Bihar BSEB Matric Mathematics cheat sheet

Euclid's division lemma: for positive integers a and b, there exist unique integers q and r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm for HCF: repeatedly replace (a, b) with (b, r) where r is the remainder of a ÷ b; the last non-zero remainder is the HCF.

Fundamental Theorem of Arithmetic: every integer greater than 1 is a unique product of primes apart from the order of the factors.

HCF(a, b) × LCM(a, b) = a × b; this holds for two positive integers only, not for three or more.

A rational number p/q in lowest terms has a terminating decimal exactly when q = 2^m × 5^n for non-negative integers m and n.

If q has a prime factor other than 2 or 5, the decimal expansion of p/q is non-terminating and repeating.

√p is irrational for every prime p; more generally √n is irrational when n is not a perfect square.

The sum or difference of a rational and an irrational number is irrational; the product of a non-zero rational and an irrational number is irrational.

Division algorithm for polynomials: p(x) = g(x) × q(x) + r(x), where either r(x) = 0 or degree r(x) < degree g(x).

Remainder theorem: the remainder when p(x) is divided by (x − a) is p(a).

Factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.

For a quadratic ax² + bx + c with zeros α and β: α + β = −b/a and αβ = c/a.

For a cubic ax³ + bx² + cx + d with zeros α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

A quadratic polynomial with given zeros α and β is k[x² − (α + β)x + αβ] for any non-zero constant k.

The graph of a quadratic polynomial is a parabola; it cuts the x-axis at two points, touches it at one point, or misses it, according as b² − 4ac is positive, zero, or negative.

A polynomial of degree n has at most n real zeros, and a polynomial's graph crosses the x-axis at each zero of odd multiplicity.

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: the pair is consistent with a unique solution when a₁/a₂ ≠ b₁/b₂ (intersecting lines).

The pair has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines).

The pair has no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines).

Graphical method: draw both lines and read the coordinates of the point of intersection as the unique solution.

Substitution method: express one variable from one equation and substitute it into the other to reduce to a single-variable equation.

Elimination method: multiply the equations to make the coefficients of one variable equal in magnitude, then add or subtract to eliminate it.

Cross-multiplication formula: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), provided a₁b₂ − a₂b₁ ≠ 0.

A pair of linear equations in two variables can have exactly one solution, no solution, or infinitely many solutions - never exactly two.

Standard form: ax² + bx + c = 0 with a ≠ 0; the discriminant is D = b² − 4ac.

D > 0 gives two distinct real roots; D = 0 gives two equal real roots; D < 0 gives no real roots.

Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.

Sum of roots = −b/a and product of roots = c/a for ax² + bx + c = 0.

Factorisation method: split the middle term bx into two terms whose coefficients multiply to ac and add to b, then group.

A quadratic equation has at most two roots; if a value satisfies the equation, it is a root and (x − that value) is a factor.

For equal roots, D = 0, so b² = 4ac; this condition is used to find unknown coefficients.

Completing the square converts ax² + bx + c = 0 into (x + b/2a)² = (b² − 4ac)/4a², which yields the quadratic formula.

An AP has a constant common difference d = aₙ₊₁ − aₙ; the general term is aₙ = a + (n − 1)d.

Sum of the first n terms: Sₙ = (n/2)[2a + (n − 1)d], or Sₙ = (n/2)(a + l) when the last term l is known.

The nth term from the end of an AP with last term l and common difference d is l − (n − 1)d.

If a, b, c are in AP, then 2b = a + c; the middle term b is the arithmetic mean of a and c.

The sum of the first n natural numbers is n(n + 1)/2.

The sum of the first n odd numbers is n², since 1, 3, 5, … is an AP with a = 1 and d = 2.

A sequence is an AP exactly when the difference of consecutive terms is the same for every pair.

To find n when the last term is given, solve a + (n − 1)d = l for n; n must be a positive integer.

Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.

Converse of BPT: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.

Two triangles are similar if their corresponding angles are equal, which forces corresponding sides to be proportional.

Similarity criteria: AA (or AAA), SSS (all three side pairs proportional), and SAS (two side pairs proportional with the included angle equal).

The ratio of areas of two similar triangles equals the square of the ratio of any pair of corresponding sides.

In a right triangle, the altitude to the hypotenuse creates two triangles similar to each other and to the original (used in Pythagoras proofs).

Pythagoras theorem: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides; the converse also holds.

In an isosceles triangle, the altitude from the vertex to the base bisects the base and the vertex angle.

Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Distance of a point (x, y) from the origin is √(x² + y²).

Section formula for dividing the join of (x₁, y₁) and (x₂, y₂) in ratio m : n internally: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Midpoint of the join of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Three points are collinear exactly when the area of the triangle they form is zero.

A point on the x-axis has coordinates (x, 0); a point on the y-axis has coordinates (0, y).

The centroid of a triangle is the average of its vertices' coordinates: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

Ratios: sin θ = perpendicular/hypotenuse, cos θ = base/hypotenuse, tan θ = perpendicular/base, with cosec, sec, cot as reciprocals.

Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Standard values: sin 0° = 0, sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1; cos values run in reverse order.

tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3; tan 90° is undefined.

Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

In height-and-distance problems, draw the right triangle, mark the known side and angle, then choose the ratio linking them to the unknown.

If the angle of elevation of the sun is 45°, the shadow of a vertical object equals its height.

A tangent to a circle is perpendicular to the radius at the point of contact.

The lengths of the two tangents drawn from an external point to a circle are equal.

The angle between two tangents from an external point is supplementary to the angle subtended by the line joining the point to the centre at the centre.

Number of tangents: none from an interior point, exactly one from a point on the circle, exactly two from an exterior point.

A parallelogram circumscribing a circle is a rhombus.

Constructions must show arcs, the centre, and the point of contact; a tangent at a point on the circle is drawn by erecting a perpendicular to the radius there.

To draw tangents from an external point P, draw the circle on OP as diameter; its intersections with the given circle are the points of contact.

Circumference of a circle = 2πr; area = πr².

Length of an arc of a sector with angle θ is (θ/360°) × 2πr.

Area of a sector = (θ/360°) × πr².

Area of the minor segment = area of sector − area of the triangle formed by the two radii and the chord.

Area of a ring (annulus) between radii R and r is π(R² − r²).

Perimeter of a sector = arc length + 2r.

If the angle of a sector is 90°, its area is one quarter of the circle's area and its arc is one quarter of the circumference.

Distance covered by a wheel in one revolution equals its circumference; number of revolutions = total distance ÷ circumference.

Cylinder: curved surface area 2πrh, total surface area 2πr(r + h), volume πr²h.

Cone: slant height l = √(r² + h²), curved surface area πrl, total surface area πr(l + r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere: curved area 2πr², total area 3πr², volume (2/3)πr³.

Frustum of a cone: volume = (1/3)πh(r₁² + r₁r₂ + r₂²) and slant height l = √[h² + (r₁ − r₂)²].

When a solid is melted and recast, volume is conserved; equate the initial and final volumes to find the unknown dimension.

When a cone, hemisphere, or cylinder is hollowed out of a solid, subtract the hollow volume from the original volume.

For a solid dipped in a cylindrical vessel, the rise in water level equals the volume of the solid divided by the base area of the vessel.

Cube: total surface area 6a² and volume a³; cuboid: total surface area 2(lb + bh + hl) and volume lbh.

Mean of raw data = sum of observations ÷ number of observations; for a frequency table, mean = Σfᵢxᵢ / Σfᵢ.

Class mark of a class is (lower limit + upper limit)/2 and is used as xᵢ in grouped data.

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class and h the class size.

Median = l + [(n/2 − cf)/f] × h, where cf is the cumulative frequency before the median class and f its frequency.

The empirical relation among the three measures is 3 Median = Mode + 2 Mean.

Probability of an event E is P(E) = number of favourable outcomes ÷ total number of equally likely outcomes.

0 ≤ P(E) ≤ 1, P(impossible event) = 0, P(sure event) = 1, and P(E) + P(not E) = 1.

The sum of probabilities of all elementary outcomes of an experiment is 1.