CBSE Class 11 Mathematics Syllabus

How does the inequality sign behave when multiplying or dividing both sides by a negative number?

The inequality sign reverses (flips direction).

How do you represent the solution of a linear inequality in two variables graphically?

As a half-plane: draw the boundary line (solid if ≤ or ≥, dashed if < or >), then shade the side containing solutions, tested with a point like (0,0).

What is the difference between a permutation and a combination?

A permutation is an arrangement where order matters; a combination is a selection where order does not matter.

State the formulas for nPr and nCr.

nPr = n! / (n−r)!; nCr = n! / [r!(n−r)!].

State the relationship between nPr and nCr.

nPr = nCr × r! (equivalently nCr = nPr / r!).

How many distinct arrangements are there of n objects where some are identical (p alike, q alike, r alike)?

n! / (p! q! r!).

State the Binomial Theorem for (a + b)ⁿ.

(a+b)ⁿ = Σ (from r=0 to n) nCr · aⁿ⁻ʳ · bʳ.

What is the general (r+1)th term in the expansion of (a + b)ⁿ?

T(r+1) = nCr · aⁿ⁻ʳ · bʳ.

How many terms are in the expansion of (a + b)ⁿ, and how do you find the middle term(s)?

There are n+1 terms. If n is even, the single middle term is the (n/2 + 1)th term; if n is odd, the two middle terms are the (n+1)/2 th and (n+3)/2 th terms.

State the formula for the nth term of an Arithmetic Progression (AP).

aₙ = a + (n−1)d, where a is the first term and d is the common difference.

State the formula for the sum of the first n terms of an AP.

Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

State the nth term and the sum of n terms of a Geometric Progression (GP).

aₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1.

What is the sum to infinity of a GP, and when does it exist?

S∞ = a/(1 − r), valid only when |r| < 1.

Define the Arithmetic Mean (AM) and Geometric Mean (GM) of two numbers a and b, and state the AM-GM inequality.

AM = (a+b)/2; GM = √(ab). For positive a, b: AM ≥ GM.

State the slope (gradient) formula for a line through points (x₁,y₁) and (x₂,y₂).

m = (y₂ − y₁) / (x₂ − x₁).

State the slope-intercept and point-slope forms of a straight line.

Slope-intercept: y = mx + c. Point-slope: y − y₁ = m(x − x₁).

What are the conditions on slopes for two lines to be parallel and perpendicular?

Parallel: m₁ = m₂. Perpendicular: m₁ · m₂ = −1.

State the formula for the distance from a point (x₁,y₁) to the line Ax + By + C = 0.

d = |Ax₁ + By₁ + C| / √(A² + B²).

State the standard equation of a circle with centre (h,k) and radius r.

(x − h)² + (y − k)² = r².

Give the standard equation of a parabola opening rightward and identify its focus and directrix.

y² = 4ax; focus at (a, 0); directrix x = −a.

State the standard equation of an ellipse (a > b) and its eccentricity.

x²/a² + y²/b² = 1; eccentricity e = √(1 − b²/a²), with 0 < e < 1.

State the standard equation of a hyperbola and its eccentricity.

x²/a² − y²/b² = 1; eccentricity e = √(1 + b²/a²), with e > 1.

State the distance formula between two points in 3D space.

d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].

State the section formula for a point dividing the join of (x₁,y₁,z₁) and (x₂,y₂,z₂) in ratio m:n internally.

((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)).

State the algebra of limits: the limit of a sum, product, and quotient of two functions.

lim(f+g) = lim f + lim g; lim(f·g) = lim f · lim g; lim(f/g) = lim f / lim g (provided lim g ≠ 0).

State the two standard trigonometric limits as x → 0.

lim(x→0) (sin x)/x = 1 and lim(x→0) (1 − cos x)/x = 0.

State the first-principles (limit) definition of the derivative of f(x).

f′(x) = lim(h→0) [f(x+h) − f(x)] / h.

State the power rule, product rule, and quotient rule for derivatives.

Power: d/dx(xⁿ)=n xⁿ⁻¹. Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v².

In mathematical reasoning, what is the negation of a statement, and how do you negate 'p and q' and 'p or q'?

Negation of a statement p reverses its truth value (~p). ~(p and q) = (~p) or (~q); ~(p or q) = (~p) and (~q).

State the formula for variance and standard deviation of n observations (ungrouped data).

Variance σ² = (1/n) Σ(xᵢ − x̄)²; standard deviation σ = √(variance), where x̄ is the mean.