Odisha BSE HSC Mathematics cheat sheet

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

Fundamental Theorem of Arithmetic: every integer greater than 1 has a prime factorisation that is unique except for the order of the prime factors.

For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

A rational number in lowest terms has a terminating decimal exactly when its denominator has only 2 and 5 as prime factors.

√p is irrational for every prime p; the sum or product of a nonzero rational and an irrational number is irrational.

A real number's decimal expansion is either terminating, repeating, or non-terminating and non-repeating; the last type is irrational.

For real x, |x| = x when x ≥ 0 and |x| = −x when x < 0; also |xy| = |x||y|.

Euclid's algorithm: replace (a, b) by (b, r) where r is the remainder of a divided by b, and continue until the remainder is 0; the last nonzero divisor is HCF.

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 ax² + bx + c = 0 (a ≠ 0), the roots have sum −b/a and product c/a.

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) when b² − 4ac ≥ 0.

For linear equations a1x + b1y = c1 and a2x + b2y = c2: unique solution if a1/a2 ≠ b1/b2; no solution if a1/a2 = b1/b2 ≠ c1/c2; infinitely many if all three ratios are equal.

AP nth term: a_n = a + (n − 1)d; sum of first n terms: S_n = n/2[2a + (n − 1)d] = n/2(a + l), where l is the last term.

To rationalise 1/(√a + √b), multiply numerator and denominator by √a − √b; the denominator becomes a − b.

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, then it is parallel to the third side.

Similar triangles have equal corresponding angles and proportional corresponding sides; valid tests include AA, SSS, and SAS similarity.

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

Pythagoras theorem: in a right triangle, hypotenuse² = base² + height²; the converse is also true.

A tangent to a circle is perpendicular to the radius at the point of contact; tangents from an external point to the same circle are equal in length.

An angle in a semicircle is a right angle; angles in the same segment are equal; opposite angles of a cyclic quadrilateral are supplementary.

Construction to divide a segment AB in ratio m:n: draw a ray AX, mark m+n equal arcs, join the last point to B, and draw a parallel through the m-th point.

In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Reciprocal relations: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ, and tan θ = sin θ/cos θ.

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

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

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

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

45°–45°–90° triangle sides are in ratio 1:1:√2; 30°–60°–90° triangle sides are in ratio 1:√3:2.

Circle: area = πr², circumference = 2πr; area of a ring with outer radius R and inner radius r is π(R² − r²).

Sector area = (θ/360)πr² and arc length = (θ/360)2πr, where θ is the central angle in degrees.

Cuboid: total surface area = 2(lb + bh + hl), volume = lbh; cube: total surface area = 6a², volume = a³.

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 = (1/3)πr²h.

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

Frustum of a cone: curved surface area = πl(R + r), total surface area = πl(R + r) + πR² + πr², volume = (1/3)πh(R² + Rr + r²).

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

Mean for grouped data: x̄ = Σf_i x_i / Σf_i, where x_i is the class mark (lower limit + upper limit)/2.

Assumed mean method: x̄ = A + (Σf_i d_i / Σf_i), where d_i = x_i − A and A is a chosen assumed mean.

Step-deviation method: x̄ = A + h(Σf_i u_i / Σf_i), where u_i = (x_i − A)/h and h is the class width.

Median of grouped data: median = l + [(n/2 − cf)/f]h, with l lower limit of median class, cf cumulative frequency before it, f its frequency, h class width.

Mode of grouped data: mode = l + [(f1 − f0)/(2f1 − f0 − f2)]h, where f1 is modal frequency and f0, f2 are adjacent class frequencies.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Probability: P(E) = favourable outcomes / total equally likely outcomes; 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).