Distance = |(1*2 + 1*3 - 5)| / √(1² + 1²) = |5 - 5| / √2 = 0/√2 = 0

Distance = |2(3) + 3(4) - 6| / √(2² + 3²) = |6 + 12 - 6| / √13 = 12/√13.

Distance = |3 + 4 - 5| / √(1² + 1²) = |2| / √2 = 2/√2 = √2.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 9 if the sum of its digits is divisible by 9.

The function is undefined at x = 2, so the domain is all real numbers except 2.

The function is undefined at x = 3, so the domain is x ≠ 3.

The expression under the square root must be non-negative, so x - 1 >= 0, hence x >= 1.

The Doppler effect refers to the change in frequency of a wave in relation to an observer moving relative to the wave source.

Dot product = 1*3 + 2*4 = 3 + 8 = 11

A . B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.

A · B = (2)(4) + (3)(-1) = 8 - 3 = 5.

The eccentricity e of a hyperbola is given by e = √(1 + b^2/a^2).

The phenomenon of two sound waves of slightly different frequencies interfering is called beats.

-NO2 is a strong electron-withdrawing group and destabilizes benzene derivatives.

-CF3 is a strong electron-withdrawing group, which destabilizes the benzene ring.

The -CH3 group is an electron-donating group and stabilizes the carbocation through hyperconjugation.

The -COOH group is a strong electron-withdrawing group, thus it destabilizes the benzene ring.

A -I group increases the acidity of a carboxylic acid by stabilizing the negative charge on the conjugate base.

A -I group increases the acidity of carboxylic acids by stabilizing the negative charge on the conjugate base.

The -NO2 group is a strong electron-withdrawing group, which destabilizes the benzene ring.

A catalyst lowers the activation energy of a reaction, thereby increasing the reaction rate.

A catalyst does not affect the equilibrium constant; it only speeds up the rate at which equilibrium is reached.

A catalyst does not affect the position of equilibrium; it only speeds up the rate at which equilibrium is reached.

Inserting a dielectric material increases the capacitance of a capacitor by reducing the electric field within it.

Inserting a dielectric material between the plates of a capacitor decreases the electric field due to polarization.

Inserting a dielectric material into a capacitor decreases the electric field inside the capacitor due to polarization.

A charged particle moving perpendicular to a magnetic field experiences a magnetic force that acts perpendicular to both the velocity and the magnetic field, causing it to move in a circular path.

A current-carrying conductor placed in a magnetic field experiences a force due to the interaction between the magnetic field and the current.

A magnetic field exerts a force only on moving charges; therefore, a stationary charge experiences no force.