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Leaving Certificate Key Facts

52 flashcards with key formulas in mathematics, physics, chemistry, and essential biology definitions for Leaving Certificate revision.

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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}

Condition for equal roots of a quadratic

The discriminant

b^{2} - 4 a c = 0

Derivative of

x^{n}

n x^{n - 1}

Product rule

(uv)^{'} = u^{'} v + u v^{'}

Chain rule

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

Integral of

x^{n}

for

n e q - 1

\frac{x ^{n + 1}}{n + 1} + c

n

th term of an arithmetic sequence

T_{n} = a + (n - 1) d

Sum of the first

n

terms of an arithmetic series

S_{n} = \frac{n}{2} (2 a + (n - 1) d)

Sum to infinity of a geometric series

S_{\infty} = \frac{a}{1 - r}

, valid when

∣ r ∣ < 1

Distance between

(x_{1}, y_{1})

and

(x_{2}, y_{2})

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Equation of a circle, centre

(h, k)

, radius

r

(x - h)^{2} + (y - k)^{2} = r^{2}

Cosine rule

c^{2} = a^{2} + b^{2} - 2 ab cos C

Modulus of

a + bi

a^{2} + b^{2}

Area of a sector (angle

θ

in radians)

\frac{1}{2} r^{2} θ

Metaphor

A direct comparison that says one thing is another, with no word such as like or as.

Simile

A comparison that uses like or as.

Dramatic irony

The audience knows something that a character does not.

Enjambment

A sentence that runs on from one line of verse to the next without a pause.

Caesura

A pause within a line of poetry.

Pathos, ethos and logos

Appeals to emotion, to the speaker's credibility and to reason.

Mitosis

One division that gives two genetically identical diploid cells, for growth and repair.

Meiosis

Two divisions that give four genetically different haploid cells, for gametes.

Enzyme

A protein that is a biological catalyst and lowers the activation energy.

Osmosis

Movement of water across a partially permeable membrane from high to low water concentration.

Aerobic respiration equation (in words)

Glucose and oxygen give carbon dioxide, water and energy as ATP.

De Moivre's Theorem for complex numbers

[r (cos θ + i sin θ)]^{n} = r^{n} (cos n θ + i sin n θ)

, fundamental for powers and

n

-th roots of complex numbers.

Derivatives of sine and cosine functions

\frac{d}{d x} (sin x) = cos x

;

\frac{d}{d x} (cos x) = - sin x

;

\frac{d}{d x} (tan x) = sec^{2} x

Derivatives of exponential and natural logarithm

\frac{d}{d x} (e^{k x}) = k e^{k x}

;

\frac{d}{d x} (ln x) = \frac{1}{x}

for

x > 0

Fundamental Theorem of Calculus

f

is continuous on

[a, b]

and

F^{'} (x) = f (x)

, then

\int_{a}^{b} f (x) d x = F (b) - F (a)

Area between two curves

y = f (x)

and

y = g (x)

A = \int_{a}^{b} [f (x) - g (x)] d x

, where

f (x) \geq g (x)

across the interval

[a, b]

Slope condition for perpendicular lines

m_{1} \times m_{2} = - 1

, or

m_{2} = - \frac{1}{m _{1}}

Perpendicular distance from

(x_{1}, y_{1})

a x + b y + c = 0

d = \frac{∣ a x _{1} + b y _{1} + c ∣}{a ^{2} + b ^{2}}

Sine rule for general triangles

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}

Trigonometric area of a triangle

Area = \frac{1}{2} ab sin C

Binomial distribution formula

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

, with mean

μ = n p

and standard deviation

σ = n p (1 - p)

Empirical rule for normal distribution

Approximately 68% of data lies within

μ \pm 1 σ

, 95% within

μ \pm 2 σ

, and 99.7% within

μ \pm 3 σ

Balanced chemical equation for photosynthesis

6 CO_{2} + 6 H_{2} O light, chlorophyll C_{6} H_{12} O_{6} + 6 O_{2}

Balanced chemical equation for aerobic respiration

C_{6} H_{12} O_{6} + 6 O_{2} \to 6 CO_{2} + 6 H_{2} O + 38 ATP

Four stages of mitosis in order

1. Prophase (chromosomes condense)
2. Metaphase (chromosomes line up at equator)
3. Anaphase (sister chromatids pulled to poles)
4. Telophase (nuclear membranes reform).

Mendel's First Law: Law of Segregation

Inherited characteristics are controlled by pairs of factors (alleles), which separate from each other during gamete formation, with each gamete carrying one allele.

Mendel's Second Law: Law of Independent Assortment

During gamete formation, alleles of two or more different genes sort into gametes independently of one another (provided genes are not linked).

DNA base pairing rules (Chargaff's rules)

Adenine (A) pairs with Thymine (T) via 2 hydrogen bonds; Cytosine (C) pairs with Guanine (G) via 3 hydrogen bonds.

Osmosis definition

The movement of water molecules from a region of higher water potential (low solute concentration) to lower water potential through a semi-permeable membrane.

Avogadro's constant and mole definition

N_{A} = 6.022 \times 1 0^{23} particles per mole

. One mole contains as many entities as there are atoms in exactly 12 g of carbon-12.

Molar volume of an ideal gas at s.t.p.

22.4 L

(

22, 400 cm^{3}

) at standard temperature (

273 K

) and pressure (

101.3 kPa

);

24.0 L

at room temperature.

Definition and formula for pH

pH = - lo g_{10} [H^{+}]

- lo g_{10} [H_{3} O^{+}]

. A change of 1 pH unit represents a tenfold change in hydrogen ion concentration.

Le Chatelier's Principle

If a system at dynamic equilibrium is subjected to a stress (change in concentration, pressure, or temperature), the system shifts in a direction that opposes the stress.

Newton's Second Law of Motion

The rate of change of momentum of an object is proportional to the net applied force and takes place in the direction of the force:

F = ma

Principle of Conservation of Momentum

In any closed system of colliding bodies, the total momentum before collision equals total momentum after collision:

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Snell's Law of Refraction

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant:

\frac{s i n i}{s i n r} = n

Boyle's Law for ideal gases

For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume:

P_{1} V_{1} = P_{2} V_{2}

Radioactive half-life (

T_{1/2}

)

The time required for half of the radioactive nuclei in a given radioactive sample to undergo decay.

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