IMU CET flashcards

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

Discriminant of a quadratic

For ax² + bx + c = 0, the discriminant D = b² - 4ac. D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives complex roots.

Sum and product of roots

If α and β are roots of ax² + bx + c = 0, then α + β = -b/a and αβ = c/a. These let you build a quadratic from its roots without solving fully.

Sine and cosine rules

In any triangle, a/sin A = b/sin B = c/sin C. The cosine rule a² = b² + c² - 2bc cos A finds a side when two sides and the included angle are known.

Derivative of xⁿ and sin x

The power rule states d/dx(xⁿ) = nx^(n-1) for any real n. The derivative of sin x is cos x, and the derivative of cos x is -sin x.

Limit of sin x over x

As x approaches zero, sin x / x approaches 1 when x is measured in radians. This standard limit underlies the derivatives of trigonometric functions.

Arithmetic progression sum

For an arithmetic progression with first term a and common difference d, the nth term is a + (n-1)d and the sum of n terms is (n/2)[2a + (n-1)d].

Archimedes' principle

A body wholly or partly immersed in a fluid experiences an upthrust equal to the weight of fluid displaced. The body floats when this upthrust equals its own weight, i.e. its density is lower.

Pressure at depth in a fluid

The absolute pressure at depth h in a fluid of density ρ is P = P₀ + ρgh, where P₀ is the pressure at the surface and g is gravitational acceleration, about 9.8 m/s².

Ohm's law and power

For a conductor at constant temperature, V = IR. Electrical power can be written P = VI, P = I²R or P = V²/R depending on which quantities are known.

Snell's law of refraction

When light crosses a boundary, n₁ sin θ₁ = n₂ sin θ₂, where n is refractive index and θ is measured from the normal. Total internal reflection occurs beyond the critical angle.

Kinematic equations

For constant acceleration a with initial velocity u, final velocity v and displacement s: v = u + at, s = ut + ½at², and v² = u² + 2as. They fail if acceleration varies.

Heat engine efficiency

The maximum (Carnot) efficiency of a heat engine is η = 1 - T_cold/T_hot, with both temperatures in kelvin. No real engine between the same reservoirs can exceed this value.

Mole and molar volume

One mole contains 6.022 × 10²³ particles, and moles equal mass divided by molar mass. At standard temperature and pressure, one mole of an ideal gas occupies about 22.4 litres.

pH scale

pH = -log[H⁺] for hydrogen ion concentration in mol/L. At 25 °C, pH + pOH = 14, so neutral water has pH 7; values below 7 are acidic and above 7 alkaline.

Oxidation and reduction

Oxidation is loss of electrons and reduction is gain (remember OIL RIG). The oxidising agent accepts electrons and is itself reduced during the reaction, and vice versa for the reducing agent.

Le Chatelier's principle

When a system at equilibrium is disturbed, it shifts to oppose the change. Raising pressure favours the side with fewer gas moles, and raising temperature favours the endothermic direction.

Ideal gas equation

PV = nRT relates pressure, volume, moles and absolute temperature, with R = 8.314 J/(mol·K). At constant n it gives the combined law P₁V₁/T₁ = P₂V₂/T₂.

Periodic trend across a period

Moving left to right across a period, nuclear charge rises while shielding stays similar, so atomic radius decreases and both ionisation energy and electronegativity increase.

Active to passive voice

The object becomes the subject, the verb becomes a form of 'be' plus the past participle, and the original subject follows 'by'. Example: 'The crew lowered the boat' becomes 'The boat was lowered by the crew'.

Reported speech tense shift

When the reporting verb is past, the reported tense usually moves back one step: present to past, past to past perfect, will to would. Time words also shift, so today becomes that day.

Affect versus effect

'Affect' is normally a verb meaning to influence, as in 'weather affects visibility'. 'Effect' is normally a noun meaning a result, as in 'the effect of the storm'.

Third conditional structure

The third conditional describes an unreal past situation: if + past perfect in the condition, then would have + past participle in the result. Example: 'If the watch had been kept, the collision would have been avoided.'

Semicolon and colon use

A semicolon joins two independent clauses that are closely related, replacing a conjunction. A colon introduces a list, an explanation or a quotation after a complete clause.

Subject-verb agreement with collectives

Collective nouns such as crew, team or staff take a singular verb when the group acts as one unit, and a plural verb when members act individually. The surrounding sentence decides which.

Average speed for equal distances

When equal distances are covered at speeds x and y, the average speed is 2xy/(x + y), the harmonic mean. It is never the simple average (x + y)/2 unless x equals y.

Successive percentage change

Two successive changes of a% and b% produce a net change of a + b + ab/100 percent. A rise of 20% then a fall of 20% therefore gives a net fall of 4%.

Combined work formula

If A alone completes a job in x days and B alone in y days, working together they finish in xy/(x + y) days, because their rates 1/x and 1/y add together.

International Date Line

The line roughly follows the 180° meridian. Crossing it eastward you subtract one day, and crossing it westward you add one day, keeping local calendar dates consistent.

Port and starboard

Looking forward from a vessel, port is the left side and starboard is the right side. At night, port is marked by a red sidelight and starboard by a green sidelight.

IALA Region A buoyage colours

In IALA Region A, used in most of the world outside the Americas, port-hand marks are red and starboard-hand marks are green when entering a harbour from seaward.

Speed, distance and time at sea

Distance in nautical miles equals speed in knots multiplied by time in hours. One knot is one nautical mile per hour, and one nautical mile is 1,852 metres.

Head-on situation under COLREGs

When two power-driven vessels meet head-on or nearly so, each alters course to starboard so they pass port side to port side. Both are give-way vessels in this situation.