Class 12 Maths questions with answers

195 NCERT Class 12 Maths questions with answers, chapter by chapter, from Relations and Functions to Probability.

How to teach Class 12 Maths · Class 12 Maths common misconceptions

• Chapter 1: Relations and Functions

  15 questions, 1 with pictures · A relation is a set of ordered pairs · Counting relations and extending one to a required type · Reflexive relations · Symmetric relations · Transitive relations · Equivalence classes and the partition they create · Equivalence relations: all three properties together · One-one (injective) functions · Onto (surjective) functions and choosing the codomain · Bijections, and what changes on a finite set · Composition of functions and invertibility

• Chapter 2: Inverse Trigonometric Functions

  15 questions, 2 with pictures · Evaluating a sum or difference of several principal values · Evaluating a trigonometric function applied to an inverse trigonometric value · Reading and identifying the graph of an inverse trigonometric function · Simplifying $\sin^{-1}(\sin x)$ and its relatives when $x$ is outside the branch · The domain and principal value branch of each inverse trigonometric function · Finding the principal value of a single inverse trigonometric ratio · Why a trigonometric function needs a restricted domain before it has an inverse · Finding the domain of an expression built on an inverse trigonometric function · Finding the range of an inverse function that has been scaled or shifted · Writing an inverse trigonometric expression in its simplest form by substitution · Solving an equation or inequality that contains inverse trigonometric functions

• Chapter 3: Matrices

  15 questions, 2 with pictures · Computing a matrix product: row against column · Constructing a matrix from a rule $a_{ij} = f(i, j)$ · Solving a matrix equation for an unknown matrix · What a matrix is: its order, its element count and the entry $a_{ij}$ · Adding, subtracting and scaling matrices · Equality of matrices, and the system of equations it produces · Types of matrices: row, column, square, diagonal, scalar, identity and zero · Powers of a matrix and equations such as $A^2 - kA + mI = O$ · Properties of matrix multiplication: what carries over from numbers and what does not · Transpose of a matrix and its properties · When a matrix product is defined, and the order of the result · Building symmetric and skew symmetric matrices, and splitting a square matrix into its two parts · Symmetric and skew symmetric matrices · Invertible matrices

• Chapter 4: Determinants

  15 questions · Collinear points and the line through two points, by determinant · Reading a zero determinant off the shape of the array · Rules the inverse obeys: products, scalars, sums and determinants · Singular or not: when $A^{-1}$ exists and how the adjoint gives it · What a determinant is, and the rule for a $2 \times 2$ · Expanding a $3 \times 3$ determinant along a row or a column · What scaling a matrix and multiplying matrices do to the determinant · Finding an unknown when a determinant is given a value · The area of a triangle as a determinant · A row against its own cofactors, and against another row's · Minors and cofactors of the elements of a determinant · The adjoint of a matrix and the identity $A(\text{adj }A) = |A|I$ · Getting $A^{-1}$ out of an equation that $A$ satisfies · Deciding whether a system is consistent · Solving a system of equations by the matrix method

• Chapter 5: Continuity and Differentiability

  15 questions · What it means for a function to be differentiable at a point · Building continuous functions: sums, quotients and composites · Choosing the constant that makes a function continuous · Differentiable implies continuous, and why the converse fails · What it means for a function to be continuous at a point · Where a function is discontinuous, and where the question does not arise · Differentiating a curve given through a parameter · The chain rule for a composite function · Derivatives of the inverse trigonometric functions · Differentiating a relation that is not solved for $y$ · Testing a function where its formula changes · The exponential and the logarithm: domains, graphs and laws · Logarithmic differentiation for a variable base and a variable exponent · The second order derivative and relations it satisfies · Differentiating $e^x$, $\log x$ and what is built from them

• Chapter 6: Application of Derivatives

  15 questions, 1 with pictures · Optimisation of a geometric quantity: boxes, rectangles, cylinders, cones and distances · The derivative as the rate of change of one quantity with respect to another · Two quantities changing with time: related rates through the Chain Rule · Two rates linked through a third: eliminating the intermediate rate · Finding the intervals on which a function increases or decreases · Functions with a gap in the domain: increasing on each piece, not across the gap · Increasing and decreasing on a bounded interval: the trigonometric cases · Finding the values of a parameter for which a function is increasing or decreasing · Increasing and decreasing functions, and the sign of the derivative · Local maxima and minima, and the critical points where they can occur · The first derivative test: reading the sign change of $f'$ through a critical point · The second derivative test, and what to do when it fails · Absolute maximum and minimum values on a closed interval · Maximum and minimum values of a function, and when they do not exist · Optimisation: two numbers with a fixed sum, and cost, revenue and profit

• Chapter 7: Integrals

  15 questions · Integration as the inverse of differentiation: the standard table, the constant of integration, and the algebra done before integrating · Integration by parts: choosing the first function, the second-function integral without a constant, and the loop that returns the original integral · Properties $P_5$-$P_7$: even and odd functions on $[-a, a]$, and $\int_0^{2a}$ in terms of $f(2a - x)$ · Substituting the inner function: linear arguments, composite arguments, and the change of variable $dx = g'(t)\,dt$ · Substitution when the derivative of the inside function is present: the $\int \frac{f'(x)}{f(x)}dx$ and $\int [f(x)]^n f'(x)\,dx$ forms · The form $\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C$: splitting the bracket into a function and its derivative · Properties $P_0$-$P_2$ and integrands with a modulus: split the interval where the expression changes sign · Substitution in a definite integral: change the limits with the variable and never return to $x$ · Substitutions that clear a radical or a fractional power: $x + 2 = t^2$, $x = t^6$, $t = \frac{1}{x}$, and rationalising · Properties $P_3$ and $P_4$: replacing $x$ by $a + b - x$ (or $a - x$), and adding the two forms of $I$ · Reducing even powers, cubes and products of sines and cosines with the double-angle, triple-angle and product-to-sum identities · A linear numerator over a quadratic or its root: $px + q = A\frac{d}{dx}(ax^2 + bx + c) + B$ · An identity that turns a trigonometric quotient or root into a standard form: half angles, $1 \pm \sin 2x$ as a perfect square, $\sec^2 - 1$, and $\sec^2 + \operatorname{cosec}^2$ splits · Completing the square: $\int \frac{dx}{ax^2 + bx + c}$ and $\int \frac{dx}{\sqrt{ax^2 + bx + c}}$ reduced to the standard forms · The six standard integrals of Section 7.4 and their use with a scaled or substituted variable

• Chapter 8: Application of Integrals

  15 questions · Area of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ by integration: $\frac{4b}{a}\int_0^a\sqrt{a^2 - x^2}\,dx = \pi ab$ · Area under a curve by vertical strips: $A = \int_a^b y\,dx$ between two ordinates · Area with horizontal strips: $A = \int_c^d x\,dy$ when the region is bounded by the $y$-axis and $y = c$, $y = d$ · A region bounded by a line and a curve, or described by inequalities: intersection points, then upper boundary minus lower on each piece · Regions below the $x$-axis: the integral is negative there, so the area is $|A_1| + A_2$ split at the crossing · Area of a circle and of its parts by integration: $4\int_0^a\sqrt{a^2 - x^2}\,dx = \pi a^2$ · Using the symmetry of a region: integrate one congruent piece and multiply by their number · Area under a modulus graph $y = |x + a|$ or $y = c + |x + a|$: write the V piecewise and cut at its vertex

• Chapter 9: Differential Equations

  15 questions · Degree of a differential equation: defined only when it is a polynomial in the derivatives, and then the power of the highest-order derivative · Order of a differential equation: the order of the highest-order derivative present, whatever surrounds it · Particular solutions of separable equations: fix the constant from the given condition after integrating, and translate a slope condition into the equation of a curve · Solutions of a differential equation: a function that satisfies it; the general solution carries as many arbitrary constants as the order, a particular solution none · Variables separable: $\frac{dy}{dx} = g(x)\,h(y)$ becomes $\int\frac{dy}{h(y)} = \int g(x)\,dx$, and the general solution is $H(y) = G(x) + C$ · A rate proportional to the quantity: $\frac{dP}{dt} = kP$ and its relatives -- continuous interest, bacteria, cooling, a changing volume · Homogeneous functions of degree $n$ -- $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ -- and the homogeneous differential equation $\frac{dy}{dx} = g\left(\frac{y}{x}\right)$ · Solving a homogeneous equation: $y = vx$ gives $\frac{dy}{dx} = v + x\frac{dv}{dx}$, then $\frac{dv}{g(v) - v} = \frac{dx}{x}$, integrate, and put back $v = \frac{y}{x}$ · First order linear equations $\frac{dy}{dx} + Py = Q$ and $\frac{dx}{dy} + P_1x = Q_1$: reading $P$ from the standard form and the integrating factor $e^{\int P\,dx}$ · Solving a linear equation: $y\,(\text{I.F.}) = \int Q\,(\text{I.F.})\,dx + C$, then divide through by the integrating factor, constant included

• Chapter 10: Vector Algebra

  15 questions · Multiplying a vector by a scalar: the magnitude of lambda a, collinear vectors and collinear points · Component form, the magnitude of a vector and the vector joining two points · The vector product in components (the determinant) and a unit vector perpendicular to two given vectors · Unit vectors, direction cosines and ratios, and a vector of a given magnitude along a direction · The section formula: internal and external division, midpoints and points on a produced segment · Magnitudes and angles from dot-product identities: expanding |a + b|^2, (a + b).(a - b) and three vectors summing to zero · Perpendicular vectors: the zero dot product test and solving for an unknown from a dot-product condition · Scalars and vectors; zero, unit, coinitial, collinear, equal and negative vectors · The scalar (dot) product: definition, component formula, the angle between two vectors and its bounds · The vector (cross) product: definition, right-hand rule, non-commutativity, unit-vector products and the Lagrange identity · Projection of a vector on another: the scalar formula, the projection vector and projections along a line or an axis · Areas of triangles and parallelograms from the vector product: sides, diagonals and vertices · The triangle and parallelogram laws: sums of sides, medians, diagonals and resultants

• Chapter 11: Three Dimensional Geometry

  15 questions · Direction angles and direction cosines of a line; $l^2 + m^2 + n^2 = 1$ · Direction ratios; direction cosines from direction ratios and from a Cartesian equation · The angle between two lines; the conditions for perpendicular and for parallel lines · Equation of a line through a point in a given direction: vector, Cartesian and parametric forms · The image (reflection) of a point in a line and the line joining a point to its image · The line through two points: direction ratios, direction cosines, collinear points and points on a segment · Parallel, intersecting and skew lines; finding the point of intersection · The shortest distance between two skew lines and between two parallel lines · A line perpendicular to two given lines: the cross product of their directions · The foot of the perpendicular from a point on a coordinate axis and the distance of the point from the axis · The foot of the perpendicular from a point to a line and the perpendicular distance · Finding a point on a line: the general point, a distance condition, a coordinate condition

• Chapter 12: Linear Programming

  15 questions, 2 with pictures · Linear programming problems and their mathematical formulation · The Corner Point Method on a bounded feasible region · The feasible region: graphing the constraints, corner points and boundedness · Unbounded feasible regions and the open half-plane test · Multiple optimal solutions and conditions on the objective coefficients

• Chapter 13: Probability

  15 questions, 1 with pictures · Bayes' theorem and reversing a conditional probability · Computing a conditional probability inside a listed sample space · Conditional probability and its defining formula · Properties of conditional probability · The multiplication rule for the probability of a joint event · Complements, "at least one", and repeated independent trials · Independent events and the product rule · Independent events are not mutually exclusive events · A partition of a sample space · The theorem of total probability · A random variable and the values it takes