Std. 11
Mathematics

An angle is directed when the sense of rotation is specified. Counter-clockwise rotation is taken as positive and clockwise rotation as negative.

A degree divides one complete revolution into 360 equal parts. A radian is the angle subtended at the centre by an arc whose length equals the radius.

An angle is in standard position when its vertex is at the origin and its initial arm lies along the positive x-axis.

For θ in radians, arc length l = rθ and the area of a sector is A = ½r²θ.

For an angle θ in standard position, sine, cosine and tangent can be understood from the coordinates of a point on the terminal arm.

The signs of sin θ, cos θ and tan θ depend on the quadrant containing the terminal arm.

Values at 0°, 30°, 45°, 60° and 90° form the basic table used repeatedly in trigonometric calculations.

sin θ and cos θ repeat after 2π, while tan θ repeats after π. Their graphs make these properties visible.

A compound angle is formed by adding or subtracting two angles. Formulae such as sin(A+B) and cos(A−B) allow a difficult angle to be broken into familiar ones.

Double-angle and allied-angle formulae connect angles such as 2θ, 90°±θ and 180°±θ with simpler trigonometric ratios.

Products of trigonometric functions can be converted into sums or differences, and sums can sometimes be converted back into products.

Trigonometric relations can be used with the geometry of a triangle to obtain useful results involving its angles and sides.

A matrix is a rectangular arrangement of numbers in rows and columns. It provides a compact way to organise and operate on data.

For [[a,b],[c,d]], the determinant is ad−bc. It is a number associated with a square matrix, not another matrix.

A non-zero determinant indicates that the corresponding square system has a unique solution in many standard applications.

Matrices can be added, subtracted and multiplied when their dimensions permit the operation.

Slope measures the rate at which y changes when x changes: m = (y₂−y₁)/(x₂−x₁), when x₁ ≠ x₂.

The same straight line can be written in slope-intercept, point-slope, two-point, intercept and general forms.

If two lines have slopes m₁ and m₂, then tan θ = |(m₂−m₁)/(1+m₁m₂)| when the denominator is non-zero.

A family of lines describes a collection satisfying a common condition; concurrent lines pass through one common point.

A circle is the locus of a point that remains at a fixed distance from a fixed point called the centre.

A circle with centre (h,k) and radius r has (x−h)² + (y−k)² = r².

A secant cuts a circle at two points, while a tangent touches it at exactly one point.

A tangent is perpendicular to the radius drawn to the point of contact. This property is central to tangent problems.

A conic is obtained by intersecting a plane with a double cone. Different positions of the plane produce different curves.

A parabola is the locus of a point whose distance from a fixed point, the focus, equals its perpendicular distance from a fixed line, the directrix.

An ellipse is the locus for which the sum of distances from two fixed points, the foci, is constant.

A hyperbola is the locus for which the absolute difference of distances from two fixed foci is constant.

Dispersion describes how widely observations are spread around a central value. Two data sets can have the same mean but very different spread.

Range is the difference between the largest and smallest values. Quartiles divide ordered data into four parts and help describe spread.

Variance measures the average squared deviation from the mean. Squaring prevents positive and negative deviations from cancelling.

Standard deviation is the positive square root of variance and is expressed in the same unit as the observations.

A random experiment has a clearly defined set of possible outcomes, but the exact outcome cannot be predicted with certainty before the experiment.

The sample space contains all possible outcomes. An event is a selected collection of outcomes from that sample space.

For equally likely outcomes, P(E) = n(E)/n(S), where n(E) is the number of favourable outcomes and n(S) is the number of outcomes in the sample space.

Conditional probability measures probability when additional information is known. Independent events do not change one another's probability.

The equation x²+1=0 has no real solution. Introducing i with i²=−1 extends the number system so such equations can be handled.

A complex number is written z = a+ib, where a is the real part and b is the imaginary part.

The conjugate of a+ib is a−ib. Its modulus is |z| = √(a²+b²), representing its distance from the origin in the complex plane.

A complex number can be represented as a point (a,b), linking algebra with geometry.

A sequence is an ordered list of numbers. A series is formed when the terms of a sequence are added.

In an A.P., the difference between consecutive terms is constant: a, a+d, a+2d, … .

In a G.P., the ratio of consecutive non-zero terms is constant: a, ar, ar², … .

For an A.P., Sₙ = n/2[2a+(n−1)d]. For a G.P. with r≠1, Sₙ = a(rⁿ−1)/(r−1).

If one task can be done in m ways and a second independent task in n ways, both together can be done in mn ways.

A permutation is an arrangement where order matters. The number of arrangements of n distinct objects taken r at a time is nPᵣ = n!/(n−r)! .

A combination is a selection where order does not matter: nCᵣ = n!/[r!(n−r)!].

When objects are arranged around a circle, rotations of the entire arrangement are treated as the same arrangement.

Induction proves a statement for every natural number by showing it is true for a starting case and that truth for n implies truth for n+1.

The theorem gives a systematic expansion of (a+b)ⁿ without multiplying every factor separately.

The (r+1)th term is Tᵣ₊₁ = nCᵣ aⁿ⁻ʳ bʳ.

When n is even there is one middle term; when n is odd there are two middle terms. Their positions follow directly from the general-term formula.

A set is a well-defined collection of distinct objects. The objects are called elements of the set.

A set can be described in roster form, by a rule, or by a suitable diagram.

A×B is the set of ordered pairs (a,b) with a∈A and b∈B. If A and B are finite, n(A×B)=n(A)n(B).

A relation from A to B is a subset of A×B. It tells us which ordered pairs are connected by the chosen rule.

A function assigns exactly one output to each input in its domain.

The domain contains allowed inputs, the codomain is the declared set of possible outputs, and the range contains outputs actually obtained.

One-one, many-one, into and onto describe how inputs and outputs are paired.

Composition applies one function after another. An inverse reverses a function when the function has the required one-to-one property.

A limit describes the value a function approaches as its input approaches a specified value, even if the function is not evaluated exactly there.

The statement limₓ→a f(x)=L means that f(x) approaches L as x approaches a.

Limits provide the foundation for continuity and differentiation.

Known limit laws allow sums, differences, products and suitable quotients to be evaluated systematically.

Informally, a function is continuous at a point if its graph has no break, jump or hole at that point.

For continuity at x=a, f(a) must exist, limₓ→a f(x) must exist, and limₓ→a f(x)=f(a).

A two-sided limit exists only when the left-hand and right-hand limits agree.

Continuity links algebraic behaviour with the visual idea of drawing a graph near the point without lifting the pencil.

The derivative measures the instantaneous rate of change of one quantity with respect to another.

At a point on a curve, the derivative gives the slope of the tangent at that point.

For f(x)=xⁿ, d(xⁿ)/dx = nxⁿ⁻¹ for the standard powers covered at this level.

Derivatives help describe rates of change, tangents and normals, and the behaviour of functions.