The weak jump shift is probably the most misused treatment/convention in duplicate bridge.
The requirements for a weak jumps shift are:
  1. A hand too weak to bid at the one level
  2. No aces and no kings outside the bid suit
  3. A singleton or void in opener's major suit or no more than three cards in partner's minor suit
Hand 1:  QJxxxx x xxx xxx
Bid 2 in response to partner's one of a suit opening bid

Hand 2:  AJxxxx xx Kxx xx
Bid 1 in response to partner's one of a suit opening bid. With seven points and an ace you should bid 1.  (If your partner opened 1 your doublton heart would be another reason to not bid 2.)

Hand 3:  Qxxxxx xxx Kxx x
If partner opens 1 bid 2.  If he/she opens 1 bid 1 - with five high card points and a singleton in a suit not bid by partner you are to strong to make a weak jump shift.   If your partner opens 1 you should bid 2 - never make a weak jump shift with two or more cards in partner's major.

Hand 4:   5 AQ1098 KJ6 AK65
If your partner responds 2 to your 1 opening bid you must pass.   Partner probably has 2 or 3 spade losers, a diamond loser or two and possibly a club loser.  Also your hearts would be of little value to partner who should have a singleton or void in hearts.  A double of a 3 or a 3 reopening bid would probably be a good bet.

The WEAK JUMP SHIFT at Bridge Guys.