MSBSHSE Solutions for Class 12 Mathematics and Statistics (Mathematics Statistics-1 (Commerce))

Mathematics & Statistics-1 (Commerce): Paper Overview and How These Solutions Are Organised

The paper you are preparing for is written on your hall ticket as Mathematics & Statistics-1 (Commerce). It is a Class 12 paper of the state board, and it is not the same as the Mathematics & Statistics papers taken by Science students. The Science stream has two separate papers (Mathematics & Statistics-1 and -2) built around vectors, three-dimensional geometry, trigonometric functions and probability distributions. Your Commerce paper replaces those with topics that support accountancy, economics and business studies: logic, matrices, calculus and its commercial applications.

All solutions on this site follow the current Balbharati textbook for this paper [verify edition year]. If your school has issued a newer reprint, compare the exercise numbering once before you rely on a page, because exercise splits sometimes shift between editions.

The textbook is organised into eight chapters. Each chapter opens with theory and solved examples, then gives you a set of numbered exercises (for example, Exercise 1.1, 1.2 and so on), and closes with a Miscellaneous Exercise that mixes questions from the whole chapter. The miscellaneous exercise is where the board-style, multi-step questions usually appear, so treat it as part of the chapter, not as optional practice.

Every solution here is written step by step, not as an answer key. Where a question needs a truth table, you will see the full table with intermediate columns. Where a sum needs the chain rule, each layer of differentiation is shown. This matters in a paper where method marks are awarded: a correct final answer with no working can lose marks, and a wrong answer with correct steps can still earn part of them.

Start with the syllabus page for HSC Commerce Mathematics & Statistics-1 to confirm which exercises are in scope, then work through the chapter pages in order.

Chapter-Wise Solutions Index for Mathematics & Statistics-1 (Commerce)

The table below lists all eight chapters of the Commerce paper in textbook order, with the number of exercises in each chapter and a direct link to that chapter's solution page. Exercise counts follow the current Balbharati textbook [verify exercise numbering against your edition, since splits sometimes shift between reprints].

Each chapter page is arranged exercise by exercise, in the same order as the textbook. Inside an exercise, questions appear with their full numbers, followed by the complete working: definitions you need, the formula used, every intermediate line, and the final answer. For a truth table question, that means all the columns. For an integration question, it means the substitution or the by-parts split shown explicitly, not just the result.

Two things to keep in mind while using the table. First, the exercise counts are a planning aid: a chapter with six exercises (Integration, Matrices) needs more days than one with two (Applications of Definite Integration), so use the numbers to divide your revision time. Second, open the chapter page only after you have attempted the sum yourself in your notebook. The links are meant to check your method, not to replace the attempt.

If you want the raw textbook before starting, download the Class 12 Commerce Mathematics & Statistics textbook PDF and keep it open beside the solution page.

Chapter 1: Mathematical Logic

Key Concepts and Exercise List

Mathematical Logic is the language the rest of the paper is written in. A statement is a sentence that is either true or false, but not both: "7 is a prime number" is a statement, "How old are you?" is not. From statements you build compound statements using logical connectives: and (∧), or (∨), not (¬), if...then (→), and if and only if (↔). Each connective gets a fixed meaning, and that meaning is shown in a truth table, which lists the truth value of a compound statement for every possible combination of its components.

Three terms follow from truth tables. A tautology is true in every row (for example, p ∨ ¬p). A contradiction is false in every row (p ∧ ¬p). A contingency is sometimes true and sometimes false. Two statements are logically equivalent if their truth tables match column for column. The chapter also covers duality (swap ∧ with ∨ and T with F), negation (including De Morgan's laws), and the related statements: converse (q → p), inverse (¬p → ¬q), and contrapositive (¬q → ¬p).

The chapter is assessed through Ex 1.1 to Ex 1.5 and Miscellaneous Exercise 1. Exercise 1.1 deals with statements and connectives, 1.2 with truth tables, 1.3 with tautology, contradiction and logical equivalence, 1.4 with duality and negation, and 1.5 with converse, inverse and contrapositive [verify exercise split against your edition]. Full worked answers for every question are on the Mathematical Logic solutions page.

Method Tip: Testing Logical Equivalence

Do not guess equivalence by reading the symbols. Build the table. If a compound statement has n distinct variables, your table needs 2ⁿ rows, so two variables give 4 rows and three variables give 8 rows. Write one column per variable, then one column per bracketed part, working from the innermost bracket outward. Finish with a column for the whole statement on each side. If the two final columns are identical, the statements are equivalent.

Typical exam question: "Show that ¬(p ∧ q) ≡ ¬p ∨ ¬q." Make five columns: p, q, p ∧ q, ¬(p ∧ q), and ¬p ∨ ¬q. Fill all four rows and point out that the last two columns match. Write the concluding line explicitly, because the marks are usually for the justification, not the table alone.

Two mistakes cost marks again and again. First, students label the converse as the contrapositive. Remember: the contrapositive flips and negates both parts (p → q becomes ¬q → ¬p) and is always equivalent to the original; the converse only swaps them and need not be. Second, a single wrong entry in an early column, such as writing F for p ∧ q when both are T, carries through and destroys the whole answer. Fill one row completely before moving to the next, and recheck the row where all variables are true.

Chapter 2: Matrices

A matrix is a rectangular arrangement of numbers in rows and columns. A matrix with m rows and n columns has order m × n. The chapter begins with types of matrices: row matrix, column matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, zero matrix, and symmetric and skew-symmetric matrices. You then learn operations: addition and subtraction (only for matrices of the same order), scalar multiplication, and matrix multiplication, where the number of columns of the first matrix must equal the number of rows of the second. Multiplication is not commutative, so AB ≠ BA in general.

After operations come transpose, determinant, adjoint and inverse. For a square matrix A, the determinant |A| is a single number. The adjoint, adj A, is the transpose of the cofactor matrix. The inverse exists only when |A| ≠ 0, and is given by A⁻¹ = (1/|A|) adj A. The chapter closes with solving a system of linear equations by the matrix method: write the system as AX = B, then X = A⁻¹B, provided A is invertible.

Key Concepts and Exercise List

The exercises are Ex 2.1 to Ex 2.6 and Miscellaneous Exercise 2. Ex 2.1 covers types of matrices and basic operations, 2.2 matrix multiplication, 2.3 transpose and determinant, 2.4 adjoint and inverse, 2.5 solving equations by the matrix method, and 2.6 mixed application problems [verify exercise split against your edition]. Step-by-step answers for every question are on the Matrices solutions page.

Method Tip: Finding the Inverse of a Matrix

Follow the same four steps every time, and show each one, because marks are given for the method:

1. Write A and find |A|. If |A| = 0, stop and state that A⁻¹ does not exist.
2. Find the cofactor of every element. The cofactor of aᵢⱼ is (−1)^(i+j) times the minor, so the sign pattern for a 3 × 3 matrix is + − + / − + − / + − +.
3. Write the cofactor matrix, then transpose it to get adj A.
4. Divide adj A by |A|: A⁻¹ = (1/|A|) adj A. Check with A A⁻¹ = I.

Two mistakes are common. First, sign errors in cofactors: a missed minus sign in the (−1)^(i+j) factor spreads into the adjoint and the final inverse. Write the sign pattern on your rough sheet before you start. Second, students compute the adjoint without checking invertibility. If |A| = 0, the inverse simply does not exist, and saying so earns the mark. A typical exam question gives a 3 × 3 matrix and asks for A⁻¹, or gives two equations and asks you to solve them by the matrix method; in both cases the cofactor step is where most marks are lost. For repeated slip-ups of this kind, see the common mistakes and revision tips page.

Chapter 3: Differentiation

Differentiation gives you the slope of a curve at any point, or the rate at which one quantity changes when another changes. In commerce, that usually means marginal cost, marginal revenue, or the rate at which profit responds to output. The chapter builds the machinery first and applies it later in Chapter 4.

It starts with the derivative from first principle: f′(x) = limh→0 [f(x + h) − f(x)] / h. You will use this definition for a few standard functions and then stop using it, because standard derivatives take over. You must know these by heart: xⁿ, eˣ, aˣ, log x, sin x, cos x, tan x, and the inverse trigonometric forms. After that come the four rules: sum and difference, product (uv)′ = u′v + uv′, quotient (u/v)′ = (u′v − uv′)/v², and chain rule. The chapter then extends differentiation to composite functions, inverse functions, logarithmic differentiation (taking log of both sides before differentiating, useful when the function is a product or power of several terms), and implicit functions, where y is not written separately as f(x) and you differentiate both sides with respect to x, adding dy/dx wherever y is differentiated.

Key Concepts and Exercise List

The exercises are Ex 3.1 to Ex 3.5 and Miscellaneous Exercise 3. Ex 3.1 covers first principle and standard derivatives, 3.2 the rules of differentiation, 3.3 composite and inverse functions, 3.4 logarithmic differentiation, and 3.5 implicit functions [verify exercise split against your edition]. Full step-by-step answers are on the Differentiation solutions page.

Method Tip: Chain Rule in Practice

For a composite function, name the outer layer and the inner layer before you differentiate anything. Take y = (3x² + 5)⁴. The outer layer is (something)⁴ and the inner layer is 3x² + 5. Differentiate the outer layer keeping the inner intact: 4(3x² + 5)³. Then multiply by the derivative of the inner layer: 6x. The answer is 24x(3x² + 5)³. Write the two layers on your rough sheet first; most chain rule errors are layer errors, not calculation errors.

Two mistakes cost marks repeatedly. First, missing negative signs: the derivative of cos x is −sin x, of cot x is −cosec²x, and of cos⁻¹x is −1/√(1 − x²). Students drop the minus and lose the whole answer. Second, incorrect product rule: in y = x² sin x, students differentiate only the first factor and write 2x sin x, forgetting the second term x² cos x. The rule needs both terms, u′v and uv′. In a 3 or 4 mark question, the examiner checks whether both terms appear, so a half-applied rule loses method marks even if the arithmetic is right. For repeated slip-ups of this kind, see the common mistakes and revision tips page.

Chapter 4: Applications of Derivatives

Chapter 3 taught you how to find a derivative. Chapter 4 asks what the derivative tells you. The sign of f′(x) decides whether a function is rising or falling, and the points where f′(x) = 0 are the candidates for a maximum or minimum. That single idea carries almost every question in this chapter.

A function is increasing on an interval if f′(x) > 0 there and decreasing if f′(x) < 0. A critical point is a value of x where f′(x) = 0 or where f′(x) does not exist. At a critical point the curve flattens, so it may turn from rising to falling (a maximum), from falling to rising (a minimum), or it may only pause and continue (a point of inflection). The second derivative separates these cases: if f″(x) < 0 at the critical point, the curve is concave down and you have a maximum; if f″(x) > 0, it is concave up and you have a minimum; if f″(x) = 0, the test fails and you must go back to the first derivative test.

The commerce questions in this chapter use cost, revenue and profit functions. If C(x) is total cost, then C′(x) is marginal cost. If R(x) is total revenue, R′(x) is marginal revenue. Profit P(x) = R(x) − C(x), and maximum profit occurs where marginal revenue equals marginal cost, that is, R′(x) = C′(x).

Key Concepts and Exercise List

The exercises are Ex 4.1 to Ex 4.4 and Miscellaneous Exercise 4. Ex 4.1 deals with increasing and decreasing functions, 4.2 with maxima and minima, 4.3 with the second derivative test, and 4.4 with applications to cost, revenue and profit [verify exercise split against your edition]. Step-by-step answers for every exercise are on the Applications of Derivatives solutions page.

Method Tip: Finding Maxima and Minima

Take a typical question: find the maximum profit if P(x) = −2x² + 40x − 150. Step one, differentiate: P′(x) = −4x + 40. Step two, set it to zero: −4x + 40 = 0, so x = 10. Step three, second derivative: P″(x) = −4, which is negative, so x = 10 gives a maximum. Step four, substitute back: P(10) = −200 + 400 − 150 = 50. Write all four steps, because the examiner usually gives marks for the derivative, the critical point and the conclusion separately.

Two errors are common. First, students find the critical point and stop, without confirming whether it is a maximum or minimum and without finding the value of the function there. The question asks for the maximum profit, not the output at which it occurs. Second, on a closed interval question, students ignore the endpoints. A maximum can sit at an endpoint even when f′(x) is not zero there, so always evaluate the function at the endpoints as well and compare. If you keep losing marks on this, the common mistakes and revision tips page lists the slips examiners see most often.

Chapter 5: Integration

Integration is differentiation run backwards. If differentiating F(x) gives f(x), then the integral of f(x) is F(x) plus a constant. That constant matters: every function of the form x² + 5, x² − 3 or x² + 100 has the same derivative, 2x, so the indefinite integral of 2x is written x² + c. The "c" is the constant of integration, and leaving it out loses a mark in almost every question.

This chapter collects the standard results you must know by heart (∫xⁿ dx, ∫(1/x) dx, ∫eˣ dx, ∫aˣ dx, the trigonometric forms) and then three techniques for turning unfamiliar integrals into those standard forms: substitution, integration by parts, and partial fractions.

Key Concepts and Exercise List

The exercises are Ex 5.1 to Ex 5.6 and Miscellaneous Exercise 5. Ex 5.1 covers standard forms, 5.2 substitution, 5.3 integration by parts, 5.4 partial fractions, and the later exercises mix the methods [verify exercise split against your edition]. Each step-by-step answer is on the Integration solutions page.

Method Tip: Choosing the Right Integration Technique

Most marks are lost not in the algebra but in picking the wrong method. Run through these checks in order:

• Does it match a standard form directly? If yes, write the answer and add c. Do not overcomplicate a one-line question.
• Is one part the derivative of another? For example, in ∫2x·cos(x²) dx, the 2x is the derivative of x². Put u = x², and the integral becomes ∫cos u du. That is substitution.
• Is it a product of two unrelated functions? For ∫x·eˣ dx, use by parts with the formula ∫u dv = uv − ∫v du, choosing u by the order inverse, logarithmic, algebraic, trigonometric, exponential (ILATE).
• Is it a rational expression with factorisable denominator? Split it into partial fractions first, then integrate each piece.

Two errors repeat every year. Forgetting "+ c" is the first. The second is a partial fraction slip: for a repeated factor like (x − 1)², you need terms A/(x − 1) and B/(x − 1)², not just one. Write the correct form before solving for A and B. If these slips keep costing you marks, the common mistakes and revision tips page lists them with fixes.

Chapter 6: Definite Integration

A definite integral has limits. Instead of producing a family of functions with "+ c", it produces a single number. The notation is ∫ab f(x) dx, where a is the lower limit and b is the upper limit. The value tells you the signed area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b.

The chapter builds the idea in two ways. First comes the definite integral as the limit of a sum: you split the interval [a, b] into n strips of equal width, add up the areas of the rectangles, and take the limit as n tends to infinity. This is the Riemann sum definition and it is usually examined as a direct "evaluate as limit of sum" question. Second comes the fundamental theorem of calculus, which is the tool you will actually use most: if F(x) is any antiderivative of f(x), then ∫ab f(x) dx = F(b) − F(a). Notice there is no constant of integration here, because it cancels in the subtraction.

Key Concepts and Exercise List

The exercises are Ex 6.1 to Ex 6.3 and Miscellaneous Exercise 6. Ex 6.1 deals with the limit of sum definition, Ex 6.2 applies the fundamental theorem to evaluate integrals directly, and Ex 6.3 works through the properties of definite integrals [verify exercise split against your edition]. Full worked answers for every exercise are on the Definite Integration solutions page.

The properties you must have at your fingertips include:

• ∫ab f(x) dx = −∫ba f(x) dx (swapping limits changes the sign)
• ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx (splitting at an interior point)
• ∫0a f(x) dx = ∫0a f(a − x) dx
• ∫−aa f(x) dx = 2∫0a f(x) dx when f is even, and 0 when f is odd

Method Tip: Using Properties of Definite Integrals

Many definite integrals look impossible until you apply a property. Take ∫0π/2 sin x / (sin x + cos x) dx. Direct integration gets messy. Apply ∫0a f(x) dx = ∫0a f(a − x) dx with a = π/2. Since sin(π/2 − x) = cos x and cos(π/2 − x) = sin x, the integral becomes ∫0π/2 cos x / (cos x + sin x) dx. Add the two forms: the numerator becomes sin x + cos x, the denominator is the same, so 2I = ∫0π/2 1 dx = π/2, giving I = π/4. No antiderivative was needed at all.

Before starting any definite integral, check whether the integrand is even or odd and whether the limits are symmetric about zero. A question like ∫−22 x³ dx is zero in one line because x³ is odd, yet students often integrate it the long way and make sign errors.

Two mistakes recur. The first is incorrect limits after substitution. If you put u = x² in a definite integral, you must also change the limits: when x = 0, u = 0; when x = 2, u = 4. Writing the u-integral with the old x-limits gives a wrong answer even if the antiderivative is correct. The fix is to write "when x = a, u = ..." under the substitution every single time. The second is ignoring properties and grinding through algebra, which wastes time and invites errors. The common mistakes and revision tips page collects these slips with short fixes.

Chapter 7: Applications of Definite Integration

Chapter 6 gave you a number. Chapter 7 tells you what that number means in a picture. The definite integral is a signed area, and once you accept that, the whole chapter becomes a geometry exercise: draw the region, decide which curve is on top, set the limits, integrate.

Key Concepts and Exercise List

The chapter has three working ideas. First, area under a single curve: the area bounded by y = f(x), the x-axis, and the lines x = a and x = b is ∫ab f(x) dx, provided f(x) does not cross the x-axis inside the interval. If it does cross, the integral counts the piece below the axis as negative, so you split the interval at the point of crossing and add the absolute values.

Second, area between two curves: if y = f(x) is the upper curve and y = g(x) the lower one between x = a and x = b, the area is ∫ab [f(x) − g(x)] dx. The limits a and b are the x-coordinates of the intersection points, found by solving f(x) = g(x).

Third, the commerce applications. Consumer surplus is the area between the demand curve and the price line, and producer surplus is the area between the price line and the supply curve [verify the exact formula and notation used in your textbook]. These questions are usually worth 3 to 4 marks and reward a correctly labelled diagram as much as the integration.

The exercises are Ex 7.1 and Ex 7.2, followed by Miscellaneous Exercise 7. Ex 7.1 covers area under a curve and area between curves; Ex 7.2 covers the commerce applications [verify exercise split against your edition]. Step-by-step answers for every sum are on the Applications of Definite Integration solutions page.

Method Tip: Setting Up Area Under Curves

Do not integrate first. Sketch first. A rough sketch takes thirty seconds and prevents most errors in this chapter.

1. Find the intersection points by solving the two equations together. Mark them on the sketch.
2. Decide which curve is above the other between those points. Test one convenient x-value in the interval if it is not obvious.
3. Write the integrand as upper minus lower, in that order.
4. Use the intersection x-coordinates as limits, then integrate.

For example, the area between y = x² and y = x is found from x² = x, giving x = 0 and x = 1. Between these values the line y = x lies above the parabola, so the area is ∫01 (x − x²) dx = 1/2 − 1/3 = 1/6 square units. Writing it the other way round gives −1/6, which is the signal that your order was wrong.

Two mistakes recur. The first is using wrong intersection points, often because a quadratic was solved hastily or a root was dropped. The fix is to substitute each root back into both equations and confirm the y-values match. The second is forgetting to subtract the lower curve, or subtracting in the wrong order, which produces a negative area. Area is a positive quantity, so a negative answer means recheck the setup, not the arithmetic. The common mistakes and revision tips page lists these slips with short fixes.

Chapter 8: Differential Equation and Applications

An equation that contains derivatives is called a differential equation. In this chapter you learn to describe such an equation by two numbers, build one from a given family of curves, and solve two standard types that appear in the exercises.

Key Concepts and Exercise List

Order is the highest derivative present in the equation. Degree is the power of that highest derivative, after the equation is cleared of radicals and fractions in the derivatives. So dy/dx + 2y = x is first order, first degree, while (d²y/dx²)³ + dy/dx = 0 is second order, third degree. Formation of a differential equation means starting from a family such as y = ax + a² and eliminating the arbitrary constant a by differentiating and substituting back.

Two solution methods carry almost all the marks. In variable separable equations you bring all y-terms to one side and all x-terms to the other, then integrate both sides and add a single constant c. In linear differential equations of the form dy/dx + Py = Q, where P and Q are functions of x only, you multiply through by an integrating factor and integrate.

The exercises run from Ex 8.1 to Ex 8.5, followed by Miscellaneous Exercise 8. Ex 8.1 deals with order, degree and formation; the middle exercises cover variable separable and linear equations; the miscellaneous exercise mixes both types [verify the exact exercise split against your edition]. Full worked answers are on the Differential Equation and Applications solutions page.

Method Tip: Solving First Order Linear Differential Equations

Before choosing a method, rewrite the equation in the standard form dy/dx + Py = Q and check that the coefficient of dy/dx is exactly 1. If it is not, divide throughout first. Then:

• Identify P and Q. Confirm P and Q contain x only, not y.
• Compute the integrating factor, IF = e^(∫P dx). Do the P-integral separately and carefully.
• Multiply the whole equation by the IF. The left side collapses to d/dx (y × IF).
• Integrate both sides with respect to x and solve for y.

A typical exam question gives dy/dx + y/x = x² and asks for the general solution. Here P = 1/x, so ∫P dx = log x and IF = e^(log x) = x. Multiplying by x gives d/dx (xy) = x³, so xy = x⁴/4 + c. Marks are usually given for the correct IF, the collapsed form, and the final integration, so show each line [verify the marks split for this question type].

Two mistakes cost marks repeatedly. The first is misclassifying order and degree, usually by counting the power of a derivative before clearing the fraction, or by treating dy/dx as degree 2 because a d²y/dx² appears elsewhere. Read the equation twice and circle the highest derivative. The second is an integration slip after separating variables, often a missing negative sign when integrating 1/y or a forgotten constant c. Differentiate your answer once as a check: it should return the original equation. The common mistakes and revision tips page collects these slips with short fixes.

How to Use These Solutions Effectively

The fastest way to waste a solution page is to read it like a story. You finish, you nod, you close the tab, and in the exam you still cannot start the sum. The solutions only work if you arrive at them with something already written on your paper.

So the order matters. First attempt the problem from the textbook on your own, even if you get stuck halfway. Give it a real try, say eight to ten minutes for a typical exercise sum. Only then open the solution. If you look first, your brain quietly convinces you that it "knew" the method, and that feeling is not the same as being able to reproduce it.

Next, compare, do not just check. Marking your final answer right or wrong tells you very little. Place your rough work next to the solution and find the exact line where the two paths separate. For example, in a linear differential equation you may have found the integrating factor correctly but multiplied Q by it wrongly. That is a different mistake from choosing the wrong method, and it needs a different fix.

Keep a small error log. One page, three columns: the question, what you did wrong, and the one-line correction. Entries like "forgot the negative sign when integrating 1/y" or "did not divide by the coefficient of dy/dx before finding P" turn into a revision sheet before the exam. Students who repeat a mistake twice usually find it in their log the second time.

Finally, after you understand a solution, close it and solve two similar sums from the same exercise without help. If those go smoothly, move on. If they do not, the method has not settled yet. The common mistakes and revision tips page has the recurring slips collected in one place, and the study plan and exam strategy page shows how to fit this routine into a weekly timetable.

Exam Pattern, Marks Distribution, and Chapter Weightage

The HSC Commerce paper for Mathematics & Statistics-1 is a written exam of 80 marks, with the remaining 20 marks coming from internal assessment [verify]. The written paper usually mixes objective questions (one mark, multiple choice or one-line), short answer questions of two to three marks, and longer questions of four to five marks [verify]. Marks are given for method as well as answer, so writing the formula, the substitution, and then the simplification earns partial credit even when the final value is wrong. A typical question such as "Find the inverse of matrix A by the adjoint method" carries four marks: one for the determinant, one for the adjoint, one for the division step, and one for the final matrix.

Chapter-wise marks are not fixed by the board in advance, and the weightage shifts a little every year [verify]. The table below gives the general pattern seen in recent papers, so treat it as guidance for planning, not as a promise.

Plan your revision in proportion to this table. Give Matrices, Differentiation, Applications of Derivatives and Integration the largest blocks of time, because together they decide a large share of the paper [verify]. Keep Logic, Definite Integration and Differential Equations for shorter but regular slots, since they are scoring once the method is clear. A practical weekly split is four days on the high-weightage chapters and two on the rest, with the seventh day for a timed section from a previous year paper. The study plan and exam strategy page turns this into a full timetable.

Textbook, Syllabus, and Previous Year Papers

Your primary source for this paper is the Balbharati textbook for Mathematics & Statistics-1 (Commerce), published by the Maharashtra State Bureau of Textbook Production and Curriculum Research [verify edition year]. Every solution on this site follows that book's exercise order, so keep the PDF open beside the solutions page. Download it from our Class 12 Commerce Mathematics & Statistics textbook PDF download page and check that the chapter numbers on the PDF match the ones you see in class. If your teacher refers to "Chapter 3, Exercise 3.2, Q7", you should land on the same question in your copy.

The syllabus document tells you what is examinable and what is not. Read it once at the start of the year and once before revision. For example, the syllabus lists the properties of definite integrals you are expected to apply, so you know exactly which ones to memorise rather than guessing. Our HSC Commerce Mathematics & Statistics-1 syllabus page mirrors the official document and flags the sub-topics that appear in almost every paper.

Previous year question papers are where the syllabus becomes real. Solve them under timed conditions: 80 marks in three hours [verify], no notes, no phone. A student who attempts a paper with the textbook open learns the chapter; a student who sits with a clock learns the exam. Start with one paper from three years ago, mark it strictly, and note which chapter lost you the most marks.

Use the two resources together. Read the chapter solutions until the method is clear, then test that method on an old paper question from the same chapter. If your inverse-by-adjoint answer is right but your working is untidy, the paper will show it, because examiners award steps. Keep a separate sheet of questions you got wrong and revisit them a week later. Solutions teach the method; papers prove you can reproduce it without help.

Frequently Asked Questions

How many chapters are there in Class 12 Mathematics & Statistics-1 (Commerce)?

The textbook has eight chapters. They run from Mathematical Logic and Matrices through to topics like calculus and its commercial applications. Each chapter page on this site lists its exercises and Miscellaneous Exercise, so you can plan revision chapter by chapter. Check your Balbharati index once to confirm the order matches your reprint [verify].

Are these solutions based on the latest Balbharati textbook?

Yes, the solutions follow the current Balbharati textbook for this paper [verify edition year]. If your school issued a newer reprint, compare the exercise numbering before relying on a page, because exercise splits sometimes shift between editions. The chapter names and overall structure stay the same, so revision planning is unaffected.

Is Mathematics & Statistics-1 (Commerce) different from the Science Mathematics paper?

Yes, they are separate papers. The Science stream has two papers built around vectors, three-dimensional geometry, trigonometric functions and probability distributions. Your Commerce paper covers logic, matrices, calculus and its commercial applications instead, topics that support accountancy, economics and business studies. Do not use Science notes for this paper, since the exercise sets differ.

How many exercises are there in Chapter 1 Mathematical Logic?

Chapter 1 is split into numbered exercises such as Exercise 1.1, 1.2 and so on, followed by a Miscellaneous Exercise that mixes questions from the whole chapter. The exact count can differ between textbook reprints [verify], so open your Balbharati copy and match the exercise numbers against the chapter page before you start.

Can I download the Class 12 Commerce Mathematics & Statistics textbook PDF?

The Balbharati textbook PDF is available from the state board's official e-textbook portal, and many schools also share it. Download the Commerce Mathematics & Statistics-1 edition, not the Science one, since the content differs. Keep the PDF handy for theory and solved examples, and use the step-by-step solutions here alongside it for practice.

Do the solutions show full steps or only final answers?

Every solution is written step by step, not as an answer key. Truth table questions show the full table with intermediate columns, and differentiation sums show each layer of the chain rule. This matters because method marks are awarded: a correct answer with no working can lose marks, while correct steps with a wrong answer can still earn part of them.