Why OpAmp is designed to have a cut-off frequency 1 kHz, but indicated Op Amp respond to sinusoidal input of 10 kHz?
Without having any more information or knowing the exact OpAmp you are talking about:
Cut-Off frequency means, that at this point the gain is decreased by 3dB in comparison to the highest gain the OpAmp can achieve (at DC and slightly above).
If the OpAmp has an open loop gain of 1,000,000 and a cut-off frequency of 1kHz, that even at one decade above that (so at 10kHz) the gain is only reduced to -23dB or about \$ \frac{1,000,000}{14.1} = 70,900 \$ (assuming a first order roll off with -20db/decade).
So as long as your 10kHz circuit is configured by feedback to a gain below 70,000 it will work just fine.

Example with the TL071:
As you can see in the plot from the datasheet from TI, the TL071 has an open loop  gain of something slightly about 90dB. The cut-off frequency (-3dB) is approximatly at 200Hz. But even at 1MHz you still have left 15dB of gain.