12th Maths One Mark Questions and Book Back Answers

About Mathematics

Hard ~360 h study

Class 12 Maths picks up where Class 11 left off and asks you to use those tools on bigger problems. You solve a system of three equations with a matrix, write a complex number in polar form and find its cube roots, and use a derivative to find the largest box you can make from a sheet of tin.

The chapters are linked. Rank of a matrix decides whether a system of equations has a solution. Complex numbers explain why a real polynomial's non-real roots come in conjugate pairs in the theory of equations. Vector products give you the equations of lines and planes in space. Differentiation and integration then take up most of Volume 2, where you find tangents, maxima and minima, areas between curves and volumes of solids of revolution.

In the public exam, knowing a formula is not enough. You have to read a problem, pick the right method and carry every step through without an arithmetic slip. The same habits help later in entrance exams and in college maths, physics and engineering courses.

What you'll learn

• Solve systems of linear equations using matrices and determinants, and decide when a system is consistent.
• Represent complex numbers in the Argand plane and find their powers and roots.
• Write and use the equations of conics, and of lines and planes in three dimensions.
• Use derivatives to find rates of change, tangents, and maximum and minimum values.
• Find areas of regions and volumes of solids of revolution using definite integrals.
• Form and solve first-order differential equations for growth, decay and similar problems.
• Construct truth tables to test whether a compound statement is a tautology or a contradiction.
• Choose a suitable probability distribution for a random experiment and find its mean and variance.

Before you start

• Class 10 and 11 algebra, including quadratic equations and basic matrices.
• Trigonometric ratios, identities and the values at standard angles.
• Sets, functions, coordinate geometry of the straight line, and Class 11 limits, differentiation and integration.

Mathematics — the syllabus in depth

What the 12 Chapters Cover

Algebra: Matrices, Complex Numbers and Equations

Chapter 1 gives four ways to solve linear systems: the inverse matrix method, Cramer's rule, Gaussian elimination and the rank method. Chapter 2 moves to complex numbers: modulus, argument, polar and Euler forms, De Moivre's theorem and finding \(n\)th roots. Chapter 3, theory of equations, covers Vieta's formulae linking roots and coefficients, conjugate pairs of roots, reciprocal equations and Descartes' rule of signs. Chapter 4 deals with inverse trigonometric functions and their principal values.

Geometry: Conics and Vectors

Chapter 5 is two-dimensional analytical geometry: circles, parabolas, ellipses and hyperbolas, with their foci, directrices, tangents and normals. Chapter 6 applies vector algebra. You use the scalar and vector triple products, then write lines and planes in both parametric and non-parametric vector form, and find angles between them, distances from a point and points of intersection.

Calculus: Rates of Change, Areas and Differential Equations

Chapter 7 uses derivatives for tangents and normals, Rolle's and Lagrange's mean value theorems, L'Hôpital's rule, increasing and decreasing functions, and maxima and minima. Chapter 8 introduces differentials, linear approximation and partial derivatives of functions of two or more variables, including Euler's theorem for homogeneous functions. Chapter 9 uses definite integrals to find the area bounded by curves and the volume of a solid of revolution. Chapter 10 forms and solves first-order differential equations and applies them to growth, decay and cooling problems.

Probability and Discrete Mathematics

Chapter 11 defines random variables, probability mass and density functions, cumulative distribution functions, mean and variance, and the Bernoulli and binomial distributions. Chapter 12 has two parts: binary operations (closure, associativity, identity and inverse) and mathematical logic, where you build truth tables for compound statements and identify tautologies and contradictions.

How to study Mathematics

Work through the chapters in order, because later ones borrow from earlier ones. Before each chapter, read the textbook's worked examples with a pen in hand, then try the exercise problems without looking. For calculus, make sure you can explain what a derivative or a definite integral means on a graph before you start the long sums.

A workable order is:

• Revise Class 11 algebra, trigonometric identities, limits and basic differentiation and integration.
• Finish Volume 1, giving extra time to conics and the vector equations of lines and planes.
• Spend the most time on Volume 2 calculus: Chapters 7 to 10.
• End with probability distributions and discrete mathematics, which are shorter but need careful practice with truth tables and distribution formulae.

Time yourself on the MCQs so you get quick at them. Keep a single sheet of formulae per chapter, such as the standard forms of conics and the integrating factor, and read it the night before a test. The summary at the end of each textbook chapter is a good place to check that sheet.

Common mistakes to avoid

• Trying to invert a singular matrix. If \(|A| = 0\), \(A^{-1}\) does not exist, so the inverse method and Cramer's rule cannot be used; switch to the rank method.
• Getting the principal argument of a complex number wrong. Work out the basic angle from \(\tan^{-1}\left|\frac{y}{x}\right|\), then use the signs of \(x\) and \(y\) to place it in the correct quadrant.
• Dropping the arbitrary constant when solving a differential equation. The general solution of a first-order equation must contain one constant, which is fixed later by the given initial condition.
• Mixing up order and degree. The order is the order of the highest derivative; the degree is the power of that highest-order derivative, and it is defined only when the equation is a polynomial in its derivatives.
• Using L'Hôpital's rule on a limit that is not indeterminate. Substitute first and apply the rule only for \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) forms, or after converting another indeterminate form into one of these.
• Assuming every inverse trigonometric function has the same range. \(\sin^{-1}x\) takes values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), but \(\cos^{-1}x\) takes values in \([0, \pi]\), and \(\tan^{-1}x\) in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).

Frequently asked questions

How do I choose between the matrix inversion method and Cramer's rule?

Both need a square coefficient matrix with a non-zero determinant, and both give the same unique solution. The inverse method solves \(AX = B\) as \(X = A^{-1}B\), which needs the adjoint. Cramer's rule gives each unknown as a ratio of determinants, such as \(x = \frac{\Delta_1}{\Delta}\). For two or three unknowns, Cramer's rule usually involves less work. If the determinant is zero, neither method applies.

Why do we study complex numbers?

An equation like \(x^2 + 1 = 0\) has no real solution, but it has the solutions \(\pm i\) in complex numbers. Every polynomial equation of degree \(n\) has \(n\) roots, counting repeated roots, once complex numbers are allowed. The polar form and De Moivre's theorem make powers and roots easy to handle, and engineers use complex numbers for alternating current circuits and waves.

What is the difference between a derivative and a differential?

The derivative \(f'(x)\) is the rate of change of \(f\) at a point, the slope of the tangent there. The differential is \(dy = f'(x)\,dx\), the change in \(y\) along the tangent line when \(x\) changes by \(dx\). For a small \(dx\) it is close to the actual change \(\Delta y\), which is why differentials are used to estimate errors and approximate values such as \(\sqrt{36.1}\).

When should I use Gaussian elimination?

Use it when you want one routine that works on any system. You reduce the augmented matrix to row-echelon form by elementary row operations and then back-substitute. Unlike Cramer's rule, it still works when the coefficient matrix is singular, and the reduced form shows at once whether the system has a unique solution, infinitely many or none.

How do I check whether a system of linear equations is consistent?

Compare the rank of the coefficient matrix \(A\) with the rank of the augmented matrix \([A|B]\). By the Rouché–Capelli theorem, the system is consistent only if the two ranks are equal. If they are equal to the number of unknowns, the solution is unique; if they are equal but less than the number of unknowns, there are infinitely many solutions. If the ranks differ, the system has no solution.

Where are conic sections used in real life?

A parabolic reflector sends light from a bulb at its focus out in a parallel beam, which is why car headlights and satellite dishes are parabolic. Planets move in elliptical orbits with the Sun at one focus. Hyperbolas are used in some radio navigation systems, and cooling towers of power plants have a hyperbolic cross-section.

What does Lagrange's Mean Value Theorem say?

If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), there is at least one \(c\) in \((a, b)\) with \(f'(c) = \frac{f(b) - f(a)}{b - a}\). In words, somewhere between \(a\) and \(b\) the tangent is parallel to the chord joining the end points. For a car, it means that at some moment the speedometer reading equals the average speed for the trip.

How is discrete mathematics connected to computer science?

Discrete mathematics studies objects that can be counted separately, not quantities that vary continuously. The logic in Chapter 12, with connectives like AND, OR and NOT and their truth tables, is the same logic used in programming conditions and in the gates of digital circuits. Binary operations such as addition modulo \(n\) also appear in computing.

What is the difference between discrete and continuous random variables?

A discrete random variable takes a finite or countable set of values, for example the number of heads when three coins are tossed. A continuous random variable can take any value in an interval, for example the lifetime of a bulb in hours. A discrete variable is described by a probability mass function; a continuous variable is described by a probability density function.

How do I solve a first-order linear differential equation?

First write it in the standard form \(\frac{dy}{dx} + Py = Q\), where \(P\) and \(Q\) are functions of \(x\). The integrating factor is \(e^{\int P\,dx}\). The solution is \(y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C\). Students often forget to divide by the coefficient of \(\frac{dy}{dx}\) before reading off \(P\), which gives a wrong integrating factor.