Studies of the Decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>K</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:math>

We report studies of the Cabibbo-suppressed decay B+/--->D(CP)K+/-, where D(CP) denotes CP eigenstates of the D0-D0; system. The analysis is based on a 29.1 fb(-1) sample collected at the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric e(+)e(-) storage ring. We measure ratios of branching fractions, relative to Cabibbo-favored B+/--->D(CP)pi(+/-), of B(B--->D1K-)/B(B--->D1pi(-))=0.125+/-0.036+/-0.010 and B(B--->D2K-)/B(B--->D2pi(-))=0.119+/-0.028+/-0.006; the index 1 (2) denotes the CP=+1 (-1) eigenstate. We also extract the partial rate asymmetries for B+/--->D(CP)K+/-, finding A(1)=0.29+/-0.26+/-0.05 and A(2)=-0.22+/-0.24+/-0.04.