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37 cardsEgzamin Maturalny (Matura Poland)

Matura Key Facts and Formulas

37 flashcards with core formulas, literary facts and dates of Polish history. Study material written for practice, not an official test.

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Quadratic formula for

a x^{2} + b x + c = 0

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Quadratic formula for

a x^{2} + b x + c = 0

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Discriminant of a quadratic equation

Δ = b^{2} - 4 a c

. If

Δ > 0

there are two real roots, if

Δ = 0

one, if

Δ < 0

none.

Vertex of

f (x) = a x^{2} + b x + c

p = - \frac{b}{2 a}

and

q = f (p) = - \frac{Δ}{4 a}

Vieta's formulas for the roots

x_{1}, x_{2}

x_{1} + x_{2} = - \frac{b}{a}

and

x_{1} \cdot x_{2} = \frac{c}{a}

n

-th term of an arithmetic sequence

a_{n} = a_{1} + (n - 1) r

Sum of the first

n

terms of an arithmetic sequence

S_{n} = \frac{a _{1} + a _{n}}{2} \cdot n

n

-th term of a geometric sequence

a_{n} = a_{1} \cdot q^{n - 1}

Sum of the first

n

terms of a geometric sequence (

q e 1

)

S_{n} = a_{1} \cdot \frac{q ^{n} - 1}{q - 1}

Sum of an infinite geometric series

S = \frac{a _{1}}{1 - q}

, which exists only when

∣ q ∣ < 1

Logarithm of a product

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y

Logarithm of a quotient

lo g_{a} \frac{x}{y} = lo g_{a} x - lo g_{a} y

Logarithm of a power

lo g_{a} x^{k} = k lo g_{a} x

Pythagorean theorem

In a right triangle,

a^{2} + b^{2} = c^{2}

, where

c

is the hypotenuse.

Law of cosines

c^{2} = a^{2} + b^{2} - 2 ab cos γ

Law of sines

\frac{a}{s i n α} = \frac{b}{s i n β} = \frac{c}{s i n γ} = 2 R

Basic trigonometric identity

sin^{2} α + cos^{2} α = 1

Distance between two points

A = (x_{1}, y_{1})

and

B = (x_{2}, y_{2})

∣ A B ∣ = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Midpoint of a segment

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Equation of a circle with centre

(a, b)

and radius

r

(x - a)^{2} + (y - b)^{2} = r^{2}

Number of combinations of

k

items from

n

(k n) = \frac{n !}{k ! ( n - k )!}

Classical probability of an event

A

P (A) = \frac{∣ A ∣}{∣Ω∣}

, the number of favourable outcomes over all equally likely outcomes.

Derivative of

x^{n}

(x^{n})^{'} = n x^{n - 1}

Volume of a cone

V = \frac{1}{3} π r^{2} h

Sum of interior angles of a polygon with

n

sides

(n - 2) \cdot 18 0^{\circ}

Polish Romanticism: key work and author

Pan Tadeusz, the national epic of Adam Mickiewicz, published in 1834.

Polish Positivism: key novel and author

The Doll (Lalka) by Boleslaw Prus, about the merchant Stanislaw Wokulski.

Polish Nobel laureates in Literature

Sienkiewicz (1905), Reymont (1924), Milosz (1980), Szymborska (1996) and Tokarczuk (2018).

Number of grammatical cases in Polish

Seven: nominative, genitive, dative, accusative, instrumental, locative and vocative.

Baptism of Poland

966, under Mieszko I of the Piast dynasty.

Battle of Grunwald

1410: Polish-Lithuanian forces defeated the Teutonic Order.

Union of Lublin

1569: Poland and Lithuania formed one Commonwealth.

Constitution of 3 May

1791: adopted by the Great Sejm.

Partitions of Poland

1772, 1793 and 1795, by Russia, Prussia and Austria.

Regained independence

11 November 1918, celebrated as Independence Day.

Start of the Second World War in Poland

1 September 1939, the German invasion; the Soviet invasion followed on 17 September.

Warsaw Uprising

Began on 1 August 1944; led by the Home Army and lasted 63 days.

Round Table talks and partly free elections

Talks in early 1989 led to the partly free election of 4 June 1989.

Egzamin Maturalny (Matura Poland) in Poland

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