BISE Lahore 12th Class Mathematics Full Book Notes — All Chapters, Questions and Answers

These BISE Lahore 12th class mathematics full book notes cover the whole 2nd Year Mathematics course for BISE Lahore students: functions and limits, differentiation, integration, analytic geometry, linear inequalities and linear programming, conic sections, and vectors. Each chapter gives a short summary, the important formulas, short questions with answers, and a long question worked out step by step, followed by a final MCQ revision section.

Important note: These are study and revision notes only. They are not official BISE Lahore material, not leaked papers, and not a guarantee of marks or questions. Keep your textbook beside you, practise every worked step with a pen, and check the final answer before moving on.

How to Use These Notes

Use these BISE Lahore 12th class mathematics full book notes in four steps. First, read the summary of a chapter. Second, copy its formulas into a small notebook. Third, answer the short questions without looking, then check the answers. Fourth, work the long question yourself and compare each step, not only the final answer.

• Write the formula first, then substitute the numbers, then simplify, then state the answer.
• For limits and derivatives, always check the domain and the conditions before you apply a rule.
• For integration, always add the constant of integration for an indefinite integral.
• Revise one chapter a day, then mix questions from earlier chapters so you do not forget them.

Quick Revision Before Chapter 1

Second Year Mathematics builds on First Year algebra, trigonometry and functions. A function assigns exactly one output to each input in its domain. A limit describes the value a function approaches, not necessarily the value it takes. A derivative measures a rate of change, and an integral reverses differentiation and measures area under a curve. Keep those three ideas in mind and the whole book becomes easier.

All Chapters at a Glance

• Chapter 1: Functions and Limits
• Chapter 2: Differentiation
• Chapter 3: Integration
• Chapter 4: Introduction to Analytic Geometry
• Chapter 5: Linear Inequalities and Linear Programming
• Chapter 6: Conic Sections
• Chapter 7: Vectors

Chapter 1: Functions and Limits

Summary: A function is a rule that gives one output for each input. Its domain is the set of allowed inputs and its range is the set of outputs. A limit, written as the limit of f(x) as x approaches a, is the value f(x) gets close to when x gets close to a. Limits are used to define continuity and derivatives.

Important formulas: If f is continuous at a, then the limit of f(x) as x approaches a equals f(a). The limit of (x squared minus a squared) divided by (x minus a), as x approaches a, equals 2a. The limit of sin x divided by x, as x approaches 0, equals 1.

Short questions: Q1. Define a function. Ans: A rule that assigns exactly one output to each input in its domain. Q2. What is the domain of f(x) equals 1 divided by (x minus 2)? Ans: All real numbers except 2. Q3. State the limit of (x squared minus 9) divided by (x minus 3) as x approaches 3. Ans: 6.

Long question with steps: Evaluate the limit of (x squared minus 4) divided by (x minus 2) as x approaches 2. Step 1: Direct substitution gives 0 divided by 0, so factorise. Step 2: x squared minus 4 equals (x minus 2)(x plus 2). Step 3: Cancel (x minus 2) for x not equal to 2, giving x plus 2. Step 4: Put x equals 2, giving 4. Answer: 4.

Chapter 2: Differentiation

Summary: The derivative of a function measures its rate of change and the slope of its tangent. Differentiation uses the power rule, product rule, quotient rule and chain rule. Derivatives also find increasing and decreasing intervals, maximum and minimum values, and rates of change.

Important formulas: If y equals x to the power n, then dy divided by dx equals n times x to the power (n minus 1). Product rule: derivative of u times v equals u times dv divided by dx plus v times du divided by dx. Quotient rule: derivative of u divided by v equals (v times du divided by dx minus u times dv divided by dx) divided by v squared. Chain rule: if y is a function of u and u is a function of x, then dy divided by dx equals dy divided by du times du divided by dx.

Short questions: Q1. What does dy divided by dx represent geometrically? Ans: The slope of the tangent to the curve at a point. Q2. Differentiate x cubed. Ans: 3x squared. Q3. When is a function increasing? Ans: When its derivative is greater than 0 on that interval.

Long question with steps: Differentiate y equals (x squared plus 1) cubed. Step 1: Let u equal x squared plus 1, so y equals u cubed. Step 2: dy divided by du equals 3u squared and du divided by dx equals 2x. Step 3: By the chain rule, dy divided by dx equals 3u squared times 2x. Step 4: Replace u, giving 6x times (x squared plus 1) squared. Answer: 6x(x squared plus 1) squared.

Chapter 3: Integration

Summary: Integration is the reverse of differentiation. An indefinite integral gives a family of functions, differing by a constant. A definite integral gives the area under a curve between two limits. Integration by parts, substitution, and partial fractions are the main methods in this chapter.

Important formulas: Integral of x to the power n dx equals x to the power (n plus 1) divided by (n plus 1), plus C, for n not equal to minus 1. Integral of 1 divided by x dx equals ln of the absolute value of x, plus C. Integration by parts: integral of u dv equals u times v minus integral of v du. Definite integral from a to b of f(x) dx equals F(b) minus F(a), where F is an antiderivative of f.

Short questions: Q1. What is the integral of 3x squared? Ans: x cubed plus C. Q2. Why is C added to an indefinite integral? Ans: Because the derivative of a constant is 0, so many functions share the same derivative. Q3. What does a definite integral represent? Ans: The signed area under the curve between the limits.

Long question with steps: Evaluate the integral of (2x plus 1) squared dx. Step 1: Expand to get 4x squared plus 4x plus 1. Step 2: Integrate term by term: integral of 4x squared dx equals (4x cubed) divided by 3; integral of 4x dx equals 2x squared; integral of 1 dx equals x. Step 3: Add C. Answer: (4x cubed divided by 3) plus 2x squared plus x plus C.

Chapter 4: Introduction to Analytic Geometry

Summary: Analytic geometry joins algebra and geometry by using coordinates. This chapter covers distance between points, section formula, slope of a line, equations of lines, and the angle between lines. A straight line is studied through its slope and intercepts.

Important formulas: Distance between (x1, y1) and (x2, y2) equals the square root of ((x2 minus x1) squared plus (y2 minus y1) squared). Slope m equals (y2 minus y1) divided by (x2 minus x1). Point-slope form: y minus y1 equals m times (x minus x1). Slope-intercept form: y equals mx plus c. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is minus 1.

Short questions: Q1. Find the slope of the line through (1, 2) and (3, 6). Ans: 2. Q2. What is the slope of a horizontal line? Ans: 0. Q3. When are two lines perpendicular? Ans: When the product of their slopes is minus 1.

Long question with steps: Find the equation of the line through (2, 3) with slope 4. Step 1: Use point-slope form y minus y1 equals m times (x minus x1). Step 2: Substitute y1 equals 3, x1 equals 2, m equals 4, giving y minus 3 equals 4(x minus 2). Step 3: Simplify to y minus 3 equals 4x minus 8. Step 4: Rearrange to y equals 4x minus 5. Answer: y equals 4x minus 5.

Chapter 5: Linear Inequalities and Linear Programming

Summary: A linear inequality in two variables divides the plane into two half-planes. Its graph is a line with one side shaded. Linear programming finds the maximum or minimum value of an objective function, subject to constraints, by checking the corner points of the feasible region.

Important formulas: The feasible region is the set of points satisfying all constraints, including x greater than or equal to 0 and y greater than or equal to 0 where required. For a linear objective function, the optimum value, if it exists, occurs at a corner point of the feasible region. To test a half-plane, put the test point (0, 0) into the inequality, unless the line passes through it.

Short questions: Q1. What is a feasible region? Ans: The region of points that satisfy all the given constraints. Q2. Where does a linear objective function attain its optimum? Ans: At a corner point of the feasible region. Q3. Which side of x plus y less than or equal to 4 contains (0, 0)? Ans: The side containing (0, 0), because 0 plus 0 is less than 4.

Long question with steps: Maximise Z equals 3x plus 2y subject to x plus y less than or equal to 4, x greater than or equal to 0, y greater than or equal to 0. Step 1: Draw the feasible region with corner points (0, 0), (4, 0) and (0, 4). Step 2: Evaluate Z at (0, 0) to get 0; at (4, 0) to get 12; at (0, 4) to get 8. Step 3: Compare the values. Answer: Maximum Z equals 12 at (4, 0).

Chapter 6: Conic Sections

Summary: Conic sections are curves obtained by cutting a cone: the circle, parabola, ellipse and hyperbola. Each has a standard equation, a centre or vertex, a focus or foci, and an eccentricity that describes its shape. Learn to recognise a conic from its equation before you start solving.

Important formulas: Circle with centre (h, k) and radius r: (x minus h) squared plus (y minus k) squared equals r squared. Parabola with vertex at the origin opening upward: x squared equals 4ay. Ellipse with centre at the origin: x squared divided by a squared plus y squared divided by b squared equals 1, for a greater than b. Hyperbola with centre at the origin: x squared divided by a squared minus y squared divided by b squared equals 1. Eccentricity is 0 for a circle, 1 for a parabola, less than 1 for an ellipse, and greater than 1 for a hyperbola.

Short questions: Q1. What is the eccentricity of a parabola? Ans: 1. Q2. Write the equation of a circle with centre (0, 0) and radius 5. Ans: x squared plus y squared equals 25. Q3. Which conic has eccentricity less than 1? Ans: An ellipse.

Long question with steps: Find the centre and radius of x squared plus y squared minus 6x plus 4y minus 12 equals 0. Step 1: Group terms: (x squared minus 6x) plus (y squared plus 4y) equals 12. Step 2: Complete the squares: (x minus 3) squared minus 9 plus (y plus 2) squared minus 4 equals 12. Step 3: Simplify to (x minus 3) squared plus (y plus 2) squared equals 25. Answer: Centre (3, minus 2), radius 5.

Chapter 7: Vectors

Summary: A vector has magnitude and direction. Vectors can be added, subtracted and multiplied by a scalar. The dot product gives a scalar and tests perpendicularity, while the cross product gives a vector perpendicular to two given vectors. Unit vectors i, j and k point along the x, y and z axes.

Important formulas: If a equals (a1, a2, a3), its magnitude is the square root of (a1 squared plus a2 squared plus a3 squared). Dot product: a dot b equals a1b1 plus a2b2 plus a3b3, which also equals |a| times |b| times cos theta, where theta is the angle between them. Two vectors are perpendicular if and only if their dot product is 0. Cross product magnitude: |a cross b| equals |a| times |b| times sin theta.

Short questions: Q1. What is the magnitude of (3, 4, 0)? Ans: 5. Q2. If a dot b equals 0, what can you say about a and b? Ans: They are perpendicular, provided neither is the zero vector. Q3. Write the unit vector along the x-axis. Ans: i.

Long question with steps: Find the angle between a equals (1, 0, 0) and b equals (1, 1, 0). Step 1: a dot b equals 1 times 1 plus 0 times 1 plus 0 times 0, which equals 1. Step 2: |a| equals 1 and |b| equals the square root of 2. Step 3: cos theta equals (a dot b) divided by (|a| times |b|), which equals 1 divided by the square root of 2. Step 4: theta equals 45 degrees. Answer: 45 degrees.

Full-Book Formula Sheet

Limits: limit of (x squared minus a squared) divided by (x minus a), as x approaches a, equals 2a. Differentiation: derivative of x to the power n equals n times x to the power (n minus 1); product, quotient and chain rules as in Chapter 2. Integration: integral of x to the power n dx equals x to the power (n plus 1) divided by (n plus 1), plus C; integration by parts as in Chapter 3. Geometry: distance formula, slope formula, point-slope form, and the parallel and perpendicular slope conditions as in Chapter 4. Linear programming: check corner points of the feasible region. Conics: standard circle, parabola, ellipse and hyperbola equations as in Chapter 6. Vectors: magnitude, dot product, and the perpendicular condition as in Chapter 7. These BISE Lahore 12th class mathematics full book notes are best revised from this sheet after you have worked one example from each chapter.

Final MCQ Revision

Try these 20 MCQs without looking at the answers. The answer key is given below the questions.

1. The limit of (x squared minus 9) divided by (x minus 3) as x approaches 3 is: (a) 3 (b) 6 (c) 9 (d) 0
2. The derivative of x to the power 4 is: (a) 4x cubed (b) x cubed (c) 4x (d) 12x squared
3. The derivative of a constant is: (a) 1 (b) 0 (c) the constant (d) x
4. The integral of 2x dx is: (a) x squared (b) x squared plus C (c) 2x squared (d) x
5. The integral of 1 divided by x dx is: (a) x (b) ln of the absolute value of x plus C (c) 1 divided by x squared (d) e to the power x
6. The slope of the line through (0, 0) and (2, 4) is: (a) 1 (b) 2 (c) 4 (d) 0
7. A line with slope 0 is: (a) vertical (b) horizontal (c) diagonal (d) undefined
8. Two lines are parallel when their slopes are: (a) opposite (b) equal (c) reciprocal (d) negative
9. The feasible region in linear programming satisfies: (a) one constraint (b) all constraints (c) no constraint (d) only x greater than 0
10. For a linear objective function, the optimum occurs at a: (a) midpoint (b) corner point (c) random point (d) centre
11. The eccentricity of a circle is: (a) 1 (b) 0 (c) 2 (d) less than 1 but not 0
12. The eccentricity of a parabola is: (a) 0 (b) 1 (c) 2 (d) 0.5
13. x squared plus y squared equals 25 is a circle of radius: (a) 25 (b) 5 (c) 10 (d) 2.5
14. The magnitude of vector (3, 4, 0) is: (a) 7 (b) 5 (c) 12 (d) 1
15. If a dot b equals 0, then a and b are: (a) parallel (b) perpendicular (c) equal (d) opposite
16. The unit vector along the y-axis is: (a) i (b) j (c) k (d) 0
17. The derivative of sin x is: (a) minus cos x (b) cos x (c) tan x (d) sec x
18. The integral of cos x dx is: (a) sin x plus C (b) minus sin x plus C (c) cos x plus C (d) tan x plus C
19. For y equals (x plus 1) squared, dy divided by dx at x equals 1 is: (a) 2 (b) 4 (c) 1 (d) 0
20. The centre of (x minus 3) squared plus (y plus 2) squared equals 25 is: (a) (3, 2) (b) (3, minus 2) (c) (minus 3, 2) (d) (0, 0)

Answer Key: 1-b, 2-a, 3-b, 4-b, 5-b, 6-b, 7-b, 8-b, 9-b, 10-b, 11-b, 12-b, 13-b, 14-b, 15-b, 16-b, 17-a, 18-a, 19-b, 20-b.

Revision Checklist

• I can evaluate a limit by factorising and can state the domain of a function.
• I can differentiate using the power, product, quotient and chain rules.
• I can integrate by expansion, substitution and parts, and I remember the constant C.
• I can find a distance, a slope, and the equation of a line in point-slope form.
• I can shade a feasible region and test its corner points for a linear programming problem.
• I can recognise a circle, parabola, ellipse and hyperbola and find a centre and radius.
• I can find a vector magnitude, a dot product, and the angle between two vectors.

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These BISE Lahore 12th class mathematics full book notes are for study and revision only. Keep your textbook beside you, practise the worked steps with a pen, and revise the formula sheet before every test.