ISC Class 11 Mathematics Syllabus

What does the discriminant b² − 4ac tell you about the roots of a quadratic?

> 0: two distinct real roots; = 0: equal (repeated) real roots; < 0: two complex conjugate roots.

For roots α and β of ax² + bx + c = 0, what are their sum and product?

Sum α + β = −b/a; Product αβ = c/a.

When solving an inequality, what happens if you multiply or divide both sides by a negative number?

The inequality sign reverses (e.g. < becomes >).

State the multiplication (fundamental) principle of counting.

If one event can occur in m ways and another in n ways, then both together can occur in m × n ways.

What is the formula for the number of permutations of n distinct objects taken r at a time?

nPr = n! / (n − r)!.

What is the formula for the number of combinations of n objects taken r at a time?

nCr = n! / [r!(n − r)!].

State the relationship between nPr and nCr.

nPr = nCr × r!.

What is the value of nCr + nC(r−1)?

(n+1)Cr (Pascal's rule).

State the Binomial Theorem for (a + b)^n where n is a positive integer.

(a+b)^n = Σ (k=0 to n) nCk · a^(n−k) · b^k.

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

T_(r+1) = nCr · a^(n−r) · b^r.

How many terms are there in the expansion of (a + b)^n?

n + 1 terms.

What is the nth term of an arithmetic progression (AP) with first term a and common difference d?

a_n = a + (n − 1)d.

What is the sum of the first n terms of an AP?

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

What is the nth term and sum to n terms of a geometric progression (GP) with first term a, ratio r?

a_n = a·r^(n−1); S_n = a(r^n − 1)/(r − 1) for r ≠ 1.

What is the sum to infinity of a GP with |r| < 1?

S_∞ = a / (1 − r).

State the relationship between the arithmetic mean (AM) and geometric mean (GM) of two positive numbers.

AM ≥ GM, with equality only when the numbers are equal.

What is the slope of a line passing 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 for two lines (slopes m₁, m₂) to be parallel or perpendicular?

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

What is the formula for the distance of a point (x₁, y₁) from 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.

What is the standard equation of a parabola opening rightward, and its focus?

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

State the standard equation of an ellipse and define eccentricity for it.

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

State the standard equation of a hyperbola and the range of its eccentricity.

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

What is the distance formula between two points in three-dimensional space?

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

What is the section formula for a point dividing the segment joining (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 two standard limits used in calculus involving sin and polynomials.

lim(x→0) sin x / x = 1; and lim(x→a) (x^n − a^n)/(x − a) = n·a^(n−1).

What is the derivative definition (first principles) of f(x)?

f'(x) = lim(h→0) [f(x+h) − f(x)] / h.

State the power rule and product rule of differentiation.

Power rule: d/dx(x^n) = n·x^(n−1). Product rule: (uv)' = u'v + uv'.

State the formula for variance of n observations using mean x̄.

σ² = (1/n) Σ(xᵢ − x̄)²; standard deviation σ = √variance.

For mutually exclusive events A and B, what is P(A ∪ B)? And for any two events?

Mutually exclusive: P(A∪B) = P(A) + P(B). General: P(A∪B) = P(A) + P(B) − P(A∩B).