Mathematics Science Group Class 10 Book PDF Download (Punjab Board)

Mathematics for Class 10 Science Group is part of the official Punjab Curriculum Authority (PCA) approved syllabus for matric (SSC Part 2). Published by Ilmi Kitab Khana, Lahore, this book follows the National Curriculum and is used in Punjab Board science schools across the province. It has thirteen units split into two sections, Algebra (Units 1-6) and Geometry (Units 7-13).

For Class 10 Science students, this book is the main reference for board exam preparation. Each unit builds theory, worked examples, and graded exercises step by step, so this page breaks down every unit with its real content and important questions. It is a useful way to revise the quadratic formula, statistics formulas, trigonometric identities, and circle theorems before the final exam.

Book Overview

Class	10 (Matric / SSC Part 2)
Subject	Mathematics (Science Group)
Board	Punjab Board (PCA, Lahore)
Publisher	Ilmi Kitab Khana, Lahore
Edition	March 2017 (1st Edition)
Total Units	13
Medium	English
Format	PDF (Free Download)

Chapter List

Unit 1: Quadratic Equations

This unit defines a quadratic equation (ax²+bx+c=0, a≠0) and pure quadratic equations, then teaches solving by factorization and by completing the square. It derives the quadratic formula and applies it to equations reducible to quadratic form, including biquadratic equations, reciprocal equations, exponential equations, and symmetric-product equations. It closes with radical equations solved by squaring, always checking for extraneous roots.

Important Questions:

• Solve 3x²-6x=x+20 by factorization. Standard form 3x²-7x-20=0 factors as (x-4)(3x+5)=0, giving x=4 or x=-5/3.
• State the quadratic formula and how it is derived. x=(-b±√(b²-4ac))/2a, derived by completing the square on ax²+bx+c=0.
• Solve x⁴-13x²+36=0. Let y=x²; y²-13y+36=0 gives (y-9)(y-4)=0, so x=±3, ±2.
• Solve √(3x+7)=2x+3 and check for extraneous roots. Squaring gives 4x²+9x+2=0, so x=-1/4 or -2; checking shows x=-2 fails, leaving {-1/4}.

Unit 2: Theory of Quadratic Equations

This unit introduces the discriminant to classify roots as rational-equal, rational-unequal, irrational, or imaginary, and covers cube roots of unity and their properties. It develops the sum and product of roots relations, symmetric functions, and forming a new quadratic from transformed roots, plus synthetic division for solving cubic and quartic equations with known roots. It ends with solving simultaneous linear-quadratic and quadratic-quadratic systems and real-life word problems.

Important Questions:

• How does the discriminant determine the nature of roots? b²-4ac>0 and a perfect square gives rational unequal roots; positive but not a perfect square gives irrational roots; zero gives equal rational roots; negative gives imaginary roots.
• If α, β are roots of ax²+bx+c=0, what are their sum and product? α+β=-b/a and αβ=c/a.
• Show the product of the three cube roots of unity is 1. 1·ω·ω²=ω³=1, verified by multiplying the three roots directly.
• Solve the system 3x+y=4 and 3x²+y²=52. Substituting y=4-3x gives x²-2x-3=0, so x=3 or -1; the solution set is {(3,-5), (-1,7)}.

Unit 3: Variations

This unit covers ratio and proportion, including third, fourth, mean, and continued proportionals, and the classical theorems of invertendo, alternendo, componendo, dividendo, and componendo-dividendo. It defines direct variation and inverse variation, extends to joint variation combining several variables, and introduces the k-method for proving conditional equalities. It finishes with applied problems such as gravitation, beam strength, and Hooke’s law that model physical relationships as variations.

Important Questions:

• Define direct and inverse variation. Direct variation means y=kx (y proportional to x); inverse variation means y=k/x, where k is the constant of variation.
• State the componendo-dividendo theorem. If a:b=c:d, then (a+b):(a-b) = (c+d):(c-d).
• A falling body’s distance d varies directly as the square of time t, with d=16 ft at t=1 sec. Find the relation. k=16, so d=16t².
• A beam 9cm wide and 12cm deep supports 1200 lb, with strength varying as breadth times depth squared. What load will a 12cm-wide, 9cm-deep beam support? k=25/27, giving a new safe load of 900 lb.

Unit 4: Partial Fractions

This unit distinguishes proper fractions from improper ones and shows how to reduce an improper fraction to a polynomial plus a proper fraction by division. It then teaches four resolution rules for decomposing a rational fraction into partial fractions depending on whether the denominator has non-repeated linear factors, repeated linear factors, non-repeated irreducible quadratic factors, or repeated irreducible quadratic factors.

Important Questions:

• Resolve (5x+4)/((x-4)(x+2)) into partial fractions. Using the zero’s method, this equals 4/(x-4) + 1/(x+2).
• What distinguishes a proper fraction from an improper fraction? A proper fraction has numerator degree less than denominator degree; an improper fraction has numerator degree equal to or greater than the denominator’s.
• What form do partial fractions take for a repeated linear factor (ax+b)ⁿ? A sum of n terms: A₁/(ax+b) + A₂/(ax+b)² + … + Aⁿ/(ax+b)ⁿ.
• Resolve 1/((x-1)²(x-2)) into partial fractions. -1/(x-1) – 1/(x-1)² + 1/(x-2).

Unit 5: Sets and Functions

This unit reviews standard number sets and set operations, union, intersection, difference, and complement, then proves commutative, associative, and distributive properties plus De Morgan’s laws using Venn diagrams. It defines ordered pairs and Cartesian products, then binary relations with domain and range, leading into the formal definition of a function. It classifies functions as into, one-one, onto, and bijective, with worked examples distinguishing each type.

Important Questions:

• State De Morgan’s Laws for sets. (A∪B)′=A′∩B′ and (A∩B)′=A′∪B′.
• If A={1,2,3,4} and B={4,5,6,7}, find A∪B and A∩B. A∪B={1,2,3,4,5,6,7}; A∩B={4}.
• Define a one-one function and an onto function. One-one: distinct domain elements have distinct images; onto: every element of the co-domain is the image of at least one domain element.
• If A={1,2,3} and B={2,5}, find A×B. {(1,2),(1,5),(2,2),(2,5),(3,2),(3,5)} — six ordered pairs.

Unit 6: Basic Statistics

This unit teaches constructing discrete and continuous frequency tables with class limits, boundaries, and midpoints, and drawing histograms, frequency polygons, and cumulative frequency polygons (ogives). It covers computing the arithmetic mean, median, mode, geometric mean, and harmonic mean for grouped and ungrouped data, plus weighted means and moving averages. It ends with measures of dispersion: range, variance, and standard deviation.

Important Questions:

• How are class boundaries obtained from class limits? By averaging the upper limit of one class and the lower limit of the next; for classes 0-9 and 10-19, the boundary is 9.5.
• Find the median of 82, 93, 86, 92, 79. Arranged: 79, 82, 86, 92, 93; with 5 values (odd), the median is the middle value, 86.
• State the formula for the mode of grouped continuous data. Mode = l + [(f1-f0)/(2f1-f0-f2)]×h, using the modal class’s lower boundary, frequencies, and class width.
• Six students scored 60, 70, 30, 90, 80, 42 in Mathematics. Find the mean and standard deviation. Mean = 62 marks; standard deviation is approximately 20.82 marks.

Unit 7: Introduction to Trigonometry

This unit covers angle measurement in the sexagesimal system (degrees, minutes, seconds) and the radian system, establishing that 180° equals π radians along with the arc-length and sector-area formulas. It defines general angles, standard position, and quadrants, then the six trigonometric ratios via the unit circle with their reciprocal identities and signs in each quadrant. It proves the three Pythagorean identities and applies trigonometry to angle-of-elevation and angle-of-depression word problems.

Important Questions:

• Convert 45.36° into degrees, minutes, seconds. 45°21′36″.
• State the degree-radian relationship and convert 15° to radians. 180° = π radians, so 15° = π/12 radians.
• A 17.9m flagpole casts a 7m shadow. Find the sun’s angle of elevation. tanα=17.9/7≈2.557, giving α≈68°40′.
• Prove the identity cotθ·secθ=cscθ. cotθ·secθ = (cosθ/sinθ)(1/cosθ) = 1/sinθ = cscθ.

Unit 8: Projection of a Side of a Triangle

This unit proves three theorems using Pythagoras’ theorem: for an obtuse-angled triangle, the square on the side opposite the obtuse angle equals the sum of the squares of the other two sides plus twice the rectangle of one side and the projection of the other; for an acute angle, the same relation holds with a minus sign; and Apollonius’ theorem on medians. These are the textbook’s version of the law of cosines, used to find unknown sides and classify triangles.

Important Questions:

• State Apollonius’ Theorem. (AB)²+(AC)²=2(BD)²+2(AD)², where AD is the median to BC and BD=DC.
• State the projection theorem for an obtuse-angled triangle. (BC)²=(AC)²+(AB)²+2·AB·AD, where AD is the projection of AC on BA produced.
• In ΔABC, AB=6cm, AC=4cm, ∠A=60°. Find BC. BC²=36+16-24=28, so BC=2√7 ≈ 5.29 cm.
• In isosceles ΔABC with AB=AC and BE⊥AC, prove BC²=2·AC·CE. Applying the acute-angle projection theorem and substituting AB=AC gives the result directly.

Unit 9: Chords of a Circle

This unit begins with basic circle vocabulary and proves that exactly one circle can pass through three non-collinear points. It then proves that a line from the centre bisecting a chord is perpendicular to it, and the converse theorems relating congruent chords to equal distances from the centre. Worked examples apply these to rectangles, parallel chords, and comparing chord lengths to the diameter.

Important Questions:

• State the theorem about a line from the centre bisecting a chord. A line drawn from the centre of a circle to bisect a chord (not a diameter) is perpendicular to that chord.
• A chord AB is 8cm long and 3cm from the centre. Find the circle’s diameter. Half-chord=4, so radius=√(3²+4²)=5cm; diameter=10cm.
• Through how many non-collinear points can exactly one circle be drawn? Three.
• Prove the diameter of a circle is its largest chord. In triangle OAB (O=centre), OA+OB>AB by the triangle inequality, and OA+OB equals the diameter, so the diameter exceeds any other chord.

Unit 10: Tangent to a Circle

This unit defines secants and tangents, then proves that a line perpendicular to a radius at its outer endpoint is tangent to the circle, and conversely that a tangent is always perpendicular to the radius at the point of contact. It proves that two tangents drawn from an external point are equal in length, and shows that when two circles touch externally or internally, the distance between their centres equals the sum or difference of their radii.

Important Questions:

• What is the relationship between a tangent and the radius at the point of contact? They are perpendicular to each other.
• Prove that two tangents from an external point to a circle are equal in length. The two right triangles formed with the centre are congruent by the H.S. postulate, so the tangent lengths are equal.
• If two circles touch externally, how is the distance between centres related to their radii? It equals the sum of the two radii.
• Two concentric circles have diameters 10cm and 5cm. Find the chord of the outer circle that touches the inner circle. Half-chord=√(5²-2.5²)≈4.33cm, so the chord is about 8.7cm.

Unit 11: Chords and Arcs

This unit proves that congruent arcs correspond to equal chords and conversely, building on circle congruence via triangle congruence at the centre. It then proves that equal chords subtend equal central angles, and the converse. Worked examples apply these results to prove that perpendicular diameters form a square and that an angle bisector from the centre bisects the arc it stands on.

Important Questions:

• What is the relationship between equal arcs and their chords? Congruent arcs have equal corresponding chords, and conversely.
• Prove that equal chords of a circle subtend equal angles at the centre. The triangles formed by the equal chords and equal radii are congruent by SAS, giving equal central angles.
• Two perpendicular diameters AC and BD are drawn in a circle. Prove ABCD is a square. The four 90° central angles create four equal arcs and equal chords, with each interior angle equal to 90°.
• Two equal diameters AB and CD intersect. Prove AD=BC. The equal diameters create symmetric equal arcs, so the chords subtending them are equal.

Unit 12: Angle in a Segment of a Circle

This unit proves the central angle theorem, that a central angle is twice the inscribed angle standing on the same arc, and uses it to prove that angles in the same segment of a circle are equal. It further shows that the angle in a semicircle is a right angle, and proves that opposite angles of a cyclic quadrilateral are supplementary. Worked examples include tangent-and-chord constructions and quadrilaterals circumscribed about a circle.

Important Questions:

• State the relationship between a central angle and an inscribed angle on the same arc. The central angle is double the inscribed angle standing on the same arc.
• What is the measure of an angle inscribed in a semicircle? A right angle, since the corresponding central angle is 180°.
• State the property of opposite angles of a cyclic quadrilateral. They are supplementary, summing to 180°.
• A chord of length 2cm in a circle of radius √2cm divides the circle into two segments. Find the angle in the larger segment. The central angle works out to 90°, so the inscribed angle in the larger segment is 45°.

Unit 13: Practical Geometry – Circles

This construction-based unit teaches locating the centre of a given circle, drawing a circle through three non-collinear points, and completing a circle from a given arc. It covers circumscribing and inscribing circles about or in a triangle (circumcircle, incircle, e-circle) and constructing equilateral triangles, squares, and hexagons inscribed in and circumscribed about a given circle. It finishes with tangent constructions from a point on or outside a circle, at a given angle, and common tangents to two circles.

Important Questions:

• How is the centre of a given circle located using straightedge and compass? Draw two chords and construct their perpendicular bisectors; the intersection point is the centre.
• What is the difference between a triangle’s circumcircle and its incircle? The circumcircle passes through all three vertices; the incircle touches all three sides, with centres found from perpendicular bisectors and angle bisectors respectively.
• What is an escribed circle (e-circle) of a triangle? A circle touching one side externally and the other two sides (produced) internally, centred where one interior and two exterior angle bisectors meet.
• Can a circle be drawn touching three converging lines? No, unlike a triangle’s incircle which requires concurrent bisectors, three lines converging to different points cannot all be tangent to one circle.

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Who Should Read This

This book is for Class 10 Science Group students preparing for the Punjab Board matric exam. It is also a good starting point for students planning to appear in entry tests like MDCAT and ECAT, since trigonometry, algebra, and statistics topics from this book form part of the test syllabus.

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Applicable Boards

This textbook is approved by the Punjab Curriculum Authority (PCA) and is used in Punjab Board schools. Students from the Federal Board (FBISE) and AJK Board can also use it, as the Science Group Mathematics syllabus is largely the same. Students from other provincial boards will find most algebra and geometry topics overlapping.

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