Maharashtra SSC Mathematics (MSBSHSE) cheat sheet

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)] / 2a.

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots (conjugate complex pair).

Sum of roots α + β = −b/a and product αβ = c/a; use these to build a quadratic from given roots: x² − (α+β)x + αβ = 0.

Cramer's rule for two variables: x = Dx/D, y = Dy/D with D = a₁b₂ − a₂b₁, Dx = c₁b₂ − c₂b₁, Dy = a₁c₂ − a₂c₁; D ≠ 0 gives a unique solution.

For equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂: a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinitely many solutions; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none; otherwise exactly one.

Arithmetic progression: nth term tn = a + (n−1)d; sum Sn = (n/2)[2a + (n−1)d] = (n/2)(a + tn).

Geometric progression: tn = a·r^(n−1); Sn = a(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = a/(1 − r) only when |r| < 1.

Factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0; remainder theorem: dividing p(x) by (x − a) leaves remainder p(a).

Index laws: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a^(−n) = 1/aⁿ.

Angles in a triangle sum to 180°; the exterior angle equals the sum of the two remote interior angles.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally, and conversely.

In similar triangles the ratio of areas equals the square of the ratio of corresponding sides; corresponding sides and perimeters are in the same ratio.

Pythagoras theorem: in a right triangle, hypotenuse² = sum of squares of the other two sides; its converse tests for a right angle.

Tangent–secant (alternate segment) theorem: the angle between a tangent and a chord through the point of contact equals the inscribed angle in the alternate segment.

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

Inscribed angle theorem: an angle inscribed in a circle equals half the central angle standing on the same arc; an angle in a semicircle is 90°.

Area formulas: triangle = ½ × base × height; circle = πr²; sector = (θ/360) × πr²; length of arc = (θ/360) × 2πr.

Volume formulas: cylinder = πr²h; cone = ⅓πr²h; sphere = (4/3)πr³; total surface area of a sphere = 4πr².

Compound interest: A = P(1 + R/100)ⁿ, where n is the number of compounding periods; CI = A − P.

Simple interest: I = PRT/100; amount = P + I.

For growth then depreciation, apply successive multipliers: final value = P(1 + r₁/100)(1 − r₂/100).

GST split on intra-state supply is CGST + SGST (each half the rate); inter-state supply carries IGST at the full rate.

Taxable value plus GST equals the invoice value; GST amount = rate% × taxable value, and CGST = SGST = half of GST for intra-state.

Percentage change = (change/original) × 100; successive percentage changes do not simply add - multiply the factors.

Profit% and loss% are calculated on cost price: profit% = (SP − CP)/CP × 100; loss% = (CP − SP)/CP × 100.

Direct variation: y = kx, so y₁/x₁ = y₂/x₂; inverse variation: y = k/x, so x₁y₁ = x₂y₂.

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

Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

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

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

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

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

Heights-and-distances method: draw the right triangle, label the known side and angle, then choose the ratio linking them (tan for opposite/adjacent, sin for opposite/hypotenuse, cos for adjacent/hypotenuse).

sin θ and cos θ each lie between −1 and 1; tan θ is undefined at 90° and 270°, and cot θ at 0° and 180°.

Mean of raw data: x̄ = (Σxᵢ)/n; for a frequency table, x̄ = (Σfᵢxᵢ)/(Σfᵢ).

Median of n ordered observations: the middle value when n is odd; the average of the two middle values when n is even.

Mode is the observation (or class) with the highest frequency; data can have more than one mode or none.

For a grouped frequency distribution, the modal class is the class with the highest frequency; mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

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

Step-deviation mean: x̄ = a + h × (Σfᵢuᵢ)/(Σfᵢ), where uᵢ = (xᵢ − a)/h and a is an assumed mean.

Less-than and more-than cumulative frequency curves (ogives) intersect at the median; each ogive plots cumulative frequency against class boundaries.

Empirical relation for moderately skewed data: Mode = 3 × Median − 2 × Mean.

Classical probability: P(A) = number of favourable outcomes / total number of equally likely outcomes.

For any event A, 0 ≤ P(A) ≤ 1; P(certain event) = 1 and P(impossible event) = 0.

Complement rule: P(A′) = 1 − P(A), where A′ is the event 'not A'.

Addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Sample space is the set of all possible outcomes; an event is a subset of the sample space.

Two events are mutually exclusive when they cannot occur together, so P(A ∩ B) = 0.

For a die, P(even) = 3/6 = ½; for a standard deck of 52 cards, P(an ace) = 4/52 = 1/13 and P(a spade) = 13/52 = ¼.

Sum of probabilities of all elementary outcomes in a sample space equals 1.