Bidding : 1S (N) - 2C - 2H (N) - 3D* - 3S (N) - 4S

* = 4th suit forcing. North has no Diamond stopper nor 3-cards Club support, so rebids his own suit. South reckons a 5-2 trump fit is good enough.

Opening Lead : ♦K (E) = King of A K J x

LTC = 6 + 7 = 13 losers

Declarer Play
4th suit forcing   -   Anticipating a 4-2 trump split
East wins the first trick with the ♦K and follows on with the Ace and Jack of Diamonds. The Jack being ruffed by Declarer.

With 6 trumps against, they are split 4-2 (or worse) 65% of the time. The same scenario applies to Opponents' Clubs. Add to this the fact that besides the trump King, there is no side entry to Dummy's Clubs and you have a very nasty situation to deal with.

Hoping to find West with the trump Jack, Declarer leads a small Spade to Dummy's King, then leads the ♠9 and (after West plays low) takes it over with his 10. It holds! Declarer now draws the remaining trumps (and yes, they do break 4-2), then plays A, K and Q Clubs from Dummy, discarding a small Heart on Dummy's Queen. Unfortunately the Clubs break 4-2 too, but 10 tricks (including A and K of Hearts) fulfill the contract.

Score = 4 x 30 pts + 500 bonus pts = 620 pts

PS
If the ♠J had been offside North could only have made his contract with both the Spade and Club suit cards of the Opponents splitting 3-3.

Bidding : 1D (E) - 1S - 2C (E) - 3H !* - 4H (E)

* = a natural bid, showing a 5-card Heart suit

LTC : 6 + 5 = 11 losers

Opening Lead : ♦J (N) = top of a doubleton, leading through Dummy's strength

Declarer (E) counts 5 possible losers : one in Clubs, two in Hearts and one or two in Spades (depending on how many he can discard or ruff in Dummy.

Declarer Play
Natural 4th suit bid   -   Indirect finesse   -   Direct finesse
Declarer wins the opening lead with his singleton ♦Q, then leads his ♣J for a direct finesse. North covers with his King (to promote his 10 !) and Dummy wins with the Ace.   Now a small trump is lead from Dummy, South plays the Queen and Declarer wins with the Ace.

The trumps are clearly split 4-1 (there would be no purpose to play South's Queen with 2 or 3 trumps), with ♥ K 10 6 remaining in North's hand. Taking them out would cost Declarer two trump tricks, losing 4 trumps in the process.

So instead (at trick 4) Declarer plays a small Spade which is ruffed in Dummy. Dummy's ♦A is now lead on which North discards another Spade. Dummy (at trick 6) then cashes his ♣Q followed by a low Club ruffed by Declarer with his ♥7.
At trick 8 Declarer leads a second small Spade for a ruff in Dummy after which the situation looks like this.

The ♦K is now lead from Dummy, on which Declarer discards his ♠Q. North ruffs, winning the trick, but can not prevent Declarer making another two tricks, fulfilling his contract.

Score = 4 x 30 trick pts + 500 bonus pts = 620 pts

Bidding : 1D (S) - 1H - 1S (S) - 3H* - 4H (S)

* = jump rebid of Hearts, showing a 6-card suit

LTC : 7 + 6 = 13 losers

Opening Lead : ♣K (E) = K from A K x x

Declarer (N) counts 5 possible loser : two in in Clubs and Hearts (if the trump break 4-1) and one in Spades.

Declarer Play
Natural jump rebid   -   Opponent not leading trumps with Q x (or J x x)   -   drawing opponents' losing trumps only
After winning the first trick, East continues with his ♣A. Normally he should lead a trump at trick 2, trying to prevent Dummy of ruffing any Clubs, but holding ♥ Q 6 and hoping Partner has ♥ J x x he does not want to jeopardy the trump trick they can make.
So after cashing his to Club winners, East leads his ♠10. Dummy refuses the finesse and wins with the Ace. Dummy now leads a small Heart for two rounds of trumps, taking out Opponents' losing trumps only.

Declarer now leads his ♦K, follow with a low Diamond to Dummy's Ace. Dummy then leads the ♦Q, which West ruffs with his trump Queen and on which Declarer discards a small Heart.
The remaining tricks are all won with Declarer's trumps and Dummy's Spade Ace and King.

Score = 10 x 30 trick pts + 50 bonus pts = 620 pts

Bidding : 1S (W) - 2C - 2H (W) - 3D* - 4C (W) - 4NT# - 5D (W) - 6C

* = 4th suit forcing. With no Diamond stopper, West shows delayed 3-card support for East's minor suit.
# = Blackwood. West has shown to have 5 Spades, at least 4 Hearts and 3 Clubs. He must therefore hold at the most a singleton Diamond!

Opening Lead : ♦A (S) See comment below.

Declarer (W) counts only 1 possible loser in Diamonds. Even a 4-0 trump split is covered.

Declarer Play
4th suit forcing   -   Blackwood Convention   -   Opponents' listening to the bidding
Normally the lead of an Ace against a Slam contract is a very bad lead, as it usually promotes an extra trick for the Declaring side. But in this special case where both E and W have shown to lack a Diamond stopper and West is marked with a singleton at best, South is wise to cash his Ace before it is too late!
The Ace wins and South exits with a trump lead. The rest is a formality. Declarer draws all trumps and cashes the remaining tricks.

Score = 6 x 20 trick pts + 750 bonus pts = 980 pts