If 2x³ + ax² + bx - 6 has a factor (2x + 1) and leaves a remainder 12 when divided by (x + 2). Calculate the values of a and b hence factorise the given expression completely.

Points A, B and C divide the line segment joining the points D(8, -4) and E(-12, 16) in four equal parts. Find the equation of the line that passes through point A and is perpendicular to DE.

Which term of the arithmetic progression 3, 15, 27, 39, …… will be 132 more than its 54th term ?

Five cards Ten, Jack, Queen, King and Ace of diamonds are shuffled face downwards. Then one card is picked at random.

(i) What is the probability that the card picked is queen ?

(ii) If a king is drawn first and put aside, what is the probability that the second card picked up is the ace ?