Comprehensive Chapter-by-Chapter Syllabus with All Models, Shortcut Tricks and Practice Problems (Classes 6 to 10)
Definition: HCF is the greatest common factor dividing numbers evenly, while LCM is the smallest multiple shared by numbers.
Shortcut Trick: For co-prime numbers, LCM is their direct product. If smaller divides larger, LCM is the larger number.
Definition: Rules to check number divisibility without full division.
Shortcut Trick: For divisibility by 72, check divisibility by both 8 and 9 simultaneously.
Definition: Integers include negative numbers, zero, and positive numbers. Absolute value represents distance from zero.
Shortcut Trick: Subtracting a negative integer is identical to adding its positive counterpart.
Definition: Closure, commutative, associative, and distributive laws over integer operations.
Shortcut Trick: Count negative signs in multiplication: even count gives positive product, odd count gives negative product.
Definition: Proper, improper, and mixed fractions and their arithmetic manipulations.
Shortcut Trick: Cross-multiply to compare fractions: for a/b and c/d, compare ad and bc products.
Definition: Converting decimals to fractions and handling repeating decimal expansions.
Shortcut Trick: For mixed recurring decimals like 0.a_bar(b), use formula: (ab - a) / 90.
Definition: Comparing quantities in same units and expressing proportionality equality.
Shortcut Trick: To divide quantity Q in ratio m:n, shares are Q*m/(m+n) and Q*n/(m+n).
Definition: Finding value of single unit to solve complex rate and work problems.
Shortcut Trick: Inverse proportion: if more workers decrease days, product of workers and days remains constant.
Definition: Complementary angles sum to 90 degrees; supplementary angles sum to 180 degrees.
Shortcut Trick: Vertically opposite angles formed by two intersecting lines are always equal.
Definition: Convex and concave polygons, interior/exterior angle sums, and diagonals count.
Shortcut Trick: Sum of exterior angles of any convex polygon is always identically 360 degrees.
Definition: Perimeter is outer boundary length; area is 2D surface coverage of rectangles and squares.
Shortcut Trick: If side of square increases by x percent, area increases by (2x + x^2/100) percent.
Definition: Calculating areas of internal and external paths running across rectangular fields.
Shortcut Trick: Cross paths running parallel to length and breadth: Area = w(L + B - w).
Definition: Recording raw data using tally counts and representing frequencies visually via picture symbols.
Shortcut Trick: Tally marks are grouped in bunches of 5 (four vertical lines with a diagonal strike through).
Definition: Arithmetic mean measures average, median measures middle value, and bar graphs display categorical data.
Shortcut Trick: To find median of n observations, arrange in ascending order and pick middle term (n+1)/2 if odd, or average of two middle terms if even.
Definition: Numbers expressed as p/q where q is not equal to 0 and integers p, q. Properties include closure, commutativity, and associativity.
Shortcut Trick: To find midway rational number between a and b, use arithmetic mean (a + b) / 2.
Definition: Infinitely many rational numbers exist between any two distinct rational numbers.
Shortcut Trick: To insert 3 numbers between a and b, use sequence: a + d, a + 2d, a + 3d.
Definition: Advanced fractional operations involving multi-story fractions and telescopic series cancellations.
Shortcut Trick: Telescopic sum: 1/(1x2) + 1/(2x3) + ... + 1/(n(n+1)) = n/(n+1).
Definition: Strict order of operations (Brackets, Of, Division, Multiplication, Addition, Subtraction) with decimals.
Shortcut Trick: Align decimal points carefully during addition/subtraction; shift decimal right equal to total zeroes in multiplier.
Definition: Rules governing exponents with integer bases and powers.
Shortcut Trick: Any non-zero base raised to power zero is identically 1 (a^0 = 1). Negative exponent inverts the base.
Definition: Expressing very large or small numbers in standard scientific notation k x 10^n (1 <= k < 10).
Shortcut Trick: To compare numbers with large powers like 2^60 and 3^40, find HCF of exponents (20), express as (2^3)^20 and (3^2)^20.
Definition: Fundamental algebraic identities for squaring binomials and difference of squares.
Shortcut Trick: To evaluate products like 103 x 97 rapidly, rewrite as (100 + 3)(100 - 3) = 100^2 - 3^2 = 9991.
Definition: Advanced manipulation of reciprocal variable expressions to find higher powers.
Shortcut Trick: For minus reciprocals: if x - 1/x = k, then x^2 + 1/x^2 = k^2 + 2 and x^3 - 1/x^3 = k^3 + 3k.
Definition: Corresponding angles, alternate interior angles, and consecutive interior angles formed by transversal cutting parallels.
Shortcut Trick: If two lines are parallel to the same line, they are parallel to each other.
Definition: Angle sum property, exterior angle theorem, and triangle side length bounds.
Shortcut Trick: Triangle inequality: sum of any two sides is greater than the third side, and difference is less than third side.
Definition: Percentage calculations, markup, discount on marked price, and successive percentage changes.
Shortcut Trick: If SP is identical for two items, one sold at x% profit and other at x% loss, overall transaction always yields net loss of (x^2)/100 percent.
Definition: Calculating interest earned over principal amount across time periods at fixed annual percentage rate.
Shortcut Trick: If a sum becomes n times in T years at simple interest, rate is given by R = 100(n - 1) / T.
Definition: Mensuration formulas for 2D geometrical shapes including circles, sectors, rhombuses, and trapeziums.
Shortcut Trick: Area of circular path (ring) with outer radius R and inner radius r is pi x (R + r)(R - r).
Definition: Total surface area, lateral surface area, and volume of 3D boxes and cubes.
Shortcut Trick: If edge of cube is increased by x%, volume increases by approximately 3x% for small increments.
Definition: Closure, commutative, associative, and distributive laws over rational numbers.
Shortcut Trick: Rearrange terms using commutative and associative properties to group fractions with common denominators before calculation.
Definition: Finding multiple rational numbers between any two numbers.
Shortcut Trick: Make denominators equal using LCM, then scale numerators by multiplying top and bottom by (n + 1).
Definition: Solving equations reducible to linear form with fractional expressions.
Shortcut Trick: Cross-multiply immediately when single fractions equal single fractions on both sides.
Definition: Translating verbal real-world statements into mathematical linear equations.
Shortcut Trick: For consecutive number problems, let numbers be x, x+1, x+2 or x, x+2, x+4 for evens/odds.
Definition: Finding exact and approximate square roots using long division and binomial expansion approximation.
Shortcut Trick: Denesting formula: sqrt((x + y) + 2*sqrt(xy)) = sqrt(x) + sqrt(y).
Definition: Cubes of numbers and extracting cube roots by grouping prime factors in triplets.
Shortcut Trick: Unit digit of cube root follows specific repeating cycles.
Definition: Properties of parallelograms, rhombuses, rectangles, squares, and trapeziums regarding sides, angles, and diagonals.
Shortcut Trick: Adjacent angles of any parallelogram are supplementary (sum to 180 degrees).
Definition: Relationship between interior and exterior angles of convex polygons.
Shortcut Trick: Number of sides n = 360 / (Exterior Angle) for regular polygons.
Definition: Factorization using common terms, regrouping, and standard algebraic identities.
Shortcut Trick: Difference of two squares: a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) = (a-b)(a+b)(a^2+b^2).
Definition: Factoring quadratic expressions by splitting middle term into two parts whose product is ac and sum is b.
Shortcut Trick: Signs of factors: if middle term is positive and constant is positive, both factors are positive.
Definition: Calculating compound interest, population growth, and depreciation models.
Shortcut Trick: For half-yearly compounding, double the time periods (2n) and halve the annual rate (r/2).
Definition: Direct variation (x/y = k) and inverse variation (x x y = k).
Shortcut Trick: In inverse proportion (men and days), increase in workers causes proportional decrease in completion time.
Definition: Curved surface area, total surface area, and volume of right circular cylinders and cones.
Shortcut Trick: Slant height of cone l = sqrt(r^2 + h^2). Cone TSA = pi x r(r + l).
Definition: Melting and recasting solid shapes where total volume remains invariant (V1 = V2).
Shortcut Trick: When melting sphere into cylinder, equate (4/3) x pi x R^3 = pi x r^2 x h directly.
Definition: Surds are irrational roots of rational numbers. Rationalization removes roots from denominators using conjugates.
Shortcut Trick: Nested surd expansion: sqrt(a + sqrt(b)) = sqrt((a + sqrt(a^2-b))/2) + sqrt((a - sqrt(a^2-b))/2).
Definition: Laws of indices with rational exponents and solving equations with variables in exponents.
Shortcut Trick: Equating bases: if a^x = a^y, then x = y (for a not equal to 0, 1, -1).
Definition: Remainder theorem states remainder when p(x) is divided by x - a is p(a). Factor theorem states x - a is factor if p(a) = 0.
Shortcut Trick: For divisor ax - b, remainder is evaluated directly at p(b/a).
Definition: Factorization and expansion formulas for cubes of binomials and trinomials.
Shortcut Trick: When evaluating sum of cubes where base sum is zero, apply the 3abc shortcut directly.
Definition: Cartesian coordinate geometry formulas for distance between points and line segment midpoints.
Shortcut Trick: To check if three points form a right-angled triangle, verify Pythagoras theorem on squared distances between vertices.
Definition: Formulas for dividing line segments in specific ratios and calculating coordinate triangle areas.
Shortcut Trick: Area of triangle with one vertex at origin (0,0), (x1, y1), and (x2, y2) is given by 0.5 x abs(x1*y2 - x2*y1).
Definition: Equations of form ax + by + c = 0 representing straight lines on Cartesian plane with infinite solutions.
Shortcut Trick: To find x-intercept, put y = 0; to find y-intercept, put x = 0.
Definition: Finding coordinate pairs satisfying two-variable linear equations and modeling real-world constraints.
Shortcut Trick: Equation of line parallel to x-axis is y = k; parallel to y-axis is x = k.
Definition: Axioms (universal truths) and postulates (geometric assumptions) governing ancient geometry.
Shortcut Trick: Euclid's Fifth Postulate: If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side.
Definition: Playfair's axiom and logical deduction of geometric theorems from postulates.
Shortcut Trick: Two distinct intersecting lines cannot be parallel to the same line.
Definition: Conditions under which two triangles are identical in shape and size.
Shortcut Trick: In isosceles triangles, angles opposite to equal sides are equal, and vice versa.
Definition: Properties relating greater sides to greater opposite angles in triangles.
Shortcut Trick: In a right-angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices.
Definition: Properties of parallelograms and the theorem regarding line segments joining side midpoints.
Shortcut Trick: Diagonals of a parallelogram bisect each other; each diagonal divides parallelogram into two congruent triangles.
Definition: Theorems concerning figures on the same base and between the same parallels.
Shortcut Trick: Area of a triangle is half the area of a parallelogram on the same base and between the same parallels.
Definition: Angles subtended by arcs, cyclic quadrilaterals opposite sum, and perpendicular from center to chord.
Shortcut Trick: Perpendicular drawn from center of circle to a chord bisects the chord.
Definition: Calculating area of triangles given lengths of all three sides without altitude.
Shortcut Trick: For equilateral triangles of side a, area simplifies to (sqrt(3)/4) x a^2.
Definition: Euclid's division lemma (a = bq + r) and Fundamental Theorem of Arithmetic for prime factorization.
Shortcut Trick: A rational number p/q has terminating decimal expansion if prime factorization of denominator q is of form 2^n x 5^m.
Definition: Relationships between zeroes and coefficients of quadratic and cubic polynomials.
Shortcut Trick: To construct cubic polynomial given roots alpha, beta, gamma: x^3 - (sum)x^2 + (sum of products)x - (product) = 0.
Definition: Consistency conditions for two-variable linear equations (unique, infinite, or no solution).
Shortcut Trick: Cross-multiplication formula for a1*x + b1*y + c1 = 0 and a2*x + b2*y + c2 = 0.
Definition: Solving quadratic equations using discriminant formula and factoring.
Shortcut Trick: If D > 0, real and distinct roots; if D = 0, real and equal roots; if D < 0, non-real complex roots.
Definition: Sequences with constant common difference d (a_n = a + (n-1)d).
Shortcut Trick: If sum of first n terms of an AP is S_n = An^2 + Bn, common difference d = 2A and first term a = A + B.
Definition: Sequences with constant common ratio r (a_n = a x r^(n-1)).
Shortcut Trick: If a, b, c are in AP, then 2b = a + c. If in GP, b^2 = ac.
Definition: Basic Proportionality Theorem (Thales theorem) and triangle similarity criteria (AAA, SSS, SAS).
Shortcut Trick: Ratio of areas of two similar triangles equals square of ratio of their corresponding sides.
Definition: Collinearity conditions, centroid formulas, and triangle area proofs.
Shortcut Trick: Three points are collinear if and only if the coordinate triangle area formed by them is identically zero.
Definition: Trigonometric ratios in right triangles and fundamental Pythagorean identities.
Shortcut Trick: For complementary angle conversions: sin(90 - theta) = cos(theta), tan(90 - theta) = cot(theta).
Definition: Solving height and distance word problems using trigonometric ratios.
Shortcut Trick: Tower shadow problem: If angles of elevation from distances a and b are complementary, tower height h = sqrt(ab).
Definition: Tangent is perpendicular to radius at point of contact; lengths of tangents from external point are equal.
Shortcut Trick: If circle is inscribed inside a quadrilateral, sum of opposite sides is equal (AB + CD = AD + BC).
Definition: Areas of circles, sectors, segments, and combinations of plane figures.
Shortcut Trick: Area of segment = Area of sector - Area of corresponding triangle (0.5 x r^2 x sin(theta)).
Definition: Calculating central tendency measures for continuous grouped frequency distributions.
Shortcut Trick: Empirical relationship connecting three measures: 3 x Median = Mode + 2 x Mean.
Definition: Measuring likelihood of events based on ratio of favorable outcomes to total equally likely outcomes.
Shortcut Trick: For rolling two dice simultaneously, maximum sum is 12 and most frequent sum is 7 (appearing 6 times out of 36 outcomes).