ISC Class 12 Mathematics Syllabus

What is the geometric meaning of a determinant of order 2 (or 3)?

Its absolute value gives the area of the parallelogram (order 2) or volume of the parallelepiped (order 3) formed by the row/column vectors.

State the formula relating the inverse of A to its adjoint.

A inverse = (1/det A) times adj(A), valid only when det A is not equal to 0.

What is the condition for a system of linear equations AX = B to have a unique solution?

det A must not equal 0 (A is non-singular); then X = A inverse times B.

When is a function f continuous at a point x = c?

When the left-hand limit equals the right-hand limit equals f(c); i.e. limit as x approaches c of f(x) = f(c).

State the definition of the derivative f'(x) as a limit.

f'(x) = limit as h approaches 0 of [f(x + h) - f(x)] / h.

State the product rule and quotient rule of differentiation.

Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v - uv') / v squared.

State the chain rule for dy/dx when y = f(u) and u = g(x).

dy/dx = (dy/du) times (du/dx).

What are the derivatives of sin x, cos x, tan x, e^x, and ln x?

d/dx sin x = cos x; cos x = -sin x; tan x = sec squared x; e^x = e^x; ln x = 1/x.

State Rolle's Theorem.

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) with f'(c) = 0.

State the Mean Value Theorem (Lagrange's).

If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a).

How do you find local maxima and minima using the first derivative test?

Find critical points where f'(x) = 0. If f' changes + to - it's a local maximum; - to + it's a local minimum.

What does the second derivative test say about a critical point c where f'(c) = 0?

If f''(c) < 0 it is a local maximum; if f''(c) > 0 it is a local minimum; if f''(c) = 0 the test is inconclusive.

What is the integral of x^n dx (n not equal to -1) and of 1/x dx?

Integral of x^n dx = x^(n+1)/(n+1) + C; integral of 1/x dx = ln|x| + C.

State the integration by parts formula.

Integral of u dv = uv minus integral of v du. Choose u by ILATE order (Inverse, Log, Algebraic, Trig, Exponential).

State the Fundamental Theorem of Calculus (Part 2).

If F is an antiderivative of f, then the definite integral from a to b of f(x) dx = F(b) - F(a).

State the property: definite integral from 0 to a of f(x) dx equals what in terms of f(a - x)?

Integral from 0 to a of f(x) dx = integral from 0 to a of f(a - x) dx.

What is the order and degree of a differential equation?

Order is the highest derivative present; degree is the power of the highest-order derivative when the equation is polynomial in derivatives.

How do you solve a variable-separable differential equation?

Write it as f(y) dy = g(x) dx, then integrate both sides separately to get the general solution.

What is the integrating factor and solution form of a linear ODE dy/dx + Py = Q?

Integrating factor IF = e^(integral of P dx). Solution: y times IF = integral of (Q times IF) dx + C.

Define a homogeneous differential equation and its standard substitution.

dy/dx = f(y/x) where f is a function of y/x. Substitute y = vx, so dy/dx = v + x(dv/dx), making it separable.

How is the dot (scalar) product of two vectors a and b defined, and what does it give?

a dot b = |a| |b| cos theta = a1b1 + a2b2 + a3b3; it is a scalar. a dot b = 0 means the vectors are perpendicular.

How is the cross (vector) product a x b defined and directed?

|a x b| = |a| |b| sin theta; it is a vector perpendicular to both a and b, with direction by the right-hand rule. Its magnitude equals the area of the parallelogram on a and b.

What is the formula for the projection of vector a on vector b?

Scalar projection = (a dot b) / |b|.

What is the scalar triple product [a b c] and its geometric meaning?

[a b c] = a dot (b x c); its absolute value is the volume of the parallelepiped. If it equals 0, the three vectors are coplanar.

Give the direction cosines relation for a line and the formula for distance between two points in 3D.

Direction cosines satisfy l squared + m squared + n squared = 1. Distance = square root of [(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2].

What is the equation of a plane in normal form and the shortest distance from a point to it?

Plane: ax + by + cz + d = 0. Distance from point (x1,y1,z1) = |ax1 + by1 + cz1 + d| / square root of (a^2 + b^2 + c^2).

In Linear Programming, define the objective function and constraints.

The objective function is the linear expression (e.g. Z = ax + by) to be maximized or minimized; constraints are the linear inequalities restricting the variables.

What is the feasible region and where do optimal solutions occur in an LPP?

The feasible region is the set of all points satisfying all constraints. The optimal value of the objective function occurs at a corner (vertex) point of the feasible region (Corner Point Theorem).

Name common types of Linear Programming problems.

Manufacturing (production) problems, diet problems, transportation problems, and allocation/optimal-mix problems.

State the formula for conditional probability P(A given B).

P(A | B) = P(A intersect B) / P(B), provided P(B) > 0.

What does it mean for events A and B to be independent, in terms of probability?

A and B are independent if P(A intersect B) = P(A) times P(B); equivalently P(A | B) = P(A).

State the Multiplication Theorem of Probability.

P(A intersect B) = P(A) times P(B | A) = P(B) times P(A | B).

State the Theorem of Total Probability.

If E1, E2, ..., En are mutually exclusive and exhaustive events, then P(A) = sum of P(Ei) times P(A | Ei).

State Bayes' Theorem.

P(Ei | A) = [P(Ei) times P(A | Ei)] / [sum over j of P(Ej) times P(A | Ej)].

What is the difference between prior and posterior probability in Bayes' Theorem?

Prior probability P(Ei) is known before the event A; posterior probability P(Ei | A) is the revised probability after observing A.

What is a probability distribution of a random variable, and what must the probabilities sum to?

It lists each value of the random variable X with its probability P(X). All probabilities are between 0 and 1 and their sum equals 1.

How is the mean (expectation) E(X) of a discrete random variable computed?

E(X) = sum of [x_i times P(x_i)] over all values.

How is the variance of a discrete random variable computed?

Var(X) = E(X squared) - [E(X)] squared = sum of [x_i squared times P(x_i)] minus the mean squared. Standard deviation is its square root.

State the conditions for a Bernoulli trials / binomial experiment.

Fixed number of trials n, each trial has two outcomes (success/failure), trials are independent, and the probability of success p is constant.

State the binomial distribution probability formula P(X = r).

P(X = r) = nCr times p^r times q^(n - r), where q = 1 - p and r = 0, 1, ..., n.

What are the mean and variance of a binomial distribution?

Mean = np; Variance = npq; Standard deviation = square root of npq. Note variance is always less than the mean since q < 1.