Class 12 Math Textbook 2026-27 (FSc/ICS Part 2) — PDF Download & Study Plan

The new book opens with graphs. Before any calculus, you learn to read a function from its picture: where it cuts the axes, where it turns, and how it moves when the equation changes.

Core topics: graphs of quadratic functions by factorization, algebraic and transcendental functions, exponential, logarithmic and hyperbolic functions, inverse functions, transformations of graphs (shifts, stretching and compression), exponential and logarithmic equations and inequalities, growth and decay problems.

Worked example: Sketch the graph of y = 3(x + 1)(x − 1).

x-intercepts: x + 1 = 0 and x − 1 = 0 give (−1, 0) and (1, 0)
Vertex: x = (−1 + 1)/2 = 0 and y = 3(1)(−1) = −3

So the vertex is (0, −3), and since a = 3 is positive the parabola opens upward.

Where students lose marks: shifting a graph the wrong way. f(x − 2) moves the graph 2 units to the right, not to the left.

Differentiation measures rate of change, and it’s the unit that decides how comfortable the rest of your year feels — integration, optimization problems, and even parts of vectors lean on differentiation fluency.

Core topics: derivatives of trigonometric, exponential and logarithmic functions, chain rule, parametric and implicit differentiation, tangent and normal, higher order derivatives, increasing and decreasing functions, maxima and minima, linear approximation and differentials.

Worked example: Differentiate: y = x² sin x

Using the product rule: d/dx(uv) = u’v + uv’ Here u = x², v = sin x → u’ = 2x, v’ = cos x

dy/dx = 2x·sin x + x²·cos x

Where students lose marks: applying the chain rule inside a product or quotient and forgetting one of the two derivatives — always write out u, v, u’, v’ separately before combining, rather than doing it in your head.

Integration reverses differentiation, and the biggest shift here is that there’s no single formula to memorize — you have to recognize which of several methods a given integral needs.

Core topics: indefinite integrals, integration by substitution, by parts and by partial fractions, definite integrals and their properties, area of plane regions, consumer’s and producer’s surplus, volumes of solids of revolution, moment of inertia.

Worked example: Evaluate: ∫ 2x cos(x²) dx

Let u = x², so du = 2x dx — which is exactly what’s sitting in the integral.

∫ cos(u) du = sin(u) + C = sin(x²) + C

Where students lose marks: trying integration by parts on something that’s really a substitution problem (or vice versa). Quick test: if you can spot a function and its derivative sitting together in the integral, try substitution first.

A differential equation contains a derivative of an unknown function. It is a full unit in the new book, and it is where integration starts doing real work: growth, decay, cooling and electric circuits.

Core topics: order and degree of a differential equation, forming differential equations, separable equations, homogeneous equations, initial value problems, applications to growth and decay, cooling, RL and RC circuits.

Worked example: Solve dy/dx = e^{x − y}.

Separate the variables: e^y dy = e^x dx
Integrate both sides: e^y = e^x + C

So y = ln(e^x + C).

Where students lose marks: forgetting the constant of integration, or adding it after taking the logarithm instead of before.

This is where algebra and geometry merge — every line and circle becomes an equation you can manipulate.

Core topics: altitudes, medians and right bisectors of a triangle, forms of the equation of a line, family of lines, concurrent lines, angle between two lines, area of a triangle, pair of straight lines through the origin.

Worked example: Find the equation of the line through (2, 3) with slope 4.

Point-slope form: y − y₁ = m(x − x₁) y − 3 = 4(x − 2) y − 3
= 4x − 8 4x − y − 5 = 0

Where students lose marks: mixing up point-slope and slope-intercept forms under exam pressure — pick one form, memorize it cold, and derive the other from it if needed.

The circle, parabola, ellipse, and hyperbola — all four are really the same idea (a plane slicing a cone) written as four different-looking equations.

Core topics: equation of a circle, tangent and normal to a circle, parabola, ellipse, hyperbola, and applications of conics.

Worked example: Identify: x² + y² − 4x + 6y − 12 = 0

Complete the square: (x² − 4x + 4) + (y² + 6y + 9) =
12 + 4 + 9 (x − 2)² + (y + 3)² = 25

This is a circle, center (2, −3), radius 5.

Where students lose marks: arithmetic slips while completing the square — always double-check by expanding your answer back out.

Kinematics is the mathematics of motion: position, velocity, acceleration and time. It overlaps with Physics, so the same practice helps in both subjects.

Core topics: displacement–time and velocity–time graphs, equations of uniformly accelerated motion, vertical motion under gravity, velocity and acceleration using differentiation and integration.

Worked example: A car moving at 36 km/h is stopped by its brakes in 10 seconds. Find the retardation and the distance covered.

36 km/h = 10 m/s, so u = 10, v = 0 and t = 10
a = (v − u)/t = (0 − 10)/10 = −1 m/s²
s = ut + ½at² = 100 − 50 = 50 m

The retardation is 1 m/s² and the car covers 50 m.

Where students lose marks: mixing units. Convert km/h to m/s before using any formula.

Some equations cannot be solved exactly. This unit teaches step-by-step methods that get as close to the answer as you need.

Core topics: graphical method, bisection method, Regula-Falsi method, Newton-Raphson method, the trapezoidal rule for definite integrals, and applied problems.

Worked example: Apply two steps of the bisection method to x² − 2 = 0 on [1, 2].

f(1) = −1 and f(2) = 2, so a root lies between 1 and 2
Midpoint 1.5: f(1.5) = 0.25, which is positive, so the root lies in [1, 1.5]
Midpoint 1.25: f(1.25) = −0.4375, which is negative, so the root lies in [1.25, 1.5]

Each step halves the interval. The exact root is √2 ≈ 1.414.

Where students lose marks: rounding too early. Keep extra decimal places during the steps and round only the final answer.

An inverse trigonometric function answers the question: which angle has this ratio? The catch is that each one gives its answer only from a fixed range.

Core topics: domains and ranges, principal values, graphs of inverse trigonometric functions, sum and difference formulas, and proofs of identities.

Worked example: Find the principal value of sin⁻¹(−1/2).

sin(π/6) = 1/2, so sin(−π/6) = −1/2
The range of sin⁻¹ is [−π/2, π/2], and −π/6 lies in it

So sin⁻¹(−1/2) = −π/6.

Where students lose marks: giving an angle outside the principal range, such as 7π/6 in this example.

A trigonometric equation usually has infinitely many solutions, because the functions repeat. This unit shows how to write all of them.

Core topics: general solutions of equations in sine, cosine and tangent, equations reducible to quadratic form, use of identities and double angles, graphical solutions, and applied problems.

Worked example: Solve sin x = 1/√2.

The reference angle is π/4, and sine is positive in the first and second quadrants
In [0, 2π]: x = π/4 or x = 3π/4

General solution: x = π/4 + 2nπ or x = 3π/4 + 2nπ, where n is an integer.

Where students lose marks: dividing both sides by sin x or cos x, which loses the solutions where that factor is zero. Factorize instead.

The last unit joins vectors with calculus. A vector valued function gives a vector for each value of t, which is how the path of a moving point is described.

Core topics: vector valued functions, differentiating them component by component, rules of differentiation for vector functions, velocity and acceleration of a moving point.

Worked example: Differentiate r(t) = t² i + sin t j.

Differentiate each component: d/dt(t²) = 2t and d/dt(sin t) = cos t

So r′(t) = 2t i + cos t j.

Where students lose marks: differentiating only one component, or dropping the unit vectors i and j from the final answer.