Mathematics 1st Year ICS Book PDF Download – (Punjab Board)

Mathematics 1st Year ICS book for Part 1 (Class 11) is published by the Punjab Education, Curriculum, Training and Assessment Authority (PECTAA). The latest 2025–26 edition is based on the Updated/Revised National Curriculum of Pakistan 2023. It covers fourteen units from Complex Numbers and Functions to Differentiation and Vectors in Space.

For ICS Part 1 students, Mathematics is the most important subject in the programme. Each unit builds on earlier concepts with clear theory, worked examples, and graded exercises. The PDF lets students revise formulas, practice numerical problems, and go over difficult topics at any time before the final board exam.

Book Overview

Class	11 (ICS Part 1 / First Year)
Subject	Mathematics
Category	ICS
Board	Punjab Board (PECTAA, Lahore)
Curriculum	NCP 2023 (Updated/Revised)
Total Chapters	14 units + Answers key
Total Pages	312
Format	PDF

Chapter List

Unit 1 – Complex Numbers

Complex numbers extend real numbers by introducing the imaginary unit i = √-1, so equations like x² + 1 = 0 finally have solutions. Students learn to write numbers as z = a + ib, find real and imaginary parts, conjugates, and add, subtract, multiply and divide complex numbers, including as ordered pairs. The chapter factorises complex polynomials using the fundamental theorem of algebra, solves quadratic equations by completing the square to get roots like 3 ± 4i, and introduces cube roots, fourth roots of unity and polar form. These ideas matter for board exams since quadratic equations with complex roots appear almost every year.

Important Questions:

• Find the multiplicative inverse of the complex number (1, 0). (1, 0), i.e., the multiplicative inverse is 1.
• Separate i/(1 + i) into real and imaginary parts (write as a simple complex number). 1/2 + (1/2)i
• Find the square root of the complex number 8 – 6i. ±(3 – i)
• Factorize a² + 4b². (a + 2bi)(a – 2bi)

Unit 2 – Functions and Graphs

Functions describe how one quantity depends on another, and this chapter builds a solid grip on domain, co-domain and range using examples like f(x) = x² – 1. Students learn to evaluate functions at given inputs, work with one-to-one, onto and bijective functions, and find where a linear function meets the axes or where a line meets another line or a parabola, using graphs to visualise solutions. The chapter also covers graphs of the square root and cube root functions and shows real-life uses, such as expressing area as a function of perimeter, useful for board exam graph questions.

Important Questions:

• If f(x) = x² – 1, find f(-3). 8
• If f(x) = √(2x + 3), find f(0). √3
• Find the domain and range of g(x) = 5 – x. Domain = R, Range = R
• Express the area A of a square as a function of its perimeter P. A = P²/16

Unit 3 – Theory of Quadratic Functions

This unit takes the quadratic function f(x) = ax² + bx + c and shows how to find its maximum or minimum value by completing the square, then sketch its parabola to locate the vertex, intercepts and axis of symmetry. Students also learn to find the inverse of a quadratic function along with its domain and range, and to solve absolute value quadratic equations and inequalities such as |x² + 1| = 5. Rational, radical and exponential equations that reduce to quadratic form are covered too, alongside real-world applications like projectile motion. These techniques are heavily tested in board exams through numerical and graph-based questions.

Important Questions:

• Find the minimum value of f(x) = x² + 6x + 13 by completing the square. 4 (at x = -3)
• Find the minimum value of f(x) = x² + 4x by completing the square. -4 (at x = -2)
• Solve the absolute value equation |x² + 1| = 5. x = ±2
• Solve the absolute value equation |x² + 5x + 4| = 0. x = -1, -4

Unit 4 – Matrices and Determinants

Matrices and determinants form one of the biggest chapters in the book, starting with matrix addition, subtraction, scalar multiplication and multiplication of two matrices, including matrices with complex entries. Students learn minors, cofactors and properties of determinants for evaluating a 3×3 determinant, then move on to row operations for finding the inverse and rank of a matrix. The unit explains how to solve systems of linear equations, both homogeneous and non-homogeneous, using matrix inversion, Cramer’s rule and Gaussian elimination, and closes with real-world uses such as encryption, geometric transformations and graphic design. Because of its length and variety, this chapter carries heavy weightage in exams.

Important Questions:

• Evaluate the determinant |1 -2 -4; 3 -1 -3; -2 3 2|. -21
• Without expansion, show that |7 8 9; 5 6 7; 2 3 4| = 0. 0 (rows are in arithmetic progression, so they are linearly dependent)
• Point A is mapped to (30, 20, -5) by the scaling matrix P = diag(-5, -5, -5). Find the coordinates of A. A(-6, -4, 1)
• A triangle has vertices A(4,1), B(-2,5), C(0,-3). Find the vertices of the triangle reflected over the y-axis. A'(-4,1), B'(2,5), C'(0,-3)

Unit 5 – Partial Fractions

Partial fractions teaches students to reverse the usual process of combining fractions, breaking a single rational expression like 2/(x² – 1) back into simpler fractions such as 1/(x-1) – 1/(x+1). The chapter covers several cases depending on the denominator’s factors: non-repeated linear factors, repeated linear factors, and non-repeated irreducible quadratic factors like x² + 1. Clear step-by-step examples show how to set up the identity, compare coefficients or substitute convenient values of x to find the unknown constants. This is a short but scoring chapter, since the same handful of methods is applied repeatedly, making it easy to master with practice before exams.

Important Questions:

• Resolve 2/(x² – 1) into partial fractions. 1/(x-1) – 1/(x+1)
• Resolve (a – b)/[(x – a)(x – b)] into partial fractions. 1/(x-a) – 1/(x-b)
• Resolve (x² + 1)/[(x + 1)(x – 1)] into partial fractions. 1 + 1/(x-1) – 1/(x+1)
• Resolve (x + 1)/(x – 1)² into partial fractions. 1/(x-1) + 2/(x-1)²

Unit 6 – Sequences and Series

Sequences and series is a long, important chapter that covers arithmetic progressions, geometric progressions and harmonic progressions along with their arithmetic, geometric and harmonic means. Students learn to find the nth term, sum of n terms, and solve problems like determining how many terms of a sequence stay below a given value or which term equals a specific number. The chapter also introduces arithmetico-geometric progressions, infinite geometric series, and sigma notation for writing sums compactly, plus real-life uses in investment planning and vehicle leasing. Given its length, this unit is a major source of numerical questions in the Punjab board exam.

Important Questions:

• Find the next four terms of the arithmetic sequence 12, 16, 20, … 24, 28, 32, 36
• Find the 15th triangular number using n(n+1)/2. 120
• Is 301 a term of the A.P. 5, 11, 17, …? No, since n = 50.33 is not a whole number
• Which term of the A.P. 3, 8, 13, … is 123? The 25th term

Unit 7 – Permutations and Combinations

Permutations and combinations build on the fundamental principle of counting to work out how many ways objects can be arranged or selected. The chapter starts with factorial notation, using expressions like 10!/(0!8!) and solving for n when (n+4)! = 3024·n!, before moving to permutations of distinct and repeated objects, circular arrangements, and combinations with their real-life uses in selecting committees or teams. Complementary combinations and the difference between arrangement (order matters) and selection (order does not matter) are explained clearly with worked examples. These counting techniques also connect directly to probability and the binomial theorem covered in the next unit.

Important Questions:

• Evaluate 10!/(0!·8!). 90
• Evaluate 12!/[3!(12-3)!]. 220
• Find n if (n + 4)! = 3024·n!. n = 5
• If 1/7! + 1/8! = x/9!, find x. x = 81

Unit 8 – Mathematical Inductions and Binomial Theorem

This unit joins two related ideas: mathematical induction, used to prove formulas true for all natural numbers, and the binomial theorem, used to expand expressions like (x + y)^n without multiplying term by term. Students practice the principle of induction on sums such as the first n odd numbers equalling n², then learn binomial expansion for positive integer, negative and fractional indices, finding specific terms, middle terms and coefficients. Pascal’s triangle gives a quick way to read off binomial coefficients, while later sections apply the binomial theorem to approximate values, find remainders, and identify the last digit of large powers, all common exam question types.

Important Questions:

• Calculate (0.97)³ using the binomial theorem. 0.912673
• Find the term independent of x in the expansion of (x – 2/x)^10. -8064
• Find the remainder when 8^100 is divided by 7 using the binomial theorem. 1
• Show that C(n,1) + C(n,2) + C(n,3) + … + C(n,n) = 2^n – 1. 2^n – 1

Unit 9 – Division of Polynomials

Division of polynomials is a short chapter that teaches long division and synthetic division, finding the quotient and remainder when dividing expressions like 3x³ – 10x² + 13x – 6 by x – 2. The remainder theorem lets students find the remainder without doing full division, simply by evaluating the polynomial at a point, while the factor theorem checks whether a given expression is a factor and helps factorise cubic polynomials completely. The chapter also connects these ideas to real-world uses in polynomial regression, signal processing and coding theory. Being compact and rule-based, it is one of the easiest chapters to score full marks in.

Important Questions:

• Use the remainder theorem to find the remainder when x² + 5x + 6 is divided by x – 2. 20
• When 4x⁴ + 2x³ + kx² + 13 is divided by x + 1, the remainder is 16. Find k. k = 1
• When x³ + x² + x + k is divided by x + 1, the remainder is 7. Find k. k = 8
• Use the factor theorem to find the zeros of B(z) = z² – z – 2. z = 2, z = -1

Unit 10 – Trigonometric Identities

Trigonometric identities begins with the distance formula and the fundamental law of trigonometry, then moves to ratios of allied angles like sin(180° + α) and cos(-θ), which let students evaluate trig ratios of angles greater than 360° or negative angles without tables. The chapter proves many identities involving sums, differences and triangle angles, and introduces double angle, half angle and triple angle identities such as sin 2θ and cos 3θ. It also shows how to express products of sines and cosines as sums or differences. Since these identities feed directly into the next two chapters on graphs and calculus, mastering them here is essential.

Important Questions:

• Without using tables, find the value of cos(-1230°). -√3/2
• Without using tables, find the value of sec(1140°). 2
• Without using tables, find the value of cos(-240°). -1/2
• Express cos 168° as a trigonometric function of a positive angle less than 45°. -cos 12°

Unit 11 – Trigonometric Functions and their Graphs

This unit looks at trigonometric functions through their domain, range, periodicity and graphs. Students learn to determine whether functions like sin²x or tan x + sec x are even, odd or neither, find periods such as 2π/5 for sin 5x, and sketch graphs of sine, cosine and tangent over given intervals. It also covers the maximum and minimum values of sinusoidal functions using amplitude, frequency and phase shift, connecting these ideas to real-world problems like solar panel angles and forces in structures. Understanding periodicity and graph behaviour here is important, since board papers often ask students to state the domain, range or period directly.

Important Questions:

• Determine whether sin²x is even, odd, or neither. Even
• Find the period of sin 5x. 2π/5
• Find the period of tan 3x. π/3
• Determine whether tan x + sec x is even, odd, or neither. Neither odd nor even

Unit 12 – Limit and Continuity

Limit and continuity introduces the idea of a function approaching a value as x approaches a point, starting with limits of sequences like (2n+3)/(n+1) and building up to limits of functions using algebraic techniques and standard trigonometric limits such as lim sin5x/x = 5. Students also learn the sandwich theorem, limits at infinity, and how to express certain limits in terms of e. The second half defines continuity and discontinuity of a function at a point, showing how to test whether a function is continuous using left-hand and right-hand limits. This chapter lays the essential groundwork for differentiation in the next unit.

Important Questions:

• Find the limit of the sequence a_n = (2n + 3)/(n + 1) as n approaches infinity. 2
• Evaluate lim(x→3) (2x + 4) using theorems of limits. 10
• Evaluate lim(x→-1) (x³ – x)/(x + 1) using algebraic techniques. 2
• Evaluate lim(x→0) (sin 5x)/x. 5

Unit 13 – Differentiation

Differentiation opens with the idea of a tangent to a curve and the derivative as the limit of a difference quotient, showing students how to find derivatives from first principles for functions like 2x² + 1 and (3x – 2)^-2. It then builds a toolkit of differentiation rules, including the power rule, sum and difference rules, and derivative of a constant multiple, so students no longer need first principles every time. Practical applications follow, such as finding the gradient and equation of a tangent line, instantaneous velocity from a position function, and rates of change in real situations. This chapter is the gateway to calculus.

Important Questions:

• Find the derivative of 2x² + 1 by definition (first principles). 4x
• Find the gradient of the curve f(x) = 3x² + 2x at x = 1. 8
• Find the equation of the tangent line to f(x) = 2x² + x at x = -1. y = -3x – 2
• Find the instantaneous velocity at t = 1 for s(t) = -16t² + 32t + 10. 0

Unit 14 – Vectors in Space

Vectors in space extends vector concepts into three dimensions using the rectangular coordinate system, covering vector addition, magnitude, direction cosines, and how to write a unit vector in a given direction, such as for v = -i + 4j – 8k. Students then learn the dot product for finding the angle between vectors and work done by a force, and the cross product for finding areas of triangles and parallelograms using its determinant formula. The chapter also explains parallel vectors, the scalar triple product and volume of a parallelepiped, plus real-world uses in physics like calculating forces and displacement between points in space.

Important Questions:

• Find the magnitude of the vector v = 3i – 2j + 6k. 7
• Find a unit vector in the direction of v = -i + 4j – 8k. (-1/9)i + (4/9)j – (8/9)k
• Find the constant a so that v = i – 3j + 4k and w = ai + 9j – 12k are parallel. a = -3
• A spacecraft moves from (120, 240, -50) to (130, 210, 80) in km. Find the magnitude of the displacement vector. 10√179 km (≈ 133.8 km)

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Download Mathematics Class 11 ICS Book PDF

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2025–26 Edition (NCP 2023 — PECTAA)

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E-Learn Punjab Edition

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Who Should Read This

ICS Part 1 (First Year) students preparing for the Punjab Board annual exam will find this book essential, since Mathematics is the core subject of ICS and it covers every topic required for the board paper. It is also useful for students planning for ECAT and other entry tests, as topics like complex numbers, matrices, sequences, and calculus are directly tested.

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Applicable Boards

PECTAA publishes this textbook for use in Punjab Board colleges. Students from the Federal Board (FBISE) and AJK Board can also use it for reference, as the ICS Mathematics syllabus is very similar across these boards. Students from Sindh and KPK boards will also find most units relevant.

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