
To reduce a Trigonometric Function into a bijection.
Graphs of Inverse Trigonometric Functions.
Properties of Inverse Trigonometric Functions.
Hyperbolic Functions:
Definition of Hyperbolic Function – Graphs.
Definition of Inverse Hyperbolic Functions – Graphs.
Addition formulas of Hyperbolic Functions.
Properties of Triangles:
Relation between sides and angles of a Triangle
Sine, Cosine, Tangent and Projection rules.
Half angle formulae and areas of a triangle
10. Vector Algebra
Addition of Vectors:
Vectors as a triad of real numbers.
Classification of vectors.
Addition of vectors.
Scalar multiplication.
Angle between two non-zero vectors.
Linear combination of vectors.
Component of a vector in three dimensions.
Vector equations of line and plane including their Cartesian equivalent forms.
Product of Vectors:
Scalar Product - Geometrical Interpretations - orthogonal projections.
Properties of dot product.
Expression of dot product in i, j, k system – Angle between two vectors.
Geometrical Vector methods.
Vector equations of plane in normal form.
Angle between two planes.
Vector product of two vectors and properties.
Vector product in i, j, k system.
Vector Areas.
Scalar Triple Product.
Vector equations of plane in different forms, skew lines, shortest distance and their Cartesian equivalents. Plane through the line of intersection of two planes, condition for coplanarity of two lines, perpendicular distance of a point from a plane, Angle between line and a plane. Cartesian equivalents of all these results
Vector Triple Product – Results
11. Calculus
Limits and Continuity:
Intervals and neighbourhoods.
Limits.
Standard Limits.
Continuity.
Differentiation:
Derivative of a function.
Elementary Properties.
Trigonometric, Inverse Trigonometric, Hyperbolic, Inverse Hyperbolic Function - Derivatives.
Methods of Differentiation.
Second Order Derivatives.
Applications of Derivatives:
Errors and approximations.
Geometrical Interpretation of a derivative.
Equations of tangents and normal’s.
Lengths of tangent, normal, sub tangent and sub normal.
Angles between two curves and condition for orthogonality of curves.
Derivative as Rate of change.
Rolle’s Theorem and Lagrange’s Mean value theorem without proofs and their geometrical interpretation.
Increasing and decreasing functions.
Maxima and Minima.
Integration:
Integration as the inverse process of differentiation- Standard forms –properties of integrals.
Method of substitution- integration of Algebraic, exponential, logarithmic, trigonometric and inverse trigonometric functions. Integration by parts.
Integration- Partial fractions method.
Reduction formulae.
Definite Integrals:
Definite Integral as the limit of sum
Interpretation of Definite Integral as an area.
Fundamental theorem of Integral Calculus.
Properties.
Reduction formulae.
Application of Definite integral to areas.
Differential equations:
Formation of differential equation-Degree and order of an ordinary differential equation.
Solving differential equation by
a) Variables separable method.
b) Homogeneous differential equation.
c) Non - Homogeneous differential equation.
d) Linear differential equations.
Here are 100 multiple-choice questions with answers, based on the mathematics study notes for Class VI to Intermediate level:
1. Arithmetic
What is the correct order of operations according to the BODMAS rule?
a) Addition, Subtraction, Multiplication, Division, Brackets, Of
b) Brackets, Of, Division, Multiplication, Addition, Subtraction
c) Brackets, Of, Multiplication, Division, Addition, Subtraction
d) Division, Multiplication, Addition, Subtraction, Brackets, Of
Answer: b
In a direct proportion, if one quantity increases, the other quantity:
a) Decreases
b) Increases
c) Remains constant
d) Becomes zero
Answer: b
If the cost price of an item is Rs. 100 and the selling price is Rs. 120, the profit is:
a) Rs. 10
b) Rs. 20
c) Rs. 220
d) Rs. 0
Answer: b
The formula for simple interest is:
a) P R T
b) (P R T) / 100
c) P + R + T
d) P(1 + R/100)^n
Answer: b
In the compound interest formula A = P(1 + R/100)^n, 'n' represents:
a) Amount
b) Principal
c) Rate of interest
d) Number of periods
Answer: d
If the ratio of two numbers is 2:3 and the first number is 10, the second number is:
a) 15
b) 20
c) 25
d) 30
Answer: a
A shopkeeper buys an article for Rs. 40 and sells it for Rs. 50. His profit percentage is:
a) 10%
b) 20%
c) 25%
d) 30%
Answer: c
A discount of 20% on a marked price of Rs. 100 is:
a) Rs. 120
b) Rs. 80
c) Rs. 20
d) Rs. 10
Answer: c
GST is an abbreviation for
a) Goods and sales tax
b) General sales tax
c) Goods and service tax
d) Government sales tax
Answer: c
The amount after 2 years for a principal of Rs. 1000 at 10% compound interest per annum is
a) Rs. 1200
b) Rs. 1210
c) Rs. 1100
d) Rs. 1120
Answer: b
2. Number System
The Hindu-Arabic system of numeration is also known as the:
a) International system
b) British system
c) Indian system
d) Metric system
Answer: c
In the number 12345, the place value of 3 is:
a) 3
b) 30
c) 300
d) 3000
Answer: c
Which of the following is a whole number?
a) -1
b) 0
c) 1/2
d) 1.5
Answer: b
A number that divides another number evenly is called a:
a) Multiple
b) Factor
c) Prime number
d) Composite number
Answer: b
A number greater than 1 that has only two factors, 1 and itself, is called a:
a) Composite number
b) Prime number
c) Even number
d) Odd number
Answer: b
Which of the following numbers is divisible by 3?
a) 121
b) 122
c) 123
d) 124
Answer: c
The largest common factor of two or more numbers is called the:
a) Lowest Common Multiple
b) Highest Common Factor
c) Prime Factorization
d) Common Multiple
Answer: b
The smallest common multiple of two or more numbers is called the:
a) Lowest Common Multiple
b) Highest Common Factor
c) Prime Factorization
d) Common Factor
Answer: a
Which of the following is an integer?
a) 1/2
b) 3.14
c) -5
d) √2
Answer: c
In the division lemma a = bq + r, the value of r is always:
a) Greater than b
b) Less than b
c) Equal to b
d) Greater than or equal to b
Answer: b
A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is called a:
a) Irrational number
b) Rational number
c) Real number
d) Complex number
Answer: b
The decimal expansion of a rational number is either:
a) Terminating or non-terminating repeating
b) Non-terminating non-repeating
c) Always terminating
d) Always non-terminating
Answer: a
The product of a number and its reciprocal is always:
a) 0
b) 1
c) The number itself
d) -1
Answer: b
The square of 5 is:
a) 10
b) 20
c) 25
d) 30
Answer: c
The square root of 25 is:
a) 5
b) 10
c) 12.5
d) 50
Answer: a
The cube of 2 is:
a) 4
b) 6
c) 8
d) 10
Answer: c
The cube root of 8 is:
a) 1
b) 2
c) 3
d) 4
Answer: b
Which of the following is an irrational number?
a) √4
b) √9
c) √16
d) √2
Answer: d
The set of all rational and irrational numbers is called
a) Integers
b) Whole numbers
c) Real numbers
d) Complex numbers
Answer: c
a^m * a^n =
a) a^(m-n)
b) a^(m+n)
c) a^(mn)
d) a^(m/n)
Answer: b
In the logarithmic expression log_b(x) = y, b is called the
a) logarithm
b) base
c) exponent
d) number
Answer: b
log_b(1) = ?
a) 0
b) 1
c) b
d) infinity
Answer: a
A set with no elements is called
a) Singleton set
b) Finite set
c) Empty set
d) Infinite set
Answer: c
A set that contains all elements under consideration is called
a) Subset
b) Universal set
c) Power set
d) Disjoint set
Answer: b
If all elements of set A are in set B, then A is a _________ of B.
a) superset
b) subset
c) equal set
d) disjoint set
Answer: b
Sets with no common elements are called
a) Equal sets
b) Disjoint sets
c) Subset
d) Overlapping sets
Answer: b
The number of elements in a set is called its
a) Power
b) Cardinality
c) Degree
d) Index
Answer: b
A U B represents
a) Intersection of A and B
b) Union of A and B
c) Difference of A and B
d) Complement of A
Answer: b
A ∩ B represents
a) Intersection of A and B
b) Union of A and B
c) Difference of A and B
d) Complement of A
Answer: a
The form of a set where all elements are listed is called
a) Set builder form
b) Roster form
c) Descriptive form
d) Tabular form
Answer: b
3. Geometry
A point has
a) Length
b) Breadth
c) Height
d) No dimensions
Answer: d
A line segment has
a) No end points
b) One end point
c) Two end points
d) Infinite end points
Answer: c
A ray has
a) No end points
b) One end point
c) Two end points
d) Infinite end points
Answer: b
Lines that do not intersect are called
a) Intersecting lines
b) Perpendicular lines
c) Parallel lines
d) Concurrent lines
Answer: c
The point where the two sides of an angle meet is called its
a) Arm
b) Vertex
c) Interior
d) Exterior
Answer: b
Angles are measured in
a) Radians
b) Degrees
c) Gradients
d) Steradians
Answer: b
An angle greater than 90 degrees and less than 180 degrees is called
a) Acute angle
b) Right angle
c) Obtuse angle
d) Straight angle
Answer: c
Two angles whose sum is 90 degrees are called
a) Supplementary angles
b) Complementary angles
c) Adjacent angles
d) Linear pair
Answer: b
A polygon with three sides is called a
a) Quadrilateral
b) Pentagon
c) Triangle
d) Hexagon
Answer: c
A triangle with all sides equal is called
a) Scalene triangle
b) Isosceles triangle
c) Equilateral triangle
d) Right-angled triangle
Answer: c
Triangles having the same shape and size are called
a) Similar triangles
b) Congruent triangles
c) Right triangles
d) Acute triangles
Answer: b
In a right-angled triangle, the side opposite the right angle is called the
a) Base b) Perpendicular c) Hypotenuse d) Altitude Answer: c
The sum of the interior angles of a triangle is
a) 90 degrees
b) 180 degrees
c) 270 degrees
d) 360 degrees
Answer: b
A quadrilateral with only one pair of parallel sides is called a
a) Parallelogram
b) Rectangle
c) Trapezium
d) Rhombus
Answer: c
A quadrilateral with opposite sides parallel and equal is called a
a) Trapezium
b) Kite
c) Parallelogram
d) Square
Answer: c
A parallelogram with all sides equal and all angles equal to 90 degrees is a
a) Rectangle
b) Rhombus
c) Square
d) Trapezium
Answer: c
The number of faces in a cube is
a) 4
b) 6
c) 8
d) 12
Answer: b
The number of edges in a cube is
a) 4
b) 6
c) 8
d) 12
Answer: d
The number of vertices in a cube is
a) 4
b) 6
c) 8
d) 12
Answer: c
Euler's formula for a polyhedron is
a) F + V = E + 2
b) F + E = V + 2
c) V + E = F + 2
d) F + V + E = 2
Answer: a
A line that intersects a circle at two points is called a
a) Tangent
b) Secant
c) Radius
d) Diameter
Answer: b
A line that touches a circle at exactly one point is called a
a) Secant
b) Tangent
c) Chord
d) Diameter
Answer: b
The longest chord of a circle is the
a) Radius
b) Diameter
c) Tangent
d) Secant
Answer: b
4. Mensuration
The perimeter of a square with side 'a' is:
a) a²
b) 4a
c) 2a
d) a/4
Answer: b
The area of a rectangle with length 'l' and width 'w' is:
a) 2(l + w)
b) l * w
c) l²
d) w²
Answer: b
The circumference of a circle with radius 'r' is:
a) πr²
b) 2πr
c) πr
d) (4/3)πr³
Answer: b
The area of a circle with radius 'r' is:
a) 2πr
b) πr
c) πr²
d) (4/3)πr³
Answer: c
The surface area of a cube with side 'a' is:
a) a³
b) 6a²
c) 4a²
d) 2a²
Answer: b
The volume of a cube with side 'a' is:
a) 6a²
b) a²
c) a³
d) 4a²
Answer: c
The surface area of a sphere with radius 'r' is:
a) (4/3)πr³
b) πr²
c) 2πr²
d) 4πr²
Answer: d
The volume of a sphere with radius 'r' is:
a) 4πr²
b) πr²h
c) (1/3)πr²h
d) (4/3)πr³
Answer: d
The curved surface area of a cylinder with radius 'r' and height 'h' is
a) πr²h
b) 2πrh
c) πr(r+h)
d) 2πr(r+h)
Answer: b
The volume of a cylinder with radius 'r' and height 'h' is
a) 2πrh
b) πr²
c) πr²h
d) 2πr(r+h)
Answer: c
The curved surface area of a cone with radius 'r' and slant height 'l' is
a) πrl
b) πr²
c) πr(l+r)
d) (1/3)πr²h
Answer: a
5. Algebra
In the expression 5x^2 + 3x - 2, the coefficient of x is:
a) 5
b) 2
c) 3
d) -2
Answer: c
A polynomial with two terms is called a
a) Monomial
b) Binomial
c) Trinomial
d) Quadratic
Answer: b
The degree of the polynomial x^3 + 2x^2 - 5x + 7 is:
a) 1
b) 2
c) 3
d) 4
Answer: c
(a + b)^2 = ?
a) a^2 + b^2
b) a^2 - b^2
c) a^2 + 2ab + b^2
d) a^2 - 2ab + b^2
Answer: c
(a - b)^2 = ?
a) a^2 + b^2
b) a^2 - b^2
c) a^2 + 2ab + b^2
d) a^2 - 2ab + b^2
Answer: d
a^2 - b^2 = ?
a) (a + b)^2
b) (a - b)^2
c) (a + b)(a - b)
d) a^2 + 2ab + b^2
Answer: c
The solution of the equation 2x + 5 = 0 is
a) 5/2
b) -5/2
c) 2/5
d) -2/5
Answer: b
The quadratic formula to find the roots of ax^2 + bx + c = 0 is
a) x = (-b ± √(b^2 + 4ac)) / 2a
b) x = (-b ± √(b^2 - 4ac)) / a
c) x = (-b ± √(b^2 - 4ac)) / 2a
d) x = (b ± √(b^2 - 4ac)) / 2a
Answer: c
The discriminant of the quadratic equation ax^2 + bx + c = 0 is given by
a) b^2 + 4ac
b) b^2 - 4ac
c) √(b^2 - 4ac)
d) -b/2a
Answer: b
If the discriminant is greater than zero, the quadratic equation has
a) No real roots
b) Two equal real roots
c) Two distinct real roots
d) Two complex roots
Answer: c
If the discriminant is equal to zero, the quadratic equation has
a) No real roots
b) Two equal real roots
c) Two distinct real roots
d) Two complex roots
Answer: b
If the discriminant is less than zero, the quadratic equation has
a) No real roots
b) Two equal real roots
c) Two distinct real roots
d) Two real roots
Answer: a
The nth term of an arithmetic progression is given by
a) a + nd
b) a + (n + 1)d
c) a + (n - 1)d
d) a - (n - 1)d
Answer: c
The sum of the first n terms of an arithmetic progression is given by
a) n/2 [a + l]
b) n/2[2a + nd]
c) n[2a + (n-1)d]
d) n/2 [2a + (n - 1)d]
Answer: d
The nth term of a geometric progression is given by
a) a * r^n
b) a * r^(n+1)
c) a * r^(n-1)
d) a * r^-n
Answer: c
A relation where every element in the domain has a unique image in the range is called a
a) Relation
b) Function
c) Sequence
d) Series
Answer: b
The inverse of the matrix A exists if and only if
a) A is a square matrix
b) det(A) = 0
c) det(A) ≠ 0
d) A is a diagonal matrix
Answer: c
If z = a + ib is a complex number, then |z| is
a) a^2 + b^2
b) √(a^2 + b^2)
c) a + b
d) a - b
Answer: b
(cos θ + i sin θ)^n =
a) cos θ + i sin θ
b) cos nθ - i sin nθ
c) cos nθ + i sin nθ
d) sin nθ + i cos nθ
Answer: c
The number of permutations of n distinct objects taken r at a time is given by
a) n!
b) r!
c) nPr = n! / (n-r)!
d) nCr = n! / (r!(n-r)!)
Answer: c
The number of combinations of n distinct objects taken r at a time is given by
a) n!
b) r!
c) nPr = n! / (n-r)!
d) nCr = n! / (r!(n-r)!)
Answer: d
In the binomial expansion of (a + b)^n, the number of terms is
a) n
b) n - 1
c) n + 1
d) 2n
Answer: c
6. Statistics
The mean of a set of data is the
a) Middle value
b) Most frequent value
c) Average value
d) Range
Answer: c
The median of a set of data is the
a) Average value
b) Middle value
c) Most frequent value
d) Range
Answer: b
The mode of a set of data is the
a) Average value
b) Middle value
c) Most frequent value
d) Range
Answer: c
The range of a set of data is the
a) Difference between the highest and lowest values b) Average of the deviations c) Square root of the variance d) Average of the squared deviations Answer: a