Maharashtra SSC Mathematics (MSBSHSE) flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)] / 2a. Substitute the coefficients directly to solve any quadratic.

Discriminant

The value D = b² − 4ac of a quadratic equation. If D > 0 there are two distinct real roots, if D = 0 the roots are real and equal, and if D < 0 there are no real roots.

Sum and product of roots

For ax² + bx + c = 0, the sum of roots α + β = −b/a and the product αβ = c/a. These let you build the equation x² − (α+β)x + αβ = 0 from given roots.

Cramer's rule

Solves two linear equations using determinants: x = Dx/D and y = Dy/D, where D = a₁b₂ − a₂b₁. A non-zero D means a unique solution exists.

Nature of linear pair solutions

For two linear equations, ratios a₁/a₂ = b₁/b₂ = c₁/c₂ give infinitely many solutions; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ give none; any other case gives exactly one.

Arithmetic progression nth term

In an AP with first term a and common difference d, the nth term is tn = a + (n−1)d. The sum of the first n terms is Sn = (n/2)(a + tn).

Geometric progression sum

For a GP with first term a and ratio r ≠ 1, Sn = a(rⁿ − 1)/(r − 1). If |r| < 1, the infinite sum S∞ = a/(1 − r).

Remainder and factor theorem

Dividing a polynomial p(x) by (x − a) leaves remainder p(a). The binomial (x − a) is a factor of p(x) exactly when p(a) equals zero.

Basic proportionality theorem

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio. The converse also holds: proportional division implies the line is parallel.

Similar triangle area ratio

In similar triangles, corresponding sides, medians and perimeters are in one ratio k, while the areas are in the ratio k². This follows from the square of the scale factor.

Pythagoras theorem

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse identifies a right angle from side lengths.

Alternate segment theorem

The angle between a tangent and a chord through the point of contact equals the inscribed angle standing on the alternate arc of that chord.

Tangent properties of a circle

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

Inscribed angle theorem

An angle inscribed in a circle equals half the central angle subtending the same arc. Consequently, an angle inscribed in a semicircle measures 90°.

Sector area and arc length

For a sector of angle θ in a circle of radius r: area = (θ/360) × πr² and arc length = (θ/360) × 2πr.

Mensuration volumes

Volume of a cylinder is πr²h, of a cone is ⅓πr²h, and of a sphere is (4/3)πr³. The total surface area of a sphere is 4πr².

Compound interest formula

Amount A = P(1 + R/100)ⁿ where P is principal, R the annual rate and n the number of years. Compound interest is A − P.

Simple interest formula

Simple interest I = PRT/100, where P is principal, R the annual rate percent and T the time in years. The amount repaid is P + I.

CGST and SGST split

For an intra-state supply, the GST rate is split equally into CGST and SGST, each half the total rate. An inter-state supply attracts the full rate as IGST.

Profit and loss percentage

Profit% = (SP − CP)/CP × 100 and loss% = (CP − SP)/CP × 100, both calculated on cost price. Selling price equals cost price times (100 ± percentage)/100.

Direct and inverse variation

In direct variation y = kx, so y₁/x₁ = y₂/x₂. In inverse variation y = k/x, so the products x₁y₁ and x₂y₂ are equal.

Trigonometric ratios

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

Reciprocal trigonometric ratios

cosec θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. These three ratios are undefined wherever their corresponding sine, cosine or tangent value is zero.

Pythagorean identities

The three fundamental identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. They hold for every angle where the ratios are defined.

Standard trigonometric values

sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cosine values run in reverse. tan 30° = 1/√3, tan 45° = 1 and tan 60° = √3.

Complementary angle relations

For complementary angles: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and similarly sec(90° − θ) = cosec θ.

Angle of elevation

The angle measured upward from the horizontal to the line of sight when an observer looks at an object above eye level. It is used in heights-and-distances problems.

Mean of grouped data

Using step deviation, x̄ = a + h × (Σfᵢuᵢ)/(Σfᵢ), where uᵢ = (xᵢ − a)/h, a is an assumed mean and h the class width.

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 and h the class width.

Mode of grouped data

Mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the frequency of the modal class and f₀, f₂ are the frequencies of adjacent classes.

Empirical mean-median-mode relation

For moderately skewed distributions, Mode = 3 × Median − 2 × Mean. It allows one of the three measures to be estimated when the other two are known.

Classical probability

P(A) = number of favourable outcomes divided by total number of equally likely outcomes. The value always lies between 0 and 1 inclusive.

Complement and addition rules

P(A′) = 1 − P(A). For any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and the last term vanishes when the events are mutually exclusive.