9th Class Mathematics English Medium 2025-26 PDF Download – PCTB

Class 9 students in Punjab can now download the latest Mathematics textbook for 2025-26 in English Medium PDF format. Published by the Punjab Curriculum and Textbook Board (PCTB) and based on the National Curriculum of Pakistan 2023, this experimental edition covers 13 units from the real number system through probability, replacing the previous 17-chapter Punjab Textbook Board syllabus with a leaner, restructured course.

For 9th class students, Mathematics forms the base for board exams and future FSc/ICS study. Each unit builds from foundational algebra and number theory through trigonometry, coordinate geometry, logic, similarity, and statistics, with worked examples and graded exercises. The PDF lets students revise formulas, practice important questions from every unit, and prepare thoroughly before the annual exam.

Book Overview

Class	9th Class
Subject	Mathematics
Medium	English Medium
Board	Punjab Curriculum and Textbook Board (PCTB)
Curriculum	National Curriculum of Pakistan 2023
Authors	Muhammad Akhtar Shirani, Madiha Mahmood, Ghulam Murtaza
Edition	Experimental Edition 2025-26
Total Chapters	13
Total Pages	288
Format	PDF (Free Download)

Chapter List

Unit 1: Real Numbers

Unit 1 defines the real number system R = Q ∪ Q′, distinguishing rational numbers (terminating or recurring decimals) from irrational numbers such as π, e, and √2 (non-terminating, non-recurring), and shows how to locate irrationals like √5 on a number line using the Pythagoras theorem. It covers converting recurring decimals to p/q form, finding rational numbers between two values, the additive, multiplicative, distributive, equality, and order properties of real numbers, laws of radicals and indices, surds, and rationalizing denominators, then applies these to temperature conversion, profit and loss, and ratio-based problems frequently tested in exams.

Important Questions:

• The sum of three consecutive integers is forty-two, find the three integers. The integers are 13, 14, and 15.
• Find two numbers whose sum is 68 and difference is 22. The numbers are 45 and 23.
• Find two rational numbers between 3 and 4. 7/2 (3.5) and 15/4 (3.75) are two rational numbers between 3 and 4.
• The temperature in Lahore was reported as high as 48°C. Using °F = (9/5)°C + 32, find the temperature on the Fahrenheit scale. The temperature is 118.4°F.

Unit 2: Logarithms

Unit 2 covers scientific notation for writing very large or small numbers as a×10^n, then introduces logarithms as the inverse of exponentiation, defining log_b(x)=y alongside its exponential form b^y=x. Students learn common logarithm (base 10) with characteristic and mantissa, using logarithm and antilogarithm tables, plus natural logarithm (base e). The four laws of logarithm—product, quotient, power, and change of base—are proved and applied to expand or condense expressions and solve equations. Real-world applications include the decibel scale, earthquake magnitude, and compound growth/decay problems, all frequently tested in board exams.

Important Questions:

• The circumference of the Earth at the equator is about 40,075,000 metres. Express this number in scientific notation. 4.0075 × 10^7 m.
• Find the value of x: log_x 64 = 3. x = 4.
• Without using a calculator, evaluate: log_2 18 − log_2 9. 1.
• The magnitude of an earthquake is given by M = log_10(A/A_o). If amplitude A is 10,000 and reference amplitude A_o is 10, what is the magnitude? M = 3.

Unit 3: Sets and Functions

Building on mathematics as the study of patterns and structures, the unit defines sets, subsets, power sets, and singleton and empty sets, then covers union, intersection, difference, and complement, verified through Venn diagrams and De Morgan’s laws for two and three sets. Students apply the inclusion-exclusion principle to solve real-world classification problems such as surveys and enrollment data. The unit then introduces Cartesian products, binary relations, domain and range, and functions — notation f(x), linear and quadratic forms, and types like into, one-one, onto, and bijective functions — all tested through numerical and diagram-based questions.

Important Questions:

• In a class of 55 students, 34 like to play cricket and 30 like to play hockey, with each student liking at least one of the two games. How many students like to play both games? 9 students like both games.
• In a group of 500 employees, 250 can speak Urdu, 150 can speak English, 50 can speak Punjabi, 40 can speak both Urdu and English, 30 can speak both English and Punjabi, and 10 can speak both Urdu and Punjabi. How many can speak all three languages? 130 employees can speak all three languages.
• Consider the function defined by f(x) = 5x + 2. If f(x) = 32, find the value of x. x = 6.
• If g(x) = 3x + 2, what is the value of g(-3)? g(-3) = -7.

Unit 4: Factorization and Algebraic Manipulation

Unit 4 opens with common factors and trinomial factoring shown concretely using algebra tiles before moving to symbolic methods, covering five factorization types: quadratics like x²+px+q and ax²+bx+c, sum/difference forms a⁴+a²b²+b⁴, expressions like (x+a)(x+b)(x+c)(x+d)+k, perfect cubes (a±b)³, and sum/difference of cubes a³±b³. It then teaches HCF and LCM of algebraic expressions by factorization and division, their relationship (LCM×HCF = product of the two expressions), and square roots of polynomials by factorization and division methods, closing with real-world cost, profit, and engineering applications.

Important Questions:

• Factorize by identifying common factors: 6x + 12? 6(x + 2).
• Factorize: x² + x − 12? (x + 4)(x − 3).
• Factorize: 125a³ − 1? (5a − 1)(25a² + 5a + 1).
• Find the HCF by factorization method of 21x²y, 35xy²? 7xy.

Unit 5: Linear Equations and Inequalities

Unit 5 covers solving linear equations like 3x – 5 = 7 with rational coefficients and showing solutions on a real number line, then moves to inequalities using greater-than, less-than, and or-equal-to symbols, noting a negative multiplier flips the order. Students graph inequalities in two variables as half-planes using an associated equation and a test point such as (0,0), then combine two inequalities to find a feasible region. Key exam terms include problem constraints, decision variables, corner points, objective function, and optimal solution, since questions ask students to maximize or minimize functions like f(x,y) = 2x + 3y at corner points.

Important Questions:

• Solve and represent the solution on a real line: 12x + 30 = -6. x = -3.
• Maximize f(x, y) = 2x + 5y subject to 2y – x ≤ 8, x – y ≤ 4, x ≥ 0, y ≥ 0. Maximum value is 92 at the corner point (16, 12).
• Solve and represent the solution on a real line: (x + 5)/3 = 1 – x. x = -1/2.
• Find the maximum value of g(x, y) = x + 4y subject to constraints x + y ≤ 4, x ≥ 0, y ≥ 0. Maximum value is 16 at the corner point (0, 4).

Unit 6: Trigonometry

Unit 6 covers angles in standard position measured in degrees and radians, including conversions between DMS and decimal degrees, co-terminal angles, and arc length/sector area formulas (ℓ = rθ, A = ½r²θ). It defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) from a right triangle, complementary angle relations, and the Pythagorean identities sin²θ+cos²θ=1, tan²θ+1=sec²θ, 1+cot²θ=cosec²θ. Students derive exact ratios for 30°, 45°, 60°, solve right triangles given limited data, and apply trigonometry to angles of elevation and depression — all frequently tested in board exams.

Important Questions:

• Find in which quadrant 65° lies and write a co-terminal angle for it. It lies in the 1st quadrant; a co-terminal angle is 425°.
• Find the value of tan 60° without using a calculator. tan 60° = √3.
• Each side of a square field is 60 m long. Find the length of the diagonal of the field. The diagonal is 60√2 m ≈ 84.85 m.
• The angle of elevation of the top of a flag post from a point on the ground 40 m away is 60°. Find the height of the post. The height is 40√3 m ≈ 69.28 m.

Unit 7: Coordinate Geometry

Unit 7 builds the coordinate plane from two perpendicular axes and shows how any point P gets an ordered pair (x, y), its abscissa and ordinate, placing it in one of four quadrants. Students derive and apply the distance formula and the midpoint formula, then move into slope (gradient) as rise over run, testing collinearity and right angles via slopes. The unit derives six equivalent equations of a straight line — slope-intercept, point-slope, two-point, symmetric, intercept, and normal form — and shows every ax + by + c = 0 represents a line, applying these to real situations like maps, bridges, and aviation headings.

Important Questions:

• What is the gradient (slope) of the line passing through the points (1, 2) and (4, 6)? 4/3.
• Find the equation of the line in the form y = mx + c that passes through the points (3, 7) and (5, 11). y = 2x + 1.
• The coordinates of two points on a road are A(3, 4) and B(7, 10). Find the midpoint of the road. (5, 7).
• Find the distance between the points A(6, 7) and B(0, -2). 3√13 units (≈10.82 units).

Unit 8: Logic

Working with Class 9 exams involves proving conditional statements, differentiating premises from conclusions, and reasoning about truth using accepted rules of algebra — this unit builds exactly that foundation. It covers mathematical statements and their truth values, the five logical operators (negation, conjunction, disjunction, implication, biconditional) with truth tables, converse/inverse/contrapositive relationships, and the distinctions between axiom, conjecture, and theorem. Students also learn to construct deductive algebraic proofs by justifying each step (multiplicative identity, distributive law, commutative law) to show LHS equals RHS, a skill directly tested in board exams.

Important Questions:

• Prove that (x + 1)² – (x – 1)² = 4x. True: LHS = (x² + 2x + 1) − (x² − 2x + 1) = 4x = RHS, hence proved.
• Contrapositive of q → p is? ~p → ~q.
• Prove that (x – 4)² + 9 = x² – 8x + 25. True: LHS = x² – 8x + 16 + 9 = x² – 8x + 25 = RHS, hence proved.
• Suppose x is an integer. If x is odd, then prove 9x + 4 is odd. True: x = 2k+1 gives 9x+4 = 18k+13 = 2(9k+6)+1, which is odd.

Unit 9: Similar Figures

Similar Figures teaches students to identify similar polygons and triangles by checking that corresponding angles are equal and corresponding sides are proportional, covering AA, SAS, and SSS similarity conditions. It builds the core relationships that the ratio of areas of similar figures equals the square of the ratio of corresponding sides, while the ratio of volumes (and mass) of similar solids equals the cube of that ratio, using scale factor “k”. The unit also covers regular polygons’ interior/exterior angles, diagonals, symmetry, and tessellation, applying these to real-life problems like shadows, architecture, tiling, and carpeting — all frequently tested in board exams.

Important Questions:

• A man who is 1.8 m tall casts a shadow 0.76 m long. At the same time a telephone pole casts a 3 m shadow. Find the height of the pole? 7.11 m.
• The areas of two similar triangles are 16 cm² and 25 cm². What is the ratio of a pair of corresponding sides? 4:5.
• The radii of two spheres are in the ratio 3:4. What is the ratio of their volumes? 27:64.
• What is the sum of the interior angles of a decagon (10-sided polygon)? 1440°.

Unit 10: Graphs of Functions

Unit 10 covers sketching and interpreting graphs of linear, quadratic, cubic, reciprocal and exponential functions, including y = ax^n for positive, negative and rational n, and y = ka^x for a > 1. Students learn to find x- and y-intercepts, identify parabola direction from the sign of a, recognise asymptotes in reciprocal graphs, and read growth/decay curves. The unit also introduces finding gradients of curves by drawing tangents (Δy/Δx) and applies graphing to real-life contexts such as salary functions, tax, and cost-revenue break-even analysis — all recurring exam topics with worked numerical examples.

Important Questions:

• What is the slope of the line y = 5x + 3? 5.
• A company manufactures school bags with cost function C(x) = 1200 + 20x and revenue function R(x) = 50x. Find the break-even point. x = 40 bags.
• A newspaper agency has a profit function p(x) = 10x − 70, where x is the number of newspapers. Find the profit for 500 newspapers. Rs. 4,930.
• A shirt manufacturer’s cost function is C(x) = 1500 + 10x + 0.2x². Find the cost of producing 200 shirts. Rs. 11,500.

Unit 11: Loci and Construction

Loci and Construction covers building triangles from given sides, angles, or side-angle combinations, including the ambiguous SSA case where two, one, or no triangles may result. Students learn to construct perpendicular bisectors, medians, angle bisectors, and altitudes of a triangle, identifying their points of concurrency — circumcentre, centroid, incentre, and orthocentre. The unit then defines loci (circle, parallel lines, perpendicular bisector, angle bisector) as sets of points satisfying distance conditions, and applies intersecting loci to real-life placement problems like water sources, cell towers, and quarantine zones — key for exam construction and word-problem questions.

Important Questions:

• Two points A and B are 8.2 cm apart. Construct the locus of points 5 cm from point A. The locus is a circle of radius 5 cm centred at A.
• Two points E and F are 5.4 cm apart. Construct a locus of a point P which moves such that it is equidistant from E and F. The locus is the perpendicular bisector of line segment EF.
• Construct a right angled triangle with measures of sides 6 cm, 8 cm and 10 cm. Triangle constructed with sides 6 cm, 8 cm, 10 cm, with the right angle formed opposite the 10 cm side.
• Construct a triangle ABC with mAB = 5.3 cm, m∠A = 30° and m∠B = 120°. Draw the locus of all points which are equidistant from A and B. Triangle ABC is constructed by the ASA method, and the required locus is the perpendicular bisector of AB.

Unit 12: Information Handling

Organizing raw numerical data into grouped frequency tables using class limits, class boundaries, and tally marks, then displaying it through histograms (including unequal class widths) and frequency polygons plotted at class midpoints is central to this unit. Students learn the three measures of central tendency — arithmetic mean (direct and short/assumed-mean methods), median (using the median-class formula with cumulative frequency), and mode (using modal-class frequencies) — for both ungrouped and grouped data, plus weighted mean for situations like budget allocation, marketing, and exam grading. These formulas, especially median and mode for grouped data, are frequently tested numerically in exams.

Important Questions:

• Following are the heights (in inches) of 12 students: 55, 53, 54, 58, 60, 61, 62, 56, 57, 52, 51, 63. Find the median height. Median = 56.5 inches.
• What is the midpoint of the class (34 – 38) in the chemistry scores frequency distribution? Midpoint = 36.
• The arithmetic mean of 45 numbers is 80. Find their sum. Sum = 3600.
• Find the mode of the frequency distribution with class intervals 5–9, 10–14, 15–19, 20–24, 25–29 and frequencies 1, 8, 18, 11, 2. Mode = 17.44.

Unit 13: Probability

Unit 13 introduces probability as the ratio of favourable outcomes to total possible outcomes, P(A) = n(A)/n(S), built on core terms such as experiment, outcome, sample space, event, and equally likely events. Students practice finding the probability of a single event using dice, coins, cards, and balls, then extend this to the complement rule P(A’) = 1 – P(A) for an event not occurring. The unit also teaches relative frequency as an estimate of probability and expected frequency (N × P(A)) through real-life data like surveys, exam scores, and defective products, skills directly tested in numerical and multiple-choice exam questions.

Important Questions:

• Arshad rolls a dice with sides labelled L, M, N, O, P, U. What is the probability that the dice lands on a consonant? 2/3.
• Aslam rolled a dice. What is the probability of getting the numbers 3 or 4? 1/3.
• The probability that Ayesha will pass the examination is 0.85. What will be the probability that Ayesha will not pass the examination? 3/20.
• An urn contains 10 red balls, 5 green balls and 8 blue balls. Find the probability of selecting a red ball at random. 10/23.

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2025-26 Edition (PCTB – National Curriculum 2023):

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Who Should Read This

This book is for 9th class students studying Mathematics in English medium under the new National Curriculum 2023. Students in PCTB affiliated schools following the new syllabus should use this textbook for their 2025-26 annual board exam preparation.

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Applicable Boards

This textbook is published by Punjab Curriculum and Textbook Board (PCTB) based on National Curriculum of Pakistan 2023. It is used in all PCTB affiliated schools in Punjab. Federal Board and other provincial boards implementing the new national curriculum may also find this book useful.

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